Normalization of Horn clauses with disequality constraints

Horn clauses constitute a convenient, Turing-powerful tool for the specification and representation of analysis problems. However, it is not possible to express disequality conditions directly and in a natural way by means of Horn clauses. This thesis therefore proposes an extension of Horn clauses with several kinds of disequality constraints. In order to provide an automatic analysis framework capable of expressing explicit disequality conditions, the considered class of Horn clauses is restricted to the class H1, for which the satisfiability problem is decidable. H1 is chosen as a large and expressive class which can approximate in a natural way any finite set of Horn clauses, making it particularly useful for approximative analyses of termmanipulating programs based on abstract interpretation. We build on an existing normalization procedure that transforms finite sets of H1-clauses to equivalent tree automata, and show that this procedure can be adapted such that finite sets of H1-clauses extended with disequality constraints are normalized into equivalent tree automata with disequality constraints. Since emptiness is decidable for these automata, this reduction proves satisfiability decidable for H1-clauses extended with disequality constraints. In particular, we provide corresponding proofs for the following extensions: disequalities between terms, disequalities between subterms specified by paths, and homdisequalities, i.e., disequalities between images of terms under a given tree homomorphism. Such hom-disequalities allow to test whether two terms are distinct modulo a semantic interpretation, enabling to neglect information that is considered irrelevant for the intended comparison. We furthermore provide an extension of tree automata with term equality and disequality constraints, generalizing the known class of tree automata with constraints between brothers of Bogaert and Tison from 1992, enjoying full Boolean closure, but still having a decidable emptiness problem — whereas emptiness for automata with path equalities has been shown undecidable by Mongy in 1981. The four main contributions of the thesis are published in [SR11], [SR12], [RS12], and [RS10], respectively.