Term rewriting systems from Church-Rosser to Knuth-Bendix and beyond

Term rewriting systems are important for computability theory of abstract data types, for automatic theorem proving, and for the foundations of functional programming. In this short survey we present, starting from first principles, several of the basic notions and facts in the area of term rewriting. Our treatment, which often will be informal, covers abstract rewriting, Combinatory Logic, orthogonal systems, strategies, critical pair completion, and some extended rewriting formats

Unique normal forms for lambda calculus with surjective pairing

AbstractWe consider the equational theory λπ of λ-calculus extended with constants π, π0, π1 and axioms for surjective pairing: π0(πXY) = X, π1(πXY) = Y, π(π0X)(π1X) = X. Two reduction systems yielding the equality of λπ are introduced; the first is not confluent and, for the second, confluence is an open problem. It is shown, however, that in both systems each term possessing a normal form has a unique normal form. Some additional properties and problems in the syntactical analysis of λπ and the corresponding reduction systems are discussed

Sequentiality in orthogonal term rewriting systems

AbstractFor orthogonal term rewriting systems Q. Huet and J.-J. Lévy have introduced the property of ‘strong sequentiality’. A strongly sequential orthogonal term rewriting system admits an efficiently computable normalizing one-step reduction strategy. As shown by Huet and Lévy, strong sequentiality is a decidable property. In this paper we present an alternative analysis of strongly sequential term rewriting systems, leading to two simplified proofs of the decidability of this property. We also compare some related notions of sequentiality that recently have been proposed