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

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

Integration of Rewriting, Narrowing, Compilation, and Heuristics for Equality Reasoning in Resolution-Based Theorem Proving

We present a framework for the integration of the Knuth-Bendix completion algorithm with narrowing methods, compiled rewrite rules, and a heuristic difference reduction mechanism for paramodulation. The possibility of embedding theory unification algorithms into this framework is outlined. Results are presented and discussed for several examples of equality reasoning problems in the context of an actual implementation of an automated theorem proving system (the Mkrp-system) and a fast C implementation of the completion procedure. The Mkrp-system is based on the clause graph resolution procedure. The thesis shows the indispensibility of the constraining effects of completion and rewriting for equality reasoning in general and quantifies the amount of speed-up caused by various enhancements of the basic method. The simplicity of the superposition inference rule allows to construct an abstract machine for completion, which is presented together with computation times for a concrete implementation

Integration of Rewriting, Narrowing, Compilation, and Heuristics for Equality Reasoning in Resolution-Based Theorem Proving

We present a framework for the integration of the Knuth-Bendix completion algorithm with narrowing methods, compiled rewrite rules, and a heuristic difference reduction mechanism for paramodulation. The possibility of embedding theory unification algorithms into this framework is outlined. Results are presented and discussed for several examples of equality reasoning problems in the context of an actual implementation of an automated theorem proving system (the Mkrp-system) and a fast C implementation of the completion procedure. The Mkrp-system is based on the clause graph resolution procedure. The thesis shows the indispensibility of the constraining effects of completion and rewriting for equality reasoning in general and quantifies the amount of speed-up caused by various enhancements of the basic method. The simplicity of the superposition inference rule allows to construct an abstract machine for completion, which is presented together with computation times for a concrete implementation

Tactics for the Improvement of Problem Formulation in Resolution-Based Theorem Proving

We transform a user-friendly formulation of aproblem to a machine-friendly one exploiting the variabilityof first-order logic to express facts. The usefulness of tacticsto improve the presentation is shown with several examples.In particular it is shown how tactical and resolution theoremproving can be combined

Presentation of Proofs in an Equational Calculus

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. Often completion and rewrite proofs are only given in the form of a program trace. 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 present a method to handle equational proofs in such systems. To this end equation solution graphs are introduced to represent paramodulation or rewrite proofs. In the process of transforming these proofs into proofs with equation chains, the inherent structure can also be extracted by exploiting topological properties similar to those of refutation graphs in the pure first order case

Tactics for the Improvement of Problem Formulation in Resolution-Based Theorem Proving

We transform a user-friendly formulation of aproblem to a machine-friendly one exploiting the variabilityof first-order logic to express facts. The usefulness of tacticsto improve the presentation is shown with several examples.In particular it is shown how tactical and resolution theoremproving can be combined

Reformulating Resolution Problems by Tactics

A straightforward formulation of a mathematical problem is mostly not ad-equate for resolution theorem proving. We present a method to optimize suchformulations by exploiting the variability of first-order logic. The optimizingtransformation is described as logic morphisms, whose operationalizations aretactics. The different behaviour of a resolution theorem prover for the sourceand target formulations is demonstrated by several examples. It is shown howtactical and resolution-style theorem proving can be combined