The complexity of logical theories

Berman, Leonard

doi:10.1016/0304-3975(80)90037-7

Cited by 127 publications

(106 citation statements)

References 3 publications

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“…Presburger arithmetic is a classical first order theory of logic, proven decidable by Presburger [40]. Bounds on the complexity of decision algorithms have been examined in the general case [8,17,22,23,38], as a function of the number of quantifier alternations [25,28,31,43,46], and for fixed (small) number of quantifier alternations [29,45].…”

Section: Property 2 Gives a Nice Geometric Characterization Of Presbumentioning

confidence: 99%

Presburger Arithmetic, Rational Generating Functions, and Quasi-Polynomials

Woods

2013

Automata, Languages, and Programming

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Abstract. Presburger arithmetic is the first-order theory of the natural numbers with addition (but no multiplication). We characterize sets that can be defined by a Presburger formula as exactly the sets whose characteristic functions can be represented by rational generating functions; a geometric characterization of such sets is also given. In addition, if p = (p1, . . . , pn) are a subset of the free variables in a Presburger formula, we can define a counting function g(p) to be the number of solutions to the formula, for a given p. We show that every counting function obtained in this way may be represented as, equivalently, either a piecewise quasi-polynomial or a rational generating function. Finally, we translate known computational complexity results into this setting and discuss open directions.

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Section: Property 2 Gives a Nice Geometric Characterization Of Presbumentioning

confidence: 99%

Presburger Arithmetic, Rational Generating Functions, and Quasi-Polynomials

Woods

2013

Automata, Languages, and Programming

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“…It is well-known that satisfiability of Presburger arithmetic is decidable in doubly exponential space. For complexity results of corresponding decision procedures, we refer to [9,8,2]. Note that the set of vectors v satisfying a Presburger formula with free variables is a semi-linear set which can be effectively computed [10,11].…”

Section: Unordered Treesmentioning

confidence: 99%

Numerical document queries

Seidl

Schwentick

Muscholl

2003

Proceedings of the Twenty-Second ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems

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A query against a database behind a site like Napster may search, e.g., for all users who have downloaded more jazz titles than pop music titles. In order to express such queries, we extend classical monadic second-order logic by Presburger predicates which pose numerical restrictions on the children (content) of an element node and provide a precise automata-theoretic characterization. While the existential fragment of the resulting logic is decidable, it turns out that satisfiability of the full logic is undecidable. Decidable satisfiability and a querying algorithm even with linear data complexity can be obtained if numerical constraints are only applied to those contents of elements where ordering is irrelevant. Finally, it is sketched how these techniques can be extended also to answer questions like, e.g., whether the total price of the jazz music downloaded so far exceeds a user's budget.

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“…It was shown 2ExpTime-hard in [FR74] and to be in 2ExpSpace in [FR79]. An exact complexity characterization is provided in [Ber80] (double exponential time on alternating Turing machines with linear amounts of alternations).…”

Section: Theorem 1 (Presburger Arithmetic Decidability)mentioning

confidence: 99%

Witness Runs for Counter Machines

Barrett

Demri

Deters

2013

Frontiers of Combining Systems

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Abstract. In this paper, we present recent results about the verification of counter machines by using decision procedures for Presburger arithmetic. We recall several known classes of counter machines for which the reachability sets are Presburger-definable as well as temporal logics with arithmetical constraints. We discuss issues related to flat counter machines, path schema enumeration, and the use of SMT solvers.

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The complexity of logical theories

Cited by 127 publications

References 3 publications

Presburger Arithmetic, Rational Generating Functions, and Quasi-Polynomials

Presburger Arithmetic, Rational Generating Functions, and Quasi-Polynomials

Numerical document queries

Witness Runs for Counter Machines

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