A Decision Procedure for the First Order Theory of Real Addition with Order

Ferrante, Jeanne; Rackoff, Charles

doi:10.1137/0204006

Cited by 151 publications

(125 citation statements)

References 2 publications

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“…However, in this situation it is important to discuss how the symbols from τ are represented in the input of the CSP. In our context, Γ is semilinear, it is natural to assume that the relations are given as quantifier-free formulas in disjunctive normal form where the coefficients are represented in binary (by the mentioned result of Ferrante and Rackoff [14], every relation with a first-order definition over (Q; +, , 1) has a definition of this form). However, there are other natural representations, and we will for infinite languages always discuss how the relations are given.…”

Section: Csp(γ )mentioning

confidence: 99%

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Tropically Convex Constraint Satisfaction

Bodirsky

Mamino

2017

Theory Comput Syst

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A bstract. A semilinear relation S ⊆ Q n is max-closed if it is preserved by taking the componentwise maximum. The constraint satisfaction problem for max-closed semilinear constraints is at least as hard as determining the winner in Mean Payoff Games, a notorious problem of open computational complexity. Mean Payoff Games are known to be in NP ∩ co-NP, which is not known for max-closed semilinear constraints. Semilinear relations that are max-closed and additionally closed under translations have been called tropically convex in the literature. One of our main results is a new duality for open tropically convex relations, which puts the CSP for tropically convex semilinear constraints in general into NP ∩ co-NP. This extends the corresponding complexity result for scheduling under and-or precedence constraints, or equivalently the maxatoms problem. To this end, we present a characterization of max-closed semilinear relations in terms of syntactically restricted first-order logic, and another characterization in terms of a finite set of relations L that allow primitive positive definitions of all other relations in the class. We also present a subclass of max-closed constraints where the CSP is in P; this class generalizes the class of max-closed constraints over finite domains, and the feasibility problem for max-closed linear inequalities. Finally, we show that the class of max-closed semilinear constraints is maximal in the sense that as soon as a single relation that is not max-closed is added to L, the CSP becomes NP-hard.

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Section: Csp(γ )mentioning

confidence: 99%

“…A relation R ⊆ Q n is semilinear if R has a first-order definition in (Q; +, , 1); equivalently, R is a finite union of finite intersections of (open or closed) linear half spaces; see Ferrante and Rackoff [14]. In this article we study the computational complexity of constraint satisfaction problems with semilinear constraints.…”

Section: Introductionmentioning

confidence: 99%

Tropically Convex Constraint Satisfaction

Bodirsky

Mamino

2017

Theory Comput Syst

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“…[Pre29,Tar51,KK67,FR75]. In this section, we establish such a property to prove the pspace upper bound of the IPC ++ -satisfiability problem.…”

Section: Quantifier Eliminationmentioning

confidence: 99%

LTL over Integer Periodicity Constraints

Demri

2004

Lecture Notes in Computer Science

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Abstract. Periodicity constraints are used in many logical formalisms, in fragments of Presburger LTL, in calendar logics, and in logics for access control, to quote a few examples. In the paper, we introduce the logic PLTL mod , an extension of Linear-Time Temporal Logic LTL with past-time operators whose atomic formulae are defined from a first-order constraint language dealing with periodicity. Although the underlying constraint language is a fragment of Presburger arithmetic shown to admit a pspace-complete satisfiability problem, we establish that PLTL mod model-checking and satisfiability problems remain in pspace as plain LTL (full Presburger LTL is known to be highly undecidable). This is particularly interesting for dealing with periodicity constraints since the language of PLTL mod has a language more concise than existing languages and the temporalization of our first-order language of periodicity constraints has the same worst case complexity as the underlying constraint language. Finally, we show examples of introduction the quantification in the logical language that provide to PLTL mod , expspacecomplete problems. As another application, we establish that the equivalence problem for extended single-string automata, known to express the equality of time granularities, is pspace-complete by designing a reduction from QBF and by using our results for PLTL mod .

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“…For linear real arithmetic, Ferrante and Racko 's method [12] and Loos and Weispfenning's method [16] belong to this class, and so does Cooper's method for Presburger arithmetic [7]. Other methods, more geometrical in kind, project conjunctions of atoms and thus need some form of conversion to DNF; such is the case of Fourier-Mozkin elimination for linear real arithmetic, and of Pugh's Omega test for Presburger arithmetic [23].…”

Section: Formulasmentioning

confidence: 99%

Quantifier Elimination by Lazy Model Enumeration

Monniaux

2010

Computer Aided Verification

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We propose a quantifier elimination scheme based on nested lazy model enumeration through SMT-solving, and projections. This scheme may be applied to any logic that fulfills certain conditions; we illustrate it for linear real arithmetic. The quantifier elimination problem for linear real arithmetic is doubly exponential in the worst case, and so is our method. We have implemented it and benchmarked it against other methods from the literature.

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A Decision Procedure for the First Order Theory of Real Addition with Order

Cited by 151 publications

References 2 publications

Tropically Convex Constraint Satisfaction

Tropically Convex Constraint Satisfaction

LTL over Integer Periodicity Constraints

Quantifier Elimination by Lazy Model Enumeration

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