Link oping Electronic Articles in
Computer and Information Science
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Foundations for the Situation
Calculus
Hector Levesque Fiora Pirri and Ray Reiter
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Hector Levesque Fiora Pirri and Ray Reiter
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Abstract
This article gives the logical foundations for the situations as histories vari
ant of the situation calculus fo cusing on the following items
The language of the situation calculus
Foundational axioms for the domain of situations
Axioms for an underlying domain theory
The syntax and semantics of the logic programming language GOLOG
Axioms for knowledge and sensing actions
Essential metatheoretic results ab out the situation calculus
Authors addresses
Hector Levesque
Department of Computer Science
University of Toronto
Toronto Canada M S A
hector ai toronto edu
Fiora Pirri
Dipartimento di Informatica e Sistemistica
Universit a degli Studi di Roma La Sapienza
Via Salaria Roma Italy
pirri assi dis uniroma it
Ray Reiter
Department of Computer Science
University of Toronto
Toronto Canada M S A reiter ai toronto edu
Intro duction
The situation calculus McCarthy has long b een a staple of AI research
but only recently have there b een attempts to carefully axiomatize it The
variant describ ed here has evolved in resp onse to the needs of the Cognitive
Rob otics Pro ject at the University of Toronto and its foundations are now
su ciently develop ed to b e gathered in one place That is what this pap er
sets out to do
The principal intuition captured by our axioms is that situations are
histories nite sequences of primitive actions and we provide a binary
constructor do a s denoting the action sequence obtained from the history
s by adding action a to it Other intuitions are certainly p ossible ab out the
nature of situations McCarthy and Hayes saw them as snapshots of a
world For the purp oses of representing narratives McCarthy and Costello
wish to express that any numb er of actions may o ccur b etween a situa
tion and its successor and therefore they do not app eal to a constructor
like do
The axioms and supp orting metatheory presented in this pap er have
provided foundations for a wide range of practical work in mo deling dynami
cal systems including planning Green Levesque control Levesque
et al De Giacomo et al simulation Kelley database up dates
Reiter Bertossi et al diagnosis of dynamical systems McIlraith
and agent programming and rob otics Jenkin et al Lesp erance et
al Burgard et al Shapiro et al Reiter
The Language of the Situation Calculus
The following description is taken from Pirri and Reiter L is a
sitcalc
second order language with equality It has three disjoint sorts action for
actions situation for situations and a catch all sort object for everything else
dep ending on the domain of application Apart from the standard alphab et
of logical symb ols we use and with the usual de nitions of a full
set of connectives and quanti ers L has the following alphab et
sitcalc
Countably in nitely many individual variable symb ols of each sort
We shall use s and a with subscripts and sup erscripts for variables
of sort situation and action resp ectively We normally use lower case
roman letters other than a s with subscripts and sup erscripts for
variables of sort object In addition b ecause L is second order
sitcalc
its alphab et includes countably in nitely many predicate variables of
all arities
Two function symb ols of sort situation
A constant symb ol S denoting the initial situation
0
A binary function symb ol do action situation situation
The intended interpretation is that do a s denotes the succes
sor situation resulting from p erforming action a in situation s
A binary predicate symb ol < situation situation de ning an order
ing relation on situations The intended interpretation of situations
0
is as action histories in which case s < s means that s is a prop er
0
subhistory of s
A binary predicate symb ol P oss action situation The intended
interpretation of P oss a s is that it is p ossible to p erform the action
a in situation s
For each n countably in nitely many predicate symb ols with ar
n
ity n and sorts action obj ect These are used to denote situation
indep endent relations like
human J ohn pr imeN umber n
mov ing Action r un per son l oc l oc etc
For each n countably in nitely many function symb ols of sort
n
action obj ect obj ect These are used to denote situation inde
p endent functions like
sq r t x heig ht M tE v er est ag ent r un per son l oc l oc etc
For each n a nite or countably in nite numb er of function
n
symb ols of sort action obj ect action These are called action
functions and are used to denote actions like pick up x mov e A B
etc In most applications there will b e just nitely many action
functions but we allow the p ossibility of an in nite numb er of them
Notice that we distinguish b etween function symb ols taking values
of sort object and those the action functions taking values of
sort action In what follows the latter will b e distinguished by the
requirement that they b e axiomatized in a particular way by what we
shall call action precondition axioms
For each n a nite or countably in nite numb er of predicate
n
symb ols with arity n and sorts action obj ect situation
These predicate symb ols are called relational uents In most ap
plications there will b e just nitely many relational uents but we
do not preclude the p ossibility of an in nite numb er of them These
are used to denote situation dep endent relations like ontabl e x s
husband M ar y J ohn s etc Notice that relational uents take just
one argument of sort situation and this is always its last argument
For each n a nite or countably in nite numb er of function
n
symb ols of sort action obj ect situation action obj ect These
function symb ols are called functional uents In most applications
there will b e just nitely many functional uents but we do not
preclude the p ossibility of an in nite numb er of them These are
used to denote situation dep endent functions like
ag e M ar y s pr imeM inister I tal y s etc
Notice that functional uents take just one argument of sort situation
and this is always its last argument
Notice that only two function symb ols of L S and do are p ermitted
sitcalc 0
to take values in sort situation
Foundational Axioms for Situations
Here we fo cus on the domain of situations The primary intuition ab out
situations that we wish to capture axiomatically is that they are nite se
quences of actions We want also to b e able to say that a certain sequence of
actions is a subsequence of another To formalize these desiderata we adopt
the following four foundational axioms for the situation calculus taken from
1
do a s do a s a a s s
1 1 2 2 1 2 1 2
P P S a s P s P do a s s P s
0
Axiom is a unique names axiom for situations two situations are
the same i they are the same sequence of actions Two situations S and
1
S may b e di erent yet assign the same truth values to all uents So a
2
situation in the version of the situation calculus presented here must not b e
identi ed with the set of uents that hold in that situation i e with a state
The prop er way to understand a situation is as a history namely a sequence
of actions two situations are equal i they denote identical histories The
second axiom is second order induction on situations The imp ortance
of induction for the situation calculus is describ ed by Reiter
There are two more axioms
s < S
0
0 0
s < do a s s v s
0 0 0
Here s v s is an abbreviation for s < s s s The relation < provides
0
an ordering relation on situations Intuitively s < s means that the action
0
sequence s can b e obtained from the sequence s by adding one or more
2
actions to the front of s
The ab ove four axioms are domain independent They provide the basic
prop erties of situations in any domain sp eci c axiomatization of particular
3
uents and actions Henceforth we shall call them
Time Pro cesses and Concurrency in the Situation
Calculus
As presented ab ove the foundational axioms for situations provide a purely
qualitative notion of time whose only temp oral concept is sequential action
o ccurrence An action o ccurs before or after another within a situation
There is no way of expressing that an action o ccurs at a particular time or
that two or more actions o ccur concurrently While we do not present them
here it is p ossible to extend these axioms for situations to accommo date
time and concurrency See for example Pinto and Reiter and
for a discussion of how to mo del pro cesses in the situation calculus using
interleaving concurrency see Reiter
1
In what follows lower case Roman characters will denote variables in formulas
Moreover free variables will always b e implicitl y universally prenex quanti ed
2
Readers familiar with the programming language LISP will have noticed that
in the situation calculus the constant S is just like NIL and do acts like cons
0
Situations are simply lists of primitive actions For example the situation term
do C do B do A S is simply an alternative syntax for the LISP list CBA
0
cons C cons B cons A nil Notice that to obtain the action history cor
resp onding to this term namely the p erformance of action A followed by B
followed by C we read this list from right to left Therefore when one reads
0
situation terms from right to left the relation s < s means that situation s is a
0
prop er subhistory of the situation s The situation calculus induction axiom
is simply the induction principle for lists If the empty list has prop erty P and if
whenever list s has prop erty P so do es cons a s then all lists have prop erty P
3
These foundational axioms are simpler than those presented by Reiter and
others These earlier axiomatizations for the situation calculus are all derivable
from the four axioms given here
Domain Axioms and Basic Theories
of Actions
Our concern here is with axiomatizations for actions and their e ects that
have a particular syntactic form These are called basic action theories and
we next describ e these
De nition The Uniform Formulas
Let b e a term of sort situation The terms of L uniform in are
sitcalc
the smallest set of terms such that
Any term that do es not mention a term of sort situation is uniform
in
is uniform in
If g is an n ary function symb ol other than do and S and t t
0 1 n
are terms uniform in whose sorts are appropriate for g then
g t t is a term uniform in
1 n
The formulas of L uniform in are the smallest set of formulas such
sitcalc
that
If t and t are terms of the same sort object or action and if they
1 2
are b oth uniform in then t t is a formula uniform in
1 2
When P is an n ary predicate symb ol of L other than P oss and
sitcalc
< and t t are terms uniform in whose sorts are appropriate
1 n
for P then P t t is a formula uniform in
1 n
Whenever U U are formulas uniform in so are U U U
1 2 1 1 2
and v U provided v is an individual variable and it is not of sort
1
situation
Thus a formula of L is uniform in i it is rst order it do es
sitcalc
not mention the predicates P oss or < it do es not quantify over variables
of sort situation it do es not mention equality on situations and whenever
it mentions a term of sort situation in the situation argument p osition of a
uent then that term is
De nition Action Precondition Axiom
An action precondition axiom of L is a sentence of the form
sitcalc
P oss A x x s x x s
1 n A 1 n
where A is an n ary action function symb ol and x x s is a
A 1 n
formula that is uniform in s and whose free variables are among x x s
1 n
For example in a blo cks world we might typically have
P oss pick up x s y hol ding y s heav y x s
The uniformity requirement on ensures that the preconditions for
A
the executability of the action A x x are determined only by the
1 n
current situation s not by any other situation
De nition Successor State Axiom
A successor state axiom for an n ary relational uent F is a
sentence of L of the form
sitcalc
F x x do a s x x a s
1 n F 1 n
where x x a s is a formula uniform in s all of whose free
F 1 n
variables are among a s x x An example of such an axiom
1 n
taken from is
br ok en x do a s
r fa dr op r x f r ag il e x s g b fa expl ode b
nexto b x s g
br ok en x s r a r epair r x
This says that x will b e broken in the successor situation do a s i x
was fragile in s and the action taking us to the successor situation was
someone r dropping x or the action was some b omb b explo ding
and b was next to x or x was already broken and the action was not
someone repairing x
As for action precondition axioms the uniformity of guarantees
F
that the truth value of F x x do a s in the successor situa
1 n
tion do a s is determined entirely by the current situation s and not
by any other situation In systems and control theory this is often
called the Markov property
A successor state axiom for an n ary functional uent f is a
sentence of L of the form
sitcalc
f x x do a s y x x y a s
1 n f 1 n
where x x y a s is a formula uniform in s all of whose
f 1 n
free variables are among x x y a s A blo cks world example
1 n
is
heig ht x do a s y a mov eT oT abl e x y
z h a mov e x z heig ht z s h y h
heig ht x s y a mov eT oT abl e x z a mov e x z
As for relational uents the uniformity of in the successor state
f
axioms for functional uents guarantees the Markov prop erty The
value of a functional uent in a successor situation is determined en
tirely by prop erties of the current situation and not by any other
situation
Following earlier ideas of Pednault Haas and Schub ert Reiter
4
shows how to solve the frame problem for deterministic actions The
resulting solution yields axioms with exactly the syntactic form of successor
state axioms
Basic Action Theories Henceforth we shall consider theories D of
L of the following forms
sitcalc
D D D D D
ss ap una S
0
where
4
This solution do es not take rami cation constraints into account but see
for p ossible ways to do this while preserving the successor state
axiom approach
are the foundational axioms for situations
D is a set of successor state axioms for functional and relational
ss
uents one for each such uent of the language L
sitcalc
D is a set of action precondition axioms one for each action function
ap
symb ol of L
sitcalc
D is the set of unique names axioms for all action function symb ols
una
of L
sitcalc
is a set of rst order sentences that are uniform in S Thus no D
0 S
0
quanti es over situations or mentions P oss < or the sentence of D
S
0
function symb ol do so that S is the only term of sort situation men
0
will function as the initial theory of tioned by these sentences D
S
0
the world i e the one we start o with b efore any actions have b een
the initial database The initial executed Often we shall call D
S
0
database may and often will contain sentences mentioning no situ
ation term at all for example unique names axioms for individuals
like J ohn M ar y or timeless facts like isM ountain M tE v er est
or dog x mammal x
De nition A basic action theory is any collection of axioms D of the
ab ove form that also satis es the following functional uent consistency
property
Whenever f is a functional uent whose successor state axiom in D is
ss
f x do a s y x y a s
f
then
j a s x y x y a s D D
f una S
0
0 0 0
y y x y a s x y a s y y
f f
This consistency prop erty provides a su cient condition for preventing
a source of inconsistency in f s successor state axiom It says that the con
ditions de ning f s value in the next situation do a s namely actually
f
de ne a value for f and that this value is unique
Metatheory for the Situation Calculus
Here we present some fundamental logical prop erties of basic action theories
These are describ ed more fully with accompanying pro ofs in
Relative Satis ability
Basic action theories enjoy an imp ortant relative satis ability prop erty
is Theorem A basic action theory D is satis able i D D
una S
0
Reduction
For the purp oses of automating deduction in the situation calculus the
foundational axioms are problematic esp ecially the induction axiom Ac
cordingly it would b e desirable to characterize broad classes of sentences
whose pro ofs need never app eal to induction or even b etter can addition
ally ignore some of the other axioms of or b est of all need never app eal
to any of the foundational axioms The main result along these lines is
the following
Theorem Reduction Theorem
is a basic action theory Suppose that D D D D D
ss ap una S
0
Suppose further that is a rst order sentence of L whose prenex
sitcalc
normal form has the form
Q s Q s s Q n
1 1 2 2 n n
where s s are variables of sort situation is quanti er free and
1 n
each Q is a sequence of zero or more quanti ers over variables of sort
i
action obj ect Then
D j i D fI nductiong j
Here D fI nductiong is D without the second order induction axiom
If does not mention the predicate symbol <
j D j i D D D D
= ss ap una S
0
Here consists of the foundational axiom together with the
=
fol lowing sentence
S do a s
0
If does not mention the predicate symbol < and it does not mention
an equality atom over terms of sort situation
j D j i D D D D
ss ap una S
0
Regression
Regression is p erhaps the single most imp ortant theorem proving mecha
nism for the situation calculus it provides a sytematic way to establish
that a basic action theory entails a so called regressable sentence
De nition The Regressable Sentences A regressable sentence of
the situation calculus is a rst order sentence W of L such that W
sitcalc
do es not quantify over situations and such that for every atom of the form
P oss mentioned by W has the form A t t for some n ary
1 n
action function symb ol A of L
sitcalc
The essence of a regressable sentence is that each of its situation terms is
ro oted at S and therefore one can tell by insp ection of such a term
0
exactly how many actions it involves It is not necessary to b e able to
tell what those actions are just how many there are In addition when a
regressable sentence mentions a P oss atom we can tell by insp ection of
that atom exactly what is the action function symb ol o ccurring in its rst
argument p osition for example that it is a mov e action
The intuition underlying regression is this Supp ose we want to prove
that a regressable sentence W is entailed by some basic action theory Sup
p ose further that W mentions a relational uent atom F t do where
F s successor state axiom is F x do a s x a s Then we can eas
F
0
ily determine a logically equivalent sentence W by substituting t
F
for F t do in W After we do so the uent atom F t do in
volving the complex situation term do has b een eliminated from W
in favour of t and this involves the simpler situation term In
F
0
this sense W is closer to the initial situation S than was W Moreover
0
this op eration can b e rep eated until the resulting goal formula mentions
only the situation term S after which intuitively it should b e su cient to
0
establish this resulting goal using only the sentences of the initial database
Regression is a mechanism that rep eatedly p erforms the ab ove reduction
starting with a goal W ultimately obtaining a logically equivalent goal W
0
whose only situation term is S We have only indicated how regression
0
works by reducing relational uent atoms in W there is an analogous way
of reducing functional uent terms
The full de nition of the regression op erator R is rather lengthy mainly
b ecause functional uents intro duce certain complications and we omit it
here See for details The principal result for regression is the following
Theorem Soundness and Completeness of Regression
Suppose W is a regressable sentence of L and D is a basic theory of
sitcalc
actions Then
R W is a sentence uniform in S
0
D j R W D j W i D
una S
0
This result shows how using regression one can reduce the problem
of proving a regressable sentence in a basic action theory to an entailment
problem relative to the initial database and the unique names axioms for
actions The foundational axioms of the situation calculus are not re
quired for establishing this entailment The action precondition and succes
sor state axioms are also not required their e ects have b een compiled into
the regressed formula Regression is the central computational mechanism
underlying implementations of the logic programming language GOLOG
and we now turn to this
GOLOG
GOLOG is a situation calculus based logic programming language for im
plementing complex b ehaviours for dynamical systems The language was
rst intro duced in and was describ ed there without pinning down
all its formal details Here we present the syntax and logical foundations
for the language most of which is taken from Pirri and Reiter
The Syntax of GOLOG Programs
For the purp oses of sp ecifying the syntax of the programming language
GOLOG we require a language that is just like L except it suppresses
sitcalc
all references to situations
De nition Situation Suppressed Terms and Formulas
The situation suppressed terms obtained from L are inductively de
sitcalc
ned by
Any variable of L of sort action or object is a situation suppressed
sitcalc
term
If f is an n ary functional uent of L and t t are
sitcalc 1 n
situation suppressed terms of sorts appropriate for the rst n argu
ments of f then f t t is a situation suppressed term
1 n
If g is an m ary non functional uent symb ol of L other than S
sitcalc 0
or do and t t are situation suppressed terms of sorts appropri
1 m
ate for the arguments of g then g t t is a situation suppressed
1 m term
The situation suppressed formulas obtained from L are inductively de
sitcalc
ned by
0
Whenever t and t are situation suppressed terms of the same sort
0
then t t is a situation suppressed formula Because situation
suppressed terms are never of sort situation the situation suppressed
formulas never mention an equality atom b etween terms of sort situ
ation
When t is a situation suppressed term of sort action then P oss t is
a situation suppressed formula
When F is an n ary relational uent symb ol of L and
sitcalc
t t are situation suppressed terms of sorts appropriate for the
1 n
rst n arguments of F then F t t is a situation suppressed
1 n
formula
When P is an m ary non uent predicate symb ol of L other than
sitcalc
< and t t are situation suppressed terms of sorts appropriate
1 m
for the arguments of P then P t t is a situation suppressed
1 m
formula
When and are situation suppressed formulas so are and
When v is a variable of L of sort action or object then v is
sitcalc
a situation suppressed formula
Therefore situation suppressed formulas are rst order never quantify over
situations never mention < nor do they ever mention terms of sort situa
tion
We can now describ e the syntax of GOLOG programs
hpr og r ami hpr imitiv e actioni j
htest conditioni j
hpr og r ami hpr og r ami j
hpr og r ami j hpr og r ami j
x hpr og r ami j
hpr ocedur e cal l i j
pro c P v hpr og r ami endPro c
1 1
pro c P v hpr og r ami endPro c
n n
hpr og r ami
Here
hpr imitiv e actioni is a situation suppressed term of sort action
htest conditioni is a situation suppressed formula
In x hpr og r ami x must b e a variable of L of sort action or
sitcalc
object
hpr ocedur e cal l i must have the syntactic form P t t where P
1 n
is an n ary predicate variable of L whose rst n arguments
sitcalc
are not of sort situation and whose last two arguments are of sort
situation Moreover t t must b e situation suppressed terms
1 n
whose sorts are appropriate for the rst n arguments of P
In a pro cedure declaration pro c P v hpr og r ami endPro c P must
b e an n ary predicate variable of L whose rst n arguments
sitcalc
are not of sort situation and whose last two arguments are of sort
situation Moreover v must b e variables of L whose sorts are
sitcalc
appropriate for the rst n arguments of P
Notice that GOLOG provides for arbitrary nesting of blo cks with pro cedure
declarations lo cal to their blo ck Other control structures e g conditionals
and while lo ops can b e de ned in terms of the ab ove constructs
def
if htest conditioni then hpr og r am i else hpr og r am i
htest conditioni hpr og r am i j htest conditioni hpr og r am i
To de ne while lo ops rst intro duce a nondeterministic iteration op erator
where hpr og r ami means do hpr og r ami or more times
def
hpr og r ami pro c P tr ue j hpr og r ami P endPro c P
Then while lo ops can b e de ned in terms of the op erator
def
while htest conditioni do hpr og r ami endWhile
htest conditioni hpr og r ami htest conditioni
Just as conventional Algol like programming languages never explicitly
refer to the state of their computation GOLOG programs never explicitly
mention situations hence the emphasis ab ove on situation suppressed terms
and formulas In de ning the semantics of such programs these situations
are taken into account as we now describ e
The Semantics of GOLOG
De nition Restoring Suppressed Situation Arguments
Whenever t is a situation suppressed term and is a term of L of sort
sitcalc
situation t denotes that term of L obtained by restoring the term
sitcalc
as the situation argument to all of the functional uents terms mentioned by
t In addition whenever is a situation suppressed formula denotes
that formula of L obtained by restoring the term as the situation
sitcalc
argument to all of the functional uent terms and all of the relational uent
atoms mentioned by
0
Next we de ne an abbreviation D o where is a program ex
0 0
pression and and are terms of sort situation D o should b e
viewed as a macro that expands into a second order situation calculus for
0
mula moreover that formula says that situation can be reached from
situation by executing some sequence of actions speci ed by Note that
our programs may b e nondeterministic that is may have multiple execu
tions terminating in di erent situations
D o is de ned inductively on the structure of its rst argument as follows
Primitive actions When is a situation suppressed term of sort
action
def
0 0
D o P oss do
Test actions When is a situation suppressed formula
def
0 0
D o
Sequence
def
0 0
D o s D o s D o s
1 2 1 2
Nondeterministic choice of two actions
def
0 0 0
D o j D o D o
1 2 1 2
Nondeterministic choice of action arguments
def
0 0
D o x x D o
Procedure cal ls For any predicate variable P of arity n whose rst
n arguments are not of sort situation and whose last two arguments
are of sort situation
def
0 0
D o P t t P t t
1 n 1 n
Blocks with local procedure declarations
D o fpro c P v endPro c
1 1 1
0
pro c P v endPro c g
n n n 0
n
def
0 0 0
s s v D o s s P v s s P P
i i i i 1 n
i=1
0
D o
0
Notice that in the de nition for pro cedure calls the actual parameters t
i
are rst evaluated with resp ect to the current situation t b efore
i
passing them to the pro cedure P so GOLOG s pro cedure invo cation is cal l
by value
Except for pro cedures the ab ove macro approach to de ning the seman
tics of GOLOG draws considerably from dynamic logic In e ect it rei es
as situations in the ob ject language of the situation calculus the p ossible
worlds with which the semantics of dynamic logic is de ned The macro de
nition for GOLOG pro cedures corresp onds to the more usual Scott Strachey
least xed point de nition in standard programming language semantics
0
With the ab ove de nition of D o in hand we are in a p osition
to de ne what we mean by the evaluation of a GOLOG program
The Evaluation of GOLOG Programs
De nition Prop er GOLOG Program
0
A GOLOG program is prop er i s and s are the only free variables in
0
dividual or predicate mentioned in D o s s In other words for each
individual variable x mentioned in x lies in the scop e of x or x in
a test condition of or x is mentioned in the scop e of a nondeterminis
tic choice op erator x Moreover for every pro cedure call P t t
1 n
mentioned in the second order variable P is b ound in by a pro cedure
declaration for P in some blo ck surrounding the pro cedure call
A prop er GOLOG program is evaluated relative to a background basic
theory of actions sp ecifying a particular application domain Sp eci cally if
D is such a basic action theory and is a prop er GOLOG program then
the evaluation of relative to D is de ned to b e the task of establishing the
following entailment
D j s D o S s
0
Any binding for the existentially quanti ed variable s obtained as a side
e ect of such a pro of constitutes an execution trace of In this resp ect
the evaluation of a GOLOG program is much like that of a Prolog goal
statement in the sense that b oth are evaluated by a theorem prover for the
purp oses of obtaining bindings to existentially quanti ed variables of the
theorem Of course in the case of GOLOG this theorem proving task is
much more daunting b ecause in general s D o S s stands for a very
0
complicated second order sentence of L and moreover D includes
sitcalc
a second order induction axiom However GOLOG enjoys a strong
Reduction prop erty that somewhat simpli es this task as we now describ e
Elimination for GOLOG
As de ned ab ove a GOLOG program is evaluated relative to a basic action
theory D One can prove that for this purp ose the foundational axioms
for situations contained in D are not needed
Theorem Elimination Theorem for GOLOG
is a basic action theory and Suppose that D D D D D
ss ap una S
0
that is a proper GOLOG program Then
j s D o S s D j s D o S s i D D D D
0 0 ss ap una S
0
Sensing and Knowledge
To talk ab out knowledge explicitly in the language of the situation calculus
we follow Mo ore and supp ose that there are situations other than S and
0
its successors that serve as epistemic alternatives to the real situations
A uent is considered to b e known in a situation s if it holds in all the
epistemic alternatives to s To allow knowledge to change as the result of
p erforming actions we imagine that certain actions return sensing results
here taken to b e binary What is known in a situation s will b e fully
determined by what was known initially the actions that led to situation
s and the sensing results of those actions The material in this section is
adapted from Lakemeyer and Levesque
The language of the epistemic situation calculus L is L augmented
e sitcalc
by two new distinguished symb ols K a unary predicate over situations
0
and SF a s a binary predicate over actions and situations Informally
K s holds when s is an epistemic alternative to S and SF a s holds
0 0
when action a returns binary sensing result in situation s
The foundational axioms for L describ e the augmented set of situations
e
First for any situation term let I nit b e the abbreviation
def
I nit s a do a s
where a and s do not app ear in The rst axiom is
do a s do a s a a s s
1 1 2 2 1 2 1 2
just as it was in and
P f s I nit s P s s a P s P do a s g s P s
which is similar to its counterpart in stating that all situations are suc
cessors of some initial one Next we need axioms that de ne the < relation
0 0
I nit s s < s
which is an analogue of the one in and
0 0
s < do a s s v s
which is unchanged Next we need to say that S and K are initial situa
0 0
tions
I nit S s K s I nit s
0 0
Finally we need to characterize the set of initial situations To ensure that
nothing need b e known ab out the uents initially we insist that there b e
initial situations to serve as p ossible epistemic alternatives for all combi
nations of uent values To keep things simple here let us assume that there
are only nitely many relational uents F F L and no functional
1 n e
uents We then use the following abbreviation
def
FV P P P P s
1 n n+1 n+2
V
n
x F x s P x
i i i i i
i=1
a P oss a s P a a SF a s P a
n+1 n+2
The next foundational axiom is then
P s I nit s FV P s
This is a variant of the existence of situations axiom rst intro duced by
Baker For some applications it will b e desirable to have additional ini
tial situations where actions pro duce arbitrary e ects Here we simply
assume that actions satisfy the successor state axioms in all epistemic alter
natives and hence we state that there are no additional initial situations
So the nal foundational axiom is
s s I nit s I nit s s s
1 2 1 2 1 2
P F V P s FV P s
1 2
We refer to these seven foundational axioms as
e
To characterize what is known we need two new categories of axioms
De nition Sensed Fluent Axiom
A sensed uent axiom of L for an action A is a sentence of the form
e
SF A x x s x x s
1 n A 1 n
where A is an n ary action function symb ol and x x s is a
A 1 n
formula that is uniform in s and whose free variables are among x x s
1 n
These axioms are used to characterize the conditions under which the action
returns the binary value For example in a dimensional rob otics domain
we might have
SF check B umper s x r obotP os x s x
as a way of saying that a bump er sensor on the rob ot returns when the
rob ot gets within units of a wall lo cated at the origin Actions that do
not involve any sensing can b e assumed to always return and hence the
would b e T RU E
De nition Initial Knowledge Axiom
An initial knowledge axiom of L is a sentence of the form
e
K s s
0
where s is a formula that is uniform in s and whose only free variable
is s
Axioms of this typ e are used to characterize the epistemic alternatives to
S For example in the dimensional rob otics world we might have
0
K s x x y r obotP os y s x y
0
as a way of saying that all the rob ot knows initially is that it is no further
than units away from the wall
Given these a know ledge action theory D is of the following form
e
D D D D D D D
sf K e e ss ap una S
0 0
where is the new set of foundational axioms D D D and D
e ss ap una S
0
are as b efore D is a set of sensed uent axioms one for each action
sf
is an initial knowledge axiom function symb ol and D
K
0
With such a theory in hand we can now explain what it means to know
a formula in an arbitrary situation by using two abbreviations First we
have
def
K P P
1 2 1 2
where the ellipsis stands for the universal closure of
fI nit s K s P s s g
2 0 1 1 2
fP s s P oss a s P oss a s
1 2 1 2
SF a s SF a s P do a s do a s g
1 2 1 2
0 0
The formula K s s can b e read as saying that s is an epistemic alter
native to situation s It is not hard to show that as a binary relation K is
required to b e transitive and Euclidean
Theorem The closure of the fol lowing two formulas is logical ly valid
K s s K s s K s s
2 1 3 2 3 1
K s s K s s K s s
2 1 3 1 3 2
With resp ect to knowledge this means that we are imagining an agent
that has p ositive and negative introsp ection The ab ove de nition of
K also inductively sp eci es the epistemic alternatives after p erforming any
sequence of actions
Theorem The closure of the fol lowing two formulas is logical ly valid
0 0
K s S K s
0 0
0 00 0 00 00
K s do a s s s do a s K s s
00 00
P oss a s P oss a s SF a s SF a s
0
So for an initial situation like S s is an epistemic alternative only when
0
0
K s holds for a non initial situation do a s an epistemic alternative is
0
00 00 00
of the form do a s where s is an alternative to s and where s and s
agree on P oss and SF
If we were to consider the formula K s as a uent we can see that
the second part of the ab ove theorem amounts to a successor state theorem
for K as it shows how to obtain the value of K do a s as a function
of K s and other prop erties of a and s Because we de ne knowledge
in terms of K b elow this means we have provided a solution to the frame
problem for knowledge a complete sp eci cation of how knowledge changes
as the result of p erforming any action
Finally we have the abbreviation
def
K now s s K s s
where is a situation suppressed formula and is any term of sort situation
This says that is known in situation if it comes out true at all epistemic
alternatives to Here for simplicity we are requiring to b e situation
suppressed a more elab orate account would allow us to express knowledge
ab out knowledge and other formulas where the mentions situations
To see how this would work in practice imagine that we have an action
theory D for the dimensional rob otics world that includes the two axioms
e
ab ove and
z r obotP os z S z
0
saying that the actual p osition of the rob ot is units away from the wall
as well as a successor state axiom for r obotP os saying that it is only a ected
by a mov e action
z r obotP os z do mov e x s r obotP os z x s
Then letting W al l C l ose stand for the abbreviation
x x y r obotP os y x y
we get the following as logical consequences of D
e
K now s W al l C l ose S
0
K now s W al l C l ose S
0
K now s W al l C l ose do mov e S
0
K now s W al l C l ose do mov e S
0
K now s W al l C l ose do check B umper do mov e S
0
K now s W al l C l ose do mov e do check B umper do mov e S
0
K now s W al l C l ose do mov e do check B umper do mov e S
0
K now s W al l C l ose do mov e do check B umper do mov e S
0
Acknowledgments
As describ ed here the formal foundations of the situation calculus have
evolved over a numb er of years and we gratefully acknowledge the con
tributions of the present emeritus and honorary memb ers of the Univer
sity of Toronto Cognitive Rob otics Group Alfredo Gabaldon Giusepp e
De Giacomo To dd Kelley Sam Kaufman Iluju Kiringai Gerhard Lake
meyer Yves Lesp erance Fangzhen Lin Je Lloyd Sheila McIlraith Daniel
Marcu Javier Pinto Shane Ruman Richard Scherl Steven Shapiro Misha
Soutchanski and Eugenia Ternovskaia We wish also to thank our research
sp onsors Agenzia Spaziale Italiana the National Science and Engineer
ing Research Council of Canada the Canadian Institute for Rob otics and
Intelligent Systems and the Centre for Information Technology of Ontario
References
A Baker A simple solution to the Yale sho oting problem In R Brach
man H J Levesque and R Reiter editors Proceedings of the First
International Conference on Principles of Know ledge Representation
and Reasoning KR pages Morgan Kaufmann Publishers
San Francisco CA
L Bertossi M Arenas and C Ferretti SCDBR An automated rea
soner for sp eci cations of database up dates Journal of Intel ligent In
formation Systems To app ear
W Burgard A B Cremers D Fox D H ahnel G Lakemeyer
D Schultz W Steiner and S Thrun The interactive museum tour
guide rob ot In Proceedings of the National Conference on Arti cial
Intel ligence AAAI pages AAAI Press MIT Press
G De Giacomo Y Lesp erance and H J Levesque Reasoning ab out
concurrent execution prioritized interrupts and exogenous actions in
the situation calculus In Proceedings of the Fourteenth International
Joint Conference on Arti cial Intel ligence pages Nagoya
Japan
R Goldblatt Logics of Time and Computation CSLI Lecture Notes
No Center for the Study of Language and Information Stanford
Univers ity Stanford CA nd edition
C C Green Theorem proving by resolution as a basis for question
answering systems In B Meltzer and D Michie editors Machine
Intel ligence pages American Elsevier New York
A R Haas The case for domain sp eci c frame axioms In F M Brown
editor The frame problem in arti cial intel ligence Proceedings of the
workshop pages Los Altos California Morgan
Kaufmann Publishers San Francisco CA
J Y Halp ern and Y O Moses A guide to completeness and complexity
for mo dal logics of knowledge and b elief Arti cial Intel ligence
M Jenkin Y Lesp erance H J Levesque F Lin J Lloyd D Marcu
R Reiter R B Scherl and K Tam A logical approach to p ortable
high level rob ot programming In Proceedings of the Tenth Australian
Joint Conference on Arti cial Intel ligence AI Perth Australia
Invited pap er
T G Kelley Mo deling complex systems in the situation calculus A
case study using the Dagstuhl steam b oiler problem In L C Aiello
J Doyle and S C Shapiro editors Principles of Know ledge Represen
tation and Reasoning Proceedings of the Fifth International Confer
ence KR pages Morgan Kaufmann Publishers San Fran cisco CA
G Lakemeyer and H Levesque AOL a logic of acting sensing know
ing and only knowing In Proc of KR Sixth International Confer
ence on Principles of Know ledge Representation and Reasoning pages
June
Y Lesp erance H J Levesque F Lin D Marcu R Reiter and
R Scherl A logical approach to high level rob ot programming a
progress rep ort In Control of the Physical World by Intel ligent Sys
tems Working Notes of the AAAI Fal l Symp
Y Lesp erance H J Levesque and R Reiter A situation calcu
lus approach to mo deling and programming agents In A Rao and
M Wo oldridge editors Foundations and Theories of Rational Agency
In press
H J Levesque What is planning in the presence of sensing In Proceed
ings of the National Conference on Arti cial Intel ligence AAAI
pages
H J Levesque R Reiter Y Lesp erance F Lin and R Scherl
GOLOG a logic programming language for dynamic domains J of
Logic Programming Special Issue on Actions
F Lin and R Reiter State constraints revisited J of Logic and
Computation special issue on actions and processes
J McCarthy Situations actions and causal laws Technical rep ort
Stanford University Reprinted in Semantic Information Pro cess
ing M Minsky ed MIT Press Cambridge Mass pp
J McCarthy and T Costello Combining narratives In A G Cohn and
L K Schub ert editors Principles of Know ledge Representation and
Reasoning Proceedings of the Sixth International Conference KR
pages Morgan Kaufmann Publishers San Francisco CA
J McCarthy and P Hayes Some philosophical problems from the
standp oint of arti cial intelligence In B Meltzer and D Michie ed
itors Machine Intel ligence pages Edinburgh University
Press Edinburgh Scotland
S A McIlraith Towards a Formal Account of Diagnostic Problem
Solving PhD thesis Department of Computer Science University of
Toronto Toronto Ontario Canada
R C Mo ore A formal theory of knowledge and action In Jerry B
Hobbs and Rob ert C Mo ore editors Formal Theories of the Com
monsense World chapter pages Ablex Publishing Corp
Norwo o d New Jersey
E P D Pednault ADL Exploring the middle ground b etween STRIPS
and the situation calculus In R J Brachman H Levesque and
R Reiter editors Proceedings of the First International Conference on
Principles of Know ledge Representation and Reasoning KR pages
Morgan Kaufmann Publishers San Francisco CA
J A Pinto Temporal Reasoning in the Situation Calculus PhD thesis
University of Toronto Department of Computer Science
F Pirri and R Reiter Some contributions to the metatheory of the
situation calculus Journal of the ACM To app ear
http www cs toronto edu cogrob o
R Reiter The frame problem in the situation calculus a simple
solution sometimes and a completeness result for goal regression
In Vladimir Lifschitz editor Arti cial Intel ligence and Mathematical
Theory of Computation Papers in Honor of John McCarthy pages
Academic Press San Diego CA
R Reiter Proving prop erties of states in the situation calculus Arti
cial Intel ligence
R Reiter On sp ecifying database up dates Journal of Logic Program
ming
R Reiter Natural actions concurrency and continuous time in the
situation calculus In L C Aiello J Doyle and S C Shapiro editors
Principles of Know ledge Representation and Reasoning Proceedings
of the Fifth International Conference KR pages Morgan
Kaufmann Publishers San Francisco CA
R Reiter Sequential temp oral GOLOG In A G Cohn and L K Schu
b ert editors Principles of Know ledge Representation and Reasoning
Proceedings of the Sixth International Conference KR pages
Morgan Kaufmann Publishers San Francisco CA
L K Schub ert Monotonic solution of the frame problem in the situa
tion calculus an e cient metho d for worlds with fully sp eci ed actions
In H E Kyb erg R P Loui and G N Carlson editors Know ledge Rep
resentation and Defeasible Reasoning pages Kluwer Academic
Press
S Shapiro Y Lesp erance and H J Levesque Goals and rational
action in the situation calculus a preliminary rep ort In Working
Notes of the AAAI Fal l Symposium on Rational Agency Concepts
Theories Models and Applications Cambridge MA
J E Stoy Denotational Semantics MIT Press