Making proofs without Modus Ponens: An introduction to the combinatorics and complexity of cut elimination

This paper is intended to provide an introduction to cut elimination which is accessible to a broad mathematical audience. Gentzen's cut elimination theorem is not as well known as it deserves to be, and it is tied to a lot of interesting mathematical structure. In particular we try to indicate some dynamical and combinatorial aspects of cut elimination, as well as its connections to complexity theory. We discuss two concrete examples where one can see the structure of short proofs with cuts, one concerning feasible numbers and the other concerning "bounded mean oscillation" from real analysis

NATURAL DEDUCTION AS HIGHER-ORDER RESOLUTION

An interactive theorem prover, Isabelle, is under development. In LCF, each inference rule is represented by one function for forwards proof and another (a tactic) for backwards proof. In Isabelle, each inference rule is represented by a Horn clause. Resolution gives both forwards and backwards proof, supporting a large class of logics. Isabelle has been used to prove theorems in Martin-L\"of's Constructive Type Theory. Quantifiers pose several difficulties: substitution, bound variables, Skolemization. Isabelle's representation of logical syntax is the typed lambda-calculus, requiring higher- order unification. It may have potential for logic programming. Depth-first subgoaling along inference rules constitutes a higher-order Prolog

Compiling Unit Clauses for the Warren Abstract Machine

This thesis describes the design, development, and installation of a computer program which compiles unit clauses generated in a Prolog-based environment at Argonne National Laboratories into Warren Abstract Machine (WAM) code. The program enhances the capabilities of the environment by providing rapid unification and subsumption tests for the very significant class of unit clauses. This should improve performance substantially for large programs that generate and use many unit clauses

A Survey of Automated Deduction

We survey research in the automation of deductive inference, from its beginnings in the early history of computing to the present day. We identify and describe the major areas of research interest and their applications. The area is characterised by its wide variety of proof methods, forms of automated deduction and applications

Fault detection and rectification algorithms in a question-answering system

A Malay proverb "jika sesat di hujung jalan, baleklah kepangkal jalan" roughly means "if you get lost at the end of the road, go back to the beginning". In going back to the beginning of the road, we learn our mistakes and hopefully will not repeat the same mistake again. Thus, this work investigates the use of formal logic as a practical tool for reasoning why we could not infer or deduce a correct answer from a question posed to a database. An extension of the Prolog interpreter is written to mechanise a theorem-proving system based on Horn clauses. This extension procedure will form the basis of the question-answering system. Both input into and output from this system is in the form of predicate calculus. This system can answer all four classes of questions as classified by Chang and Lee (1973). [Continues.