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

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

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

A Many-Sorted Calculus with Polymorphic Functions Based On Resolution And Paramodulation

A many-sorted first order calculus, called ΣRP, whose well formed formulas are sorted (typed) clauses and whose inference rules are factorization, resolution, paramodulation and weakening is extended to a many sorted calculus ΣRP* with polymorphic functions (overloading). It is assumed that the sort structure is a finite partially ordered set with a greatest element. It is shown, that this extended calculus is sound and complete, provided the functional reflexivity axioms are present. It is also shown, that unification of terms containing polymorphic functions is in general finitary, i.e. the set of most general unifiers may contain more than one element, but at most finitely many. We give a natural condition for the signature (the sort structure), such that the set of most general unifiers is always at most a singleton provided this condition holds

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

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

Mechanical Generation of Sorts in Clause Sets

The algorithm SOGEN is described, which transforms a SIG-sorted clause set CS into a SIG'-sorted clause set CS', where the output clause set is smaller, but the sort structure is more sophisticated. This produced clause set is the input for our Theorem Prover. which has ΣRP* , an extension of ΣRP as its basic deductive calculus. Both calculi have resolution and paramodulation as their basic Operations. We prove that the transformation induced by SOGEN does not affect unsatisfiability, respectively satisfiability, of the clause set

A mechanization of sorted higher-order logic based on the resolution principle

The usage of sorts in first-order automated deduction has brought greater conciseness of representation and a considerable gain in efficiency by reducing the search spaces involved. This suggests that sort information can be employed in higher-order theorem proving with similar results. This thesis develops a sorted higher-order logic SUM HOL suitable for automatic theorem proving applications. SUM HOL is based on a sorted Lambda-calculus SUM A->, which is obtained by extending Church\u27;s simply typed Lambda-calculus by a higher-order sort concept including term declarations and functional base sorts. The term declaration mechanism studied here is powerful enough to allow convenient formalization of a large body of mathematics, since it offers natural primitives for domains and codomains of functions, and allows to treat function restriction. Furthermore, it subsumes most other mechanisms for the declaration of sort information known from the literature, and can thus serve as a general framework for the study of sorted higher-order logics. For instance, the term declaration mechanism of SUM HOL subsumes the subsorting mechanism as a derived notion, and hence justifies our special form of subsort inference. We present sets of transformations for sorted higher-order unification and pre-unification, and prove the nondeterministic completeness of the algorithm induced by these transformations. The main technical difficulty of unification in ! is that the analysis of general bindings is much more involved than in the unsorted case, since in the presence of term declarations well-sortedness is not a structural property. This difficulty is overcome by a structure theorem that links the structure of a formula to the structure of its sorting derivation. We develop two notions of set-theoretic semantics for SUM HOL. General SUM-models are a direct generalization of Henkin\u27;s general models to the sorted setting. Since no known machine-oriented calculus can adequately mechanize full extensionality, we generalize general SUM-models further to SUM-model structures, which allow full extensionality to fail. The notions of SUM-model structures and general SUM-models allow us to prove model existence theorems for them. These model-theoretic variants of Andrews unifying principle for type theory\u27; can be used as a powerful tool in completeness proofs of higher-order calculi. Finally, we use our pre-unification algorithms as a central inference procedure for a sorted higherorder resolution calculus in the spirit of Huet\u27;s Constrained Resolution. This calculus is proven sound and complete with respect to our semantics. It differs from Huet\u27;s calculus by allowing early unification strategies and using variable dependencies. For the completeness proof we make use of our model existence theorem, and prove a strong lifting lemma

A mechanization of sorted higher-order logic based on the resolution principle

The usage of sorts in first-order automated deduction has brought greater conciseness of representation and a considerable gain in efficiency by reducing the search spaces involved. This suggests that sort information can be employed in higher-order theorem proving with similar results. This thesis develops a sorted higher-order logic SUM HOL suitable for automatic theorem proving applications. SUM HOL is based on a sorted Lambda-calculus SUM A->, which is obtained by extending Church';s simply typed Lambda-calculus by a higher-order sort concept including term declarations and functional base sorts. The term declaration mechanism studied here is powerful enough to allow convenient formalization of a large body of mathematics, since it offers natural primitives for domains and codomains of functions, and allows to treat function restriction. Furthermore, it subsumes most other mechanisms for the declaration of sort information known from the literature, and can thus serve as a general framework for the study of sorted higher-order logics. For instance, the term declaration mechanism of SUM HOL subsumes the subsorting mechanism as a derived notion, and hence justifies our special form of subsort inference. We present sets of transformations for sorted higher-order unification and pre-unification, and prove the nondeterministic completeness of the algorithm induced by these transformations. The main technical difficulty of unification in ! is that the analysis of general bindings is much more involved than in the unsorted case, since in the presence of term declarations well-sortedness is not a structural property. This difficulty is overcome by a structure theorem that links the structure of a formula to the structure of its sorting derivation. We develop two notions of set-theoretic semantics for SUM HOL. General SUM-models are a direct generalization of Henkin';s general models to the sorted setting. Since no known machine-oriented calculus can adequately mechanize full extensionality, we generalize general SUM-models further to SUM-model structures, which allow full extensionality to fail. The notions of SUM-model structures and general SUM-models allow us to prove model existence theorems for them. These model-theoretic variants of Andrews unifying principle for type theory'; can be used as a powerful tool in completeness proofs of higher-order calculi. Finally, we use our pre-unification algorithms as a central inference procedure for a sorted higherorder resolution calculus in the spirit of Huet';s Constrained Resolution. This calculus is proven sound and complete with respect to our semantics. It differs from Huet';s calculus by allowing early unification strategies and using variable dependencies. For the completeness proof we make use of our model existence theorem, and prove a strong lifting lemma