A refinement of strong sequentiality for term rewriting with constructors

Abstract

The notion of call-by-need evaluation has turned out to be very fruitful when applied to dialects of the λ-calculus (Henderson and Morris, 1976 , in “Proc. 3rd ACM Symp. on the Principles of Programming Languages, Atlanta; Kahn and Mac Queen, 1977 , in “IFIP 77”, North-Holland, Amsterdam; Turner, 1979 , Software Practice and Experience, Vol. 9; Vuillemin, 1974 , J. Comput. Systems Sci. , 9 , 332–354; Wadsworth, 1971 , Ph.D. dissertation, Oxford University, England). The analogous idea of sequentiality for term rewriting systems described by firstorder equations has been considered by , in “Proc. 5th ACM Symp. on the Principles of Programming Languages, San Antonio”, 1984, in “Proc. 11th ACM Symp. on the Principles of Programming Languages, Salt Lake City) and Huet-Levy (1979, Technical Report No. 359, INRIA, Le Chesney, France), of which the latter is generally considered to be the most complete theoretical treatment of the subject to date. Huet-Levy (1979) defined the notion of strong sequentiality to describe the class of linear term rewriting systems for which call-by-need computation is practical. This paper introduces an improved version of strong sequentiality called left sequentiality. Unlike strong sequentiality, left sequentiality is based on possible rather than arbitrary (and often impossible) sequences of reductions. We show that left sequentiality is more general than strong sequentiality when applied to individual terms, but is equivalent to the latter when cosidered as a property of admissible sets of left-hand sides for systems of equations. Huet-Levy (1979) showed that there are safe redex selection algorithms, i.e., algorithms deriving normal forms whenever possible, for systems based on strongly sequential sets of left-hand sides. We show that there is no algorithm which is safe for all systems based on a set of left-hand sides if that set is not left sequential. In other words, left sequentiality is not only sufficient, it is also necessary for safe computation based on the analysis of left-hand sides alone.