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 RInformation Invariants in Rob otics Articial Intel li gence In press Vol Jan Im 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 Oce of Sp onsored Research the Mathematical Sciences Institute Intel Corp oration and ATT 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 Tomas LozanoPerez 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 sideeect 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 computationtheoretic 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 Articial Intel ligence Bruce Randall Donald Ithaca and Palo Alto Contents Part I State Communication and SideEects 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 NonGeometric Formulation of Information Invariants F Provable Information Invariants with Performance Measures F Kino dynamics and TradeOs Glossary of Symbols Section Page Denition 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 conguration R ship x x ships p osition h ships heading angle b etween h and the goal direction r N direction of North H the lighthouse b eacon sensor L lighthouse Rs 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 commL 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 mbS maximum bandwidth of S A commb datapath with bandwidth log b commS datapath with bandwidth mbS 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 subsection explaining or containing that symbol Section Page Denition 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 ecient 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 conguration space S situated sensor system p ermutation of an immersion S S p ermutation of a sensor system