Martin Mundhenk

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.

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!

THE COMPLEXITY OF SATISFIABILITY FOR FRAGMENTS OF CTL AND CTL

The satisfiability problems for and are known to be EXPTIME-complete, resp. 2EXPTIME-complete (Fischer and Ladner (1979), Vardi and Stockmeyer (1985)). For fragments that use less temporal or propositional operators, the complexity may decrease. This paper undertakes a systematic study of satisfiability for -and -formulae over restricted sets of propositional and temporal operators. We show that restricting the temporal operators yields satisfiability problems complete for 2EXPTIME, EXPTIME, PSPACE, and NP. Restricting the propositional operators either does not change the complexity (as determined by the temporal operators), or yields very low complexity like NC1, TC0, or NLOGTIME.

The tractability of model checking for LTL: The good, the bad, and the ugly fragments

In a seminal paper from 1985, Sistla and Clarke showed that the model-checking problem for Linear Temporal Logic (LTL) is either NP-complete or PSPACE-complete, depending on the set of temporal operators used. If in contrast, the set of propositional operators is restricted, the complexity may decrease. This article systematically studies the model-checking problem for LTL formulae over restricted sets of propositional and temporal operators. For almost all combinations of temporal and propositional operators, we determine whether the model-checking problem is tractable (in PTIME) or intractable (NP-hard). We then focus on the tractable cases, showing that they all are NL-complete or even logspace solvable. This leads to a surprising gap in complexity between tractable and intractable cases. It is worth noting that our analysis covers an infinite set of problems, since there are infinitely many sets of propositional operators.

On Bounded Truth-Table, Conjunctive, and Randomized Reductions to Sparse Sets

In this paper we study the consequences of the existence of sparse hard sets for NP and other complexity classes under certain types of deterministic, randomized, and nondeterministic reductions. We show that if an NP-complete set is bounded truth-table reducible to some set that conjunctively reduces to a sparse set then P=NP. We next show that if an NP-complete set is bounded truth-table reducible to some set that randomly reduces (via a co-rp reduction) to some set that conjunctively reduces to a sparse set then RP=NP. Finally, we prove that if a coNP-complete set reduces via a nondeterministic polynomial time many-one reduction to a co-sparse set then PH=Θ 2 p . On the other hand, we show that nondeterministic polynomial time many-one reductions to sparse sets are as powerful as nondeterministic Turing reductions to sparse sets.

The Complexity of Satisfiability for Fragments of Hybrid Logic--Part I

The satisfiability problem of hybrid logics with the downarrow binder is known to be undecidable. This initiated a research program on decidable and tractable fragments. In this paper, we investigate the effect of restricting the propositional part of the language on decidability and on the complexity of the satisfiability problem over arbitrary, transitive, total frames, and frames based on equivalence relations. We also consider different sets of modal and hybrid operators. We trace the border of decidability and give the precise complexity of most fragments, in particular for all fragments including negation. For the monotone fragments, we are able to distinguish the easy from the hard cases, depending on the allowed set of operators.

Model Checking CTL is Almost Always Inherently Sequential

The model checking problem for CTL is known to be P-complete (Clarke, Emerson, and Sistla (1986), see Schnoebelen (2002)). We consider fragments of CTL obtained by restricting the use of temporal modalities or the use of negations---restrictions already studied for LTL by Sistla and Clarke (1985) and Markey (2004).For all these fragments, except for the trivial case without any temporal operator, we systematically prove model checking to be either inherently sequential (P-complete) or very efficiently parallelizable (LOGCFL-complete). For most fragments, however, model checking for CTL is already P-complete. Hence our results indicate that in most applications, approaching CTL model checking by parallelism will not result in the desired speed up. We also completely determine the complexity of the model checking problem for all fragments of the extensions ECTL, CTL+, and ECTL+.

The Tractability of Model-checking for LTL: The Good, the Bad, and the Ugly Fragments

In a seminal paper from 1985, Sistla and Clarke showed that the model-checking problem for Linear Temporal Logic (LTL) is either NP-complete or PSPACE-complete, depending on the set of temporal operators used. If, in contrast, the set of propositional operators is restricted, the complexity may decrease. This paper systematically studies the model-checking problem for LTL formulae over restricted sets of propositional and temporal operators. For almost all combinations of temporal and propositional operators, we determine whether the model-checking problem is tractable (in P) or intractable (NP-hard). We then focus on the tractable cases, showing that they all are NL-complete or even logspace solvable. This leads to a surprising gap in complexity between tractable and intractable cases. It is worth noting that our analysis covers an infinite set of problems, since there are infinitely many sets of propositional operators.

Upper bounds for the complexity of sparse and tally descriptions

We investigate the complexity of computing small descriptions for sets in various reduction classes to sparse sets. For example, we show that if a setA and its complement conjunctively reduce to some sparse set, then they also are conjunctively reducible to a P(A ⊕ SAT)-printable tally set. As a consequence, the class IC[log,poly] of sets with low instance complexity is contained in theEL1Σ-level of the extended low hierarchy. By refining our techniques, we also show that all word-decreasing self-reducible sets in IC[log, poly] are in NP ∩ co-NP and therefore low for NP. We derive similar results for sets inRdp(SPARSE) andRhdp (Rcp(SPARSE)), as well as in some nondeterministic reduction classes to sparse sets.