# Quantifiers and Theories A Lazy Approach

Sammanfattning: In this thesis we study Automated Theorem Proving (ATP) as well as Satisfiability Modulo Theories (SMT) and present lazy strategies for improving reasoning within these areas. A lazy strategy works by simplifying a problem, and gradually refines the abstraction only when necessary. Often, a solution can be found without solving the full problem, saving valuable resources. We formulate our contributions within two key challenges in ATP and SMT: theory and quantifier reasoning.Many problems need first-order reasoning modulo a theory, i.e., reasoning where symbols in formulas are interpreted according to some background theory. In software verification, which often involves conditions over machine arithmetic, bit-vectors as well as floating-point numbers play an important role. Finding methods for how to reason with these theories in an efficient manner is therefore an important task. In this thesis we present a lazy method for handling bit-vector constraints as well as bit-vector interpolation, which improves performance and produces simpler interpolants. Moreover, a modular approximation framework is described, which allows for high-level description of lazy strategies applicable to a multitude of theories. Such a strategy is combined with a back-end, creating an approximating SMT solver. We use floating-point arithmetic as an illustrating use case, showing how the lazy strategy can improve the overall efficiency.The quantifier is a language construct which allows for making statements about one or all objects of some universe. However, with great power comes great cost - reasoning with quantifiers is very hard. Many decision problems involving quantifiers are not decidable, e.g., validity of first-order logic. Intricate strategies are needed to handle formulas with quantifiers, especially in combination with theory reasoning. We present a new restricted form of unification, by lazy expansion of the domain of substitution, as well as efficient procedures to solve it. This is incorporated into a complete and sound sequent calculus for the combination of the theory of equality and quantifiers.

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