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 th...

We introduce the first program synthesis engine implemented inside an SMT solver. We present an approach that extracts solution functions from unsatisfiability proofs of the negated form of synthesis conjectures. We also discuss novel counterexample-guided techniques for quantifier instantiation that we use to make finding such proofs practically feasible. A particularly important class of spec...

We introduce the first program synthesis engine implemented inside an SMT solver. We present an approach that extracts solution functions from unsatisfiability proofs of the negated form of synthesis conjectures. We also discuss novel counterexample-guided techniques for quantifier instantiation that we use to make finding such proofs practically feasible. A particularly important class of spec...

Satisfiability modulo theory solvers are increasingly being used to solve quantified formulas over structures such as integers and term algebras. Quantifier instantiation combined with ground decision procedure alone is insufficient to prove many formulas of interest in such cases. We present a set of techniques that introduce inductive reasoning into SMT solving algorithms that is sound with r...

We introduce the first program synthesis engine implemented inside an SMT solver. We present an approach that extracts solution functions from unsatisfiability proofs of the negated form of synthesis conjectures. We also discuss novel counterexample-guided techniques for quantifier instantiation that we use to make finding such proofs practically feasible. A particularly important class of spec...

Satisfiability Modulo Theories (SMT) solvers have emerged as prominent tools in formal methods applications. While originally targeted towards quantifier-free inputs, SMT solvers are now often used for handling quantified formulas in automated theorem proving and software verification applications. The most common technique for handling quantified formulas in modern SMT solvers in quantifier in...

Satisfiability modulo theory solvers are increasingly being used to solve quantified formulas over structures such as integers and term algebras. Quantifier instantiation combined with ground decision procedure alone is insufficient to prove many formulas of interest in such cases. We present a set of techniques that introduce inductive reasoning into SMT solving algorithms that is sound with r...

Universal algebra is often known within computer science in the guise of algebraic specification or equational logic. In 1963, it was given a category theoretic characterisation in terms of what are now called Lawvere theories. Unlike operations and equations, a Lawvere theory is uniquely determined by its category of models. Except for a caveat about nullary operations, the notion of Lawvere t...

Universal algebra is often known within computer science in the guise of algebraic specification or equational logic. In 1963, it was given a category theoretic characterisation in terms of what are now called Lawvere theories. Unlike operations and equations, a Lawvere theory is uniquely determined by its category of models. Except for a caveat about nullary operations, the notion of Lawvere t...

Lambda theories are equational extensions of the untyped lambda calculus that are closed under derivation. The set of lambda theories is naturally equipped with a structure of complete lattice, where the meet of a family of lambda theories is their intersection, and the join is the least lambda theory containing their union. In this paper we study the structure of the lattice of lambda theories...