The Markgraf Karl Refutation Procedure PLL: a First-Order Language for an Automated Theorem Prover

The PREDICATE LOGIC LANGUAGE (PLL), a formal language in which first-order predicate logic formulas are formulated, is described. In PLL axioms and theorems are represented which are given to the MARKGRAF KARL REFUTATION PROCEDURE. Certain expressions of PLL which reflect the special facilities of this system are exhibited, viz. - an inference mechanism based on a many-sorted calculus, - the incorporation of special axioms into the inference mechanism, and - the control of the inference mechanism using special derivation strategies

Solving Equality Reasoning Problems with a Connection Graph Theorem Prover

The integration of a Knuth-Bendix completion algorithm into a paramodulation theorem prover on the basis of a connection graph resolution procedure is presented. The Knuth-Bendix completion idea is compared to a decomposition approach, and some ideas to handle conditional equations are discussed. The contents of this paper is not intended to present new material on term rewriting, instead it is more a pleading for the usage of completion ideas in automated deduction. It records our experience with an actual implementation of a hybrid system, where a completion procedure was imbedded into a connection graph theorem prover, the MKRP-system, with satisfactory positive results

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

Ein mehrsortiger Resolutionskalkül mit Paramodulation

Der Resolutionskalkül mit Paramodulationsregel wird zu einem mehrsortigen Kalkül erweitert. Als Grundlage für das automatische Beweisen erhält man mit diesem Kalkül einen stark reduzierten Suchraum und einfachere Beweise. Die Vollständigkeit, die Korrektheit und der Sortensatz, der den neuen Kalkül mit seinem einsortigen Gegenstück in Beziehung setzt, werden bewiesen. Ergebnisse über Grundtermersetzungen und Unifikation in einem mehrsortigen Kalkül werden vorgestellt. Die Implementierung eines automatischen Beweisers für den neuen Kalkül wird beschrieben. Die Vorteile der Methode werden anhand ausgewählter Beispiele belegt.The resolution calculus with paramodulationrule 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 groundterm 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. The advantages of the proposed method is verified by certain examples

Opening the AC-Unification Race

This note reports about the implementation of AC-unification algorithms, based on the variable-abstraction method of Stickel and on the constant-abstraction method of Livesey, Siekmann, and Herold. We give a set of 105 benchmark examples and compare execution times for implementations of the two approaches. This documents for other researchers what we consider to be the state-of-the-art performance for elementary AC-unification problems

Transformation of Refutation Graphs into Natural Deduction Proofs

Most computer generated proofs are stated in abstract representations not normally used by mathematicians. We describe a procedure to transform proofs represented as abstract refutation graphss into natural deduction proofs. The emphasis of this investigation is more on stylistic aspects rather than theoretical issues. In particular the topological properties of refutation graphs can be successfully exploited in order to obtain structured proofs

Some Aspects of Analogy in Mathematical Reasoning

An important research problem is the incorporation of “declarative” knowledge into an automated theorem prover that can be utilized in the search for a proof. An interesting proposal in this direction is Alan Bundy’s approach of using explicit proof plans that encapsulate the general form of a proof and is instantiated into a particular proof for the case at hand. We give some examples that show how a “declarative” highlevel description of a proof can be used to find proofs of apparently “similiar” theorems by analogy. This “analogical” information is used to select the appropriate axioms from the database so that the theorem can be proved. This information is also used to adjust some options of a resolution theorem prover. In order to get a powerful tool it is necessary to develop an epistemologically appropriate language to describe proofs, for which a large set of examples should be used as a testbed. We present some ideas in this direction

Proof Transformation with Built-in Equality Predicate

One of the main reasons why computer generated proofs are not widely accepted is often their complexity and incomprehensibility. Especially proofs of mathematical theorems with equations are normally presented in an inadequate and not intuitive way. This is even more of a problem for the presentation of inferences drawn by automated reasoning components in other AI systems. For first order logic, proof transformation procedures have been designed in order to structure proofs and state them in a formalism that is more familiar to human mathematicians. In this report we generalize these approaches, so that proofs involving equational reasoning can also be handled. To this end extended refutation graphs are introduced to represent combined resolution and paramodulation proofs. In the process of transforming these proofs into natural deduction proofs with equality, the inherent structure can also be extracted by exploiting topological properties of refutation graphs

Some basic notions of first-order unification theory

This report does not contain much novel material, but collects the basic notions and the most frequently used lemmata and theorems of first order unification theory. It is restricted to the case of free terms (i.e. no defining equations)