The Tractability of Model-Checking for LTL: The Good, the Bad, and the Ugly Fragments (original) (raw)
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Electronic Colloquium on Computational Complexity, 2006
In a seminal paper from 1985, Sistla and Clarke showed that satisfiability 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 undertakes a systematic study of satisfiability for LTL formulae over restricted sets of propositional and temporal operators. Since every propositional operator corresponds to a Boolean function, there exist infinitely many propositional operators. In order to systematically cover all possible sets of them, we use Post's lattice. With its help, we determine the computational complexity of LTL satisfiability for all combinations of temporal operators and all but two classes of propositional functions. Each of these infinitely many problems is shown to be either PSPACE-complete, NP-complete, or in P.
On the complexity of computing minimal unsatisfiable LTL formulas
We show that (1) the Minimal False QCNF search-problem (MF-search) and the Minimal Unsatisfiable LTL formula search problem (MU-search) are FPSPACE complete because of the very expressive power of QBF/LTL, (2) we extend the PSPACE-hardness of the MF decision problem to the MU decision problem. As a consequence, we deduce a positive answer to the open question of PSPACE hardness of the inherent Vacuity Checking problem. We even show that the Inherent Non Vacuous formula search problem is also FPSPACEcomplete.
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Proceedings of the Annual Conference of the South African Institute of Computer Scientists and Information Technologists, 2018
It is known [4] that both satisfiability and model-checking problems for propositional Linear-time Temporal Logic, LTL, with only a single propositional variable in the language are PSPACE-complete, which coincides with the complexity of these problems for LTL with an arbitrary number of propositional variables [14]. In the present paper, we show that the same result can be obtained by modifying the original proof of PSPACE-hardness for LTL from [14]; i.e., we show how to modify the construction from [14] to model the computations of polynomially-space bound Turing machines using only formulas of one variable. We believe that our alternative proof of the results from [4] gives additional insight into the semantic and computational properties of LTL.
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Several propositional fragments have been considered so far as target languages for knowledge compilation and used for improving computational tasks from major AI areas (like inference, diagnosis and planning); among them are the ordered binary decision diagrams, prime implicates, prime implicants, "formulae" in decomposable negation normal form. On the other hand, the validity problem val(QPROP P S ) for Quantified Boolean Formulae (QBF) has been acknowledged for the past few years as an important issue for AI, and many solvers have been designed. In this paper, the complexity of restrictions of the validity problem for QBF obtained by imposing the matrix of the input QBF to belong to propositional fragments used as target languages for compilation, is identified. It turns out that this problem remains hard (PSPACE-complete) even under severe restrictions on the matrix of the input. Nevertheless some tractable restrictions are pointed out.
The Complexity of Propositional Linear Temporal Logics in Simple Cases
Information and Computation, 2002
It is well-known that model checking and satisfiability for PLTL are PSPACE-complete. By contrast, very little is known about whether there exist some interesting fragments of PLTL with a lower worst-case complexity. Such results would help understand why PLTL model checkers are successfully used in practice.
The Complexity of Clausal Fragments of LTL
Lecture Notes in Computer Science, 2013
We introduce and investigate a number of fragments of propositional temporal logic LTL over the flow of time (Z, <). The fragments are defined in terms of the available temporal operators and the structure of the clausal normal form of the temporal formulas. We determine the computational complexity of the satisfiability problem for each of the fragments, which ranges from NLogSpace to PTime, NP and PSpace.
Complexity results for quantified boolean formulae based on complete propositional languages
2006
Several propositional fragments have been considered so far as target languages for knowledge compilation and used for improving computational tasks from major AI areas (like inference, diagnosis and planning); among them are the ordered binary decision diagrams, prime implicates, prime implicants, "formulae" in decomposable negation normal form. On the other hand, the validity problem val(QPROP P S ) for Quantified Boolean Formulae (QBF) has been acknowledged for the past few years as an important issue for AI, and many solvers have been designed. In this paper, the complexity of restrictions of the validity problem for QBF obtained by imposing the matrix of the input QBF to belong to propositional fragments used as target languages for compilation, is identified. It turns out that this problem remains hard (PSPACE-complete) even under severe restrictions on the matrix of the input. Nevertheless some tractable restrictions are pointed out.
Time-space lower bounds for satisfiability
Journal of the ACM ( …, 2005
We establish the first polynomial time-space lower bounds for satisfiability on general models of computation. We show that for any constant c less than the golden ratio there exists a positive constant d such that no deterministic random-access Turing machine can solve satisfiability in time n c and space n d , where d approaches 1 when c does. On conondeterministic instead of deterministic machines, we prove the same for any constant c less than √ 2. Our lower bounds apply to nondeterministic linear time and almost all natural NP-complete problems known. In fact, they even apply to the class of languages that can be solved on a nondeterministic machine in linear time and space n 1/c .
2002
We study the problem exact 3-SAT (X3SAT) for propositional formulas in conjunctive normal form (CNF) C = c 1 _ _ c m each clause c i containing at most three literals. Then C 2 X3SAT i there is a truth assignment t : V n ! f0; 1g; s.t. exactly one literal in each clause is set to true (1). The corresponding decision problem also is called X3SAT. As is well known problem X3SAT is NP-complete [5] and also XSAT as a generalization is NP-complete. By adapting and improving branching techniques like that in [4] we are able to prove that XSAT 2 O(2 thereby improving a more recent result found in [1] stating XSAT 2 O(2 ).
… of Computer Science, 1999. 40th Annual …, 1999
We show that non-deterministic time N T IM E(n) is not contained in deterministic time n 2−ǫ and poly-logarithmic space, for any ǫ > 0. This implies that (infinitely often) satisfiability cannot be solved in time O(n 2−ǫ ) and polylogarithmic space. A similar result is presented for uniform circuits.