Judy Goldsmith

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.

The computational complexity of dominance and consistency in CP-Nets

We investigate the computational complexity of testing dominance and consistency in CP-nets. Up until now, the complexity of dominance has been determined only for restricted classes in which the dependency graph of the CP-net is acyclic. However, there are preferences of interest that define cyclic dependency graphs; these are modeled with general CP-nets. We show here that both dominance and consistency testing for general CP-nets are PSPACE-complete. The reductions used in the proofs are from STRIPS planning, and thus establish strong connections between both areas.

Complexity of finite-horizon Markov decision process problems

Controlled stochastic systems occur in science engineering, manufacturing, social sciences, and many other cntexts. If the systems is modeled as a Markov decision process (MDP) and will run ad infinitum, the optimal control policy can be computed in polynomial time using linear programming. The problems considered here assume that the time that the process will run is finite, and based on the size of the input. There are mny factors that compound the complexity of computing the optimal policy. For instance, there are many factors that compound the complexity of this computation. For instance, if the controller does not have complete information about the state of the system, or if the system is represented in some very succint manner, the optimal policy is provably not computable in time polynomial in the size of the input. We analyze the computational complexity of evaluating policies and of determining whether a sufficiently good policy exists for a MDP, based on a number of confounding factors, including the observability of the system state; the succinctness of the representation; the type of policy; even the number of actions relative to the number of states. In almost every case, we show that the decision problem is complete for some known complexity class. Some of these results are familiar from work by Papadimitriou and Tsitsiklis and others, but some, such as our PL-completeness proofs, are surprising. We include proofs of completeness for natural problems in the as yet little-studied classes NPPP.

Limited nondeterminism

Yes, the lucky 13th column is here, and it is a guest column written by J. Goldsmith, M. Levy, and M. Mundhenk on the topic of limited nondeterminism---classes and hierarchies derived when nondeterminism itself is viewed as a quantifiable resource (as it indeed is!).Coming up in the Complexity Theory Column in the very special 100th issue of SIGACT News: A forum on the future of complexity theory. Many of the fields leading lights share their exciting insights on what lies ahead, so please be there in three!

Preference Handling for Artificial Intelligence

This article explains the benefits of preferences for AI systems and draws a picture of current AI research on preference handling. It thus provides an introduction to the topics covered by this special issue on preference handling.

Three results on the polynomial isomorphism of complete sets

This paper proves three results relating to the isomorphism question for NP-complete sets. Result 1: We construct an oracle A such that SATA is ≤mP- complete for NPA and all ≤mP-complete sets for NPA are pA- isomorphic to SATA. Result 2: We construct a time function T(n) such that DTIME(T(n)) contains btt-complete sets, which are many-one equivalent, but are not p-isomorphic. The proof of this result has two corollaries: 1) There is an oracle, D, such that NPD contains non-p-isomorophic ≤m(D),P-complete sets. 2) There is a ≤mP-degree that contains non-p-isomorphic sets. Result 3: We show that no simple modification of the diagonalization argument used by Ko, Long and Du can be used to produce sets that are both EXPtime-complete w.r.t, polynomial many-one reducibility and not p-isomorphic.

Near-testable sets

In this paper a new property of sets, near-testability, is introduced. A set S is near-testable

Semistructured probabilistic databases

(S \in NT)

Learning CP-net preferences online from user queries

if the membership relation for all immediate neighbors is polynomially computable; i.e., if the function

Updates and Uncertainty in CP-Nets

t(x) = \chi _S (x) + \chi _S (x - 1)(\bmod 2)