On Information Invariants in
Rob otics
Bruce Randall Donald
Computer Science Department
Cornell University
Ithaca New York
January
c
Please reference this document as follows
Donald B R Information Invariants in Rob otics Arti cial Intel li
gence In press Vol Jan
I m revising this paper into a longer book the paper represents
about one quarter of the total length
This b o ok describ es research done in the Rob otics and Vision Lab oratory at
Cornell University Supp ort for our rob otics research is provided in part by the
National Science Foundation under grants No IRI IRI IRI
and by a Presidential Young Investigator award to Bruce Donald and
in part by the Air Force O ce of Sp onsored Research the Mathematical Sciences
Institute Intel Corp oration and AT T Bell lab oratories
Acknowledgments This b o ok could never have b een written without discus
sions and help from Jim Jennings Mike Erdmann Dexter Kozen Je Ko echling
Tom as Lozano P erez Daniela Rus Pat Xavier and Jonathan Rees I am very
grateful to all of them for their generosity with their time and ideas The rob ots
and exp erimental devices describ ed herein were built in our lab by Jim Jennings
Russell Brown Jonathan Rees Craig Becker Mark Battisti Kevin Newman Dave
Manzanares and Greg Whelan these ideas could never have come to light without
their help and exp eriments I would furthermore like to thank Mike Erdmann Jim
Jennings Jonathan Rees John Canny Ronitt Rubinfeld Sundar Narasimhan and
Amy Briggs for providing invaluable comments and suggestions on drafts of this
b o ok Thanks to Loretta Pompilio for drawing the illustration in gure Debbie
Lee Smith and Amy Briggs drew the rest of the gures for this b o ok and I am very
grateful to them for their help I am grateful to Je Ko echling Mike Erdmann
and Randy Brost for explaining to me how lighthouses ADFs and VORs work
This b o ok was improved by incorp orating suggestions made at the Workshop on
Computational Theories of Interaction and Agency organized at the University of
Chicago by Phil Agre and Tim Converse I would like to thank Phil Agre Stan
Rosenschein Yves Lesp erance Brian Smith Ian Horswill and all members of the
workshop for their comments and suggestions I would like to thank the anony
mous referees of my pap ers for their comments and suggestions I am grateful to
Phil Agre who carefully edited the long pap er this b o ok is based up on and made
many invaluable suggestions on presentation
Preface
This monograph discusses the problem of determining the information
requirements to p erform rob ot tasks using the concept of information in
variants It represents our attempt to characterize a family of complicated
and subtle issues concerned with measuring rob ot task complexity
We discuss several measures for the information complexity of a task
a How much internal state should the rob ot retain b How many co op
erating agents are required and how much communication b etween them is
necessary c How can the rob ot change side e ect the environment in
order to record state or sensory information to p erform a task d How
much information is provided by sensors and e How much computation is
required by the rob ot We consider how one might develop a kind of calcu
lus on a e in order to compare the p ower of sensor systems analytically
To this end we attempt to develop a notion of information invariants We
develop a theory whereby one sensor can b e reduced to another much
in the spirit of computation theoretic reductions by adding deleting and
reallo cating a e among collab orating autonomous agents
This prosp ectus is based closely on a pap er of mine to app ear in Arti cial
Intel ligence
Bruce Randall Donald
Ithaca and Palo Alto
Contents
Part I State Communication and Side E ects
Introduction
Research Contributions and Applications
Examples
A Following Task
A Metho d of Inquiry
Details of the Following task
The Power of the Compass
The Power of Randomization
What do es a compass give you
Discussion Measuring Information
Part I I Sensors and Computation
Sensors
The Radial Sensor
Lighthouses Beacons Ships and Airplanes
Resources
Reduction of Sensors
Comparing the Power of Sensors
Sensor Reduction
A Reduction by Adding a Compass
Reduction using Permutation and Communication
Installation Notes
Calibration Complexity
Comments on Power
Output Communication
A Hierarchy of Sensors
Information Invariants
On The Semantics of Situated Sensor Systems
Situated Sensor Systems
Pointed Sensor Systems
Co designation Basic Concepts
Combining Sensor Systems
The General Case
Co designation Constraints
Example The Basic Idea
Example continued A Formal Treatment
The Top of Equation
The Bottom of Equation The Sensor System comm
Bandwidth and Output Vertices
Calibration Complexity and Co designation
Nonco designation Constraints and Parametric Co designation Constraints
Generality and Co designation
More General Co designation Relations
The Semantics of Co designation Constraints
The Semantics of Permutation
The Semantics of Reductions
Weak Transitivity
Strong Transitivity for Simple Sensor Systems
A Hierarchy of Reductions
A Partial Order on Simple Sensor Systems
Computational Prop erties
Algebraic Sensor Systems
Computing the Reductions and
Unsituated Permutation
Example of Unsituated Permutation
Application and Exp eriments
DJR Use Circuits and Reductions to Analyze Information Invariants
Conclusions
Future Research
References
App endices
A Algebraic Decision Pro cedures
A Application Computational Calibration Complexity
A Application Simulation Functions
A Vertex versus Graph Permutations
A Application Parametric Co designation Constraints
A Application Universal Reductions
B Relativized Information Complexity
C Distributive Prop erties
C Combination of Output Vertices
C Output Permutation
C Discussion
D On Alternate Geometric Mo dels of Information Invariants
E A Non Geometric Formulation of Information Invariants
F Provable Information Invariants with Performance Measures
F Kino dynamics and Trade O s
Glossary of Symbols
Section Page De nition Figure
App endix equation
R real numbers
S unit circle
p p tra jectories
S Q Q simulates S
combination of sensor systems
E the radial sensor
G the goal con guration
R ship
x x ship s p osition
h ship s heading
angle b etween h and the goal direction
r
N direction of North
H the lighthouse b eacon sensor
L lighthouse
R s b earing from L
g rotating green light
w ashing white light
white bit white light sensor
green bit green light sensor
time clo ck
orientation orientation sensor
h generalized compass installed on R
R
p sensed p osition
comm communication primitive
comm L R info communicate info from L to R
comm datapath lab eled
r r
S Q sensor systems
b output of a sensor S
b number of values b can take on
mb S maximum bandwidth of S A
comm b datapath with bandwidth log b
comm S datapath with bandwidth mb S A
E radial sensor installed at G
G
H lighthouse sensor installed at G
G
For some symbols the rst page reference p oints to the b egining of the sub section explaining or
containing that symbol
Section Page De nition Figure
App endix equation
vertex p ermutation
H p ermutation of H
H p ermutation of H
G
G
simulation and domination
wire reduction
wire e cient reduction
k wire reduction
k
reduction using global communication
reduction using p olynomial communication
P
G V E a graph with vertices V and edges E
d number of vertices in V
S U V W sensor systems
immersions
lab elling function
C con guration space
S situated sensor system
p ermutation of an immersion
S S p ermutation of a sensor system
G p ointed immersion
S p ointed sensor system
G
S p ointed p ermutation
G
extension of a partial immersion
extension of a p ermutation
u v output vertex
o o
diagonal
ij
T s a predicate
ex extensions of
vertex p ermutations of
graph p ermutations of A
im image of
simulation function
U
quanti er
D D D s a co designation constraints
U V VU
r dimension of C
c
degree b ound
n simulation complexity
n s a co designation complexity
D
edge p ermutation A
U U graph p ermutation of U U A
O A d group of orthogonal matrices A
cl clone function A A
Part I State Communication and
Side E ects
Introduction
As its title suggests this b o ok investigates the information requirements
for rob ot tasks Our work takes as its inspiration the information invari
ants that Erdmann introduced to the rob otics community in Erd
although rigorous examples of information invariants can b e found in the
theoretical literature from as far back as see for example BK Koz
Part I of this b o ok develops the basic concepts and to ols b ehind informa
tion invariants in plain language Therein we develop a number of motivating
examples In part I I we provide a fairly detailed analysis In particular we
admit more sophisticated mo dels of sensors and computation This analysis
will call for some machinery whose complexity is b est deferred until that
time
A central theme to previous work see the survey article Don for a
detailed review has b een to determine what information is required to solve
a task and to direct a rob ot s actions to acquire that information to solve
it Key questions concern
What information is needed by a particular rob ot to accomplish a par
ticular task
How may the rob ot acquire such information
What prop erties of the world have a great e ect on the fragility of a
rob ot plan program
Erdmann introduced the notion of measuring task complexity in bit seconds the ex
ample is imp ortant but somewhat complicated the interested reader is referred to Erd
What are the capabilities of a given rob ot in a given environment or
class of environments
These questions can b e di cult Structured environments such as those
found around industrial rob ots contribute towards simplifying the rob ot s
task b ecause a great amount of information is enco ded often implicitly into
b oth the environment and the rob ot s control program These enco dings and
their e ects are di cult to measure We wish to quantify the information
enco ded in the assumption that say the mechanics are quasi static or that
the environment is not dynamic In addition to determining how much infor
mation is enco ded in the assumptions we may ask the converse how much
information must the control system or planner compute Successful ma
nipulation strategies often exploit prop erties of the external physical world
eg compliance to reduce uncertainty and hence gain information Often
such strategies exploit mechanical computation in which the mechanics of
the task circumscrib es the p ossible outcomes of an action by dint of physi
cal laws Executing such strategies may require little or no computation in
contrast planning or simulating these strategies may b e computationally ex
p ensive Since during execution we may witness very little computation in
the sense of algorithm traditional techniques from computer science have
b een di cult to apply in obtaining meaningful upp er and lower b ounds on
the true task complexity We hop e that a theory of information invariants
can b e used to measure the sensitivity of plans to particular assumptions
ab out the world and to minimize those assumptions where p ossible
We would like to develop a notion of information invariants for charac
terizing sensors tasks and the complexity of rob otics op erations We may
view information invariants as a mapping from tasks or sensors to some mea
sure of information The idea is that this measure characterizes the intrinsic
information required to p erform the task if you will a measure of com
plexity For example in computational geometry a successful measure has
b een developed for characterizing input sizes and upp er and lower b ounds
for geometric algorithms Unfortunately this measure seems less relevant in
rob otics although it remains a useful to ol Its apparent diminished relevance
in embedded systems re ects a change in the scienti c culture This change
represents a paradigm shift from o ine to online algorithms Increasingly
rob otics researchers doubt that we may reasonably assume a strictly o ine
paradigm For example in the o ine mo del we might assume that the rob ot
on b o oting reads a geometric mo del of the world from a disk and pro ceeds
to plan As an alternative we would also like to consider online paradigms
where the rob ot investigates the world and incrementally builds data struc
tures that in some sense represent the external environment Typically online
agents are not assumed to have an a priori world mo del when the task b egins
Instead as time evolves the task e ectively forces the agent to move sense
and p erhaps build data structures to represent the world From the online
viewp oint o ine questions such as what is the complexity of plan construc
tion for a known environment given an a priori world mo del often app ear
secondary if not arti cial In part I of this b o ok we describ e two working
rob ots Tommy and Lily which may b e viewed as online rob ots We discuss
their capabilities and how they are programmed We also consider formal
mo dels of online rob ots foregrounding the situated automata of BK The
examples in part I link our work to the recent but intense interest in online
paradigms for situated autonomous agents In particular we discuss what
kind of data structures rob ots can build to represent the environment We
also discuss the externalization of state and the distribution of state through
a system of spatially separated agents
We b elieve it is pro table to explore online paradigms for autonomous
agents and sensorimotor systems However the framework remains to b e
extended in certain crucial directions In particular sensing has never b een
carefully considered or mo deled in the online paradigm The chief lacuna
in the armamentarium of devices for analyzing online strategies is a princi
pled theory of sensori computational systems We attempt to ll this gap
in part I I where we provide a theory of situated sensor systems We argue
this framework is natural for answering certain kinds of imp ortant questions
ab out sensors Our theory is intended to reveal a system s information in
variants When a measure of intrinsic information invariants can b e found
then it leads naturally to a measure of hardness or di culty If these notions
are truly intrinsic then these invariants could serve as lower b ounds in
rob otics in the same way that lower b ounds have b een developed in com
puter science
In our quest for a measure of the intrinsic information requirements of
a task we are inspired by Erdmann s monograph on sensor design Erd
Also we note that many interesting lower b ounds in the complexity theoretic
sense have b een obtained for motion planning questions see eg Reif HSS
Nat CR see eg Don Can Bri for upp er b ounds Rosenschein has de
veloped a theory of synthetic automata which explore the world and build
data structures that are faithful to it Ros His theory is set in a logical
framework where sensors are logical predicates Perhaps our theory could b e
viewed as a geometric attack on a similar problem This work was inspired
by the theoretical attack on p erceptual equivalence b egun by DJ and by
the exp erimental studies of JR Horswill Hors has developed a semantics
for sensory systems that mo dels and quanti es the kinds of assumptions a
sensori computational program makes ab out its environment He also gives
source to source transformations on sensori computational circuits In ad
dition to the work discussed here in Section for a detailed bibliographic essay
on previous research on the geometric theory of planning under uncertainty
see eg Don or Don
The goals outlined here are ambitious and we have only taken a small
step towards them The questions ab ove provide the setting for our inquiry
but we are far from answering them This b o ok is intended to raise issues
concerning information invariants survey some relevant literature and to ols
and take a rst stab at a theory Part I of this b o ok Sections provides
some practical and theoretical motivations for our approach In part I I Sec
tions we describ e one particular and very op erational theory
This theory contains a notion of sensor equivalence together with a notion
of reductions that may b e p erformed b etween sensors Part II contains an
example which is intended to illustrate the p otential of a such a theory We
make an analogy b etween our reductions and the reductions used in com
plexity theory Readers interested esp ecially in the four questions ab ove will
nd a discussion of installation complexity and the role of calibration in
comparing sensors in Section b elow Section discusses the seman
tics of sensor systems precisely as such this section is mathematically formal
and contains a number of claims and lemmata This formalism is used to
explore some prop erties of what we call situated sensor systems We also
examine the semantics of our reductions The results of Section are
then used in Section to derive algebraic algorithms for reducing one
sensor to another
Research Contributions and Applications
Rob ot builders make claims ab out rob ot p erformance and resource consump
tion In general it is hard to verify these claims and compare the systems I
really think that the key issue is that two rob ot programs or sensor systems
for similar or even identical tasks may lo ok very di erent Part I of this
b o ok attempts to demonstrate how very di erent systems can accomplish
similar tasks We also discuss why it is hard to compare the p ower of
such systems The examples in part I are distinguished in that they p ermit
relatively crisp analytical comparisons We present these examples so as to
demonstrate the standard of crispness to which we aspire these are the kinds
of theorems ab out information tradeo s that we b elieve can b e proved for
sensorimotor systems The analyses in part I are illuminating but ad hoc
In part I I we present our theory which represents a systematic attempt to
make such comparisons based on geometric and physical reasoning Finally
we try to op erationalize our analysis by making it computational we give
e ective alb eit theoretical pro cedures for computing our comparisons Our
algorithms are exact and combinatorially precise
We wish to rigorously compare embedded sensori computational systems
To do so we de ne a reduction that attempts to quantify when we can
e ciently build one sensor system out of another that is build one sensor
using the comp onents of another Hence we write A B when we can
build system A out of system B without adding to o much stu The last is
analogous to without adding much information complexity Our measure
of information complexity is relativized b oth to the information complexity
of the sensori computational comp onents of B and to the bandwidth of A
This relativization circumvents some tricky problems in measuring sensor
complexity In this sense our comp onents are analogous to oracles in the
theory of computation Hence we write A B if we can build a senso
rimotor system that simulates A using the comp onents of B plus a little
rewiring A and B are mo deled as circuits with wires datapaths con
necting their internal comp onents However our sensori computational sys
tems di er from computation theoretic CT circuits in that their spatial
con guration i e the spatial lo cation of each comp onent is as imp ortant
as their connectivity
We develop some formal concepts to facilitate the analysis Permutation
mo dels the p ermissible ways to reallo cate and reuse resources in building
another sensor Intuitively it captures the notion of rep ositioning resources
such as the active and passive comp onents of sensor systems e g infra red
emitters and detectors Geometric codesignation constraints further restrict
the range of admissible p ermutations I e we do not allow arbitrary re
lo cation instead we can constrain resources to b e installed at the same
lo cation such as on a rob ot or at a goal Output communication formalizes
our notion of a little bit of rewiring When resources are p ermuted they
must b e reconnected using wires or data paths If we separate previously
colo cated resources we will usually need to add a communication mecha
nism to connect the now spatially separate comp onents Like CT reductions
A B de nes an e cient transformation on sensors that takes B to A
However we can give a generic algorithm for synthesizing our reductions
whereas no such algorithm can exist for CT Whether such reductions are
widely useful or whether there exist b etter reductions is op en however we try
to demonstrate the p otential usefulness b oth through examples and through
general claims on algorithmic tractability We also give a hierarchy of re
ductions ordered on p ower so that the strength of our transformations can
b e quanti ed
We foresee the following p otential for application of these ideas
Comparison Given two sensori computational systems A and B we
can ask which is more p owerful in the sense of A B ab ove
Transformation We can also ask Can B b e transformed into A
Design Supp ose we are given a sp eci cation for A and a bag of
parts for B The bag of parts consists of b oxes and wires Each b ox
is a sensori computational comp onent black b ox that computes a
function of i its spatial lo cation or p ose and ii its inputs The
wires have di erent bandwidths and they can ho ok the b oxes to
gether Then our algorithms decide can we embed the comp onents
of B so as to satisfy the sp eci cation of A The algorithms also give
the embedding that is how the b oxes should b e placed in the world
and how they should b e wired together Hence we can ask can the
sp eci cation of A b e implemented using the bag of parts B
Universal Reduction Consider application ab ove Supp ose that in
addition to the sp eci cation for A we are given an enco ding of A as
a bag of parts and an embedding to implement that sp eci cation
Supp ose further that A B Since this reduction is relativized b oth to
For example no algorithm exists to decide the existence of a linear space or log space
p olynomial time Turing computable etc reduction b etween two CT problems
A and to B it measures the p ower of the comp onents of A relative to
the comp onents in B By universally quantifying over the con guration
of A we can ask can the comp onents of B always do the job of the
comp onents of A
Our work represents a rst stab at these problems and there are a number
of issues that our formalism do es not currently consider We discuss and
acknowledge these issues in Section
Examples
A Following Task
A Metho d of Inquiry
To introduce our ideas we consider a task involving two autonomous mobile
rob ots One rob ot must follow the other Now many issues related to
information invariants can b e investigated in the setting of a single agent
We wish however to relate our discussion to the results of Blum and Kozen
in Section b elow who consider multiple agents Second one of our ideas
is that by spatially distributing resources among collab orating agents the
information characteristics of a task are made explicit That is by asking
How can this task be performed by a team of robots one may highlight the
information structure In rob otics the evidence for this is so far largely
anecdotal In computer science often one learns a lot ab out the structure of
an algorithmic problem by parallelizing it we would eventually like to argue
that a similar metho dology is useful in rob otics
Here is a simple preview of how we will pro ceed We rst note that it is
p ossible to write a servo lo op by which a mobile rob ot can track follow a
nearby moving ob ject using sonar sensing for range calculations and servo
ing so as to maintain a constant nominal following distance A rob ot running
this program will follow a nearby ob ject In particular it will not prefer
any particular kind of ob ject to track If we wish to program a task where
one rob ot follows another we may consider adding lo cal infra red communi
cation b etween the rob ots enabling them to transmit and receive messages
This kind of communication allows one rob ot to lead and the other to follow
It provides an exp erimental setting in which to investigate the concept of
information invariants
Details of the Following task
We now discuss the task of following in some more detail Consider two
autonomous mobile rob ots such as those describ ed in RD The rob ots we
have in mind are the Cornell mobile rob ots RD but the details of their
construction are not imp ortant The rob ots can move ab out by controlling
motors attached to wheels The rob ots are autonomous and equipp ed with
a ring of simple Polaroid ultrasonic sonar sensors Each rob ot has an
onboard pro cessor for control and programming
We wish to consider a task in which one rob ot called Lily must follow
another rob ot called Tommy It is p ossible to write such a control lo op using
only sonar readings and p osition force control alone
We now augment the rob ots describ ed in RD as follows This descrip
tion characterizes the rob ots in our lab We equip each rob ot with infra
red mo dems sensors arrayed in a ring ab out the rob ot b o dy Each mo dem
consists of an emitter detector pair When transmitting or receiving each
mo dem essentially functions like the remote control for home appliances eg
TV s Exp eriments with our initial design Don seemed to indicate that
the communication bandwidth we could exp ect was roughly baud feet
That is at a distance of fo ot b etween Lily and Tommy we could exp ect to
communicate at baud at feet the reliable communication rate drops
to baud and so forth
We pause for a moment to note that this simple exp erimentally determined
quantity is our rst example of an information invariant
Now mo dem i is mounted so as to b e at a xed angle from the front
of the rob ot base and hence it is at a xed angle from the direction of
i
forward motion which is de ned to b e
The IR mo dems can time slice b etween collision detection and communication more
over nearby mo dems on the same rob ot can stagger their broadcasts so as not to
interfere with each other
Now supp ose that Tommy is traveling at a
commanded sp eed of v note v need not b e p os
itive For the task of Following each mo dem
panel i on Tommy transmits a unique identi er
eg Tommy the angle and the sp eed v That
i
is he transmits the following triple h id v i
i
In this task Lily transmits the same infor
mation with a di erent id of course This means
Figure The Cornell mobile
that when the rob ots are in communication each
rob ot Tommy Note mounted
top to b ottom on the cylindrical
can detect the p osition using sonars and IR s
enclosure the ring of sonars the
the heading and the name of the other rob ot In
IR Mo dems and the bump
sensors Lily is very similar
e ect each rob ot can construct a virtual radar
screen like those used by air tra c controllers
on which it notes other rob ots their p osition and heading as well as ob
stacles and features of the environment The screen see g is in lo cal
co ordinates for each rob ot It is imp ortant to realize that although g
lo oks like a pair of maps in fact each is simply a lo cal reconstruction
of sensor data Moreover these lo cal maps are up dated at each iteration
through the servo lo op and so little retained state is necessary
Now rob otics employs the notion of con guration space LoP to describ e
control and planning algorithms The con guration of one of our rob ots is its
p osition and heading Con guration space is the set of all con gurations In
our case the con guration space of one rob ot is the space R S A related
notion is state space which is the space of con gurations and velocities of
The identi er is necessary for applications involving more than two rob ots Also using
the id a rob ot can disambiguate other rob ots broadcasts from its own IR broadcast eg
re ections o white walls
This data is noisy but since an adequate servo lo op for following can b e constructed
using sonars alone RD the IR s only add information to the task The IR information
do es not measurably slow down the rob ot since the IR pro cessing is distributed and is
not done by the Scheme controller
In the language of DJ the sonar sensors plus the IR communication represent
concrete sensors out of which the virtual sensors shown in g can b e constructed The
construction essentially involves adding the IR information ab ove to the servo lo op for
following using sonar given in RD The details are not particularly imp ortant to this
discussion
See Lat for a go o d introduction
v
T
v w
o o
w
L
Lily
Tommy
Figure The radar screens of Tommy and Lily Tommy T is approaching a
wall on his right at sp eed v while Lily L follows at sp eed w
the rob ot After some re ection it may b e seen that g is a geometric
depiction of a state space for the rob ot task of following it is actually a
representation of the mutual con guration spaces of the rob ots Dep ending
on where the rob ots are in g each must take a di erent control servo
action The p oints where one rob ot takes the same parameterized action
may b e group ed together to form an equivalence class Essentially we parti
tion the state space in g into regions where the same action is required
This is a common way of synthesizing a a feedback control lo op See g
The p oint is that in this analysis we may ask What state must the robot
Lily retain After some thought the answer is very little since the radar
screens in g may b e drawn again from new sensor readings at each
iteration That is no state must b e retained b etween servo lo op iterations
b ecause in an iteration we only need some lo cal state to pro cess the sensor
information and draw the information in g We do not address whatever
state Tommy would need to gure out where to lead only how he should
mo dify his control so as not to lose Lily One consequence of this kind
of stateless following is that if communication is broken or one rob ot is
obscured from the other then the rob ots have no provision no information
v
v
I w
I
w
w
C
F
w C
Tommy
Tommy
Figure The statespace radar screen of Tommy is partitioned to indicate the control
for Lily For the task of following we could partition Lily s screen instead but this
is clearer for exp osition On the left is Lily s direction control and the regions are F
follow C correct and I intercept The commanded motion direction is shown as
an arrow On the right is Lily s sp eed control with w b eing very slow w fast and
w w w w This control partition is conditioned on Tommy s sp eed v
from the past on which to base a strategy to reacquire contact They can
certainly go into a search mo de but this mo de is stateless in the sense that
it is not based on retained state data built up from b efore b efore the break
in contact In short at one time step Lily and Tommy wake up and lo ok
at their radar screens Based on what they see they act If one cannot see
the other p erhaps it can b egin a search or broadcast a cry for help This
is an essential feature of statelessness or reactivity Let us call a situation
in which the rob ots maintain communication preserving the control loop If
they break communication it breaks the control lo op
Now supp ose that Tommy has to go around a wall as in g Supp ose
Tommy has a geometric mo del of the wall from a map or through recon
struction Then it is not hard for Tommy to calculate that if he takes a quick
turn around the wall as shown in tra jectory p that the line of sight b e
tween the rob ots may b e broken Since Lily is stateless as describ ed ab ove
when communication is broken the following task will fail unless Lily can
reacquire Tommy It is di cult to write a general such search reacquire
pro cedure and it would certainly delay the task
For this reason we may prefer Tommy to predict when line of sight com
L T
p p
0
Figure Following around a wall The shorter path p is quicker by t than p but it
cannot b e executed without more communication or state
munication would b e broken and to prefer a tra jectory like p g When
executed slowly enough such tra jectories as p will allow the rob ots to main
tain communication and hence allow the following task to pro ceed However
there is a cost for example we may reasonably assume that taking p will
take t longer than p Now let p denote the tra jectory that follows the
same path as p but slowed down so it takes the same time as p It might
also b e reasonable to assume that if Tommy slowed down enough to follow
p the rob ots could also maintain communication
Hence in this example the quantity t is a measure of the cost of
maintaining communication It is a kind of invariant But we can b e more
precise
In particular Tommy has more choices to preserve the control lo op The
distance at which Lily servos to Tommy is controlled by a constant which
we will call the fol lowing distance d Hence Tommy could transmit an
additional message to Lily containing the a new following distance d The
meaning of this message would b e tighten up that is to tell Lily to servo
at a closer distance Note that the message h heel d i essentially enco des
a plan D a new servo lo op for Lily In this case Lily will servo to follow
Tommy at the closer distance d which will successfully p ermit the rob ots to
navigate p while maintaining contact
Another p ossibility is that we could allow Lily to retain some state and
allow Tommy to broadcast an encoding of the tra jectory p This enco ding
could b e via p oints on the path or a control program essentially by trans
mitting the message h p i Tommy transmits a plan a motion plan for Lily
In this case after losing contact with Tommy Lily will follow the path or
So p is the time rescaled tra jectory from p DX
For an explicit use of this constant in an actual servo lo op see for example RD
plan p op en lo op until Tommy is reacquired
In b oth these cases we must allow Lily to retain enough state to store
d or p Since Lily already stores some value for d see RD we need
merely replace that However the storage for the plan or path p could
b e signi cant dep ending on the detail
Finally we could imagine a scenario where Lily retains some amount of
state over time to track Tommy For example by observing Tommy s tra
jectory b efore the break in communication it may b e p ossible to extrap olate
future p ositions one could for example use forward pro jections Erd or
a kalman lter Based on these extrap olations Lily could seek Tommy in
the region of highest exp ectation I will not detail this metho d here but it
is not to o di cult to see that it requires some amount of state for Lily to do
this computation and let us call this amount s
There is a trade o b etween execution time t communication trans
mitting h d i or h p i and internal state storage for p or s What is this
relationship Here is a conjecture one would like to prove ab out this rela
tionship For a path or a control program p or D we denote its information
complexity by jpj For example jpj could measure the number of via p oints
on p times their bit complexity the number of bits required to enco de a single
p oint
Idea There is an information invariant c for the task of fol lowing whose
units are bit seconds In particular
c jpj t jD j t st
p s
D
where t t and t are the execution times for the three strategies above
p s
D
Equation should b e interpreted as a lower b ound like the Heisenberg
principle It is no coincidence that Erdmann s information invariants are also
in bit seconds An information invariant such as quanti es the tradeo
b etween sp eed communication and storage Currently to prove such crisp
results we must rst make a number of assumptions ab out dynamics and
geometry see app endix F Moreover the metho ds we describ e b elow
typically yield results using order notation big oh O or big theta
instead of strict equality
One example of provable information invariants is given in the kinody
namic literature CDRX DX DX This work is concerned with provable
planning algorithms for rob ots with dynamics We give some details in ap
p endix F Here we note that Xavier in Xa DX developed trade o s
similar in avor to Equation Both Erdmann and Xavier obtain trade
o s b etween information and execution sp eed Their metho ds app ear to
require a p erformance measure eg the cost of a control strategy One
might view our work and also BK b elow as investigating information in
variants in the absence of a p erformance measure In this case we cannot
directly measure absolute information complexity in bit seconds Instead
we develop a way to relativize or reduce one sensori computational system
to another in order to quantify their relative p ower See app endix F for
more details on information invariants with p erformance measures
To summarize the ambition of this work is to de ne the notions in
Idea so they can b e measured directly Previous work Erd Xa DX
has required a p erformance measure in order to obtain a common currency
for information invariance In order not to use this crutch we rst de ne a
set of transformations on sensori computational systems Second we prop ose
understanding the information invariants in terms of what these transforma
tions preserve
The Power of the Compass
In Blum and Kozen wrote a ground breaking pap er on maze searching
automata BK Koz This section is devoted to a discussion of their
pap er On The Power of the Compass BK and we interpret their results in
the context of autonomous mobile rob ots and information invariants The
reader is urged to consult the clear and readable pap er BK for more details
In we p osed the following question with Jim Jennings
Question DJ Let us consider a rational reconstruction of mobile robot
programming There is a task we wish the mobile robot to perform and the task
is speci ed in terms of external e g human speci ed perceptual categories For
example these terms might be concepts like wall do or hallway or Professor
Hop croft The task may be speci ed in these terms by imagining the robot has
virtual sensors which can recognize these objects e g a wal l sensor and their
parameters e g length orientation etc Now of course the physical robot is
not equipped with such sensors but instead is armed with certain concrete physical
sensors plus the power to retain history and to compute The task level program
ming problem lies in implementing the virtual sensors in terms of the concrete robot
capabilities We imagine this implementation as a tree of computation in which
the vertices are control and sensing actions computation and state retention A
particular kind of state consists of geometric constructions in short we imagine
the mobile robot as an automaton connected to physical sensors and actuators
which can move and interrogate the world through its sensors while taking notes
by making geometric constructions on scratch paper But what should these con
structions be What program runs on the robot How may these computation trees
be synthesized
Let us consider this question of state namely what should the rob ot
record on its scratch pap er In rob otics the answer is frequently either
nothing i e the rob ot is reactive and should not build any representa
tions or a map namely the rob ot should build a geometric mo del of
the entire environment In particular even schemes such as LS require a
worst case linear amount of storage in the geometric complexity n of the
environment Can one do b etter Is there a su cient representation that is
b etween and O n
Blum and Kozen provide precise answers to these questions in the setting
of theoretical situated automata This section didactically adopts the
rhetorical we to compactly interpret their results While these results are
theoretical we b elieve they provide insight into the question ab ove
We de ne a maze to b e a nite two dimensional obstructed checkerboard
A nite automaton DFA in the maze may in addition to its automaton
transitions transit on each move to an adjacent unobstructed square in the
N S E or W direction We say an automaton can search a maze if eventually
it will visit each square It need not halt and it may revisit squares Hence
this kind of searching is the theoretical analog of the exploration task
that many mo dern mobile rob ots are programmed to p erform However note
that in this entire section there is no control or sensing uncertainty
We can consider augmenting an automaton with a single counter using
this counter it can record state Two counters would not b e an interesting
enhancement b ecause then we obtain the p ower of a Turing machine
A counter is like a register A DFA with a counter can keep a count in the register
increment or decrement it and test for zero A single counter DFA introduced by Fi
in can b e viewed as a sp ecial kind of push down stack automaton PDA that
has only one stack symbol except for a top of the stack marker This means we should
not exp ect a single counter machine to b e more p owerful than a PDA which in turn is
We say two or more automata search a maze together as follows The
automata move synchronously in lo ck step This synchronization could b e
e ected using global control or with synchronized clo cks When two au
tomata land on the same square each transmits its internal state to the
other
Finally we may externalize and distribute the state Instead of a counter
we may consider equipping an automaton with pebbles which it can drop
and pick up Each p ebble is uniquely identi able to any automaton in the
maze On moving to a square an automaton senses what p ebbles are on the
square plus what p ebbles it is carrying It may then drop or pick up any
p ebbles
Hence a pure automaton is a theoretical mo del of a reactive rob ot like
creature Many simple physical rob ot controllers are based on DFA s The
exchange of state b etween two automata mo dels lo cal communication b e
tween autonomous agents The p ebbles mo del the b eacons often used by
mobile rob ots or more generally the ability to side e ect the environment
as opp osed to the rob ot s internal state in order to p erform tasks Fi
nally the single counter mo dels a limited form of state storage It is much
more restrictive than the tap e of a Turing machine I b elieve that quanti
fying communication b etween collab orating mobile rob ots is a fundamental
information theoretic question In manipulation the ability to structure the
environment through the actions of the rob ot see eg Don or the me
chanics of the task see eg Mas seems a fundamental paradigm How do
these techniques compare in p ower
We call automata with these extra features enhanced and we will assume
that automata are not enhanced unless noted Given these assumptions
Blum and Kozen demonstrate the following results First they note a result
of Budach that a single automaton cannot search all mazes Next they
prove the following
There are two unenhanced automata that together can search all
mazes
considerably weaker than a Turing machine see eg HU Ch The pro of that a two
counter DFA can simulate a Turing machine was rst given by Papert and McNaughton
in Min but shorter pro ofs are now given in many textb o oks for example see HU
Thm
See BK for references
There is a two pebble automaton that can search all mazes
There is a one counter automaton that can search all mazes
These results are crisp information invariants It is clear that a Turing
machine could build a p erfect map of the maze that would b e linear in the
size of the maze This they term the na ve linear space algorithm This is
the theoretical analog of most map building mobile rob ots even those that
build top ological maps still build a linear space geometric data structure
on their scratch pap er But implies that there is a log space algorithm
to search mazes that is using only an amount of storage that is logarithmic
in the complexity of the world the maze can b e searched This is a precise
answer to part of our question
However the p oints also demonstrate interesting information invari
ants demonstrates the equivalence in the sense of information of
b eacons and communication Hence side e ecting the environment is equiv
alent to collab orating with an autonomous co agent The equivalence of
and to suggests an equivalence in this case and a tradeo in general
b etween communication state and side e ecting the environment Hence we
may credit BK with a excellent example of information invariance
The Power of Randomization
Erdmann s PhD thesis is an investigation of the p ower of randomization
in rob otic strategies Erd The idea is similar to that of randomized
algorithms by p ermitting the rob ot to randomly p erturb initial conditions
the environment its own internal state or to randomly choose among ac
tions one may enhance the p erformance and capabilities of rob ots and derive
Here is the idea First BK show how to write a program whereby an unenhanced
DFA can traverse the b oundary of any single connected comp onent of obstacle squares
Now supp ose the DFA could remember the southwesternmost corner in a lexicographic
order of the obstacle Next BK show how all the free space can then b e systematicically
searched It su ces for a DFA with a single counter to record the y co ordinate y of
min
this corner We now imagine simulating this algorithm as e ciently as p ossible using
a Turing machine and we measure the bit complexity If there are n free squares in the
environment then y n and the algorithm consumes O log n bits of storage For
min
details see BK
probabilistic b ounds on exp ected p erformance This lesson should not b e
lost in the context of the information invariants ab ove For example as Erd
mann p oints out one nite automaton can search any maze if we p ermit it
to randomly select among the unobstructed directions The probability that
such an automaton will eventually visit any particular maze square is one
Randomization also helps in nite D mazes see Section for more on
the problems that deterministic as opp osed to randomized nite automata
have in searching D mazes although the exp ected time for the search in
creases some
These observations ab out randomizing automata can b e even extended to
unbounded mazes the mazes we have considered are nite However in a D
unbounded maze although the automaton will eventually visit any particular
maze square with probability one the exp ected time to visit it is in nite In
D however things are worse in D unbounded mazes the probability that
any given cub e will b e visited drops from one to ab out
What do es a compass give you
Thus we have given precise examples of information invariants for tasks or
for one task namely searching or exploration However it may b e less
clear what the information invariants for a sensor would b e Again Blum
and Kozen provide a fundamental insight We motivate their result with the
following
Question Suppose we have two mobile robots Tommy and Lily con
gured as described in Section Suppose we put a ux gate magnetic
compass on Lily but not on Tommy How much more powerful has Lily
become What tasks can Lily now perform that Tommy cannot
Now any rob ot engineer knows compasses are useful But what we want
in answer to question is a precise provable answer Happily in the case
where the compass is relatively accurate BK provide some insight
While the p ower of randomization has long b een known in the context of algorithms
for maze exploration Erdmann was able to lift these results to the rob otics domain In
particular one challenge was to consider continuous state spaces as opp osed to graphs
In considering how a very accurate sensor can aid a rob ot in accomplishing a task
this metho dology is closely allied with Erdmann s work on developing minimal sensors
Erd
Consider an automaton of any kind in a maze Such an automaton
e ectively has a compass since it can tell N S E W apart That is on landing
on a square it can interrogate the neighboring N S E W squares to nd out
which are unobstructed and it can then accurately move one square in any
unobstructed compass direction
By contrast consider an automaton in a graph that need not b e a maze
Such an automaton has no compass on landing on a vertex there are some
number g of edges leading to free other vertices and the automaton
must choose one
Hence as Blum and Kozen p oint out Mazes and regular planar graphs
appear similar on the surface but in fact di er substantial ly The primary
di erence is that an automaton in a maze has a compass it can distinguish
N S E W A compass can provide the automaton with valuable information
as shown by the second of our results BK Recall p oint in Section
Blum and Kozen show that in contrast to no two automata together can
search all nite planar cubic graphs in a cubic graph all vertices have degree
g They then prove no three automata su ce Later Kozen showed
that four automata do not su ce Koz Moreover if we relax the planarity
assumption but restrict our cubic graphs to b e D mazes it is known that
no nite set of nite automata can search all such nite D mazes BS
Hence BK Koz provide a lower b ound to the question What informa
tion do es a compass provide We close by mentioning that in the avor of
Section there is a large literature on randomized search algorithms for
graphs As in Section randomization can improve the capability and
p erformance of the search automata
Discussion Measuring
Information
We have describ ed the basic to ols and concepts b ehind information in
variants We illustrated by example how such invariants can b e analyzed and
derived We made a conceptual connection b etween information invariants
and trade o s In previous work tradeo s arose naturally in kino dynamic
situations in which p erformance measures planning complexity and robust
ness in the sense of resistance to control uncertainty are traded o We
noted that Erdmann s invariants are of this ilk Erd
However without a p erformance cost measure it is more di cult to
develop information invariants We b elieve measures of information com
plexity are fundamentally di erent from performance measures Our interest
here is in the former we will not discuss p erformance measures again until
app endix F Here are some measures of the information complexity of a
rob otic task a How much internal state should the robot retain b How
many cooperating agents are required and how much communication between
them is necessary and c How can the robot change side e ect the envi
ronment in order to record state or sensory information to perform a task
Examples of these categories include a space considerations for computer
memory b lo cal IR communication b etween collab orating autonomous mo
bile rob ots and c dropable b eacons With regard to a we note that of
course memory chips are cheap but in the mobile rob ot design space most
investigations seem to fall at the ends of the design sp ectrum For exam
ple near reactive systems use almost no state while map builders and
mo del based approaches use a very large linear amount Natara jan Nat
has considered an invariant complexity measure analogous to b namely the
number of rob ot hands required to p erform an assembly task This quan
ti es the interference kinematics of the assembly task and assumes global
synchronous control With regard to c the most easily imagined physi
cal realization consists of co ded IR b eacons however external side e ects
could b e as exotic as chalking notes on the environment as parking p olice do
on tires or assembling a collection of ob jects into a con guration of lower
entropy and hence greater information Calibration is an imp ortant form
of external state which we explore in part I I
In part I we exploited automata theoretic results to explore invariants
that trade o internal state communication and external state While part I
concentrates on information invariants for tasks we did touch on how infor
mation invariants for sensors can b e integrated into the discussion In partic
ular we reviewed a precise way to measure the information that a compass
gives an autonomous mobile rob ot Somewhat surprisingly trading o the
measures a c prove su cient to quantify the information a compass sup
plies
The compass invariant illustrates the kind of result that we would like
to prove for more general sensors That is we could add a fourth measure
d How much information is provided by sensors While the examples we
presented are p erhaps didactically satisfying we must introduce some more
machinery in order to extend our discussion to include two additional imp or
tant measures of the information complexity of a rob otic task d and e
How much computation is required of the robot In part I I we explore these
issues in some detail In particular we describ e how one might develop a kind
of calculus on measures a e in order to compare the p ower of sensor
systems analytically To this end we develop a theory whereby one sensori
computational system can b e reduced to another much in the spirit of
computation theoretic reductions by adding deleting and reallo cating a
e among collab orating autonomous agents
Part II Sensors and Computation
Sensors
Intuitively we can imagine a sensor system b eing implemented as a tree
of sensori computational elements in which the vertices are controllers and
sensors computing devices and state elements Such a system is called a
virtual sensor by DJ In a virtual sensor outputs are computed from the
outputs of other sensors in the same device Given two sensor systems E
and H we would like to b e able to quantify the information the sensors
provide In particular supp ose E and H are di erent implementations in
a sense we shall so on make precise of sup er cially similar sensor systems We
would like to b e able to determine whether the two systems are equivalent
in the sense that that they deliver equivalent information that is whether
H More generally we would like to b e able to write an equation E
like
H E
where we can rigorously sp ecify what b ox we need to add to H to make
sensor E For example the b ox could represent some new sensing or some
computation on existing sensory and stored data In part I I we discuss some
metho ds for achieving these goals To illustrate our techniques we describ e
two sensors the radial sensor Erd and the beacon or lighthouse sensor
We then develop metho ds to compare the sensors and their information in
variants These sensors b ear some relation to the compass discussed in part I
it is our goal here to quantify this relationship precisely In the b eginning
we will allow informal de nitions which su ce for building intuition The
following concepts will b e de ned precisely in section the term simu
late the output of a sensor a sensori computational resource the relation
and the op erator We b egin as follows
De nition Informal For two sensor systems S and Q we say Q sim
ulates S if the output of Q is the same as the output of S In this case we
Q write S
The op erator in Equation represents adding something to H
Informally this something is what we would like to call a resource later
is an equivalence relation in Section We will later see that
Here is a preview of the formalism we will develop We view sensor sys
tems as circuits We mo del these circuits as graphs Vertices corresp ond
to di erent sensori computational comp onents of the system what we will
call resources b elow Edges corresp ond to data paths through which
information passes Di erent embeddings of these graphs corresp ond to dif
ferent spatial allo cation of the resources We also p ermit resources to b e
colocated This requires that we consider graph immersions as well as graph
embeddings Immersions are like embeddings but they need not b e injective
Under this mo del the concepts ab ove are easily formalized For example
the op eration turns out to b e like taking the union of two graphs
One key idea involves asking What information is added or lost in
a sensor system when we change its immersion and What information
is preserved under al l immersions Our goal will b e to determine what
classes of immersions preserve information Sections explore this
idea through an example
The Radial Sensor
We b egin with a didactic example In Erd Erdmann demonstrates a
metho d for synthesizing sensors from task sp eci cations The sensors have
the prop erty of b eing optimal or minimal in the sense that they convey
exactly the information required for the control system to p erform the task
For our purp oses it is su cient to examine a particular sensor called the
radial sensor which is the output of one of his examples The radial sensor
arises by considering manipulation strategies in which the rob ot must achieve
a goal despite uncertainty
De nition is formalized in Section
The radial sensor works as follows Consider a small rob ot in the plane
Supp ose there is a goal region G which is a small disc in the plane See g
The rob ot is at some con guration x R and at some heading h S
Both these state variables are unknown to the rob ot The rob ot can only
command relative motions relative to the lo cal co ordinate system sp eci ed
by x h Thus it would command a velocity v and the rob ot would move
in relative direction which is global direction h The radial sensor
returns the angle which is the angle b etween h and the ray b etween x and
r
to reduce its distance to the the goal The rob ot need only command v
r
goal This example easily generalizes to the case where there is uncertainty
in the rob ot s control system that is the aim of v see LMT Erd It
is plausible and indeed Erdmann proves that this sensor is necessary and
su cient to write a feedback lo op that provably attains the goal
To summarize the radial sensor returns information that enco des the
relative heading of the goal G relative to the rob ot s current heading h
r
See g We emphasize that the radial sensor do es not reveal the con gu
ration x h of the rob ot b eyond this We will not describ e p ossible physical
implementations of the radial sensor but see Erd for a discussion
Lighthouses Beacons Ships and Airplanes
We now describ e another sensor Our goal is to compare this sensor to
the radial sensor using information invariants See g We call this a
lighthouse sensor system We call this a sensor system since as describ ed
it involves two physically separated agents We motivate this sensor as
follows Consider two mobile rob ots which we denote L and R see g
L will b e the lighthouse b eacon and R will b e the ship The rob ots
live in the plane In introducing the lighthouse system we will informally
introduce machinery to describ e sensori computational resources
In the language of DJ the p erceptual equivalence classes for this sensor are the rays
emanating at x
Erdmann emphasizes the sp ecial cases where the rob ot always knows its heading or
where the rob ot s heading is always xed say due North so that h is always identically
zero In these cases the radial sensor returns the global heading to the goal This sp ecial
case arises in the domain of manipulation with a rob ot arm which of course is why it is
natural for Erdmann s theory The radial sensor we present is just slightly generalized for
the mobile rob ot domain
h
x
r
R
G
Figure The Radial Sensor E showing heading h and relative goal direction
r
Resources
Now to analyze the information invariants we must b e careful ab out the
implementation of the sensor system and in particular we must b e careful
to count how resources a e Section are consumed and allo cated
much the same way that one must b e careful in p erforming a complexity
analysis for an algorithm Let us catalog the following kinds of resources
Emitters On L there are two lights which we call physical emitters
g that rotates at a constant angular There is a unidirectional green light
velocity That is the green light shines along a ray that is anchored at its
h
R
white
Concrete sensors
green
time
Virtual sensor
construct orientation sensor out of time
and the b eacons
virtual sensors
define orientation
pi
time beacons white green
N
g
physical emitters
L
w
Figure The b eacon sensor H which is based on the same principle employed by
lighthouses
origin at L The ray sweeps rotates ab out L The green light can only b e
seen by p oints on that ray Second there is an omnidirectional white light
w that ashes whenever the green light is p ointing due North That is the
white light can b e seen from all directions
Concrete Sensors On R there is is a photo electric sensor that detects
when a white light illuminates R Another sensor detects green light There
is also a clo ck on R
Computation There is a computer on R that we can program in Scheme
following RD The concrete sensors ab ove are interfaced to Scheme via li
brary functions as in RD The functions white and green are of
type unit bool and return t when light is sensed and f otherwise
The clo ck is available as the function time which returns the time mea
sured in small units We can measure the time and space requirements of a
computation using standard techniques Furthermore we may quantify the
amount of sensor information consumed by counting the number of calls to
white green and time and the number of bits returned
Now here is how lighthouses work See g The ship R times the
p erio d t b etween white ashes Then it measures the time t b etween a
w
white ash and the next green ash Clearly the angle of the rob ot the
angle b etween North and the ray from L to R can computed as t t
w
Assuming the ship is moving slowly relative to t
w
Virtual Sensors We can implement this as a virtual sensor DJ called
orientation shown immediately b elow The orientation sensor is sp ec
i ed as a computation that i calls concrete sensors ii retains some lo cal
state T and iii do es some computation etc It is easy to measure
the time and space requirements of the circuit that computes Hence
we can implement certain virtual sensors to compute orientation We detail
this implementation b elow
Given the resources ab ove we can implement the following virtual sensors
on R
Virtual sensor
construct orientation sensor out of time
and the b eacons
define orientation
pi
time beacons white green
time beacons white white
We must make some assumptions to prove this real time program is correct For
example we must assume the clo ck and the pro cessor are very fast relative to the green
light and the ship
time b etween b eacons
event and event are type unit bool
define time beacons event event
sleep until event
let T time
sleep until event
time T
utility in scheme RD
sleep until waits until thunk returns t
and then returns
define sleep until thunk
Resources R does not have Let us contrast our exemplar rob ot ship R
with an enhanced version R that corresp onds to a real ship navigating at
sea using lighthouse sensors We should not confuse R with a real ship A
real ship R has a map on which are lo cated a priori features including a
p oint which R will assume corresp onds to the lo cation of L True North is
indicated on the map R computes as ab ove see g and draws a ray
on the map anchored at L that is degrees from North R now knows
that it is on that ray In addition to p ossessing a map and knowing the map
co ordinates of L a real ship often has a compass In the rob otics domain
orientation o dometry could approximate an accurate compass Real ships
also have communication devices like radios We observe communication
resources compare roughly to b in Section Our unenhanced rob ot R
however is not a real ship and it has none of these resources
Mo dern aircraft navigate using two sensors similar to the radial and light
house sensors An Automatic Direction Finder ADF is a radial sensor An
ADF is simply a needle that p oints to a ground radio transmitter in relative
airplane co ordinates You do not need to know where you are or which way
you are headed You simply make the needle p oint straight ahead by turning
the airplane So it is a radial sensor and you track into the goal A VOR
VHF Omnirange is a lighthouse sensor The VOR ground transmitter has
the equivalent of a green and white light arrangement The radio receiver in
the plane deco des it and then tells you the radial direction from the trans
mitter in global co ordinates Then if you actually want to y to the VOR
Ob jects of type unit bool are called boolean thunks
you have to have a compass lo ok at it and turn the plane to y in the same
direction as your radio indicates The VOR uses a clo ck just like in the
lighthouse The green emitter in the VOR rotates at Hz and the white
North light ashes times a second The receiver in the plane deco des
the di erence just like in the lighthouse example to give a direction VORs
do not use light but they broadcast in the Megahertz range instead of the
visual range
To follow a radial sensor you only need to make the source b e straight
ahead of you to follow a lighthouse sensor you need a compass The radial
sensor is in lo cal co ordinates and the lighthouse sensor is in global co ordi
nates
The ADF requires fewer instruments but pilots tend to use the VOR
Why Because that way you can lo ok up your p osition on a chart which
is often what you care ab out one VOR gives you a line two give you your
lo cation But if you just want to get somewhere all you need is the ADF
There are some other reasons for using VORs such as the fact that VORs are VHF
while ADFs are LF MF so ADF reception gets blo cked by thunderstorms while VOR
reception do es ne On the other hand VORs require line of sight whereas ADFs will
work over the horizon
Reduction of Sensors
Comparing the Power of Sensors
Let us call the radial sensor E and the unenhanced lighthouse system H
The sensors are of course sup er cially similar b oth have comp onents at
two spatially separated lo cations Both sensors measure angles Of course
they measure di erent angles We cannot transform the information deliv
ered by H into the information sp eci cation of E without consuming more
resources These sensors deliver incomparable information in that neither
delivers strictly more information than the other
We wish to b e able to compare two sensors even when they deliver incom
parable information To do this we introduce a mechanism called reduction
which allows us to compare the p ower of two sensor systems such as E and
H Hence even though neither E nor H delivers strictly more information
they are comparable under a partial order induced by our reduction
Sensor Reduction
The analytic goal of sensor reduction is to b e able to write equations like
Equation The op erational goal is to build one sensor out of another and
to measure the p ower of the construction by a careful accounting for the
resources we add To illustrate the concept we give two ways of constructing
sensor E from sensor H First following Section we assume that R is
lo cated at x R and has heading h S However R cannot sense these
N
h
R
h
r x
R
N
L G
Figure Reduction using a compass h
R
state variables and it do es not know its con guration x h Before we b egin
we stress the following our goal is to change sensor H by adding resources
so as to simulate sensor E We have accomplished this task when R knows
the angle which is shown in gs and
r
A Reduction by Adding a Compass
We sketch a way to construct sensor E from H This way is easy since it
involves adding a p owerful resource namely a compass to H We will mo del
this reduction as a function s from sensors to sensors The reduction contains
the following steps which we denote s s and s see g
s We place the b eacon L at the goal G
s We add a concrete sensor called a compass to R The compass
senses the heading h
s The devices on R compute using the function orientation
ab ove and then compute h See g
r
The reduction also adds a small amount of computation but only a con
stant amount two subtractions We handle this by de ning the compass
to include this computation Sp eci cally we de ne a sensor h to b e a de
R
vice that i computes the heading h ii takes the output value of from
orientation as an input and iii outputs as sp eci ed in step s h
r
R
could b e implemented by a compass plus a small circuit to compute the
value given h and The subscript R of h denotes that it is installed on
r
R
R We will continue to refer call h a compass even though it is really a
R
compass plus a small amount of computation
In this reduction all the changes are made to R L remains the same
Now recall Equation Intuitively we can substitute h for the b ox in
R
this Equation and de ne the op erator to enco de how h is added to H
R
as sp eci ed in steps s s ab ove
Reduction using Permutation and Communication
The reduction in Section requires adding new resources the compass
h The next reduction we consider involves two new concepts The rst
R
is permutation and it involves redistributing resources in a sensor system
without consuming new resources Surprisingly a redistribution of resources
can add information to the system In order for p ermutation to add informa
tion it is necessary for the sensor system to b e spatially distributed as for
example H is see g When p ermutation gains information it may b e
viewed as a way of arranging resources in a con guration of lower entropy
The second concept is communication It measures resource b in Sec
tion We consider adding communication primitives of the form comm L R
info which indicates that L sends message info to R Like p ermutation
In using the term compass we make no commitment to a particular technology
for implementation such as sensing magnetic elds In particular the compass is
an orientation sensor that could in principle b e implemented using o dometry or dead
reckoning plus some initial calibration Moreover North N can b e any xed direction
for our purp oses and need not b e true North In the language of LMT the compass
senses the pro jection of a p erfect p osition sensor p R S onto S
communication only makes sense in a spatially distributed sensor system
That is b ecause spatially colo cated comp onents can communicate for free
in our mo del only external datapaths add information complexity to the
system Internal datapaths have the same spatial source and destination
External datapaths have a di erent spatial source and destination Hence
p ermutation alone can change the information complexity of a system by
externalizing internal datapaths To analyze a system like H we view it
as a system comp osed of autonomous collab orating agents L and R each
of which has certain resources The comm primitive ab ove we view as
shared b etween L and R We measure communication by counting the num
b er of agents and the bits required to transmit info This is the only kind of
communication we will consider here i e L R and so we will henceforth
abbreviate it by comm info
Given these concepts we can sketch another reduction y See g The
reduction contains the following steps which we denote y y and so forth
y As b efore we place L at the goal G
g
y We move the physical emitters from L to R i e we mount them
w
on the rob ot North for the emitters should b e installed in the direction
of R s heading That is the white light ashes when the green light passes
the lo cal to R North which is de ned to b e the rob ot s heading h
y We move the concrete sensors green white and time from
R to L
y We move the virtual sensor orientation co ded ab ove to L That
is now this program will run on L
See g Given y y by calling the pro cedure orientation
L can now compute the value of the angle shown in the gure However
r
although L now knows R do es not We solve this problem by allowing
r
L to communicate the value to R using the comm primitive describ ed
r
ab ove
y L communicates the value of to R using the primitive comm
r r
Note that the p ermutation steps y y require no new resources
They merely require p ermuting the sensors and emitters We do not view the
relo cation of the virtual sensor as moving the computer to L Instead we
view the virtual sensor orientation as a computational circuit we move
that circuit to L
N h
r
x
g
R
physical emitters
w
comm
r
white
Concrete sensors
green
time
Virtual sensor
construct orientation sensor out of time
and the b eacons
L G
virtual sensors
define orientation
pi
r
time beacons white green
Figure Reduction using Permutation and Communication
Installation Notes
Crucial to installing a sensor is describing how the various physical resources
should b e lined up We call these alignments calibrations Since these cali
brations constrain the spatial relationships among the various resources as
opp osed to leaving them arbitrary they e ectively add information to the
system A calibration is some spatial relationship that is lo cked into place
at the outset This relationship may or may not change over time Even
when it do es change the initial calibration may still add information to the
system since the system can measure relative distances to the initial setting
Hence calibration introduces an invariant that p ersists at b est for the life
time of the system For example by eliminating uncertainty at installation
we p erform a kind of calibration thereby eradicating that uncertainty for
the duration of the calibration Hence calibration can displace the task of
dealing with sensor uncertainty from the execution phase to the installation
or layout phase The purp ose of this section is to introduce formal means
for describing these calibrations which we call instal lation notes To make
this more concrete let us consider the calibrations necessary to p ermute
and install sensor system H in the two reductions s Section and y
Section
The installation notes are numbered I I and so forth
step s and Note I step y The installation notes for steps Note I
s and y are identical When installing L at G we must make sure that L
and G line up p erfectly otherwise the angle measured will not b e exactly
r
Note I When installing the physical emitters on L we must make sure
that North for the emitters line up p erfectly with true North Compare
Note I b elow
Note I step s When installing the compass we must make sure that
it lines up p erfectly with the heading of the rob ot
Note I step y We want the white light to ash when the green light
passes through R s heading h Hence when installing the physical emitters
on R we must make sure that relative North for the emitters line up
p erfectly with the rob ot s heading h
Calibration Complexity
It is di cult to precisely measure the information gained in calibration How
ever we note the following First the calibrations in I I and I each add
an equivalent amount of information to the system each installation re
quires calibration of two degree of freedom DOF systems each of which
has con guration space S Hence we say that I I and I are equivalent
instal lation calibrations
Now let us consider calibrations I and I ab ove This installation requires
a careful calibration of two DOF systems To calibrate H so that at L is
lo cated at the p oint G clearly adds information More precisely note that we
have so far considered the radial sensor E at a xed goal G in the plane Let
This section devolves to a suggestion of Mike Erdmann Erd for which we are
very grateful
us denote this particular installation by E More generally for a p oint y in
G
the plane we make the dep endence explicit by writing E thus we obtain a
y
family of sensors f E g parameterized by y R
y
Similarly let us denote by H the sensor system H installed so that L
y
is lo cated at the p oint y Now our goal is to approximate one particular
E using some H Clearly we could consider the case G y however in
y
G
sp ecifying E we sp ecify G and so this information is given That is it
G
is no more work to lo cate H at G than to lo cate E at G and the latter
is unavoidable it is the only way to implement E Hence we should b e
G
allowed to do at least this much work in installing H In other words merely
in order to sp ecify the sensor task it is necessary to calibrate a DOF system
to G there is a sense in which the problem of approximating E cannot b e
sp eci ed without calibrating to some y R This argument is similar to
saying that certain algorithms must at least read all their input In this case
we say that the calibrations I and I are necessary to specify the sensor E
That is the calibration required to install H is necessary to specify E
G G
When the calibration parameter the subscript G in this case is understo o d
we will drop it
De nition Informal Consider two sensor systems S and Q When S
and Q require equivalent instal lation calibrations and when the calibrations
required to instal l Q are necessary to specify S we say that S dominates Q
in calibration complexity
In Section we describ ed a reduction using a compass that yields a
new sensor system from H In Section we describ ed a reduction using
p ermutation and communication obtaining a di erent new sensor system
from H From the preceding discussion Section we conclude that E
dominates b oth of these new sensor systems in calibration complexity
Now it is clear that calibration is a source of information We view
calibration as a measure of the external state see resource c Section
required for the task Quantifying external state is tricky since the time at
which the resource is allo cated eg the time of calibration may b e much
earlier than the time of the task execution We developed the relatively
sophisticated p ersp ective of calibration complexity in this Section precisely
to deal with this problem Finally it is worth noting the sp ecial role of time
in this analysis in that calibration and execution may b e distant in time
We found it surprising that time would app ear so crucial not only here but
also in the virtual sensor orientation
Comments on Power
The reduction in Section requires adding an orientation sensor which
may b e implemented using a compass or o dometry The reduction in Sec
tion requires p ermuting resources sensors and emitters It also re
quires adding communication since L must now communicate to R
r
Let H denote the p ermutation of H describ ed in steps y y in
Section Thus in H L has not b een assigned any particular lo cation
and while L knows R do es not By installing H so that L is assigned the
r
Now recall the orientation sensor lo cation G we obtain a sensor called H
G
h for R describ ed in Section Thus in the language of Equation
R
we have sketched how
E H h
G G R
comm H E
r
G
G
Equation holds for all G The op erator denotes combining the
two sensor subsystems If this sounds somewhat op erational we will give
a more analytic discussion b elow in Section and a formal de nition in
section where we describ e the semantics of our sensor mo del in detail
Output Communication
The term comm in Equation says that we p ermit the p ermuted sys
r
tem H to route the information from one subsystem of H to another
r
G G
spatially removed subsystem these subsystems happ en to b e L and R in
our case First note that is exactly the desired output of the sensor
r
E Hence the term comm denotes an internal rerouting L R of this
r
G
information within the p ermuted sensor system H Let us generalize this
G
construction
De nition Let b be a variable that ranges over al l possible values that
a sensor system can compute We call b the output of the system Let b
be the number of values b can take on and de ne log b to be both the size
of b and the output size of the sensor The output size is an upper bound
on the bit complexity of b For example if b takes on integer values in the
range q then b q and log b log q In our example is the
r
output of E the quantity log is the output size of E Now suppose
r
G G
the information b is communicated over a datapath e We wil l assume that
the information is communicated repeatedly without loss of generality we
take the unit of time to be the interval of the occasion to communicate the
information Thus we can take the size of the output b to be the bandwidth
of e
To return to our example it is clear that we can make the p ermuted sensor
system H satisfy the information sp eci cation of E if we merely add one
G
G
internal re routing op eration of bandwidth log In this case we say we
r
have added output communication to the p ermuted sensor system
More precisely let S b e a sensor system with output b Let Q b e another
sensor system We imagine Q as a circuit embedded in say the plane
Let comm b b e a sensor system with one datapath e that has band
width log b Then adding output communication to Q can b e viewed as
the following transformation on sensor systems Q Q comm b The
transformation is parameterized by the bandwidth of S The b ounded
bandwidth datapath e can b e spliced into Q anywhere We note that this
transformation can b e comp osed with p ermutation in either order
Q Q Q comm b
Q Q comm b Q comm b
We give a fully formal graph theoretic mo del of this transformation in sec
tion
To b orrow a Unix metaphor this transformation allows the system to do an internal
rcp but not rpc that is it can copy information b etween subsystems but it cannot
request arbitrary remote evaluations
A Hierarchy of Sensors
The examples ab ove illustrate a general principle This principle is anal
ogous to the notion of reduction in the theory of computation We would
like our notion of reduction to do work analogous to the work done by
computation theoretic reductions Consider two sensor systems S and Q
Recall the de nitions of simulation De nition and calibration complex
ity from Section
De nition We de ne the internal resp external bandwidth of a sen
sor system S to be the greatest bandwidth of any internal resp external edge
in S The output size of S is given by De nition We de ne the maxi
mum bandwidth mb S to be the greater of the internal bandwidth external
bandwidth and the output size of S We call a sensor system monotonic if
its internal and external bandwidths are bounded above by its output size
De nition We write S Q when
Q Q simulates S S
S dominates Q in calibration complexity and
mb Q is bounded above by mb S
Calibration exploits external state De nition allows us to order systems
on how much information this external state from calibration yields We
will complete the formalization and analysis of calibration complexity later
in Sections and App endix A Here is the basic idea Calibration
complexity measures how much information we add to a sensor system when
we install and calibrate it Installing a sensor system may require physically
establishing some spatial relation b etween two comp onents of the system
In this case we say the two comp onents codesignate by the spatial relation
More generally we may have to establish a relation b etween a comp onent
and a reference frame in the world Most generally when we compare two
sensor systems S and Q we typically must install and calibrate them in
some appropriate relative con guration again in a spatial relation When
all these relations are in equalities of con guration we say the system is
simple When all the relations are semi algebraic s a we say the system is
algebraically codesignated
Now let Q denote a p ermutation of sensor system Q as describ ed in
Section For a formal de nition see De nition
De nition We write S Q if there exists some permutation Q of
sensor system Q such that S Q
Recall the meaning of comm info from Sections and Finally
De nition Given two sensor systems S and Q choose b such that
log b mb S We say S is e ciently reducible to Q if
S Q comm b
In this case we write S Q
For monotonic sensor systems it su ces to take b to b e the output of
S see App endix A This sp ecial case motivates the construction on the
r h s of where we add output communication to the sensor system Q
Section
We now recap a couple of crisp results using reductions
Claim a E H h and
G G R
comm b E H
r
G
G
Proof Recall the discussion from Section on calibration complexity To
obtain a we use the reduction that employs a compass Section The
pro of of b obtains by the reduction using p ermutation and communication
u
Section t
Now recall Equation The relation E H h which derives from
G G R
the compass reduction in Section do es not imply e cient reducibility
since adding a new concrete sensor h is to o p owerful to imply e cient
R
reducibility However by the reduction in Section
Prop osition Erdmann s radial sensor E is e ciently reducible to the
lighthouse sensor system H that is E H
Proof Recall from Equation that E H comm and that is
r r
G
G
the output of E From this and claim b we conclude that E H
G
u
t
Information Invariants
The relation de nes a hierarchy of sensors Compare the p erceptual
lattice of DJ who prop ose a geometric program for the analysis and syn
thesis of sensors based on their p erceptual equivalence classes The relation
orders sensor systems on the complexity of their information invariants
At this p oint it would b e useful to review the particular information
invariants in our example Here is the basic idea The invariants may b e
is an equivalence relation analyzed by rst examining Equation Since
we obtain the p eculiar equation
H H h comm
r
G R
G
Now what exactly do es Equation mean We understand that at
present this equation is not yet formal Our goal is to understand this
intriguing result To do so we must give a formal account of the colo cation
of resources Here is a general idea of how we will pro ceed
Recall the transformation describ ed in Section and De nition
where we added output communication to a sensor system Recalling that
It is p ossible to develop a geometric account of information invariance by pursuing
the direction of DJ For more on this connection see app endix C The account we
give in section is also geometric but with a di erent avor App endix C deals
with the geometry of lattices where an element of the lattice represents essentially a
knowledge state In section we examine di erent immersions of sensor systems
Permutations or automorphisms of the function space of immersions that preserve
the sensor functionality are viewed as a kind of information preserving transformation
and hence a mo del of information invariance
h denotes the compass at rst glance we would app ear to obtain the fol
R
lowing information invariant a compass is equivalent to permutation plus
output communication This idea is tantalizing b ecause it seems to de ne an
information equivalence b etween normally unapp osed categories it yields an
information invariant relating sensors communication and resource p ermu
tation The invariant is valid However it app ears that this invariant is
critically conditioned on the type of information b eing rerouted by the output
communication Output communication p ermits us to transform b etween lo
cal and global co ordinates however if some form of orientation sensing at
L is not present b efore the output communication step then no amount of
p ermutation and communication can simulate a global compass In sec
tion we address the generality of Equation There we mo del the
colo cation of resources as geometric codesignation constraints This colo ca
tion can b e mo deled as a quotient map and in section we discuss its
relationship to information invariance
In the language of DJ communication and p ermutation p ermit us to map b etween
the p erceptual equivalence classes PEC s of E the rays describ ed in Section and
the PEC s of H
On The Semantics of Situated
Sensor Systems
In this section we formalize our mo del of sensor systems We give formal
de nitions of the reductions using p ermutation and by combining sensor
systems and adding resources Section discusses the semantics of
sensor systems precisely as such this section is mathematically formal and
contains a number of claims and lemmata This formalism is used to explore
some prop erties of what we call situated sensor systems We also examine
the semantics of our reductions The results of Section are then
used in Section to derive algebraic algorithms for reducing one sensor
to another Below we use the term sensor system to mean sensori
computational system where it is melli uous
Situated Sensor Systems
We formalize our mo del of sensor systems using a concept similar to the
communication graph from distributed systems FLM
De nition A lab elled graph G is a directed graph V E with vertices
V and edges E together with a labelling function that assigns a label to
each vertex and edge Where there is no ambiguity we denote the labelling
function by
De nition A sensor system S is represented by a labelled graph V E
Each vertex is labelled with a comp onent Each edge is labelled with a con
nection
In Section we de ned comp onents and connections op erationally
We now give a formal de nition Comp onents and connections are de ned
by their simulation functions Simulation functions describ e the b ehavior of
b oth comp onents and connections
Consider a comp onent v asso ciated with vertex v To simulate a com
p onent we need to know i its inputs and ii its con guration Supp ose a
comp onent has r inputs and s outputs each of which lies in some space R
Let C b e the con guration space of the comp onent A simulation function
r s
for a comp onent v is a map R C R
v
Now we connect the comp onents together Assume for a moment that all
the comp onents have the same input output structure as ab ove i e that
v
r and s are xed throughout the system but that the comp onents themselves
may p erform di erent functions We mo del an edge e b etween vertices v
and u by its lab el e b and by a pair of integers i j log b is the
bandwidth of the edge Section and the index i resp j sp eci es to
which of the r outputs of v resp s inputs of u we attach e i r
and j s
Now a simulation function for this edge e is taken to b e a function
e
R R We will usually restrict the edge functions to b e the identify function
but they also check for bandwidth i e that the transmitted data has size
no greater than log b
We also de ne a resource called the output device Each sensor sys
tem must have exactly one vertex with this lab el called the ouput vertex
The output vertex of the sensor system is where the output of the sensor
is measured The simulation function for the output device is the identity
function but the output value of this device de nes the output value of the
sensor system In the examples introduced in Section the radial sensor
E lighthouse sensor system H and the p ermuted lighthouse sensor H we
lo cate the output vertex on the ship at R
r s
Comp onents that retain state can b e mo deled by a function R C S R S
v
where S is a store that records the state For example a state element with k bits of state
k
would b e mo deled with S f g Alternatively S can b e absorb ed as a factor subspace
in the con guration space of the comp onent
A simulation function for an entire sensor system U then is a col
U
lection of comp onent simulation functions such as and edge simulation
v
functions such as The function simulates all the comp onent simula
e U
tion functions in the correct con guration and simulates routing the data
b etween them using the edge simulation functions We adopt the conven
tion that two comp onents can communicate without an explicit connection
when they are spatially colo cated When all these comp onent and edge func
tions are semi algebraic then the sensor simulation function is also semi
U
algebraic see Section These concepts will b e used to implement our
notions of a sp eci cation for a sensor system Section application
and universal reductions App endix A
De nition Consider a sensor system U with simulation function
U
The output value of U at a particular con guration is the value computes
U
for that con guration Hence the output value of U is a function of U s
con guration
The notions output value and output De nition are related as fol
lows The output of U is a variable that ranges over al l possible output values
of U Given another sensor system V we say the output of U is the same
as the output of V when and are identical
U V
Under this mo del we can simulate trees of embedded sensorimotor com
putation It is also p ossible in principle to simulate more general graphs and
systems with state but in this case the value at the output vertex may vary
over time even for a xed con guration In this case we need some explicit
notion of time and blo cking to mo del the a synchronous arrival of data at
a comp onent Such extensions are considered in Jen for now we restrict
our attention to trees which su ce to mo del our examples In general our
discussion is restricted to consider one clo ck tick however generalizations
are p ossible to consider the time varying b ehavior of the system Jen
Let us relate these new de nitions to the examples from part I Examples
of comp onents are given by the resources describ ed in Section Con
nections are like data paths in that they carry information a connection s
Note the sensor system H comm in Equation is e ectively a tree and not
r
G
a graph even though there is data ow b oth from R to L and L to R This is b ecause the
output vertex u on R do es not feed back into the system
o
lab el represents the information that will b e sent along that path Connec
tions carry data b etween comp onents One common connection is sp eci ed
using the comm info primitive de ned in Section For example recall
the the p ermuted sensor system H introduced in Section Next recall
Equation
E H h
G G R
H comm E
r
G
G
Consider the sensor system sp eci ed by the b ottom r h s of Equation
H comm
r
G
In the graph representation of the edge from the virtual orientation
sensor at G to the output device at R is lab elled
r
Now for each vertex v in V we assume there is a con guration space C
v
A p oint in this space C represents a p ossible con guration of the comp onent
v
Some comp onents have con gurations that change during the op eration of
the system for example in the lighthouse sensor system all comp onents
mounted on the ship change con guration as the ship moves Others are
g
installed at xed con gurations For example the emitters in the light
w
house example are installed at a sp eci c p osition L and orientation the
white light ashes when the green light p oints North So the con guration
space C for these emitters is R S For convenience let us assume that
all comp onents have the same con guration space C and so C C for all
v
v V
To summarize a component is a primitive device that computes a a
function of i its inputs and ii its con guration z C Each comp onent is
installed at a vertex of communication graph with d vertices whose edges are
the connections describ ed ab ove The graph is immersed in a con guration
d
space C and the con guration z of a comp onent is the con guration of
its vertex More generally comp onents can b e actuators An actuator is
a comp onent whose output forces the con guration of the graph to change
or evolve through a dynamics equation If the con guration of the entire
d
graph is z z z z C then the dynamics equation mo dels
d
a mapping from the actuator comp onent v s output at z to the tangent
d
space T C to the con guration space See DJR Jen for more discussion
z
of actuators
Now we give
De nition A situated or immersed sensor system S is a sensor sys
tem S V E together with an immersion V C of the vertices If
v V then we call v the con guration of the vertex v When there is no
ambiguity we also call v the con guration of the comp onent v
A situated sensor system is mo deled by an immersed graph If the map
in De nition is injective then we call an embedding Immersions need
not b e injective In particular in order to colo cate vertices it is necessary
for immersions to b e non injective
In De nition the immersion may b e a partial as opp osed to total
function indicating that we do not sp ecify the spatial con guration of those
comp onents whose vertices are outside the domain of the immersion We
denote the domain of a partial immersion V C by C We denote
its image by im
is a di erent Example H is a situated sensor system H H
G
G
H immersion of the same sensor system H and so H
G
This example illustrates a general concept permutation of a situated
sensor system corresp onds to the choice of a di erent immersion with the
same domain Formally
De nition Let S S be a situated sensor system A p ermutation
S of S is a situated sensor system S such that the domain C of
C of are the same and the domain
Furthermore for technical reasons we also p ermit a p ermutation to
change which vertex has the output device lab el See App endix C
We can now formalize De nition to say precisely what it means for
two partially situated sensor systems to b e equivalent
Technically there are two kinds of p ermutation De nition is called vertex per
mutation in App endix A we discuss a more general mo del called graph permutation
Vertex p ermutation su ces for all examples in this b o ok but our results go through for
graph p ermutation as well
De nition Formalized Given two sensor systems S and Q we say Q
simulates S if the output of Q is the same as the output of S In this case
Q More generally suppose we write we write S
U S
for two situated sensor systems Equation is clearly wel l de n ed when
and are total Now suppose that and are partial leaving unspeci ed
the con gurations of components v of S and u of U Then Equation
is taken to mean that U simulates S for any con guration of v and
u
For De nition in the case where say is partial we op erationalize
Equation by rewriting it as a statement ab out all extensions of
That is we de ne ex to b e the set of all extensions of Then we write
ex Equation holds with bars placed over the immersions We
treat similarly with an inner universal quanti er although co designation
constraints Sections allow us to make the choice of extension
of dep end on the extension that is b ound by the outer quanti er For
example De nition b ecomes for all con gurations x C of v for all
con gurations y D x of u Equation holds Here D x is a set in C
S S
that varies with x the function D mo dels the co designation constraints
S
De nition can b e generalized to any number of unbound vertices see
Equation in Section
De nition uses a strong notion of simulation in which the outputs
of the sensor systems must b e identical A weaker notion which merely
requires the same equilibrium b ehavior is introduced in Section
Pointed Sensor Systems
Supp ose we wish to consider a sensor system S V E where one comp o
nent v for v V is in a particular con guration G C This corresp onds
to immersion via the partial function with domain f v g and range f G g
We may abbreviate the situated system S by writing S to distinguish
G
it from the un situated system S This is the notation we use in Section
and after Of course for this notation to capture all the information ab ove
ab out v we must sp ecify the preimage of G under but we did that in
Section when we wrote down
let us denote by H the sensor system H instal led with L G
G
We now explain the notation used in example First we formalize
our discussion of S ab ove
G
De nition A Pointed Immersion of a sensor system S V E is a
pair G where V C is an immersion of the vertices of S and G
im G is called the base p oint An extension of a partial pointed immersion
G is any total pointed immersion G where is an extension of
De nition A Pointed Sensor System is a triple S G where S
is a situated sensor system and G is a pointed immersion De nition
of S We abbreviate S G by S
G
Hence H in example is a p ointed sensor system Next
G
De nition A Pointed Permutation of a sensor system S is a pointed
sensor system S G where is a permutation of
Hence H in example is a p ointed p ermutation of the p ointed sensor
G
is a p ointed p ermutation of S then S is a system H In general if S
G G
G
G
p ointed p ermutation of S
G
Co designation Basic Concepts
If we view the con gurations of comp onents in a sensory system as vari
ables then convention gives a default for determining which variables
are free and which are b ound Here is another view
The partial immersion sp eci es which variables are sp ecialized to b e con
stants These are the vertices in the domain of the immersion Their con
gurations corresp ond to b ound variables constants The con guration
More precisely we must write down that the preimage of G under the immersion
contains v
G is called weakly pointed if is partial and G is not necessarily contained in im
variables for vertices outside the domain of the immersion are not yet sp e
cialized and hence are free
We now have two concepts to de ne and investigate First we show how
to sp ecify systems which contain some constant con guration variables After
that we must nd a way to make two free variables codesignate see Cha
Two vertices r and u codesignate under an immersion when r u
More generally r and u co designate under di erent immersions and when
v u We now pro ceed with these two tasks
Recall our example of a p ointed sensor system S from Section ab ove
G
Recall S S G and S V E The domain of is the single vertex
G
v V Now to continue supp ose that r V is the vertex of comp onent
r and that r v so that do es not sp ecify how to immerse r Consider
a di erent sensor system U with at least one vertex u We wish to consider
combining U and S by saying something like this
Immerse S with vertex v at G Now vertex r of S will b e somewhere
say R but we want to immerse U so that u is at R also
Hence we don t care where R is save that we wish to colo cate r and u
To do this we make r and u co designate under the immersions of S and U
We call this a codesignation constraint after Cha Here is how we may say
this more precisely
Let S denote sensor system S immersed with vertex v at G as
G
ab ove Immerse the rest of S in any consistent manner and denote
this immersion by Thus is the extension of so that the restriction
of to f v g is identical to Now let R C b e the con guration
j
f v g
of r under i e R r Denote by the partial immersion of
U de ned as follows sends vertex u of U to R Note that G is a
constant and R is a free variable in the sense that R dep ends on
which extension of we choose whereas G do es not
In Equations we abbreviated this construction as follows
S U
G R
which is short for S U with and de ned as ab ove Note that
is not su cient to sp ecify the desired partial immersion unless we also
note that the preimage under the immersion of G contains vertex v of
S and that
r R u
represents a co designation constraint we will de ne such constraints
formally b elow in Section We must also sp ecify that G is a constant
and R is a free variable The notation explained in is used in the b o dy
of the b o ok for example in Equation
It remains for us to de ne precisely the op erator we just used and we
do so b elow in De nitions b elow
Combining Sensor Systems
The op erator is de ned on two graphs as a way of taking their union
Sp eci cally
De nition Consider two graphs G V E and G V E We
de ne the combination G G of G and G as fol lows
G G V V E E
We may de ne on sensor systems De nition by lifting the de
nition for graphs We may de ne on two immersed graphs whenever the
immersions are compatible An immersion of G and an immersion of G
are said to b e compatible when the two immersions agree on the intersection
V V for total immersions or more generally on C C for partial
functions Given this de nition we have
Claim The operator de ned in De nition is associative and
commutative
u
Proof De nitional t
The General Case
Let S and U b e two situated sensor systems Let V denote the
vertices of S and U the vertices of U Our notation ab ove S U H h
G R
G R
etc is e ective when the image of each partial immersion is a singleton eg
V f G g and U f R g In these cases it su ces to abbreviate
S S and U U
G R
and to sp ecify which if any of the con gurations G and R is constant and
which if any is free We now generalize this notation for more complicated
partial immersions
Supp ose S and U have compatible partial immersions Now
V and U which need not b e singletons in general represent the
constant con guration bindings of vertices analogous to the singleton G
ab ove We now consider co designation constraints All the co designation
constraints we have seen so far in section have this form each was
a pair v u V U A co designation constraint is compatible with the
immersions and if one of the following is true
v is not in the domain C of
u is not in the domain C of
v u
This de nition is not quite general enough we must also b e able to sp ecify
a that two vertices of U resp V co designate this means two comp onents
of S must b e colo cated b we must also b e able to sp ecify that that two ver
tices not co designate for example that v u The general de nition
is complicated and is given in De nition b elow
However putting o the formal de nitions for a moment we can see
what a combined sensor system really is In summary the immersions
and sp ecify which comp onent con gurations are to b e held constant The
co designation constraints sp ecify which comp onents are to b e co lo cated
De nition Let S and U be two situated sensor systems with
compatible partial immersions The combined sensor system
S U
is speci ed by together with a set of codesignation constraints compati
ble with and We say the combination is de ned when the partial
immersions and are compatible
Now consider two sensor systems S and U Both have output vertices
say v and u resp If v u then this vertex remains the output vertex
o o o o
of S U In the case where v u we must naturally sp ecify which is the
o o
unique output vertex of the new combined sensor system By convention
we will declare it to b e either v or u we must say which We adopt
o o
one default convention for this choice in Section For more on output
vertices see App endices C C
De nition sp ecializes to the particular cases such as Equation we
have considered by appropriate choice of partial immersions and co designa
tion constraints To illustrate these choices we give an example b elow in
section The op erator is asso ciative and commutative see claim
and App endix A
Co designation Constraints
Throughout this section we let S and U b e two situated sensor
systems with compatible partial immersions V C and U C
De nition De ne the partial immersion as fol lows
C
x if x V
x
x if x U
We say the map is de ned when the partial immersions and
are compatible
De nition A co designation constraint is a pair x y V U
De nition We say a codesignation constraint x y is compatible with
the partial immersions and if one of the fol lowing is true
x is not in the domain C of
y is not in the domain C of
x y
This is not a severe restriction when we are considering p ermutations like S U of
S U See App endix C
Noncodesignation constraints are mo deled symmetrically to co designation
constraints A co designation constraint x y indicates that we require that
for any total immersion that extends
x y
holds A noncodesignation constraint requires inequality instead of equality
in
Example The Basic Idea
As an example let us interpret Equation We give it again
E H h
G G R
comm H E
r
G
G
Recall E and H are situated sensor systems E is the radial sensor
G G G
g
lo cated lo cated at G R H is the lighthouse sensor with the emitters
G
w
at G and oriented Northward
When H is situated at G as ab ove to obtain H the immersion is partial
G
leaving the p osition R of the ship unsp eci ed in H h denotes the compass
G R
installed at R calibrated towards North Equation top holds for any
ship s p osition R so long as the sensor system h is co lo cated at R Compare
R
the r h s of Equation to As in in Equation once the
preimages under the immersion of G and R are sp eci ed the immersion of
the combined sensor system b ecomes clear
Now H de nes a new immersion of H by new we mean di erent from
G
H The immersion dep ends on R but the equation holds for any R
G
comm de nes a graph with exactly one edge e e is an edge with lab el
r
e from the virtual sensor orientation to the ship the output
r
vertex at R Thus e is an edge b etween two vertices of H or H but
G
note that e is not part of the graph H nor H e is only present in the
G
combination H comm
r
G
Finally by convention Equation by itself only holds for G But we
sp ecify in the sentence b elow Equation that it holds for any G This is
equivalent to placing the symbols G b efore Equation This e ectively
frees G The app earance of G as a subscript on the l h s and b oth r h s
of Equation indicates a co designation constraint
Example continued A Formal Treatment
The Top of Equation
We now rewrite Equation using the general notation of section In this
example we do not explicitly consider orientation of comp onents However
the discussion can b e generalized by taking the con gurations G and R to
lie in the con guration space R S
Let b e a partial immersion of E Let b e a partial immersion of E
G
that installs it at G so that E E
G
G
Let b e a partial immersion of H Let b e a partial immersion of H
G
g
at G so that H H We will de ne that installs the emitters
G
G
w
co designation constraints so that all the concrete and virtual sensors are
installed on the ship i e at R
Let v and v b e the vertices of H such that v g and v w
Let u u b e the vertices of H corresp onding to the concrete and
k
virtual sensors describ ed in Section In particular u is the vertex of
the virtual sensor orientation
Let u b e the output vertex of H
o
Let b e a partial immersion of the compass h Let w b e the vertex of
the compass in h Then we can rewrite the top of Equation as
E H h top
G G
together with the co designation constraints
f u u g
i
i k
v v u u u w
o
A careful analysis will show that while it is necessary that the rotating emitter g b e
lo cated at G the omnidirectional w can b e anywhere Hence the co designation constraint
v v is unnecessary However by removing it we are left with the problem of synchro
g and w Either we must add communication or else calibrate the emitters and nizing
give w a clo ck These issues complicate the example and so we will not deal with them
further
The Bottom of Equation The Sensor System comm
Now H denotes a di erent immersion of H Call this immersion Let
G
denote the partial immersion that installs the concrete and virtual sensors at
G We will de ne co designation constraints so that the emitters are installed
on the ship We must now precisely de ne what comm means
We can b e sure of getting the semantics of comm correct by treating
it as a sensor system in its own right alb eit a small one Now comm
r
de nes the graph with vertices f u u g and a single edge e u u with
o o
e We observe that the transformation on sensor systems whereby
r
we add output communication Section and De nition implies the
following
The head vertex u of the edge e u u is de ned to be the
o o
output vertex of the sensor system comm
r
Our mo del of communication is fairly abstract External communication
is probably not p ossible without some form of bu ering by either the sender
or the receiver comm should include this bu er to b e more realistic ab out
mo deling internal state
Hence the b ottom half of Equation may b e written
comm b ot H E
r G
G
together with the co designation constraints
f u u g
i
i k
v v
Hence the b ottom co designation constraints for b ot are di erent
from the top co designation constraints for top in that in the b ottom
constraints w do es not app ear since it is asso ciated with the compass
Second in the b ottom equation the output vertex is not constrained to b e
colo cated with the virtual sensor orientation Thus the co designation
constraint u u disapp ears
o
In this example the vertices of comm are also vertices of H but more generally
the vertex sets can b e disjoint
Bandwidth and Output Vertices
We have de ned comm as a graph with a single edge e The argument
parameter b to comm b determines the bandwidth of e Thus for example
comm b sp eci es a graph with one edge e whose lab el is b This sp eci es that
the edge is a datapath that can carry information b if b requires k log b
bits to enco de then k is the bandwidth of e
Now recall the discussion on how to choose output vertices in combined
sensor systems Section Here Section Equation b ot we have
u as the output vertex of b oth H and comm and so it unambiguously
o r
remains the output vertex of the combined system H comm More
r
generally we adopt the following
Convention Let S be a sensor system Unless otherwise stated we
take the output vertex of the combined sensor system S comm to be the
output vertex u of comm
o
For more on bandwidth see App endix A for more on output vertices
under p ermutation see App endices C C
Calibration Complexity and Co designation
The size of the set or number of co designation constraints is one
measure of calibration complexity see Section However this should
b e only part of the measure One reason that the number of co designation
constraints alone is not a go o d measure is that one sensor system say H
for argument could have a single comp onent that functions in the place of
several colo cated comp onents in another sensor system say V For exam
ple we could build a sensor V as follows Consider the emitter g in H
Break up the emitter g into all its tiny wires p ower supply laments ro
tating actuator etc All these comp onents must then b e colo cated This
would result in more co designation constraints for V than for H and thus a
spuriously high measure of calibration or installation complexity
Instead in order to measure calibration complexity we should compare
size using something like order Big Oh O notation This is the basic
idea we use but there are some additional subtleties that we defer to ap
p endix A There we prop ose a measure of calibration complexity that is
more reasonable This measure retains however one useful prop erty it is
easy to compute it in fact like size ab ove it can b e computed in the same
time it takes to read the input
Nonco designation Constraints and Parametric Co designa
tion Constraints
To complete our mo del for this example we must also introduce nonco des
ignation constraints so that G R this is necessary for our sensors to work
Supp ose the radial sensor E has two vertices t and t where t is the output
o o
vertex and t is the central vertex of E this is the vertex lo cated at G in
g The nonco designation constraints for b oth top and b ot are
f u v t t g
o
The former is a constraint on H and H The latter is a constraint on E
Finally we require the codesignation constraint
t u
o o
Equation is called a parametric co designation constraint it ensures that
for all extensions resp we have of and and u
o
G G G G G
G G
u Parametric co designation constraints are discussed further t
o o
G
G
in App endix A
This completes our detailed discussion of the sensor systems in Equa
tion The example is designed to explain most facets of our theory in
a simple setting Let us sketch how to make this analysis computationally
e ective We choose two arbitrary p oints G and R in C We b egin with
the two p ointed immersions and with domains f t t g and f v u g
o
G G
resp So is total and is partial These functions and the desired
G G
p ermutation are
G
t t u v
o
G R
G
R G
G
G R
G
We want our analysis to b e true for any R and G with R G and
not just the ones we chose To do this we in e ect wish to universally
quantify over R and G and treat these con gurations as variables To do this
carefully and computationally requires the quanti cation machinery from
Section Here we give the basic idea Now after our rst use of
Equation we wrote
Equation holds for al l G
This sentence e ectively adds G to the front of Equation and
hence to Equations top and b ot We call this freeing G To obtain
this e ect we rewrite Equations top and b ot as follows Remove the
G subscripts that is replace by any immersion of E Similarly replace
G
by See Section for more details We have chosen by and
G
G
this notation b ecause our constructions are parameterized by the task and
the task is sp eci ed by G The notation leaves this parameterization explicit
As we shall see b elow p erhaps the cleanest way to mo del this example is to
treat all the sensor systems as initially unsituated yet resp ecting all the
non co designation constraints ab ove This may b e done using the to ols
developed in the sequel Sections
Generality and Co designation
Consider a sensor system S with d vertices V immersed via a map
d
V C The con guration space of this sensor system can b e viewed as C
d
since any immersion can b e represented as a p oint in C Consider a
co designation constraint u v for u v V This sp eci es a new immersion
d d
of S in a quotient C u v of C in which the images of u and v are
identi ed This quotient construction can b e used to analyze information
equivalence in certain cases We give an example b elow
In Section we discussed how general Equations and are
We can now address this question more precisely by noting that the top
and b ottom of Equation have di erent co designation constraints This
means that equivalence only holds under the appropriate spatial identi ca
tions Recall each co designation constraint sp eci es such an identi cation
Hence Equation is a relation that holds only on a quotient of con g
uration space It is analogous to a pro jective invariant in geometry an
invariant relation that holds for pro jective space but not for a ne space
V d
This just says that the function space C is isomorphic to C
To see this analogy recall that for example real pro jective space RP is
obtained as a quotient of real Euclidean space R by identifying all nonzero
p oints on a line through the origin to a single p oint There exist pro jective
relationships in RP for example invariants in pro jective geometry that
do not hold in R In our case it seems that by investigating the structure
of these quotient relations one may measure the generality of information
invariants and more generally information preserving transformations eg
reductions and immersions on sensor systems
It is interesting to note that the geometric structure of nonco designation
constraints is di erent from the quotient construction given ab ove The
d d
quotient construction can b e viewed as follows Let C C b e
d d
the pro jection of C onto C This map mo dels the quotient construction
d d
since C is isomorphic to C u v Hence mo dels the identi cation
e
of u and v then induces a new immersion
d
C
y
e
C u v
One the other hand nonco designation constraints are essentially a kind
of genericity requirement To see this let us assume that u and v are the
rst and second of the d vertices of V We then consider an immersion to
b e generic when it sends u and v to di erent values De ne the diagonal
f z z C j z C g Then the nonco designation constraint insists
that we avoid the embedded diagonal that is we must have an immersion
d
C C
Combining and gives the general form for the con guration space
of the sensor
More General Co designation Relations
The Semantics of Co designation Constraints
The co designation constraints we have encountered so far mo del the neces
sary equality of images of vertices under immersions For example
u v
for some particular u U and v V
U
C
V
Let us call this simple kind of co designation constraints in equality
co designation constraints
More generally we could consider relations of the form The three p oints
z u and v are colinear or u is within distance d of v etc
This other kind of co designation constraints could b e called general codes
ignation relations We could mo del such a relation as follows consider a
triple u v where is a semi algebraic s a predicate on C C So far
in considering equality co designation constraints all the predicates we have
used have b een diagonals
x y i x y
This choice explicitly enco des the assumption that all working sensor
con gurations can b e sp eci ed using colo cation or noncolo cation For ex
ample for the lighthouse sensor H it is necessary for the green and white
g
lights to b e colo cated Similarly the sensor only works when the ship
w
R is not at G These statements give geometric constraints on the sensor
semantics the non co designation constraints sp ecify what non colo cations
must o ccur for the sensor to function prop erly Hence equality co designa
tion constraints such as enco de the assumption that the only geometric
characteristic that a ects sensor semantics is the colo cation of comp onents
Obviously this is not true for all sensors but it is true for the sensors we have
considered in this b o ok We call such sensors simple and they are worth a
de nition De nition b elow
More generally we could in principle require general co designation re
lations to hold b etween comp onent con gurations or more generally it
may b e true that there exist relationships other than in equality that must
For a nonco designation constraint we complement the diagonal
hold for the sensors to function prop erly In this b o ok we primarily discuss
simple sensor systems and only in Sections do we consider the
rami cations of such extensions However we feel our framework could and
should b e extended to handle at least restricted algebraic co designation To
see how this would go assume for a moment semi algebraic s a predicates
for general co designation relations The e ect of general co designation re
lations would b e geometrically as follows First for a nonco designation
constraints the forbidden diagonal would generalize to b e an arbitrary va
d
riety Y in C Y would b e characterized by some p olynomial inequalities and
immersions Y would b e forbidden For general co designation relations
d
we would construct a quotient whereby p oints in C would b e identi ed via
an algebraic map a p olynomial equation The geometry of such spaces can
b e complicated however from a theoretical p oint of view a line of attack
can b e seen
We can summarize this discussion with a de nition that captures the kind
of sensor systems this b o ok addresses
De nition A sensor system that can be speci ed using only a nite
number of equality codesignation and noncodesignation constraints is called
simple A sensor system that can be speci ed using only a nite number of
semi algebraic predicates in its general codesignation and noncodesignation
constraints is called algebraically co designated
Since is algebraically co designated all simple systems are algebraic
co designated We consider only simple sensor systems in Sections
However the algorithms in Section apply to all algebraically co desig
nated systems
The Semantics of Permutation
The semantics of p ermutations is intimately b ound up in the semantics of
co designation We now discuss the connection The results of this section
not only clarify our semantics but also lead to a computational result which
we describ e later in Section
The meaning of a p ermutation see De nition is clear for a totally
situated sensor system i e a sensor system with a total immersion Recall
from section that we can view an immersion and its p ermutation
d
as elements of the con guration space C Now supp ose for a moment
d
that for every immersion C it is p ossible to choose a p ermutation
d
satisfying De nition Imagine that for each C we build a
d
sequence of such choices f g C where This
i
i
de nes a map
d d d
C C C
Hence a p ermutation can b e viewed as a way of p ermuting the comp o
nents of a sensori computational system or it may b e viewed as a kind of
automorphism of sensor con guration space
Now supp ose we now allow to b e a partial immersion Then by a
p ermutation of we mean a di erent partial immersion with the same
domain the de nition still applies
Permutations of a partial immersion have a structure that is related to
co designation constraints in that each can b e characterized geometrically
d
via regions in C Consider a partial immersion Given we can de ne
the set of extensions of
d
ex f C j g
j
C
d
which is a region in C A p ermutation of corresp onds to selecting a
d
new region ex of C with this prop erty
T T
C C C C
ex
ex
Now it would b e convenient if we could treat the regions ex and ex
d
like equivalence classes in C That way we could view and as the
generators of di erent classes of immersions A partial function then corre
d
sp onds to a region in C and p ermutation corresp onds to choice of a di erent
d
region in C To take this view we need the following
Prop osition Let be a permutation of Then ex and ex are
disjoint unless
d
We defer the necessity of quotienting C and removing diagonals until Section
The choice will not in general b e unique
Proof Let ex ex Since is an extension of b oth and we
have
j j
C
C
But is a p ermutation of which implies that and have the same
u
domain De nition Since C C therefore t
Let denote all p ermutations of Essentially prop tells us
d
that the map ex f Regions in C g has an injection like prop erty
the images of distinct p ermutations under ex do not intersect The map ex
also has a surjection like prop erty which we characterize as follows
Claim Let V C where is partial and is total Then there
exists a permutation of such that is an extension of
u
t Proof Take
j
C
Prop osition Fix a partial immersion The images of ex
d d
f Regions in C g cover C that is
d
ex C
u
Proof Immediate from claim t
We can summarize this as follows we have viewed p ermutation as a
bijective self map of It is equivalent to view p ermutation as a bijective
self map of the disjoint equivalence classes
f ex g
d
for all p ermutations of in C This viewp oint is justi ed by the
following two claims
Prop osition The map
d
p C
s t ex
is wel l de ned
Proof Observe that p see claim The map p is de ned
j
C
d
for every C by props and That p is uniquely de ned
u
by we see from prop t
Now supp ose the domain C of contains k vertices k d We
can represent any p ermutation of by the k images z z of the
k
vertices of C under That is we can represent any such p ermutation
k k
by a p oint in C Conversely any p oint in C de nes a p ermutation
Lemma The fol lowing properties hold
k
C
The map p is a projection and we can give it in C coordinates as
d k
p C C
z z z z z
k d k
d
Let be a permutation of Then ex C is a cylinder over
k
C and ex p
The map p is a quotient map
d k
C p C
u
Proof De nitional t
Finally we note that our discussion of p ermutation for partially immersed
sensor systems can b e sp ecialized to p ointed sensor systems and p ointed
p ermutation with the same base p oint If is a p ointed p ermutation of
with p oint G then the classes ex and ex have these additional prop erties
see De nition
T T
G im im G im im
ex ex
We use to denote isomorphism
For p ointed sensor systems the surjection like prop erties props and only
hold for the class of p ointed immersions with the same base p oint
Thus for partially immersed systems we have a handle on p ermutation
and now we know more precisely what the di erence b etween eg H and
G
is see Section in terms of p ermutation Permutation corresp onds H
G
d
to choosing a di erent equivalence class of C For most of this b o ok we
examine a sp ecial case where the sensor systems are partially situated that
is the domains of the immersions are non empty A p owerful generalization
is given in Section where the sensor systems can b e unsituated This
will allow us to understand the unsituated sensor system H precisely as a
p ermutation of the unsituated system H
The Semantics of Reductions
Recall the de nition of e ciently reducible De nition To explore this
notion we rst turn to the question of whether or not the relation in
De nition is transitive
Consider three sensor systems U V and W and their p ermutations
Sensor System Vertices Immersion Permutation Permutation
U U f u u g U U
V V f v v g V V V V
W W f w w g W W W W W W
If is transitive then if U V and V W then U W We
explore when this prop erty holds From De nitions and we can see
that dominance in calibration complexity De nition is transitive and
so we will concentrate here on the less obvious asp ects of transitivity To
simplify the discussion we only deal with co designation constraints but the
argument generalizes mutatis mutandis for nonco designation constraints
Weak Transitivity
First let us observe that always ob eys a prop erty that is like transitivity
but weaker We now elab orate Supp ose U V Then De nition
there exists some p ermutation V V of V such that U V see
De nition for the de nition of So we have
Other p ermutations are p ossible only a couple are shown
See Sections and A for more on computational calibration complexity
U V
Now supp ose V W Then there exists a p ermutation W W
such that
V W
From and and the de nition of De nitions we have
U W
and therefore U W This prop erty we call weak transitivity
Strong Transitivity for Simple Sensor Systems
Simple sensor systems De nition ob ey strong transitivity so long as
all p ermutations are chosen to ob ey their co designation constraints Supp ose
U V and W are all simple If is transitive then if U V and V W
then U W In other words
Supp ose U V and V W Then there exist p ermutations V
V of V and W W of W such that
U V
and
V W
Compare with Then if is transitive then there exists
another p ermutation W W of W such that
U W
Strong transitivity is a much stricter condition than weak transitivity
It requires that we b e able to comp ose the immersions and to
somehow construct the immersion This may not in general b e p ossible
However it is p ossible for simple sensor systems in which only equality co des
ignation constraints are employed to sp ecify the system De nition
In order for strong transitivity to hold we must make sure that b oth the
p ermutations and for V and V resp ect the co designation constraints
for V s semantics This is b ecause we cannot exp ect any p ermutation of W
to simulate U if either or are faulty con gurations of V We call an
immersion of V valid if resp ects the co designation constraints for V
This corresp onds to restricting to the valid regions of sensor con guration
d
space C as in Sections and We call a p ermutation V V of
V valid if its immersion is valid In this case we also say that the sensor
system V is valid
Lemma The relation De nition is transitive for valid simple
sensor systems De nition
Proof Assume there exist valid p ermutations and so that
and hold as ab ove We construct an immersion so that holds
The picture we have is as follows
V
y y
U C
x x
W
Consider g Certain vertices for example v and u are colo cated
Co designation implies colo cation but the converse is not necessarily true In
constructing a new immersion we must simulate all colo cations b ecause that
way we will b e sure to repro duce all co designation constraints accurately in
the new immersion Because only colo cation a ects sensor semantics for
simple sensor systems De nition this su ces to ensure that the new
immersion preserves the sensor semantics In short colo cation is evidence
for co designation
We want to construct as follows see g
W C
w v if v w
i j j i
u v w
U
W
V
u
v
w
W
v
v
w
w
V
Figure The situated sensor systems U U V V V V W
W and W W for lemma The vertices of U V and W are U
f u u g V f v v g and W f w w g resp Not all vertices are shown
w v u and w v u v w and v
w
The general form of the set of colo cations that must simulate is
W V This construction is general and can b e expressed as follows
Let
f W V C
w w
i i
The map f is almost the map we want When the image of f is a one
p oint set f z g we de ne w z If w V is not a singleton
i i
see g then we have a choice in the construction of In this case
we know that w f w Since f w is nite we can enumerate all
i i i
u
p ossible candidates for one of them will b e the correct one t
We note that our pro of is not constructive we only prove there exists
u v
U
V
u
v
V
v
w
W v
w
Figure The case where w is not a singleton in this case it is f v v g
i
V In this example v v w Now we note that v and v colo cate under
but not under However this di erence cannot b e semantic i e it cannot a ect the
sensor function since we assume that b oth p ermutations are chosen to b e valid w r t the
co designation constraints for V In other words v v is not a co designation constraint
for V in this example
a p ermutation W However we can give a pro cedure for enumerating the
nite number of candidates for the p ermutation It is p ossible to check
which is the correct one by applying the results of the next section Sec
tion
We do not b elieve that the relation holds for arbitrary algebraically
co designated sensors This is b ecause the algebraic constraints may b e in
compatible It would b e of interest to nd a restricted class that is larger
than equality co designation for which transitivity holds
A Hierarchy of Reductions
We now use our study of s transitivity to understand the reduction Def
inition Now even when is transitive it app ears that is not To
see this supp ose that A B and B C Then it app ears that to reduce
A to B we require one extra wire namely comm A and that to reduce
B to C we could require another extra wire comm B and therefore in
the worst case to reduce A to C we could require two extra wires That is
it could b e that A cannot reduce to C with fewer than two extra wires We
have yet to nd a non arti cial example of this lower b ound but it app ears
to indicate that is not transitive even for simple sensor systems
Let us summarize The reduction De nition is a wire reduc
tion It do es not app ear to b e transitive The reduction De nition is
a wire reduction It is transitive for simple sensor systems lemma
We could analogously de ne a wire or more generally any k wire reduction
by mo difying Equation in De nition to
k
S Q k comm b
k times
z
where k comm b denotes comm b comm b
Since this suggests a hierarchy of reductions indexed by k
In general we have the following
De nition We say a relation is transitive when x y and y z
always implies x z To distinguish this from graded transitivity below
we call this elementary transitivity when necessary
X X
We say a map F N with F i is a graded relation on
i
X X when each is a relation on X X We also write F as f g
i i
i N
We say that F has graded transitivity or is graded transitive if the
fol lowing property holds For every x y z X if x y and y z then
i j
x z
i j
Clearly the k wire reductions f g form a graded relation
i
i N
I would like to thank Ronitt Rubinfeld for contributing key insights to this discussion
of k wire reductions
Elementary transitivity is the sense used in lemma
Corollary a The wire reduction called in De nition
is elementary transitive for simple sensor systems
b The k wire reductions f g are graded transitive De nition
i
i N
for simple sensor systems
Proof a is de nitional from lemma To see b we use lemma and
recall De nition and that the op erator is asso ciative and commutative
claim To complete the argument we also need a technical lemma
u
given by the distributive prop erty of prop C t
We call the k wire reductions f g a hierarchy of reductions We say
i
i N
such a hierarchy i e any graded relation on X collapses if it is isomorphic
to an elementary relation i e to a single subset of X
Corollary The hierarchy of k wire reductions k on simple sensor
systems collapses if is elementary transitive on simple sensor systems
Proof Supp ose X Z k To collapse the hierarchy it su ces to
k
show that X Z This follows from lemma by observing that the
op erator is commutative and asso ciative and by the distributive prop erty
of prop C
Now construct k sensor systems Y Z i comm b where log b
i X X
mb X for i k Hence each of the i extra wires in Y has band
i
width log b see Sections and De nition to see that this
X
yields su cient bandwidth see De nition So there exist k simple
sensor systems Y Y Y with k more wires than Z resp such
k
that X Y Y Y Z Recall that and
k k
u
observe that If and are transitive then X Z t
For monotonic sensor systems we can simply take b to b e the output
X
of X see Section Cor is stated for simple sensor systems but
it holds for the more general algebraic systems in which case each Y is
i
algebraic but not necesssarily simple
See app endix A
A Partial Order on Simple Sensor Systems
In this section all sensor systems are assumed to b e simple
De nition We write U V if there exists some integer k such that
U V
k
As a reduction corresp onds to adding an arbitrary amount of global
p oint to point communication It is easy to see that is elementary tran
sitive for simple sensor systems
In a multi agent sensor system it makes sense to allow the size i e
number of comp onents of the system to grow and to consider reductions
parameterized by that size For example given a sensor system U we can
use the notation i U to denote i copies of U Now even if for another
sensor system V we have U V it is unlikely that we will have i U i V
for all i N However it is easy to see the following
u
Claim If U V then for any i j N i U j V t
k
The family f i U g is just one example of such a system we could
i N
imagine other examples where the number of comp onents number of agents
or number of sensors varies with i Our emphasis has changed slightly from
the preceding Before we asked what k N su ces such that U V
k
Now we ask to nd that k as a function of the size of U and V
Now we might deem it unfair to add arbitrary communication to the
system Let us instead consider adding only a p olynomial amount of com
munication In De nition U and V are data and q is a xed p olynomial
n is the size of U and V eg take n to b e the total number of comp onents
q n a function of n is the amount of communication su cient to reduce
U to V
De nition Let U and V be sensor systems We write U V if there
P
exists some xed polynomial function q n of the size n of U and V such that
U V for al l sizes n
q n
So the assertion U V is a statement ab out a family of sensor
P
systems It says that U reduces to V by p ermuting V and adding an amount
of communication that is p olynomial in the size of U and V In particular
note that if U V then for any i j N i U j V However we can
P P
say something stronger
Lemma Completeness of Polynomial Communication U V if
P
and only if U V
Proof If is trivial we show the only if direction If U and V have at most
n vertices then global p oint to point communication can b e implemented by
adding O n new datapaths Hence it is always true that U V Any
O n
additional communication would b e sup er uous and would not add p ower to
u
the system t
It follows that is elementary transitive on simple sensor systems
P
Therefore it is a partial order on simple sensor systems
Computational Prop erties
In this section we give a computational mo del of simulation De ni
tion and discuss an algorithm for deciding the relations and This
section relies heavily on the results of the previous section Readers
unfamiliar with algebraic decision pro cedures may wish to consult the review
in app endix where we review some basic facts ab out semi algebraic sets
This section also yields b ene ts in terms of clarity For example p ointed im
mersions are a somewhat awkward way to sp ecify co designation constraints
the machinery of this section enables us to disp ense with them in an elegant
matter
Algebraic Sensor Systems
The algorithms in this section are algebraic and use the theory of real
closed elds In the rst order theory of real closed elds we can quantify
over real variables but not over functions This might seem to imply that we
cannot quantify over immersions of sensor systems since these immersions
are functions However since our immersions have a nite domain each
immersion function can b e represented as a p oint in a con guration space
d
C Therefore we can quantify over them in our algebraic theory We now
pro ceed to use this fact
De nition We say a function is semi algebraic when its graph is semi
algebraic
Consider a situated sensor system U and for the moment assume
that is a valid immersion that is semi algebraic s a and total Let us
de ne the size d of U to b e the number of vertices in U Now
d
De nition A simulation function for U is a map C R
U U
where R is a ring We call the value R of on a sensor con gu
U U
ration to be the sensor value or output value at
Simulation functions compute the value of the sensor given a con gura
tion of the sensor The idea is that we can apply a simulation function to
determine what value the sensor will return what the sensor will compute
in con guration De nition also formalizes our notion of a sp eci
cation for a sensor system alluded to in the context of design Section
application See Section and app endix A for more on simulation
functions
Example Recall the lighthouse sensor system H g A simu
lation function for H computes the value of We imagine works
H H
by simulating the orientation sensor see Section Other equiva
lent simulations are also possible equivalent means they compute the same
value for For example let x h R S be the con guration of the
ship R Let L R S be the con guration of the lighthouse Then
tan x L We note that this simulation function is not algebraic
because arctangent is not algebraic See example below
Now if the con guration space C is algebraic then so is the function
d
space C Hence every immersion of U with algebraic co ordinates can b e
d
represented as an algebraic p oint in C So is algebraic exactly when it
can b e represented as such an algebraic p oint
d
Now let T b e a predicate on C in the theory of real closed elds Then
T is either true or false and we can decide it by applying T to
Next supp ose we now p ermit to b e partial We call a partial func
tion semi algebraic when its restriction to its domain C is semi
j
C
d
algebraic If is semi algebraic then the set of its extensions ex C is
also semi algebraic We then observe that the expression denoting for all
extensions resp there exists an extension of T holds namely
ex T
is also semi algebraic f g To quantify over all extensions of
we simply quantify over the con gurations of the vertices outside the domain
of By Section we can also guess p ermutations of that is it
is p ossible to existentially quantify over p ermutations and hence to decide
sentences of the form
T
which means there exists a p ermutation of such that for any extension
of T holds That is
ex T
To guess a p ermutation of we existentially quantify over the con gurations
of vertices inside the domain of
Example Let C be an algebraic con guration space Let V be a set of
three vertices V f v v v g Now we can encode any algebraic func
tion V C semi algebraically eg by a set of three ordered pairs
f v z v z v z g where v z i Let us call such a
i i
s a representation of by the name z z z
z z z f V C j v z i g
i i
Now consider a partial immersion V C with domain f v g such
that v G where G is algebraic We can encode as
z z G z z
We can also encode the extensions and permutations of semi algebraically
Speci cally we can encode any permutation of by a single point z the
image of v we can encode any extension of by a pair z z the
images of v and v respectively
Thus we can rewrite as
z z z T z z z
We call the existential quanti cation guessing since deciding a predicate in the
existential theory of the reals is like guessing a witness to make the predicate true
If C has dimension r then the formula is a Tarski sentence in r
c c
variables
We summarize
Prop osition If V C is a semi algebraic partial function then
the set ex s extensions and the set s permutations are also
u
semi algebraic t
To guess a valid p ermutation De nition we restrict the con g
urations to lie within the algebraic co designation constraints as describ ed
in Sections and We are simply using algebraic decision pro ce
dures to make these choices e ective Any s a co designation constraints for
an algebraically co designated sensor system can b e represented by a s a set
r
D R The structure of the region D esp ecially in relation to the region
ex is discussed in Sections and We must restrict our choice of
p ermutation to D To guess a valid p ermutation we mo dify to b e
D ex T
De nition We call a sensor system U algebraic if it is algebraically
codesignated De nition has an algebraic con guration space C and
a semi algebraic algebraic simulation function def
U
Example Recall example above The simulation function in
H
ex is not algebraic However we can de ne a semi algebraic simu
lation function that encodes the same information and is adequate in the
sense that we can use it to compare the sensor H s function to another ori
entation sensor The algebraic simulation function we give now is adequate
to decide the relation
To construct an algebraic version of we use a simple trick from cal
H
culus also used in kinematics see for example DKM Let be a con
tan guration of sensor system H g De ne q where
H
see ex and q denotes which quadrant R is in relative
H
encodes the same information as but it is semi algebraic to L
H
H
We wil l not prove is algebraic but here is a brief argument Substitute
H
Then we have sin u u and cos u u u tan
and our simulation function is a rational function By clearing denominators
we obtain an algebraic function See DKM for details Essential ly the graph
is a s a set in correspondence with graph of the non algebraic map of
H
H
The correspondence is given by u
Computing the Reductions and
Now supp ose we have two algebraic sensor systems U and V We wish to
decide whether U V If U U and V V then we wish to
decide whether there exists a p ermutation such that
U V
Here in Section the relation is used as in De nition That is
we wish to decide the following assume that and are partial
ex ex
U V
Equation do es not incorp orate the co designation constraints Since
U and V are algebraically co designated their co designation constraints may
d
b e represented as semi algebraic sets D D and D in C So
U V VU
b ecomes
D ex D ex D
U V
V U VU
Note that V s co designation constraints dep end on that is the s a set
D is a s a function of This technicality is necessary to allow for
VU
su cient generality in sp ecifying co designation and is explained further in
app endix A
Using Grigoryev s algorithm thm A we can decide in the fol
lowing time We use to compute the time b ound Let n b e the size of
the simulation functions and Let r b e the dimension of C Let n
U V
c
D
b e the size of the s a predicates for the co designation constraints D D
U V
and D In the outer existential quanti er binds some number k d
VU
vertices of V that are in the domain of The inner universal quantifer binds
the remaining d k vertices of V The middle universal quanti er binds up to
d vertices of U Hence we see there are at most r r d variables and there
c
are a alternations Let us treat the maximum degree as a constant
The predicate has size m n n Therefore we can decide in time
D
a
O r O r d
c
m n n
D
De nition Consider an algebraic sensor system U with d vertices
Recall we call d the size of U We call the size n of a sensor simulation
function the simulation complexity We call the size n of the codesigna
U
D
tion constraints for U the co designation complexity We call U small if n
O
and n are only polynomially large in d i e n n d
D D
Now let us assume that it is p ossible to compute dominance in calibration
complexity see De nition in a time that much faster than see
app endix A for how Then we see the following
Lemma There is an algorithm for deciding the relation De ni
tion for algebraic sensor systems It runs in time polynomial in the
simulation and codesignation complexity n n and sub doubly exponen
D
tial in the size of the sensor systems That is if the system has size d the
time complexity is
O
r d
c
n n
D
where r is the dimension of the con guration space for a single component
c
u
t
Corollary For smal l sensor systems De nition of size d there
is an algorithm to decide the relation in time
O
r d
c
d
u
t
Recall all small systems are algebraic
Corollary For algebraic sensor systems the relation De nition
can be decided in the same time bounds as in lemma and cor
Proof Consider deciding S Q comm b as in De nition Recall
the de nition of compatibility for partial immersions Section We rst
observe that p ermutation the op eration and combination the op era
tion commute for compatible partial immersions This is formalized as
the distributive prop erty shown in prop C We have already shown
how to guess a p ermutation Q of Q Our arguments ab ove for guessing
extensions and p ermutations can b e generalized mutatis mutandis to com
pute the combination De nition of two algebraic sensor systems Since
comm b is a constant sized sensor system vertices one edge with only a
constant number of co designation constraints at most we may guess how
to combine it with a p ermutation Q of Q within the same time b ounds given
in lemma and cor To complete the pro of we require a technical
argument given in app endix A on how to simulate a p ermuted sensor
u
system t
See app endix A
Unsituated Permutation
In Section we examined a sp ecial case where U and V are par
tially situated that is the domains of and are non empty We now
give a p owerful generalization in which the sensor systems can b e unsitu
ated Using the ideas in Sections and we can give an abstract
version of p ermutation that is applicable to partially immersed sensor sys
tems with co designation constraints Each set of co designation constraints
de nes a di erent arrangement in the space of all immersions Each cell in
d
the arrangement in turn corresp onds to a region in C
Permutation corresp onds to selecting a di erent family of immersions
while resp ecting the co designation constraints Since this corresp onds to
d
choosing a di erent region of C the picture of abstract p ermutation is really
not that di erent from the computational mo del of situated p ermutations
discussed in Section Supp ose a simple sensor system U has d vertices
two of which are u and v When there is a co designation constraint for u
and v we write that the relation u v must hold This relation induces
d d
a quotient structure on C and the corresp onding quotient map C
d
C u v identi es the two vertices u and v Similarly we can mo del a
d
non co designation constraint as a diagonal C that must b e avoided
d
Abstract p ermutation of U can b e viewed as follows Let D C u
U
d
v D is the quotient of C under For a partial immersion to b e
U
chosen compatibly with the co designation constraints we view p ermutation
as a bijective self map of the disjoint equivalence classes
f ex g
Thus in general the group structure for the p ermutation must resp ect the
quotient structure for co designation corresp ondingly we call such p ermuta
tions valid Below we de ne the diagonal precisely
Now an unsituated sensor system U could b e mo deled using a partial im
d
mersion with an empty domain In this case ex C and Equation
sp ecializes to the single equivalence class f D g In this singular case we
U
can take several di erent approaches to de ning unsituated p ermutation i
Although consistent with situated p ermuta We may de ne that
tion i is not very useful We choose a di erent de nition For unsituated
p ermutation we rede ne and ex in the sp ecial case where has
an empty domain ii When U is simple we may de ne to b e the set
d
of colo cations of vertices of U That is let x x b e a p oint in C and
d
th
de ne the ij diagonal f x x j x x g De ne p ermutation
ij d i j
d
as a bijective self map of the cells in the arrangement generated by all
d
such diagonals f g So is an arrangement in C of com
ij i j d
dr
c
plexity O d ex is a cell in the arrangement and ex
is a representative of the equiva is a witness p oint in that cell Hence
As in situated p ermutation unsituated p ermutation can lence class ex
b e viewed as a self map of the cells f ex g or equivalently as a self map of
g Perhaps the cleanest way to mo del our main examples the witnesses f
is to treat all the sensor systems as initially unsituated yet resp ecting all
the non co designation constraints This may b e done by algebraically
sp ecifying all the co designation constraints letting the domain of each
immersion b e empty using ii ab ove choose unsituated p ermutations
that resp ect the co designation constraints The metho ds of Section
can b e extended to guess unsituated p ermutations In our examples each
guess i e each unsituated p ermutation corresp onds to a choice of which
vertices to colo cate All our computational results including our b ounds
in Section can b e shown to hold for unsituated p ermutation by a simple
extension of the arguments ab ove
The co designation relation u v the quotient map the non co designation relation
and de nition ii of unsituated p ermutation can all b e extended to algebraic sensor
systems using the metho ds of Section
Example of Unsituated Permutation
Unsituated p ermutation is quite p owerful Consider deciding Equation
in this example we only consider vertex p ermutation of simple sensor sys
tems In particular we want to see that makes sense for unsituated
p ermutation when we replace by by etc to obtain
D ex D ex D
U V
V U VU
With situated p ermutation we are restricted to rst choosing the
partial immersion and thereby xing a number of vertices of S Next we
can p ermute U to b e near these vertices this corresp onds to the choice of
This pro cess gets the colo cations right but at the cost of generality
we would know that for any top ologically equivalent choice of we can
choose a p ermutation such that holds For simple sensor systems
top ologically equivalent means with the same vertex colo cations
Unsituated p ermutation allows us to do precisely what we want In
d
place of a partial immersion for S we b egin with a witness p oint C
represents an equivalence class ex of immersions all of which colo cate
the same vertices as So says which vertices should b e colo cated but
not where Now given the outer existential quanti er in chooses an
of represents an equivalence class ex of U unsituated p ermutation
do es The immersions of U all of which colo cate the same vertices of U as
d
other disjoint equivalence classes are also subsets of C each equivalence
class colo cates di erent vertices of U and the set of all such classes is
Choice of selects which vertices of U to colo cate The co des
ignation constraint D then enforces that when measuring the outputs of
S
S and U we install them in the same place More sp eci cally given
determines as data determines which vertices of S to colo cate choice of
which vertices of U are colo cated construction of D determines which
S
vertices of U and S are colo cated Most sp eci cally given the con guration
d
of S D in turn de nes a region D in the con guration space C
S S
of U This region constraints the necessary coplacements of U relative to
S
pushing direction
C line of pushing
COF
B
y L
0
a x pushing
R direction
do forever
let measure torque
COF
push y
B cond zero
negative move x
y
L
positive move x
b
R
x
Figure Proto col P for a two ngered
gripp er
Figure a the two nger pushing task vs b the two
rob ot pushing task The goal is to push the blo ck B in a straight
line
Application and Exp eriments
This section will b e expanded and revised
We now describ e an application of the theory in this b o ok presented
in DJR This work is still preliminary but we describ e it here to give some
feeling for the p otential of our theory The pap er DJR relies heavily on the
results and metho ds introduced here DJR quantify a new resource f
How much information is encoded or provided by the task mechanics The
theme of exploiting task mechanics is imp ortant in previous work One
could de ne exploiting task mechanics for rob ot manipulation as taking
advantage of the mechanical and physical laws that govern how objects move
For example see the discussion of Mas EM Erd in part I
and change Currently in our framework the mechanics are embedded in the
geometry of the system In DJR we developed information invariants that
explicitly trade o f with resources a e from the abstract in the style
of the preceding Developing such invariants is quite challenging We close
with an example This example op ens up a host of new issues see DJR for
details
Fig a depicts a two nger planar pushing task The goal is to push
the b ox B in a straight line pure translation The two ngers f and f
are rigidly connected for example they could b e the ngers of a parallel jaw
gripp er One complication involves the micro mechanical variations in the
slip of the b ox on the table This phenomenon is very hard to mo del and
hence it is di cult to predict the results of a one ngered push we will only
obtain a straight line tra jectory when the center of friction COF lies on the
line of pushing However with a two ngered push the b ox will translate
in a straight line so long as the COF lies b etween the ngers The nice
thing ab out this strategy is that the COF can move some and the ngers
can keep pushing since we only need ensure the COF lies in some region
C see g a instead of on a line Second if the COF moves outside C
then the ngers can move sideways to capture it again For example DJR
implemented following control lo op on our Puma Sense the reaction torque
about the point in g a If push forward in direction y If
move the ngers in x else move the ngers in x See g
From the mechanics p ersp ective it might app ear we are done However
it is di cult to overstate how critically the control lo op g relies on
global communication and control Now consider the analogous pushing
task in g b Each nger is replaced by an autonomous mobile rob ot with
only lo cal communication con gured as describ ed in Section of part I
Each rob ot has a ring of one bit contact bump sensors In addition by
examining the servo loop in RD it is clear that we can compute a measure
of applied force by observing the applied p ower the p osition and velocity of
the rob ot and the contact sensors
Pushing Reorientation
Task Task
global co ordination
Now we ask how can the system in g b ap
R and control P
proximate the pushing strategy g ab ove We
lo cal IR communication
P R partial synchrony
observe the following Each rob ot can compute its ap
MPMD
plied force and contact mo de and communicate these
uniform SPMD
P R asynchronous
data to the other The rob ots together must p erform
no explicit
a control strategy move iny move in x etc Since
communication
the rob ots are not rigidly linked there are ve qual
Figure Summary
itative choices on how to implement a move in x
of parallel manipulation
proto cols from DJR
Our exp eriments suggest these strategies are aided by
the ability to sense the b ox s surface normal and to
compliantly align to it The IR Mo dem mechanism describ ed in part I allows
the communication of the following information each rob ot s identity orien
tation and sp eed In addition here are several kinds of information a rob ot
might transmit for the pushing task whether it is in contact with the b ox
the contact b earing where the contact is on the bump er ring the p ower
b eing applied to the motors and the lo cal surface normal of the b ox Next
a rob ot could communicate the message Do this strategy or else I
am ab out to do this strategy Finally the rob ots may have to transmit
communication primitives like Wait and Acknowledged
While it is p ossible to sp ecify and indeed implement su cient communi
cation to p erform this task robustly it is di cult to convince oneself that
some particular communication scheme is optimal or indeed even necessary
In DJR we analyze information invariants for manipulation tasks using
the formalism presented here For example it is clear the surface normal
computation requires some internal state and the compliant align can b e
viewed as consuming external state or as temp orary calibration Communi
cation app ears fundamental to p erforming the task in g b So we ask
what communication is necessary b etween the rob ots to accomplish the
rob ot pushing task How many messages and what information is required
In DJR we use the metho ds introduced here to compare and contrast push
ing proto cols and to answer these questions First we precisely describ e
two manipulation tasks for co op erating mobile rob ots that can push large
heavy ob jects One task is shown in g b the other in g More
sp eci cally we ask Can al l explicit local and global communication between
the agents be removed from a family of pushing protocols for these tasks
DJR answer in the a rmative a surprising result by using the general
metho ds introduced here for analyzing information invariants
DJR Use Circuits and Reductions to Analyze
Information Invariants
In DJR we develop and analyze synchronous and asynchronous manipula
tion proto cols for a small team of co op erating mobile rob ots than can push
large b oxes The b oxes are typically several rob ot diameters wide and
times the mass of a single rob ot although the rob ots have also pushed couches
that are heavier p erhaps times the mass and rob ot diameters in
size We build on the ground breaking work of Mason EM and others
on planar sensorless manipulation Our work di ers from previous work on
pushing in several ways First the rob ots and b oxes are on a similar dynamic
and spatial scale Second a single rob ot is not always strong enough to move
the b ox by itself sp eci cally its strength dep ends on the e ective lever
arm Third we do not assume the rob ots are globally co ordinated and con
trolled More precisely we rst develop proto cols based on the assumption
that lo cal communication is p ossible and then we subsequently remove that
communication via a series of source to source transformations on the pro
to cols Fourth our proto cols assume neither that the rob ot has a geometric
mo del of the b ox nor that the rst moment of the friction distribution the
COF is known Instead the rob ot combines sensorimotor exp eriments and
manipulation strategies to infer the necessary information the exp eriments
have the avor of JR Finally the pushing literature generally regards the
pushers as moving kinematic constraints In our case b ecause i there are
at least two rob ot pushers and ii the rob ots are less massive than the b ox
the rob ots are really force appliers in a system with signi cant friction
Of course our proto cols rely on a number of assumptions in order to
work We use the theory of information invariants developed here to reveal
these assumptions and exp ose the information structure of the task We b e
lieve our theory has implications for the parallelization of manipulation tasks
on spatially distributed teams of co op erating rob ots To develop a parallel
manipulation strategy rst we start with a p erfectly synchronous proto col
This question was rst p osed as an op en problem in a draft of this b o ok
with global co ordination and control Next in distributing it among co op
erating spatially separated agents we relax it to an MPMD proto col with
lo cal communication and partial synchrony Finally we remove all explicit
communication The nal proto cols are asynchronous and essentially uni
form or SPMD the same program runs on each rob ot Ultimately the
rob ots must b e viewed as communicating implicitly through the task dynam
ics and this implicit communication confers a certain degree of synchrony
on our proto cols Because it is b oth di cult and imp ortant to analyze the
information content of this implicit communication and synchronization we
b elieve that using our theory of information invariants is justi ed
The manipulation proto cols in DJR are rst mo deled as circuits us
ing the formalism developed in Section Source to source transforma
tions on these proto cols are then represented as circuit transformations The
circuit transformations are mo deled using the reductions describ ed in this
b o ok For the task in g b DJR consider three pushing proto cols P
P and P and their interreducibility under In particular we trans
form an MPMD pushing proto col P with explicit IR communication to
an asynchronous SPMD proto col P with no explicit communication This
transformation is then analyzed as an instance of reducing the latter to the
former using There are several things we have learned We can deter
mine a lot ab out the information structure of a task by i parallelizing it
and ii attempting to replace explicit communication with communication
through the world through the task dynamics Communication through
the world takes place when a rob ot changes the environment and that change
can b e sensed by another rob ot For example proto col P uses explicit com
munication and proto col P makes use of an enco ding in the task mechanics
of the same information Our approach of quantifying the information com
plexity in the task mechanics involves viewing the world dynamics as a set
of mechanically implemented registers and data paths This p ermits
certain kinds of de facto communication b etween spatially separated rob ots
DJR also consider three proto cols R R and R for a reorientation
task see g A transformational approach to developing these proto
cols is viewed as a series of reductions The nal proto col R has several
advantages over the initial proto cols R and R Using proto col R two
rob ots instead of three su ce to rotate the b ox The proto col is uniform
SPMD MPMD Single Multiple Program Multiple Data
SPMD in that the same program including the same termination predi
cate runs on b oth rob ots More interesting in R it is no longer necessary
for the rob ots to have an a priori geometric mo del of the b ox whereas such
a mo del is required for R and R
In terms of program development synchrony and communication we
have a corresp ondence b etween these proto cols shown in g We b e
lieve that a metho dology for developing co ordinated manipulation proto cols
is emerging based on the to ols describ ed here This metho dology helps
transform an o ine synchronous manipulation strategy eg P or R with
global co ordination and control into an online asynchronous distributed
strategy P or R for the same task
Developing Parallel Manipulation Proto cols DJR
Start with a sensorless EM or near sensorless JR Rus manipulation
proto col requiring global co ordination of several agents eg parallel
jaw ngers or ngers of a dextrous hand
Distribute the proto col over spatially separated agents Synchronize
and co ordinate control using explicit lo cal communication
De ne virtual sensors for the quantities step measures
Implement each virtual sensor using concrete sensors on mechanical
observables
Transform the communication b etween two agents L and R into shared
data structures
Implement the shared data structures as mechanical registers
Our circuits mo del the proto cols in the steps ab ove Our reductions mo del
the transformations b etween steps By the results of Section these
reductions can b e e ectively computed Therefore in principle the trans
formations in DJR could b e synthesized automatically We b elieve that
our metho ds are useful for developing parallel manipulation proto cols We
We use the term in the sense of DJ others particularly Henderson have used similar
concepts See Section for examples of virtual and concrete sensors
y
Y
x
X
a The b ox motion b The rob ot motions
Figure The task is to rotate the b ox by a
sp eci ed angular amount Here we illustrate the
b ox b eing rotated by three co op erating
autonomous agents a The motion of the b ox
viewed in world co ordinates b The relative
motion of the pushing rob ots viewed in a system of
co ordinates xed on the b ox The arrows illustrate
the direction of the applied forces From DJR
have implemented and tested our asynchronous distributed SPMD manipu
lation proto cols using Tommy and Lily and found them robust and e cient
See DJR for a full discussion
Conclusions
In this b o ok we suggested a theory of information invariance that includes
sensors and computation Our results generalize the work of BK rst we
consider fairly detailed yet abstract mo dels of physical autonomous mobile
rob ots second we consider generalizations and variations on compasses and
orientation sensors third we develop a generalized and strati ed theory of
the p ower of such sensori computational devices As such p erhaps our
work could b e called On the generalized power of generalized compasses
We think that information invariants can serve as a framework in which
to measure the capabilities of rob ot systems to quantify their p ower and to
reduce their fragility with resp ect to assumptions that are engineered into
the control system or the environment We b elieve that the equivalences that
can b e derived b etween communication internal state external state com
putation and sensors can prove valuable in determining what information
is required to solve a task and how to direct a rob ot s actions to acquire
that information to solve it Our work prop oses a b eachhead on informa
tion invariance from which we hop e such goals may b e obtained There are
several things we have learned First we were surprised by how imp ortant
time and communication b ecome in invariant analysis Much insight can b e
gained by asking How can this sensor be simulated by a simpler system with a
clock resp communication Time based sensors are ubiquitous in mo dern
aircraft navigation systems compare Section In DMEs distance
measuring equipment a ground station and the plane talk to each other and
measure di erences in timing pulses to estimate their distance apart GPS
which was approved in July for use in airplanes also op erates on timing
principles
Rob ot builders make claims ab out rob ot p erformance and resource con
sumption In general it is hard to verify these claims and compare the
systems One reason is calibration pre calibration can add a great deal of
information to the system In order to quantify the use of external state
we suggested a theory of calibration complexity Our theory represents a sys
tematic attempt to make such comparisons based on geometric and physical
reasoning Finally we try to op erationalize our analysis by making it compu
tational we give e ective alb eit theoretical pro cedures for computing our
comparisons Our algorithms are exact and combinatorially precise
Our reduction De nition attempts to quantify when we can ef
ciently build one sensor out of another that is build one sensor using the
comp onents of another Hence we write A B when we can build A out of
B without adding to o much stu The last is analogous to without adding
much information complexity Our measure of information complexity is
relativized b oth to the information complexity of the sensori computational
comp onents of B and to the bandwidth of A This relativization circumvents
some tricky problems in measuring sensor complexity see App endix A In
this sense our comp onents are analogous to oracles in the theory of com
putation Hence we write A B if we can build a sensor that simulates A
using the comp onents of B plus a little rewiring A and B are mo deled as
circuits with wires datapaths connecting their internal comp onents How
ever our sensori computational systems di er from computation theoretic
CT circuits in that their spatial con guration i e the spatial lo cation
of each comp onent is as imp ortant as their connectivity
Permutation mo dels the p ermissible ways to reallo cate and reuse re
sources in building another sensor Codesignation constraints further restrict
the range of admissible p ermutations Output communication formalizes our
notion of a little bit of rewiring Like CT reductions A B de nes an
e cient transformation on sensors that takes B to A However we give a
generic algorithm for synthesizing our reductions whereas no such algorithm
can exist for CT Whether such reductions are widely useful or whether
there exist b etter reductions eg our k wire reductions in Section
is op en however in our lab oratory we are using to design manipulation
For example no algorithm exists to decide the existence of a linear space or log space
p olynomial time Turing computable etc reduction b etween two CT problems
proto cols DJR for multiple mobile rob ots We also give a hierarchy of
reductions ordered on p ower so that the strength of our transformations
can b e quanti ed See App endix A for a discussion of universal reduc
tion as p er Section See App endices A and C for more on relativized
information complexity
Our work raises a number of questions For example can rob ots exter
nalize or record state in the world The answer dep ends not only on the
environment but also up on the dynamics A juggling rob ot probably cannot
On a conveyor b elt it may b e p ossible supp ose bad parts are reoriented
so that they may b e removed later However it is certainly p ossible during
quasi static manipulation by a single agent In moving towards multi agent
tasks and at least partially dynamic tasks we are attempting to investigate
this question in b oth an exp erimental and theoretical setting We discuss
these issues further in DJR
By analogy with CT reductions we may de ne an equivalence relation
such that A B when A B and B A We may also ask do es a
k k
k k
given class of sensori computational systems contain complete circuits to
which any member of the class may b e reduced Note that the relation
k
holds b etween any two complete circuits
Weaker forms of sensori computational equivalence are p ossible If we
de ne the state of a sensor system U to b e a pair z b where z is the con
guration of the system and b is the output value at z we can examine the
equilibrium b ehavior of U as it evolves in state space Recall the De ni
tion let us call this strong simulation By analogy let us say that a
system U weakly simulates another system V when U and V have identical
forward attracting compact limit sets in state space If we replace strong
in De nition with weak simulation all of our structural simulation
results go through mutatis mutandis The computational results also go
through if we can compute limit sets and their prop erties a di cult prob
lem in general Failing this if we can derive the prop erties of limit sets by
hand then in principle reductions using weak simulation instead of strong
can also b e calculated by hand simulation
Finally can we record programs in the world in the same way we may
externalize state Is there a universal manipulation circuit which can read
these programs and p erform the correct strategy to accomplish a task Such
I am grateful to Dan Ko ditschek who has suggested this formalism in his pap ers
a mechanism might lead to a rob ot which could infer the correct manipulation
action by p erforming sensori motor exp eriments
Future Research
This b o ok represents a rst stab at several di cult problems and there are
a number of issues that our formalism do es not currently consider We now
acknowledge some of these issues here
Our theory allows us to compare members of a certain class of sensor
systems and moreover to transform one system into another However
it do es not p ermit one to judge which system is simpler or b etter or
cheaper In particular for a given measurement problem it do es not p er
mit a simplest sensor system to b e identi ed There are several reasons
for this The rst is that there are inherent limitations on comparing abso
lute sensor complexity and these problems represent structural barriers to
obtaining go o d notions of b etter or simpler The theory is designed in
part to get around some of these limitations We discuss these problems
which are quite deep in app endix A at some length Second such com
parisons would require an explicit p erformance measure We discussed such
measures as sp eed or execution time in Section In app endix F we
argue that such p erformance measures allow us to apply kinodynamic analysis
to ols DX Xa There is no doubt that external p erformance measures such
as simpler and b etter and cheaper could b e used with our framework
but we don t know what exactly these measures are It app ears that e cient
algorithms for exploiting these measures will have to take advantage of their
structure
Instead of investigating p erformance measures we have argued that it is
very hard to even measure or compare the p ower of sensorimotor systems
To address this problem we developed our reductions To make our stance
clear consider as an analogy the theory of computation CT CT do es not
tell us which algorithms are more simple but it do es tell us which are
more p owerful i e which can compute more In our theory as in CT
we can de ne transformations or reductions that we consider fair and
then discuss equivalence of systems up to these transformations Now in
CT given p erformance measures eg asymptotic complexity we can also
compare the p erformance of algorithms although there are many di erent
measures to choose from But in CT faster do es not necessarily mean
simpler in any sense Our reductions are analogous to CT reductions
Execution time or sp eed is analogous to computational complexity Finally
as in CT notions of simplicity are orthogonal to notions of either reduction
or p erformance
However the notion of p erformance measures op ens up a host of prac
tical issues Certainly some simple scheme of lo oking up the cost of
comp onents in a table could b e used in conjunction with our system An
instrumentation engineer confronted by a problem where one measurement
strategy is ine ective may choose to measure some other prop erty to solve
the problem rather than recon gure the sensori computational comp onents of
the system for example measuring the temp erature rather than the pressure
of a xed volume of gas This approach is not envisaged by our theorems
although the p ower of two given strategies could b e compared Furthermore
distinct measurement strategies have costs other than those considered here
for example the cost of transducers the e ect on the measurement noise
of measuring one observable and inferring another from its value noise prop
erties of transducers and common mo de e ects for example in p ositioning
strain gauges These issues should b e considered in future work
There is much to b e done Our mo del of reduction is very op erational
and others should b e attempted In addition to measuring the information
complexity of communication it may b e valuable to quantify the distance
messages must b e sent Similarly it may make sense to measure the size
of a resource p ermutation or how far resources are moved All these ideas
remain to b e explored Finally we have approached this problem by inves
tigating information invariance that is the kind of information preserving
equivalences that can b e derived among systems containing the resources a
e Section An alternative would b e to lo ok at information variance
that is it would b e valuable to have a truly uniform measure of information
that would apply across heterogenous resource categories
In the app endices we present a number of imp ortant extensions and
attempt to address some of the issues raised in this section
I would like to thank several anonymous referees for suggesting these issues and the
wording to describ e them
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APPENDICES
A
Algebraic Decision Pro cedures
The algorithms in section are algebraic and use the theory of real
closed elds for an introduction to algebraic decision pro cedures see for
example the classic pap er of BKR or discussions in b o oks such as DKM
ch CLS In Section we reduce our computational problem to
deciding the truth of a Tarski formula Tar the algebraic algorithms can
then decide such a sentence Tarski s language is also called the language of
semi algebraic s a sets Such sets are real semi algebraic varieties de ned
by p olynomial equalities and inequalities where the p olynomial co e cients
are algebraic numbers A Tarski formula is a logical sentence that quanti es
existentially or universally over each of the real variables A typical Tarski
formula might b e
x y z w
xy w
xw x w
z w x y y x
Also called Tarski s Language or the rst order language of algebra and geometry
One common mathematical term is the rst order theory of real closed elds
More generally we can think of a Tarski sentence as
x x x
r r
s x x R
r
C
s x x R
r
C
C
m
s x x R
m r m
where each is a quanti er each R is a real relation and C C are
i j m
logical connectives A quanti er is either or and it quanti es over a
i
real variable x A real relation is a relation among real values and is one
i
of or A logical connective is one of or Each s s is a
m
p olynomial in R x x and so is a sentence in r variables We call
r
r
the set Y R de ned by or a semi algebraic set and conversely
r
a set Y R is called semi algebraic if it can b e written in a form like
The set Y is called algebraic if the only real relation we require is equality
The b o olean characteristic function T of a semi algebraic set such as
Y is de ned as
T x x
r
s x x R
r
C
s x x R
r
C
C
m
s x x R
m r m
T is called a semi algebraic predicate Hence can b e written
T x x x x x
r r r
So and can b e built out of these
Let b e a s a predicate Let x denote x x and for a quanti er
r
let x denote x x If T is a s a predicate for the s a set Y
r Y
we will abbreviate the sentence x T x x as follows
Y
x Y x
Given this convention a little manipulation shows that as a consequence
the formula x T x x is therefore equivalent to
Y
x Y x
Let b e the a total degree b ound for the p olynomials s s in
m
We call the number of p olynomials m the size of the Tarski sentence
and of the s a predicate T in Observe however that to calculate a
b ound O r m on the number of terms in we would employ the degree
b ound and the number of variables r as well
Now it is remarkable that one can decide such sentences in complete
generality although Tarski s original algorithm Tar was non elementary
this b ound has b een improved by a chain of researchers since then For
example BKR showed how to decide the rst order theory of real closed
O m
O m
and space In elds with a purely algebraic algorithm in time
Section we use this result
Theorem A Grigoryev Gri Sentences in the theory of real closed
elds can be decided in time doubly exponential only in the number of quan
ti er alternations More speci cally the truth of a Tarski sentence for m
polynomials of degree in r variables where a r is the number of quan
ti er alternations in the prenex form of the formula can be decided in time
a
O r
m
u
Proof See Gri t
Tarski developed this algorithm around but it was not published until later
A Application Computational Calibration Com
plexity
Recall the discussion in Section We wish to develop an algorithm for
deciding the relation b etween sensor systems Comparing the calibration
complexity De nitions of two sensor systems seems easier than the
issues of immersion and simulation b ecause the calibration complexity do es
not change with the immersion so long as the immersion resp ects the co desig
nation constraints The essential idea b ehind computing calibration complex
ity is to measure the complexity of the co designation constraints that sp ecify
a sensor system One measure of course is the number of co designation
constraints but other measures such as the degree and the quanti cation
are also imp ortant Using the algebraic metho ds from Section we can
develop to ols to measure the complexity of algebraic relations such as those
encountered in algebraically co designated sensor systems De nition
Now to decide the relation we must b e able to decide dominance
in calibration complexity see De nition We prop ose to measure cali
bration and installation complexity by the complexity of the co designation
constraints In general one may measure the complexity of the co designa
tion constraints by comparing the complexity of the semi algebraic varieties
that the algebraic co designations sp ecify One way to do this is to count the
number degree quanti cation and dimension of the semi algebraic co des
ignation constraints This gives numbers for m a and r for an algebraic
complexity measure such as for example Equation can then b e
used as a measure of the sensor s calibration complexity These b ounds can
then b e compared using big Oh O notation to determine which sensor
dominates in terms of calibration complexity The comparison can done in
essentially the same time it takes to read the input and the time required is
very small compared to the time for the algebraic simulation
Some of the complexity in our theory results from a decision to pro
ceed through an abstract de nition of a sensor system indep endent of the
underlying con guration space and then to map that system into a partic
ular space One may ask whether this approach though it mirrors much
of mo dern geometry is essential to the results obtained We b elieve that it
would b e p ossible to start with an a priori con guration space see Equa
tion instead of constructing it as a quotient of set di erences This
would eliminate some of the technical baggage required co designation non
co designation and so forth However it app ears that this approach would
leave unanswered the question of measuring the complexity of the underlying
con guration space and hence it would not yield a computational theory of
calibration complexity
A Application Simulation Functions
Recall the discussion of simulation functions for comp onents edges and sen
sor systems on page Section We now discuss simulation functions
and their enco dings It is imp ortant that simulation functions work on p er
muted sensor systems Here is how this might b e accomplished
A Vertex versus Graph Permutations
We now consider two orthogonal kinds of p ermutation In b oth mo dels
the vertex and edge lab els v and e never change The rst mo del is
called vertex permutation and is given in De nition In this mo del
the vertices can move and they drag the comp onents and wires with them
That is the vertices move under p ermutation and as they move the edges
follow Vertex p ermutation su ces for all reductions in this b o ok and the
machinery in Sections and su ces to compute the reductions
and
We can also consider an alternate mo del called edge permutation where
the edge connectivity changes An edge p ermutation can b e mo deled as
follows Consider a graph with vertices V and edges E Start with any
bijection V V We call an edge permutation since it induces the
restriction map E E on the edge set E An edge p ermutation
j
E
says nothing ab out the immersion of a graph
We can also comp ose the mo dels We de ne a graph permutation to b e a
vertex p ermutation followed by an edge p ermutation In a graph p ermuta
tion the vertices and the edges move indep endently That is vertices may
move but in addition the edge connectivity may change To illustrate the
di erent mo dels consider a sensor system U with three vertices f v v v g
with lab els v B i U has one edge e v v of bandwidth
i i
k that connects B to B So the B are the comp onents of the system and
i
e is a datapath A vertex p ermutation U of U would move the vertices and
therefore the comp onents spatially but in U e would still connect v and
v and therefore B and B An edge p ermutation of U would change
the edge connectivity So for example an edge p ermutation U could b e
a graph with one edge e v v connecting v to v and hence B
to B But in U no edge would connect v and v Finally consider a
graph p ermutation U of U Supp ose U U that is U is the vertex
p ermutation U followed by the edge p ermutation ab ove U has the same
edge connectivity as U However in U the vertices are immersed as in
U
To summarize let U b e a situated sensor system A graph p ermu
tation of U is given by U U where is a vertex
p ermutation and is an edge p ermutation
So vertex p ermutation preserves the graph top ology whereas edge p er
mutation can move the edges around Edge p ermutation p ermits arbitrary
rewiring using existing edges It cannot add new edges nor can it change
their bandwidth Although vertex p ermutation su ces for all the exam
ples in this b o ok graph p ermutation is useful and required in DJR
Graph p ermutation is also required for some of the applications discussed
in Section particularly design and universal reduction see Ap
p endix A Here we will content ourselves with answering two questions
i if we p ermit graph p ermutation do es it change our complexity b ounds
and ii do es graph p ermutation give us a more p owerful reduction
We rst turn to question i Fortunately we can extend our computa
tional results to graph p ermutation without di culty To do this we mo del
a graph p ermutation of a sensor system U as a vertex p ermutation of U
followed by an edge p ermutation of U Using this scheme we can com
pute all our reductions etc within the same time b ounds given
in lemma and cor p ermitting graph p ermutation in place of ver
tex p ermutation throughout Our other lemmas also go through mutatis
mutandis
We now elab orate An adjacency matrix for a sensor system with d ver
tices is a d d binary matrix An adjacency matrix with bandwidth has
non negative integer entries An entry of b in row v column u sp eci es a
directed edge of bandwidth log b b etween vertices v and u Given an
edge p ermutation we can construct a new adjacency matrix and the edge
This representation is not hard to extend to comp onents with multiple inputs and
outputs using an r d sd matrix
simulation functions such as in Section can b e constructed from the
e
adjacency matrix Now we may view the edges data paths in our sensor
system as part of its con guration Hence in di erent con gurations the
system may have di erent wiring diagrams di erent edges We now con
sider this such con gurations and the resulting con guration space In
particular we wish to demonstrate their algebraicity
Consider a sensor system U with d vertices V and O d edges E When
we p ermit graph p ermutation a con guration of this system can b e sp eci ed
by a pair were V C is an immersion De nition of U and
is an edge p ermutation As we have discussed lives in the con guration
d
space C What ab out is a member of the p ermutation group on d
elements can b e mo deled as a d d binary matrix called a permutation
matrix Every p ermutation matrix has a single in each row and column
the other entries b eing zero Let Z denote the eld Z Then the space
of p ermutation matrices is O Z d the O rthogonal group of d d binary
matrices Each element is an orthogonal matrix with determinant
Every rewiring of U using only existing edges is enco ded by a p ermu
tation O Z d So to mo del vertex p ermutation plus rewiring we
d d
extend our usual sensor con guration space from C to C O Z d It
is not hard to extend this mo del to add one extra wire output communi
cation or several extra wires for k wire reductions Section The
space O Z d is algebraic and the computation of edge simulation functions
from adjacency matrices is s a
Now how exp ensive it is to compute the reductions and using graph
p ermutation Perhaps surprisingly even with this extended con guration
space which has dimension d r d instead of r d we still obtain the same
c c
complexity b ounds given in lemma and cor so long as r and s are
O
d r d
c
constants This is b ecause see Equations n is still
O
r d
c
n
We now address question ii do es graph p ermutation give us a more
p owerful reduction In answer we show the following
Lemma A The Clone Lemma Graph permutation can be simulated
Another way to see this is as follows even if we try each of the d edge p ermutations
this additional d factor is absorb ed by the O in the second exp onent
using vertex permutation preceded by a linear time and linear space trans
formation of the sensor system
Proof Given a sensor system U we clone all its vertices and attach the
edges to the clones The cloned system simulates the original when each ver
tex is colo cated with its clone Comp onents remain asso ciated with original
vertices We can move an edge indep endently of the comp onents it originally
connected by moving its vertices which are clones Any graph p ermutation
of U can b e simulated by a vertex p ermutation of the cloned system
More sp eci cally Given a graph G V E with lab elling function
we construct a new graph G V E with lab elling function Let the
cloning function cl V V b e an injective map from V into a universe of
vertices V such that cl V V We lift cl to V and then restrict it to E
to obtain cl E cl V as follows If e u v then cl e cl u cl v
Edge lab els are de ned as follows cl e e
Finally we de ne V V cl V and E cl E We de ne the lab elling
function on V as follows v v when v V Otherwise v
returns the identity comp onent which can b e simulated as the identity
function
Supp ose U has d jV j vertices and jE j edges This transformation adds
u
only d vertices and can b e computed in time and space O d jE j t
Let us denote by cl U the linear space clone transformation of U de
scrib ed in lemma A Now consider any k wire reduction Section
k
We see that
Corollary A Let k N Suppose that for two sensor systems U and V
we have V U using graph permutation Then V cl U using only
k k
u
vertex permutation t
See App endix C
The pro of can b e strengthened as follows Recall that two comp onents can commu
nicate without an explicit connection when they are spatially colo cated Therefore the
0
pro of go es through even if cloned vertices have no asso ciated comp onents that is v
for v V This version has the app eal of changing the enco ding without adding additional
physical resources
Class Edge Permutation
In practice we wish to imp ose some restrictions on edge and graph p ermuta
tion For example supp ose we have a sensor system U containing two co op er
ating and communicating mobile rob ots L and R The sensori computational
systems for L and R are mo deled as circuits The datapaths in the system
in addition to bandwidth have a type of the form source destination
where b oth source and destination f L R g When p ermuting the edges
of U to obtain U it makes sense to p ermute only edges of the same type
More generally we may segregate the edge types into two classes internal
edges L L and R R and external edges L R and L R In constructing
U we may use an internal edge of su cient bandwidth to connect any
two comp onents where source destination External edges of su cient
bandwidth can b e used when source destination Hence in class edge
p ermutation we p ermute edges within a class Class edge p ermutation leaves
unchanged the complexity b ounds and the lemmas of Section A
In this example maintaining exactly two physical lo cations can b e done
using simple co designation constraints More generally we take source
destination C
A Application Parametric Co designation Constraints
Recall Equation in which we formulated the sensor reduction problem
as a s a decision pro cedure We now discuss some technical details of this
equation using the notation and hypotheses of Section
In order to allow for su cient generality we must p ermit V s co desig
nation constraints to dep end on U s con guration That is the s a set
D is a s a function of Recall that U denotes the sensor system
VU
U installed at con guration Now given that sensor system U is at con g
uration we are interested in whether or not sensor system V can simulate
U but only when V s con guration satis es some constraint D
VU
that depends on That is we are interested in the question
Do es V simulate U given that lies in D
VU
For example consider the reduction in prop osition Here sp eci es
among other things the ship s con guration x h in the radial sensor E
In particular we do not care what happ ens when D
VU
We think of x h as one co ordinate of The parametric co designation
constraint D is used to ensure that the corresp onding ship in the light
H
house sensor H is also placed at x h The question Can H simulate E
only makes sense given that i H and E are b oth installed at G and ii
the ships in H and E are in the same con guration Static co designation
constraints that are invariant with ensure i whereas parametric co desig
nation constraints that vary with ensure ii This could b e implemented
as follows Let resp b e the pro jection of E s resp H s con gura
E x H x
tion that returns the ship s con guration So in particular x h
E x
These pro jections are clearly semi algebraic functions Then this asp ect of
the parametric co designation constraint D could b e implemented as
H
D
H
H x E x
The fact that we have an equality constraint in re ects the fact
that E and H are simple sensor systems De nition In general for
arbitrary algebraic sensors systems D could sp ecify a more complicated
H
s a relation b etween and
Formally parametric co designation constraints as D and D see
H
VU
Equation can b e mo deled as parametric s a sets see Can
De nition A Canny A parametrically de ned semi algebraic set D
is de ned as fol lows D is a s a set which is a function of some argument
Hence there is an implicitly de ned s a predicate T z which is true
D
i z D Now let Y be a s a set with predicate T So when we write
Y
D Y we mean z T z T z which gives us a s a predicate
D Y
which is true of those values of such that D Y
D
A Application Universal Reductions
We can now use the to ols from App endix A A to develop an algorithm
for universal reduction application of Section Universal reduction
requires graph p ermutation see App endix A
Let U and V b e sensor systems Supp ose we are given a sp eci cation for
U and a bag of parts for V The sp ec as usual is enco ded as a simulation
function as describ ed in Section We are also given a simulation
U
function for V The bag of parts consists of b oxes and wires Each b ox is
V
a sensori computational comp onent blo ck b ox that computes a function
of a its spatial lo cation or p ose and b its inputs The wires have di erent
bandwidths and they can ho ok the b oxes together Recall we are given a
simulation function for each comp onent v and a simulation function
v
for each edge e indeed this is how the global simulation functions
e U
and are enco ded see Section Then our algorithms ab ove decide
V
can we immerse the comp onents of V so as to satisfy the sp ec of U The
algorithms also give the immersion that is how the b oxes should b e placed
in the world and how they should b e wired together Hence we can ask
can the sp ec of U b e implemented using the bag of parts V
Now supp ose that in addition to the sp ec for U we are given an enco ding
of U as a bag of parts and an immersion to implement that sp ec Supp ose
further that U V Since this reduction is relativized b oth to U and to V
it measures the p ower of the comp onents of U relative to the comp onents
in V By universally quantifying over the con guration of U we can ask
can the comp onents of V always do the job of the comp onents of U
More sp eci cally Let b e a con guration of the sensori computational
system U Let U U b e a graph p ermutation of U App endix A
Let denote the set of all graph p ermutations of so if U has d ver
tices then O Z d Thus and enco des the
spatial immersion of U as well as its wiring connectivity By Section
and App endix A is s a
Similarly let b e a con guration of V Hence we can decide the Tarski
sentence
D
VU
U V
where D is a parametric s a co designation constraint App endix A
VU
When Equation holds we say that U universal ly reduces to V or that
there is a universal reduction from U to V Hence is p ossible to compute uni
versal reductions algebraically With the notation and hypotheses as ab ove
throughout App endix the time complexity of deciding is given by
Equation which b ecomes
a
O r O r d d
c
m n n
D
O
r d
c
Equation is still n n Hence we have that
D
Corollary A Universal reductions Equation can be computed in
u
the same time bounds given in Equations t
B
Relativized Information
Complexity
Let us sp ecialize de nition to monotonic sensor systems
De nition Monotonic Consider two monotonic sensor systems S
and Q and let b be the output of sensor S We say S is e ciently reducible
to Q if
S Q comm b
In this case we write S Q
For the sensors we have considered their complexity could essentially
b e characterized using the size log b of the output b We now generalize
this de nition slightly Our motivation is as follows There are sensor sys
tems whose complexity cannot b e well characterized by the number of bits
of output For example consider a grandmother sensor Such a sen
sor lo oks at a visual eld and outputs one bit returning t if the visual
eld contains a grandmother and f if it do esn t Now one view of the
sensor interpretation problem is that of information reduction and identi
cation compare DJ which discusses hierarchies of sensor information
However consider a somewhat di erent p ersp ective that views sensors as
model matchers So imagine a computational pro cess that calculates the
probability P G V of G grandmother given V the visual eld i e
This discussion devolves to a suggestion of Sundar Narasimhan Nar for which we
are very grateful
the probability that G is in the data the visual eld itself The sensor in
the former case is something sp eci c only to detecting grandmothers while
the latter prefers to see a grandmother as the mo del that b est explains the
current data The latter is a pro cess that computes over mo del classes For
example this sensor might output Tiger when given a fuzzy picture that
is b est explained as a tiger
In short one may view a sensor system as storing prior distributions
These distributions bias it toward a xed set of mo del classes In principle
the stored distributions may b e viewed either as calibration or internal state
To quantify the absolute information complexity of a sensor system we need
to measure the information complexity of mo del classes stored in the prior
distribution of the sensor This could b e very di cult
Instead we prop ose to measure a quantity called the maximum bandwidth
of a sensor system Intuitively this quantity is the maximum over all internal
and external edge bandwidths data paths That is
De nition Part One We de ne the internal resp external band
width of a sensor system S to be the greatest bandwidth of any internal
resp external edge in S The output size of S is given by De nition
We de ne the maximum bandwidth mb S to be the greater of the internal
bandwidth external bandwidth and the output size of S
The maximum bandwidth is an upp er b ound on the relative intrinsic out
put complexity relativized to the information complexity of the comp onents
Sections and We explore this notion brie y b elow
Maximum bandwidth is a measure of internal information complexity
The bandwidth is a measure of information complexity only relative to the
sensori computational comp onents of the system For example imagine that
we had a sensor system with a single comp onent that outputs one bit when
it recognizes a complicated mo del say a grandmother The only data path
in the system has bandwidth one bit b ecause the single comp onent in the
system is very p owerful So even though the maximum bandwidth is small
the absolute information complexity may b e large
Now one may ask why prefer one mo del over another and there can b e many answers
Nar advocates Minimum Description Length or MDL This theory attempts to minimize
L M L D M where L M is the length of mo del and L D M is the length of the
data given that the mo del is minimal
So some sensors are black b oxes We call a sensor system a black box if it
is enco ded as a single comp onent The only measure of bandwidth we have
for a black b ox is its output size For example Erdmann s radial sensor E
Section is essentially a black b ox plus output communication
More generally we call a sensor system monotonic if its internal and
external bandwidths are b ounded ab ove by its output size So black b ox
sensors are trivially monotonic All the sensor systems in this b o ok are
monotonic But some of the systems in our forthcoming work DJR are
not
In light of this discussion we now give a generalized de nition of the
reduction using relativized information complexity
First let S b e a monotonic sensor system with output b as in De ni
tion In this case we de ne comm S to b e comm b
More generally for p ossibly non monotonic sensors we will let comm S
k
b e comm where k is the relative intrinsic output complexity of S Mea
suring this k in general is di cult but we will treat the maximum band
width De nition of S as an upp er b ound on k Finally we generalize
De nition to non monotonic sensor systems as follows
De nition Generalized Consider two sensor systems S and Q We
say S is e ciently reducible to Q if
S Q comm S
In this case we write S Q
C
Distributive Prop erties
In this app endix we prove some technical prop erties ab out the p ermu
tation of partial immersions These prop erties are algebraic and we call
them the distributive prop erties First we consider pure p ermutation
and combination i e without output vertices as in De nition Then
in C C we generalize to include p ermutation and combination of output
vertices Recall the de nition of compatibility for partial immersions Sec
tion
De nition C Let and be compatible partial immersions We say the
permutations and are compatible p ermutations of and if and
are also compatible
We would like to show that for immersions combination and p ermutation
commute That is for two compatible partial immersions and if and
are compatible p ermutations then
In answer we can now show the following
Claim C Consider two compatible partial immersions and together
with two compatible permutations and Then
Let Then there exists such that
Proof First let b e a p ermutation of Let and
j
C
Then is a p ermutation of and is a p ermutation of
j
C
and
Conversely supp ose and are compatible p ermutations of and
Then we observe that since the domains of and resp and are
identical therefore the domains of and are identical Hence
u
is a p ermutation of t
Next we ask for sensor systems do combination and p ermutation com
mute That is for two sensor systems S and U is it true that
S U S U
whenever is de ned see De nition
In answer we show the following
Prop osition C Consider two sensor systems S and U as above Assume
their immersions are compatible so that S U is de ned Then
Let S and U be compatible permutations of S and U Then S U
is a permutation of S U
Let S U be a permutation of S U Then there exist compatible
permutations S and U of S and U resp such that S U
S U
Proof Let S S U U S S and U U and apply
u
claim C t
C Combination of Output Vertices
Recall the de nition of combination in Section There we considered two
sensor systems S and U Both have output vertices say v and u resp
o o
When we combine the two sensor systems S and U to form S U we must
sp ecify the unique output vertex of the new combined sensor system We
now show how to choose output vertices in a consistent manner so that the
combination op eration remains asso ciative and commutative
First we view each sensor system as a pointed graph a graph with one
distinguished vertex called the output vertex We de ne on two p ointed
graphs in such a manner as to pro duce a new p ointed graph For example let
G u b e a p ointed graph with output vertex u Let G u b e another
p ointed graph Then
G u G u G G u u
where G G denotes combination De nition The output vertex
u u is de ned as follows Let V b e the universe of all p ossible vertices
So for any graph G with vertices and edges V E we have V V We
i i i i
insist that V have a total order De ne u u min u u
It is easy to see that under this de nition the op eration on p ointed
graphs is b oth asso ciative and commutative
C Output Permutation
Recall De nition There we also p ermited a p ermutation to change
which vertex has the output device lab el This kind of p ermutation is
not required for the monotonic sensor systems App endix A considered in
this b o ok but it is needed for the general theory and it is used explicitly
in DJR We formalize this notion here
We de ne an op eration called output permutation on p ointed graphs Ap
p endix C The e ect of this op eration is to choose a new distinguished
vertex For example for a graph G with distinguished p oint u we could
o
choose a new distinguished vertex u We represent this op eration by
G u G u
o
We call G u an output permutation of G u
o
Now following App endix A let us call our existing notion of p ermu
tation De nition by the name vertex p ermutation to distinguish it
from output p ermutation It is p ossible to comp ose output p ermutations
and vertex p ermutations We adopt
We must b e careful not to confuse a pointed graph with a pointed sensor system
De nition
Convention C We use the term p ermutation to include both output per
mutations and vertex permutations Similarly we wil l use the operator for
any permutation
This convention is necessary to make combination and p ermutation com
mute in general
C Discussion
In this app endix A A we have made sure that combination the
op eration and p ermutation the op eration commute So for example for
any sensor system S have ensured that S comm S comm i e
we can do the p ermutation and combination in any order Second we have
ensured that the combination op eration is commutative and asso ciative
Third in De nition for the reduction see generalized De nition
we have given the single edge e in comm enough bandwidth so that it still
works when we switch it e around using p ermutation Hence the sensor
system Q comm S in Equation may b e implemented as the sensor
system Q p ermuted in an arbitrary way plus one extra data path whose
bandwidth is that of the largest ow in S
D
On Alternate Geometric
Mo dels of Information
Invariants
We have presented a geometric mo del of information invariants I am
grateful to John Canny and Jim Jennings for suggesting that I provide an
abstract example of information invariants using the language and con
cepts developed in DJ The resulting mo del is somewhat di erent in avor
from that of section
Here is a alternate geometric mo del for an example of information in
variance Let U b e an arrangement of p erceptual equivalence classes as
in DJ A simple control strategy may b e mo deled as a subgraph of the
RR graph DJ on U Now consider the lattice of p erceptual equivalence
classes formed by xing the task environment and varying the sensing map
as in DJ Let U and V b e two arrangements of p erceptual equivalence
classes in the lattice Then there is an information invariant for U and V
when they have a common coarsening W together with a control strategy
on W Note that by construction this control strategy agrees on the overlap
of U and V
This example is simple it remains to develop and exploit this geometric
mo del for other kinds of information invariants
A coarsening of U and V is a partition W such that b oth U and V are ner than W
E
A Non Geometric Formulation
of Information Invariants
There are several places where we have exploited the geometric structure
of rob otics problems in constructing our framework First our sensors are
geometrical in that they measure geometric quantities Second the con
guration of a sensor is geometrical in that each comp onent is physically
placed and oriented in physical space
It is of some interest to derive an abstract version of our framework in
which geometry plays no role Such a framework would b e something like
a logical framework
It is not hard to formulate our approach in a geometry free manner First
one would say that the value or the output of a sensor is simply a value in
some set Next one would replace the con guration space C of a comp onent
by any set of the form
C f z j z is a lo cation g
C can b e taken to have no structure whatso ever All the de nitions construc
tions and pro ofs of section then go through mutatis mutandis there
is no geometry anywhere In particular our formerly geometric co desig
nation constraints now reduce to Chapman s prop ositional co designation
constraints Cha
I am grateful to Stan Rosenschein for encouraging me to develop this generalization
It is now worth asking what are the implications for section It is
easy to extend the de nition of a simulation function for a sensor system
U
d
U one obtains a set map C R where C is as in and R is an
U
arbitrary set At this p oint we lose the algebraic prop erties we exploited to
derive the algorithms of section Hence our algorithms do not obtain
when we remove the geometric structure In particular we lose our main
computational result lemma It seems plausible however that other
deductive mechanisms might b e used instead to obtain similar results in
the abstract non geometric case
F
Provable Information
Invariants with Performance
Measures
F Kino dynamics and Trade O s
It is p ossible to develop provable information invariants in the sp ecial case
where we have p erformance measures Consider once again the information
invariants discussed ab ove in Section That these invariants Equation
are related to kino dynamics CDRX DX DX should come as no surprise
since the execution time for a control strategy is taken as cost In Xa Pat
Xavier introduced a new algorithmic mechanism for measuring kino dynamic
trade o s see DX for a brief description These techniques were used
to quantify the trade o s b etween planning complexity executor complexity
and safety clearance Essentially Xavier considers how closely one
T
can approximate an optimal time tra jectory and how much safety in
S
the sense of headway is required to execute the approximate solution with
an uncertain control system Xavier obtained equicomplexity curves in
the plane These curves may b e interpreted as follows For a xed
T S
complexity r which may b e equivalently viewed as i the running time
of the planner ii the space requirements of the planner or iii the dis
cretization density of the phase space for the dynamical system representing
the rob ot Xavier s planner obtains a kino dynamic solution which satis es
a one parameter family of approximations of the form
f
r
T S
where f is a function conditioned on complexity r Hence represents
r
an information invariant as well and if we view the following distance d
as b eing similar to the clearance parameter such kino dynamic metho ds
S
app ear attractive We b elieve that these metho ds could b e used to prove
information invariants like while they require sp eci c assumptions ab out
the dynamics and geometry they are quite general in principle Pursuing
such theorems is a fruitful line of future research
Kino dynamic trade o s are one source of information invariants and one
may even nd provable rigorous characterizations for information questions
therein eg DX Xa However there is something a bit disatisfying ab out
this line of attack First it makes controls not sensing the senior partner
much in the same way that in the LMT theory see Don recognizabil
ity is a second class citizen compared with reachability In LMT this is a
consequence of a bias towards sensorless manipulation EM in kino dynam
ics it is a consequence of mo del based control Second kino dynamics relies
on a measure of cost in this case time and hence the results emphasize
p erformance not comp etence