A many-sorted calculus based on resolution and paramodulation

The first-order calculus whose well formed formulas are clauses and whose sole inference rules are factorization, resolution and paramodulation is extended to a many-sorted calculus. As a basis for Automated Theorem Proving, this many-sorted calculus leads to a remarkable reduction of the search space and also to simpler proofs. Soundness and completeness of the new calculus and the Sort-Theorem, which relates the many-sorted calculus to its one-sorted counterpart, are shown. In addition results about term rewriting and unification in a many-sorted calculus are obtained. The practical consequences for an implementation of an automated theorem prover based on the many-sorted calculus are described

Representation, Verification, and Visualization of Tarskian Interpretations for Typed First-order Logic

peer reviewedThis paper describes a new format for representing Tarskian-style interpretations for formulae in typed first-order logic, using the TPTP TF0 language. It further describes a technique and an implemented tool for verifying models using this representation, and a tool for visualizing interpretations. The research contributes to the advancement of automated reasoning technology for model finding, which has several applications, including verification

Constructive problem solving : a model construction approach towards configuration

In this paper we give a formalisation of configuration as the task to construct for a given specification, which is understood as a finite set of logical formulas, a model that satisfies the specification. In this approach, a specification consists of two parts. One part describes the domain, the possible components, and their interdependencies. The other part specifies the particular object that is to be configured. The language that is used to represent knowledge about configuration problems integrates three sublanguages that allow one to express constraints, to build up taxonomies, and to define rules. We give a sound calculus by which one can compute solutions to configuration problems if they exist and that allows one to recognize that a specification is inconsistent. In particular, the calculus can be used in order to check whether a given configuration satisfies the specification

Concept logics

Concept languages (as used in BACK, KL-ONE, KRYPTON, LOOM) are employed as knowledge representation formalisms in Artificial Intelligence. Their main purpose is to represent the generic concepts and the taxonomical hierarchies of the domain to be modeled. This paper addresses the combination of the fast taxonomical reasoning algorithms (e.g. subsumption, the classifier etc.) that come with these languages and reasoning in first order predicate logic. The interface between these two different modes of reasoning is accomplished by a new rule of inference, called constrained resolution. Correctness, completeness as well as the decidability of the constraints (in a restricted constraint language) are shown

A resolution principle for clauses with constraints

We introduce a general scheme for handling clauses whose variables are constrained by an underlying constraint theory. In general, constraints can be seen as quantifier restrictions as they filter out the values that can be assigned to the variables of a clause (or an arbitrary formulae with restricted universal or existential quantifier) in any of the models of the constraint theory. We present a resolution principle for clauses with constraints, where unification is replaced by testing constraints for satisfiability over the constraint theory. We show that this constrained resolution is sound and complete in that a set of clauses with constraints is unsatisfiable over the constraint theory if we can deduce a constrained empty clause for each model of the constraint theory, such that the empty clauses constraint is satisfiable in that model. We show also that we cannot require a better result in general, but we discuss certain tractable cases, where we need at most finitely many such empty clauses or even better only one of them as it is known in classical resolution, sorted resolution or resolution with theory unification

Unification in sort theories and its applications

In this article I investigate the properties of unification in sort theories. The usual notion of a sort consisting of a sort symbol is extended to a set of sort symbols. In this language sorted unification in elementary sort theories is of unification type finitary. The rules of standard unification with the addition of four sorted rules form the new sorted unification algorithm. The algorithm is proved sound and complete. The rule based form of the algorithm is not suitable for an implementation because there is no control and the used data structures are weak. Therefore we transform the algorithm into a deterministic sorted unification procedure. For the procedure sorted unification in pseudo-linear sort theories is proved decidable. The notions of a sort and a sort theory are developed in a way such that a standard calculus can be turned into a sorted calculus by replacing standard unification with sorted unification. To this end sorts may denote the empty set. Sort theories may contain clauses with more than one declaration and may change dynamically during the deduction process. The applicability of the approach is exemplified for the resolution and the tableau calculus

Taxonomic Reasoning and Lexical Semantics

Taxonomic reasoning is used in many applications, including many-sorted logic, knowledge bases, document retrieval, and natural language processing. These various applications have been dealt with independently. Because they have so much in common, a general approach to taxonomic reasoning would seem to be justified. This paper presents a theory of lexical semantics as an example of such a general approach. The theory defines a representation and an algebra for that representation. The operations of the algebra are inherently parallel, making them well matched to the capabilities of modern computer systems