2,867 research outputs found

    Bounds on the Automata Size for Presburger Arithmetic

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    Automata provide a decision procedure for Presburger arithmetic. However, until now only crude lower and upper bounds were known on the sizes of the automata produced by this approach. In this paper, we prove an upper bound on the the number of states of the minimal deterministic automaton for a Presburger arithmetic formula. This bound depends on the length of the formula and the quantifiers occurring in the formula. The upper bound is established by comparing the automata for Presburger arithmetic formulas with the formulas produced by a quantifier elimination method. We also show that our bound is tight, even for nondeterministic automata. Moreover, we provide optimal automata constructions for linear equations and inequations

    Quantifier elimination for the reals with a predicate for the powers of two

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    In 1985, van den Dries showed that the theory of the reals with a predicate for the integer powers of two admits quantifier elimination in an expanded language, and is hence decidable. He gave a model-theoretic argument, which provides no apparent bounds on the complexity of a decision procedure. We provide a syntactic argument that yields a procedure that is primitive recursive, although not elementary. In particular, we show that it is possible to eliminate a single block of existential quantifiers in time 2O(n)02^0_{O(n)}, where nn is the length of the input formula and 2kx2_k^x denotes kk-fold iterated exponentiation

    Partial Quantifier Elimination

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    We consider the problem of Partial Quantifier Elimination (PQE). Given formula exists(X)[F(X,Y) & G(X,Y)], where F, G are in conjunctive normal form, the PQE problem is to find a formula F*(Y) such that F* & exists(X)[G] is logically equivalent to exists(X)[F & G]. We solve the PQE problem by generating and adding to F clauses over the free variables that make the clauses of F with quantified variables redundant. The traditional Quantifier Elimination problem (QE) is a special case of PQE where G is empty so all clauses of the input formula with quantified variables need to be made redundant. The importance of PQE is twofold. First, many problems are more naturally formulated in terms of PQE rather than QE. Second, in many cases PQE can be solved more efficiently than QE. We describe a PQE algorithm based on the machinery of dependency sequents and give experimental results showing the promise of PQE

    A Survey of Satisfiability Modulo Theory

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    Satisfiability modulo theory (SMT) consists in testing the satisfiability of first-order formulas over linear integer or real arithmetic, or other theories. In this survey, we explain the combination of propositional satisfiability and decision procedures for conjunctions known as DPLL(T), and the alternative "natural domain" approaches. We also cover quantifiers, Craig interpolants, polynomial arithmetic, and how SMT solvers are used in automated software analysis.Comment: Computer Algebra in Scientific Computing, Sep 2016, Bucharest, Romania. 201
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