## Section: Scientific Foundations

### Automated Deduction

The main goal is to prove the validity of assertions obtained from program analysis. To this end, we develop techniques and automated deduction systems based on rewriting and constraint solving. The verification of recursive data structures relies on inductive reasoning or the manipulation of equations and it also exploits some form of reasoning modulo properties of selected operators (such as associativity and/or commutativity).

Rewriting, which allows us to simplify expressions and formulae, is a key ingredient for the effectiveness of many state-of-the-art automated reasoning systems. Furthermore, a well-founded rewriting relation can be also exploited to implement reasoning by induction. This observation forms the basis of our approach to inductive reasoning, with high degree of automation and the possibility to refute false conjectures.

The constraints are the key ingredient to postpone the activity of solving complex symbolic problems until it is really necessary. They also allow us to increase the expressivity of the specification language and to refine theorem-proving strategies. As an example of this, the handling of constraints for unification problems or for the orientation of equalities in the presence of interpreted operators (e.g., commutativity and/or associativity function symbols) will possibly yield shorter automated proofs.

Finally, decision procedures are being considered as a key ingredient for the successful application of automated reasoning systems to verification problems. A decision procedure is an algorithm capable of efficiently deciding whether formulae from certain theories (such as Presburger arithmetic, lists, arrays, and their combination) are valid or not. We develop techniques to build and combine decision procedures for the domains which are relevant to verification problems. We also perform experimental evaluation of the proposed techniques by combining propositional reasoning (implemented by means of Boolean solvers – Binary Decision Diagrams or SAT solvers) and decision procedures, and their extensions to semi-decision procedures for handling larger (possibly undecidable) fragments of first-order logic.

We investigate techniques to incorporate the use of decision procedures in the model-checking of infinite state systems. The state of such systems is described by the models of theories specifying data types (such as integers or arrays) and their behavior is identified by (possibly infinite) sequences of these models which share the interpretation of the symbols interpreted in the theories (e.g., the addition over the integers). In this context, checking if a system satisfies a certain property may be reduced to checking the satisfiability of a formula in the theory obtained as the combination of the theories describing the sequence of states in the computation. To solve this problem, it is crucial to develop new combination methods for non-disjoint unions of theories.