The Computational Complexity of Probabilistic Planning

We examine the computational complexity of testing and finding small plans in probabilistic planning domains with both flat and propositional representations. The complexity of plan evaluation and existence varies with the plan type sought; we examine totally ordered plans, acyclic plans, and looping plans, and partially ordered plans under three natural definitions of plan value. We show that problems of interest are complete for a variety of complexity classes: PL, P, NP, co-NP, PP, NPPP, co-NPPP, and PSPACE. In the process of proving that certain planning problems are complete for NPPP, we introduce a new basic NPPP -complete problem, E-MAJSAT, which generalizes the standard Boolean satisfiability problem to computations involving probabilistic quantities; our results suggest that the development of good heuristics for E-MAJSAT could be important for the creation of efficient algorithms for a wide variety of problems.

[1]  Janos Simon.  On some central problems in computational complexity , 1975 .

[2]  John Gill,et al.  Computational Complexity of Probabilistic Turing Machines , 1977, SIAM J. Comput..

[3]  Allan Borodin,et al.  Parallel Computation for Well-Endowed Rings and Space-Bounded Probabilistic Machines , 1984, Inf. Control..

[4]  Hermann Jung.  On Probabilistic Time and Space , 1985, ICALP.

[5]  David Chapman,et al.  Planning for Conjunctive Goals , 1987, Artif. Intell..

[6]  John N. Tsitsiklis,et al.  The Complexity of Markov Decision Processes , 1987, Math. Oper. Res..

[7]  Richard E. Ladner.  Polynomial Space Counting Problems , 1989, SIAM J. Comput..

[8]  John L. Bresina,et al.  Anytime Synthetic Projection: Maximizing the Probability of Goal Satisfaction , 1990, AAAI.

[9]  David A. McAllester,et al.  Systematic Nonlinear Planning , 1991, AAAI.

[10]  Seinosuke Toda,et al.  PP is as Hard as the Polynomial-Time Hierarchy , 1991, SIAM J. Comput..

[11]  Jacobo Torán,et al.  Complexity classes defined by counting quantifiers , 1991, JACM.

[12]  W. Lovejoy.  A survey of algorithmic methods for partially observed Markov decision processes , 1991 .

[13]  V Vinay.  Counting auxiliary pushdown automata and semi-unbounded arithmetic circuits , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[14]  Anne Condon,et al.  The Complexity of Stochastic Games , 1992, Inf. Comput..

[15]  Carme Àlvarez,et al.  A Very Hard log-Space Counting Class , 1993, Theor. Comput. Sci..

[16]  Dan Roth,et al.  On the Hardness of Approximate Reasoning , 1993, IJCAI.

[17]  Todd Michael Mansell,et al.  A method for Planning Given Uncertain and Incomplete Information , 1993, UAI.

[18]  Reid G. Simmons,et al.  Risk-Sensitive Planning with Probabilistic Decision Graphs , 1994, KR.

[19]  Gerald Tesauro,et al.  TD-Gammon, a Self-Teaching Backgammon Program, Achieves Master-Level Play , 1994, Neural Computation.

[20]  Leslie Pack Kaelbling,et al.  Acting Optimally in Partially Observable Stochastic Domains , 1994, AAAI.

[21]  Robert P. Goldman,et al.  Epsilon-Safe Planning , 1994, UAI.

[22]  Daniel S. Weld,et al.  Probabilistic Planning with Information Gathering and Contingent Execution , 1994, AIPS.

[23]  Martin L. Puterman,et al.  Markov Decision Processes: Discrete Stochastic Dynamic Programming , 1994 .

[24]  Tom Bylander,et al.  The Computational Complexity of Propositional STRIPS Planning , 1994, Artif. Intell..

[25]  Nicholas Kushmerick,et al.  An Algorithm for Probabilistic Planning , 1995, Artif. Intell..

[26]  David E. Smith,et al.  Representation and Evaluation of Plans with Loops , 1995 .

[27]  Bernhard Nebel,et al.  COMPLEXITY RESULTS FOR SAS+ PLANNING , 1995, Comput. Intell..

[28]  V. S. Subrahmanian,et al.  Complexity, Decidability and Undecidability Results for Domain-Independent Planning , 1995, Artif. Intell..

[29]  Andrew G. Barto,et al.  Improving Elevator Performance Using Reinforcement Learning , 1995, NIPS.

[30]  Craig Boutilier,et al.  Exploiting Structure in Policy Construction , 1995, IJCAI.

[31]  Christer Bäckström.  Expressive Equivalence of Planning Formalisms , 1995, Artif. Intell..

[32]  Craig Boutilier,et al.  Computing Optimal Policies for Partially Observable Decision Processes Using Compact Representations , 1996, AAAI/IAAI, Vol. 2.

[33]  Eric Allender,et al.  Relationships Among PL, #L, and the Determinant , 1996, RAIRO Theor. Informatics Appl..

[34]  T. Dean,et al.  Generating optimal policies for high-level plans with conditional branches and loops , 1996 .

[35]  Craig Boutilier,et al.  Approximate Value Trees in Structured Dynamic Programming , 1996, ICML.

[36]  Eric Allender,et al.  The Complexity of Policy Evaluation for Finite-Horizon Partially-Observable Markov Decision Processes , 1997, MFCS.

[37]  E. Allender,et al.  Encyclopaedia of Complexity Results for Finite-Horizon Markov Decision Process Problems , 1997 .

[38]  Craig Boutilier,et al.  Abstraction and Approximate Decision-Theoretic Planning , 1997, Artif. Intell..

[39]  Michael L. Littman,et al.  The Complexity of Plan Existence and Evaluation in Probabilistic Domains , 1997, UAI.

[40]  Michael L. Littman,et al.  Probabilistic Propositional Planning: Representations and Complexity , 1997, AAAI/IAAI.

[41]  Michael L. Littman,et al.  MAXPLAN: A New Approach to Probabilistic Planning , 1998, AIPS.

[42]  Leslie Pack Kaelbling,et al.  Planning and Acting in Partially Observable Stochastic Domains , 1998, Artif. Intell..

[43]  Michael L. Littman,et al.  Using Caching to Solve Larger Probabilistic Planning Problems , 1998, AAAI/IAAI.

[44]  Shlomo Zilberstein,et al.  Finite-memory control of partially observable systems , 1998 .

[45]  Craig Boutilier,et al.  Decision-Theoretic Planning: Structural Assumptions and Computational Leverage , 1999, J. Artif. Intell. Res..