Contributions to the compositional semantics of first-order predicate logic

Henkin, Monk and Tarski gave a compositional semantics for first-order predicate logic. We extend this work by including function symbols in the language and by giving the denotation of the atomic formula as a composition of the denotations of its predicate symbol and of its tuple of arguments. In addition we give the denotation of a term as a composition of the denotations of its function symbol and of its tuple of arguments.Comment: 14 pages, 1 figur

Relational Semantics for Databases and Predicate Calculus

The relational data model requires a theory of relations in which tuples are not only many-sorted, but can also have indexes that are not necessarily numerical. In this paper we develop such a theory and define operations on relations that are adequate for database use. The operations are similar to those of Codd's relational algebra, but differ in being based on a mathematically adequate theory of relations. The semantics of predicate calculus, being oriented toward the concept of satisfiability, is not suitable for relational databases. We develop an alternative semantics that assigns relations as meaning to formulas with free variables. This semantics makes the classical predicate calculus suitable as a query language for relational databases.Comment: 18 pages, 8 figures. arXiv admin note: text overlap with arXiv:cs/060703

Compositional Semantics for the Procedural Interpretation of Logic

The composition of logic programs out of clauses has been studied semantically, but not the composition of a single clause out of its components. Structurally, a logic program can be regarded as a sentence in clausal form. In his procedural interpretation of logic programs, Kowalski has shown that a positive Horn clause can be viewed as a procedure in the programming sense. This interpretation suggests a composition operator for logic programs, the one where a clause results from composing a head with a body. In this paper we give more detail to the procedural interpretation by giving an algebraic characterization of Kowalski’s composition. In addition, we give algebraic characterizations of the composition of goals in a procedure body and for the composition of the predicate symbol with the argument tuple within a goal. A starting point for the semantic operator corresponding to composition of goals is provided by Tarski’s cylindric algebra semantics for first-order predicate logic. Tarski’s construction is briefly reviewed and suitably modified. The additional semantic operators are shown to be correct with respect to the fixpoint semantics of the logic program as a whole.