Effective Algorithms for the Satisfiability of Quantifier-Free Formulas Over Linear Real and Integer Arithmetic

A core technique of modern tools for formally reasoning about computing systems is generating and dispatching queries to automated theorem provers, including Satisfiability Modulo Theories (SMT) provers. SMT provers aim at the tight integration of decision procedures for propositional satisfiability and decision procedures for fixed first-order theories – known as theory solvers. This thesis presents several advancements in the design and implementation of theory solvers for quantifier-free linear real, integer, and mixed integer and real arithmetic. These are implemented within the SMT system CVC4. We begin by formally describing the Satisfiability Modulo Theories problem and the role of theory solvers within CVC4. We discuss known techniques for building solvers for quantifier-free linear real, integer, and mixed integer and real arithmetic around the Simplex for SMT algorithm. We give several small improvements to theory solvers using this algorithm and describe the implementation and theory of this algorithm in detail. To extend the class of problems that the theory solver can robustly support, we borrow and adapt several techniques from linear programming (LP) and mixed integer programming (MIP) solvers which come from the tradition of optimization. We propose a new decision procedure for quantifier-free linear real arithmetic that replaces the Simplex for SMT algorithm with a variant of the Simplex algorithm that performs a form of optimization – minimizing the sum of infeasibilties. In this thesis, we additionally describe techniques for leveraging LP and MIP solvers to improve the performance of SMT solvers without compromising correctness. Previous efforts to leverage such solvers in the context of SMT have concluded that in addition to being potentially unsound, such solvers are too heavyweight to compete in the context of SMT. We present an empirical comparison against other state-of-theart SMT tools to demonstrate the effectiveness of the proposed solutions.