this paper we concentrate on a specification language that can be of use in both cases. We present a sound and complete Gentzen-style deduction system for a logic that can be best (though rather informally) described as first order logic with inclusion and a let construct binding variables in nondeterministic terms. We should also remark here that this paper presents a non-trivial firstorder extension of a calculus presented in [BK 95a]. It contains a detailed analysis of the phenomena related to empty carriers in multi-sorted environment. Also, we do not place any restrictions on the models, and allow for both empty carriers and partial functions. One may observe at this point that instead of multialgebras we might have used relational algebras. However, we prefer functions for the same reason that functions are present in first order logic -- they are more intuitive in applications and have properties that make their use easier. Moreover, as it will be explained in the last section, we need "ordinary" algebras to be a special case of the formalism we develop.

. We consider reasoning and rewriting with set-relations: inclusion, nonempty intersection and singleton identity, each of which satisfies only two among the three properties of the equivalence relations. The paper presents a complete inference system which is a generalization of ordered paramodulation and superposition calculi. Notions of rewriting proof and confluent rule system are defined for such nonequivalence relations. Together with the notions of forcing and redundancy they are applied in the completeness proof. Ground completeness cannot be lifted to the nonground case because substitution for variables is restricted to deterministic terms. To overcome the problems of restricted substitutivity and hidden (in relations) existential quantification, unification is defined as a three step process: substitution of determistic terms, introduction of bindings and "on-line" skolemisation. The inference rules based on this unification derive non-ground clauses even from the ground one...

The paper presents a variant of first order logic for specifying nondeterministic software. Models of the logics are multialgebras, i.e. multi-sorted algebras with set-valued operations, together with multi-sorted valuations of variables. We allow empty carrier sets but the valuations are kept total. Terms are interpreted as sets and the usual set of algebraic terms is extended by an additional let construct used for limiting nondeterminism. Atomic formulae are of the form t 1 ! t 2 where ! is a rewrite operator, corresponding semantically to inclusion. For the