The Complexity of the Satisfiability Problem for Krom Formulas (original) (raw)
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On the computational complexity of satisfiability in propositional logics of programs
Theoretical Computer Science, 1982
The satisfiability problems of propositional algorithmic logic and propositional dynamic logic are shown to be complete in the classes of languages accepted in polynomial space by the Ijeterministic and alternating Turing machines respectively. Explicit upper and lower bounds on. 1:he space complexity ase calculated. Exponential lower bounds on the space co;Bplexity of tht: !;atisfiability problems of rthese logics extended by adding a certain program connective are proved.
On complexity reduction of Σ 1 formulas
Archive for Mathematical Logic, 2003
For a fixed q ∈ N and a given 1 definition φ (d, x), where d is a parameter, we construct a model M of I 0 + ¬ exp and a non standard d ∈ M such that in M either φ has no witness smaller than d or φ is equivalent to a formula ϕ (d, x) having no more than q alternations of blocks of quantifiers.
2005
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.
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.
… 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.
On the combinatorial and algebraic complexity of quantifier elimination
Journal of the ACM, 1996
In this paper, a new algorithm for performing quantifier elimination from first order formulas over real closed fields in given. This algorithm improves the complexity of the asymptotically fastest algorithm for this problem, known to this data. A new feature of this algorithm is that the role of the algebraic part (the dependence on the degrees of the imput polynomials) and the combinatorial part (the dependence on the number of polynomials) are sparated. Another new feature is that the degrees of the polynomials in the equivalent quantifier-free formula that is output, are independent of the number of input polynomials. As special cases of this algorithm new and improved algorithms for deciding a sentence in the first order theory over real closed fields, and also for solving the existential problem in the first order theory over real closed fields, are obtained.
On the Complexity of SAT (Revised)
2008
We show 1 that non-deterministic time N T IM E(n) is not contained in deterministic time n √ 2− and polylogarithmic space, for any > 0. This implies that (infinitely often) satisfiability cannot be solved in time O(n √ 2− ) and poly-logarithmic space. A similar result is presented for uniform circuits.
The quantifier complexity of polynomial-size iterated definitions in first-order logic
Mathematical Logic Quarterly, 2010
We refine the constructions of Ferrante-Rackoff and Solovay on iterated definitions in first-order logic and their expressibility with polynomial size formulas. These constructions introduce additional quantifiers; however, we show that these extra quantifiers range over only finite sets and can be eliminated. We prove optimal upper and lower bounds on the quantifier complexity of polynomial size formulas obtained from the iterated definitions. In the quantifier-free case and in the case of purely existential or universal quantifiers, we show that Ω(n/ log n) quantifiers are necessary and sufficient. The last lower bounds are obtained with the aid of the Yao-Håstad switching lemma.
Complexity and structural heuristics for propositional and quantified satisfiability
2007
Decision procedures for various logics are used as general-purpose solvers in computer science. A particularly popular choice is propositional logic, which is simultaneously powerful enough to model problems in many application domains, including formal verification and planning, while at the same time simple enough to be efficiently solved for many practical cases. Similarly, there are also recent interests in using QBF, an extension of propositional logic, as a modeling language to be used in a similar fashion. The hope is that QBF, being a more powerful language, can compactly encode, and in turn, be used to solve, a larger range of applications. Still, propositional logic and QBF are respectively complete for the complexity classes NP and PSPACE, thus, both can be theoretically considered intractable. A popular hypothesis is that realworld problems contain underlying structure that can be exploited by the decision procedures. In this dissertation, we study the impact of structural constraints (in the Acknowledgement First of all, I would like to thank my advisor Prof. Moshe Y. Vardi for the support and guidance provided through out my six years at Rice. You have advised me on not just the technical developments and approaches, but also the methodology, presentation, and insight needed to do research. And of course, the financial support too. Also, I would like to thank the other members of my defense committee, Professors Devika Subramanian, Walid Taha, and Kartik Mohanram for their help. I have taken classes under all of you and I learned much, especially on looking at the area of computer science at a wider angle than what I would originally do.
The Complexity of Generalized Satisfiability for Linear Temporal Logic
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.