Disunification in ACI 1 Theories ∗ Agostino Dovier Carla Piazza Universit`a di Udine Dip. di Matematica e Informatica Via Le Scienze 206 33100 Udine (Italy) dovier|piazza@dimi.uniud.it Enrico Pontelli New Mexico State University Dept. Computer Science Box 30001, Dept. CS Las Cruces, NM 88003 (USA) epontell@cs.nmsu.edu Abstract Disunification is the problem of deciding satisfiability of a system of equations and dise- quations with respect to a given equational theory. In this paper we study the disunification problem in the context of ACI 1 equational theories. This problem is of great importance, for instance, for the development of constraint solvers over sets. Its solution opens new possibil- ities for developing automatic tools for static analysis and software verification. In this work we provide a characterization of the interpretation structures suitable to model the axioms in ACI 1 theories. The satisfiability problem is solved using known techniques for the equality constraints and novel methodologies to transform disequation constraints to manageable solved forms. We propose four solved forms, each offering an increasingly more precise description of the set of solutions. For each solved form we provide the corresponding rewriting algorithm and we study the time complexity of the transformation. Remarkably, two of the solved forms can be computed and tested in polynomial time. All these solved forms are adequate to be used in the context of a Constraint Logic Programming system—e.g., they do not introduce universal quantifications, as instead happens in some of the existing solved forms for disunification prob- lems. Keywords: Equational theories, disunification, ACI, complexity, CLP , Sets. 1 Introduction Equational theories are first-order theories whose axioms are universally quantified equations be- tween first-order terms [38]. A non-empty equational theory E forces certain classes of syntactically different terms to be interpreted as the same object in any model of E. For instance, if E contains the axiom X + Y = Y + X (for the sake of simplicity, we omit the universal quantification in the description of axioms), then the terms a + b and b + a have to be interpreted in the same way in any model of E. On the other hand, equational theories are not sufficiently strong to state that two terms must be distinguished in all the models of E. As a matter of fact, the 1-element structure 1 —the structure which maps all terms to the unique domain element—is a model of any equa- tional theory, and in 1 any constraint of the form s = t is unsatisfiable. If a “wider” structure is ∗ A preliminary version of this paper, entitled ACI1 Constraints by A. Dovier, C. Piazza, E. Pontelli, and G. Rossi, appeared in Danny De Schreye ed., Proceedings of the 1999 International Conference on Logic Programming. Pages 573–587, The MIT Press, 1999. 1