This article discusses the two incarnations of Otter entered in the CADE-13 Automated Theorem Proving Competition. Also presented are some historical background, a summary of applications that have led to new results in mathematics and logic, and a general discussion of Otter.

Introduction Many researchers who study the theoretical aspects of inference systems believe that if inference rule A is complete and more restrictive than inference rule B, then the use of A will lead more quickly to proofs than will the use of B. The literature contains statements of the sort "our rule is complete and it heavily prunes the search space; therefore it is efficient". 2 These positions are highly questionable and indicate that the authors have little or no experience with the practical use of automated inference systems. Restrictive rules (1) can block short, easy-to-find proofs, (2) can block proofs involving simple clauses, the type of clause on which many practical searches focus, (3) can require weakening of redundancy control such as subsumption and demodulation, and (4) can require the use of complex checks in deciding whether such rules should be applied. The only way to determ

Abstract. The prototype of a content based search engine for mathematical knowledge supporting a small set of queries requiring matching and/or typing operations is described. The prototype — called Whelp — exploits a metadata approach for indexing the information that looks far more flexible than traditional indexing techniques for structured expressions like substitution, discrimination, or context trees. The prototype has been instantiated to the standard library of the Coq proof assistant extended with many user contributions. 1

Abstract. We propose several methods for writing efficient subsumption procedures for non-unit clauses, tested in practice as parts incorporated into the Gandalf family of theorem provers. Versions of Gandalf exist for classical logic, first order intuitionistic logic and type theory. Subsumption is one of the most important techniques for cutting down search space in resolution theorem proving. However, for many problem categories most of the proof search time is spent on subsumption. While acceptable efficiency has been achieved for subsuming unit clauses (see [7], [2]), the nonunit subsumption tends to slow provers down prohibitively. We propose several methods for writing efficient subsumption procedures for non-unit clauses, succesfully tested in practice as parts built into the Gandalf family of theorem provers: – ordering literals according to a certain subsumption measure – indexing first two literals of each nonunit clause – pre-computed properties of terms, literals and clauses – a hierarchy of fast filters for clause-to-clause subsumption – combining subsumption with clause simplification – linear search among the strongly reduced number of candidates for back subsumption The presented methods for substitution were among the key techniques enabling the classical version of Gandalf to win the MIX division of the CASC-14 prover contest in 1997. The approach of the paper is purely empirical, presenting the methods and bringing some statistical evidence. 1 Gandalf Family of Provers Before continuing with the details of the subsumption methods we will present an overview of the Gandalf family of provers. We use the name Gandalf for the interdependent, code-sharing, resolution-based automated theorem provers we are developing: a resolution prover for first-order intuitionistic logic Tammet [9], for a fragment of Martin-Löf’s type theory Tammet [10] and for first-order

Abstract. The paper describes an innovative technique for efficient retrieval of mathematical statements from large repositories, developing and substantially improving the metadata-based approach introduced in [13]. 1

We introduce the distributed theorem prover Peers-mcd for networks of workstations. Peers-mcd is the parallelization of the Argonne prover EQP, according to our Clause-Diffusion methodology for distributed deduction. The new features of Peers-mcd include the AGO (Ancestor-Graph Oriented) heuristic criteria for subdividing the search space among parallel processes. We report the performance of Peers-mcd on several experiments, including problems which require days of sequential computation. In these experiments Peersmcd achieves considerable, sometime super-linear, speed-up over EQP. We analyze these results by examining several statistics produced by the provers. The analysis shows that the AGO criteria partitions the search space effectively, enabling Peers-mcd to achieve super-linear speed-up by parallel search. 1 Introduction Distributed deduction is concerned with the problem of proving difficult theorems by distributing the work among networked computers. The motivation is to st...

Indexing data structures have a crucial impact on the performance of automated theorem provers. Examples are discrimination trees, which are like tries where terms are seen as strings and common prefixes are shared, and substitution trees, where terms keep their tree structure and all common contexts can be shared. Here we describe a new indexing data structure, called context trees, where, by means of a limited kind of context variables, also common subterms can be shared, even if they occur below di#erent function symbols. Apart from introducing the concept, we also provide evidence for its practical value. We describe an implementation of context trees based on Curry terms and on an extension of substitution trees with equality constraints, where one also does not distinguish between internal and external variables.

. This note presents purely mechanical proofs of the Levi commutator problem in group theory. The problem was solved first by using the theorem prover EQP, developed by William McCune at the Argonne National Laboratory. The fastest proof was found by using Peers-mcd, the Clause-Diffusion parallelization of EQP, developed by the author at the University of Iowa. 1 The Levi commutator problem The Levi commutator problem is an equational problem in group theory. Given the axioms for a group with product and identity e e x ' x x \Gamma1 x ' e (x y) z ' x (y z) the commutator is a binary operator [ ; ] defined by: [x; y] ' x \Gamma1 y \Gamma1 x y: The Levi commutator problem consists in proving that x [y; z] ' [y; z] x , [[x; y]; z] ' [x; [y; z]] that is, x [y; z] ' [y; z] x holds if an only if the commutator is associative. A textbook proof of this theorem can be found in [10]. In the input to the theorem provers, the group axioms and the commutator definition are...

Machine [War83] implementation for Prolog) are stored in an array similar to the WAM heap. It is an array of pairs h tag, address i, where tag can be ref or struct, that is, a function symbol f. The field address contains a heap address. Terms are stored on the heap as in the WAM: each function symbol of arity n is followed by n contiguous ref positions pointing to its arguments. Each uninstantiated variable corresponds to a ref position pointing to itself. For example, the heap below at the left contains f(x; g(x); g(x); y) at the address 20: . . . . . . 20 f 21 ref 21 22 ref 30 23 ref 30 24 ref 24 . . . . . . 30 g 31 ref 21 . . . . . . Note that in such a representation the whole term needs not to be contiguous, and that common subterms ---not only variables--- can be shared, like the subterm g(x) at position 30. Moreover, unlike it happens in other term representations, matching and unification operations do not need to deal with a partial substitution: du...

Abstract. A huge amount of mathematical knowledge is nowadays available on the World Wide Web. Many different solutions and technologies for searching that knowledge have been developed as well. We present the state of the art of searching mathematics on the web, giving some insight on future developments in this area. 1.