3 research outputs found
Solving quantified linear arithmetic by counterexample-guided instantiation
This paper presents a framework to derive instantiation-based decision procedures for satisfiability of quantified formulas in first-order theories, including its correctness, implementation, and evaluation. Using this framework we derive decision procedures for linear real arithmetic (LRA) and linear integer arithmetic (LIA) formulas with one quantifier alternation. We discuss extensions of these techniques for handling mixed real and integer arithmetic, and to formulas with arbitrary quantifier alternations. For the latter, we use a novel strategy that handles quantified formulas that are not in prenex normal form, which has advantages with respect to existing approaches. All of these techniques can be integrated within the solving architecture used by typical SMT solvers. Experimental results on standardized benchmarks from model checking, static analysis, and synthesis show that our implementation in the SMT solver CVC4 outperforms existing tools for quantified linear arithmetic
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Linear Arithmetic Satisfiability Via Strategy Improvement
Satisfiability-checking of formulas in the theory of linear rational arithmetic (LRA) has broad applications including program verification and synthesis. Satisfiability Modulo Theories (SMT) solvers are effective at checking satisfiability of the ground fragment of LRA, but applying them to quantified formulas requires a costly quantifier elimination step. This article presents a novel decision procedure for LRA that leverages SMT solvers for the ground fragment of LRA, but avoids explicit quantifier elimination. The intuition behind the algorithm stems from an interpretation of a quantified formula as a game between two players, whose goals are to prove that the formula is either satisfiable or not. The algorithm synthesizes a winning strategy for one of the players by iteratively improving candidate strategies for both. Experimental results demonstrate that the proposed procedure is competitive with existing solvers