Quantifier (logic)

Summary

In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier \forall in the first order formula \forall x P(x) expresses that everything in the domain satisfies the property denoted by P. On the other hand, the existential quantifier \exists in the formula \exists x P(x) expresses that there exists something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable. The most commonly used quantifiers are \forall and \exists. These quantifiers are standardly defined as duals; in classical logic, they are interdefinable using negation. They can also be used to define more complex quantifiers, as in th

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https://en.wikipedia.org/wiki/Quantifier_(logic)

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On Sets with Cardinality Constraints in Satisfiability Modulo Theories

Viktor Kuncak, Robin Steiger, Philippe Paul Henri Suter

Boolean Algebra with Presburger Arithmetic (BAPA) is a decidable logic that can express constraints on sets of elements and their cardinalities. Problems from verification of complex properties of software often contain fragments that belong to quantifier-free BAPA (QFBAPA). Deciding the satisfiability of QFBAPA formulas has been shown to be NP-complete using an eager reduction to quantifier-free Presburger arithmetic that exploits a sparse-solution property. In contrast to many other NP-complete problems (such as quantifier-free first-order logic or linear arithmetic), the applications of QFBAPA to a broader set of problems has so far been hindered by the lack of an efficient implementation that can be used alongside other efficient decision procedures. We overcome these limitations by extending the efficient SMT solver Z3 with the ability to reason about cardinality constraints. Our implementation uses the DPLL(T) mechanism of Z3 to reason about the top-level propositional structure of a QFBAPA formula, improving the efficiency compared to previous implementations. Moreover, we present a new algorithm for automated decomposition of QFBAPA formulas. Our algorithm alleviates the exponential explosion of considering all Venn regions, significantly improving the tractability of formulas with many set variables. Because it is implemented as a theory plugin, our implementation enables Z3 to prove formulas that use QFBAPA constructs alongside constructs from other theories that Z3 supports (e.g. linear arithmetic, uninterpreted function symbols, algebraic data types), as well as in formulas with quantifiers. We have applied our implementation to verification of functional programs; we show it can automatically prove formulas that no automated approach was reported to be able to prove before.

2010

NP Satisfiability for Arrays as Powers

Viktor Kuncak, Rodrigo Raya

We show that the satisfiability problem for the quantifier-free theory of product structures with the equicardinality relation is inNP. As an application, we extend the combinatory array logic fragmentto handle cardinality constraints. The resulting fragment is independentof the base element and index set theories.

Springer, Cham2022

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Boolean Algebra with Presburger Arithmetic (BAPA) is a decidable logic that can express constraints on sets of elements and their cardinalities. Problems from verification of complex properties of software often contain fragments that belong to quantifier-free BAPA (QFBAPA). in contrast to many other NP-complete problems (such as quantifier-free first-order logic or linear arithmetic), the applications of QFBAPA to a broader set of problems has so far been hindered by the lack of an efficient implementation that can be used alongside other efficient decision procedures. We overcome these limitations by extending the efficient SMT solver Z3 with the ability to reason about cardinality (QFBAPA) constraints. Our implementation uses the DPLL(T) mechanism of Z3 to reason about the top-level propositional structure of a QFBAPA formula, improving the efficiency compared to previous implementations. Moreover, we present a new algorithm for automatically decomposing QFBAPA formulas. Our algorithm alleviates the exponential explosion of considering all Venn regions, significantly improving the tractability of formulas with many set variables. Because it is implemented as a theory plugin, our implementation enables Z3 to prove formulas that use QFBAPA constructs with constructs from other theories that Z3 supports, as well as with quantifiers. We have applied our implementation to the verification of functional programs; we show it can automatically prove formulas that no automated approach was reported to be able to prove before.

Springer-Verlag New York, Ms Ingrid Cunningham, 175 Fifth Ave, New York, Ny 10010 Usa2011