Safety-critical systems, especially those found in avionics and automotive industries, rely on machine arithmetic to perform their tasks: integer arithmetic, fixed-point arithmetic or floating-point arithmetic (FPA). Machine arithmetic exhibits subtle differences in behavior compared to the ideal mathematical arithmetic, due to fixed-size representation in memory. Failure of safety-critical systems is unacceptable, due to high-stakes involving human lives or huge amounts of money, time and effort. By formally proving properties of systems, we can be assured that they meet safety requirements. However, to prove such properties it is necessary to reason about machine arithmetic. SMT techniques for machine arithmetic are lacking scalability. This thesis presents approaches that augment or complement existing SMT techniques for machine arithmetic. In this thesis, we explore approximations as a means of augmenting existing decision procedures. A general approximation refinement framework is presented, along with its implementation called UppSAT. The framework solves a sequence of approximations. Initially very crude, these approximations are fairly easy to solve. Results of solving approximate constraints are used to either reconstruct a solution of original constraints, obtain a proof of unsatisfiability or to refine the approximation. The framework preserves soundness, completeness, and termination of the underlying decision procedure, guaranteeing that eventually, either a solution is found or a proof that solution does not exist. We evaluate the impact of approximations implemented in the UppSAT framework on the state-of-the-art in SMT for floating-point arithmetic. A novel method to reason about the theory of fixed-width bit-vectors called mcBV is presented. It is an instantiation of the model constructing satisfiability calculus, mcSAT, and uses a new lazy representation of bit-vectors that allows both bit- and word-level reasoning. It uses a greedy explanation generalization mechanism capable of more general learning compared to traditional approaches. Evaluation of mcBV shows that it can outperform bit-blasting on several classes of problems.

Jednou z nejsložitěji verifikovaných vlastností programů v oblasti formální analýzy je živost. K jedné z metod ověřujících tuto vlastnost patří i dokazování neukončitelnosti programů. Naše práce popisuje návrh a implementaci dvou algoritmů ověřujících neukončitelnost. Inspirujeme se již existujícími přístupy, jako jsou rekurentní množiny a nadaproximace cyklů s využitím invariantů ve tvaru rekurentních relací. Hlavní výzvu pro nás představovalo přizpůsobení těchto algoritmů SSA (single static assignment) reprezentaci použité v 2LS a jejich celková integrace v našem frameworku. Vzpomínané přístupy se nám podařilo spojit do analýzy neukončitelnosti, která dosahuje nejlepší výsledky v porovnání s existujícími nástroji, které byly srovnané na soutěži SV-COMP 2017.