This report provides a detailed description of the TPTP Problem Library for automated theorem proving systems. The library is available via Internet, and forms a common basis for development of and experimentation with automated theorem provers. This report provides: ffl the motivations for building the library; ffl a discussion of the inadequacies of previous problem collections, and how these have been resolved in the TPTP; ffl a description of the library structure, including overview information; ffl descriptions of supplementary utility programs; ffl guidelines for obtaining and using the library; Contents 1 Introduction 2 1.1 Previous Problem Collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 What is Required? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Inside the TPTP 6 2.1 The TPTP Domain Structure . . . . . . . . . . . . . . . . . . . . . ...

Theorem provers based on model elimination have exhibited extremely high inference rates but have lacked a redundancy control mechanism such as subsumption. In this paper we report on work done to modify a model elimination theorem prover using two techniques, caching and lemmaizing, that have reduced by more than an order of magnitude the time required to find proofs of several problems and that have enabled the prover to prove theorems previously unobtainable by top-down model elimination theorem provers.

PROTEIN (PROver with a Theory Extension INterface) is a PTTPbased first order theorem prover over built-in theories. Besides various standardrefinements known for model elimination, PROTEIN also offers a variant of model elimination for case-based reasoning and which does not need contrapositives.

We give modifications of model elimination which do not necessitate the use of contrapositives. These restart model elimination calculi are proven sound and complete and their implementation by PTTP is depicted. The corresponding proof procedures are evaluated by a number of runtime experiments and they are compared to other well known provers. Finally we relate our results to other calculi, namely the connection method, modified problem reduction format and Near-Horn Prolog.

We design new inference systems for total orderings by applying rewrite techniques to chaining calculi. Equality relations may either be specified axiomatically or built into the deductive calculus via paramodulation or superposition. We demonstrate that our inference systems are compatible with a concept of (global) redundancy for clauses and inferences that covers such widely used simplification techniques as tautology deletion, subsumption, and demodulation. A key to the practicality of chaining techniques is the extent to which so-called variable chainings can be restricted. Syntactic ordering restrictions on terms and the rewrite techniques which account for their completeness considerably restrict variable chaining. We show that variable elimination is an admissible simplification techniques within our redundancy framework, and that consequently for dense total orderings without endpoints no variable chaining is needed at all.

We propose a method for combining the clause linking theorem proving method with theorem proving methods based on orderings. This may be useful for incorporating term-rewriting based approaches into clause linking. In this way, some of the propositional inefficiencies of ordering-based approaches may be overcome, while at the same time incorporating the advantages of ordering methods into clause linking. The combination also provides a natural way to combine resolution on non-ground clauses, with the clause linking method, which is essentially a ground method. We describe the method, prove completeness, and show that the enumeration part of clause linking with semantics can be reduced to polynomial time in certain cases. We analyze the complexity of the proposed method, and also give some plausibility arguments concerning its expected performance. 1 Introduction There are at least two basic approaches to the study of automated deduction. One approach concentrates on solving...

A notion of embedding appropriate to higher-order syntax is described. This provides a representation of annotated formulae in terms of the difference between pairs of formulae. We define substitution and unification for such annotated terms. Using this representation of annotated terms, the proof search guidance technique of rippling can be extended to higher-order theorems. We illustrate this by several examples based on an implementation of these ideas in Prolog.

When the Model Elimination (ME) procedure was rst proposed, a notion of lemma was put forth as a promising augmentation to the basic complete proof procedure. Here the lemmas that are used are also discovered by the procedure in the same proof run. Several implementations of ME now exist but only a 1970's implementation explicitly examined this lemma mechanism, with indi erent results. We report on the successful use of lemmas using the METEOR implementation of ME. Not only does the lemma device permit METEOR to obtain proofs not otherwise obtainable by METEOR, or any other ME prover not using lemmas, but some well-known challenge problems are solved. We discuss several of these more di cult problems, including two challenge problems for uniform general-purpose provers, where METEOR was rst in obtaining the proof. The problems are not selected simply to show o the lemma device, but rather to understand it better. Thus, we choose problems with widely di erent characteristics, including one where very few lemmas are created automatically, the opposite of normal behavior. This selection points out the potential of, and the problems with, lemma use. The biggest problem normally is the selection of appropriate lemmas to retain from the large number generated. 1

In this paper we describe the theorem prover METEOR which is a high-performance Model Elimination prover running in sequential, parallel and distributed computing environments. METEOR has a very high inference rate, but as is the case with better chess-playing programs speed alone is not sufficient when exploring large search spaces; intelligent search is necessary. We describe modifications to traditional iterative deepening search mechanisms whose implementation in METEOR result in performance improvements of several orders of magnitude and that have permitted the discovery of proofs unobtainable by top-down Model Elimination provers. 1 Introduction Model Elimination (ME) [Lov68, Lov69, Lov78] is the basis for the underlying inference mechanism of several high-performance theorem provers. The design of these provers is adapted from the architecture of the WAM (Warren Abstract Machine) [War83] --- the de facto standard for efficient Prolog implementations. Such provers includ...

. Rippling is a type of rewriting developed in inductive theorem proving for removing differences between terms; the induction conclusion is annotated to mark its differences from the induction hypothesis and rippling attempts to move these differences. Until now rippling has been primarily employed in proofs where there is a single induction hypothesis. This paper describes an extension to rippling to deal with theorems with multiple hypotheses. Such theorems arise, for instance, when reasoning about data-structures like trees with multiple recursive arguments. The essential idea is to colour the annotation, with each colour corresponding to a different hypothesis. The annotation of rewrite rules used in rippling is similarly generalized so that rules propagate colours through terms. This annotation guides search so that rewrite rules are only applied if they reduce the differences between the conclusion and some of the hypotheses. We have tested this implementation on a number of pro...