Abstract
We determine the complexity of counting models of bounded size of specifications expressed in linear-time temporal logic. Counting word-models is #P-complete, if the bound is given in unary, and as hard as counting accepting runs of nondeterministic polynomial space Turing machines, if the bound is given in binary. Counting tree-models is as hard as counting accepting runs of nondeterministic exponential time Turing machines, if the bound is given in unary. For a binary encoding of the bound, the problem is at least as hard as counting accepting runs of nondeterministic exponential space Turing machines, and not harder than counting accepting runs of nondeterministic doubly-exponential time Turing machines. Finally, counting arbitrary transition systems satisfying a formula is #P-hard and not harder than counting accepting runs of nondeterministic polynomial time Turing machines with a PSPACE oracle, if the bound is given in unary. If the bound is given in binary, then counting arbitrary models is as hard as counting accepting runs of nondeterministic exponential time Turing machines.
Highlights
Model counting, the problem of computing the number of models of a logical formula, generalizes the satisfiability problem and has diverse applications: many probabilistic inference problems, such as Bayesian net reasoning [13], and planning problems, such as computing the robustness of plans in incomplete domains [15], can be formulated as model counting problems of propositional logic
Linear-time Temporal Logic (LTL) model counting asks for computing the number of transition systems that satisfy a given LTL formula
The hardness proof is similar to the one for Theorem 2: for a nondeterministic polynomial space Turing machine M bounded by a polynomial p(n) and an input word w we can define a formula φwM in the same way as in Theorem 2
Summary
The problem of computing the number of models of a logical formula, generalizes the satisfiability problem and has diverse applications: many probabilistic inference problems, such as Bayesian net reasoning [13], and planning problems, such as computing the robustness of plans in incomplete domains [15], can be formulated as model counting problems of propositional logic. The high complexity in the formula is, not a major concern in practice, since specifications are typically small while models are large (cf the state-space explosion problem) We complement these algorithms by analyzing the computational complexity of the model counting problems for full LTL by placing the problems into counting complexity classes. A nondeterministic polynomial time Turing machine M (with or without oracle) has at most O(2p(n)) different runs on inputs of length n for some polynomial p This means that there is an exponential upper bound on functions in #P and in #oC for every C. An LTL tautology has exponentially many word-models of length k and more than doublyexponentially many tree-models of height k This means, that no function in any of the counting classes defined above can capture the counting problems for LTL. All proofs omitted due to space restrictions can be found in the full version [19]
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