Probabilistic Algorithms for Deciding Equivalence of Straight-Line Programs

Abstract

Let Q be any algebraic structure and [formula omitted] the set of all total programs over Q using the instruction set [formula omitted]. (A program is total if no division by zero occurs during any computation ) Let the equivalence problem for [formula omitted] be the problem of deciding for two given programs in [formula omitted] whether or not they compute the same funcuon The following results are proved: (1) If Q is an inftmte field (e.g, the rauonal numbers or the complex numbers), then the equwalence problem for [formula omitted] is probabilistlcally decidable in polynomml time. The result also holds for programs with no dwlslon instructions and Q an infimte integral domain (e.g., the integers). (2) If Q is a finite field, or if Q is a fimte set of integers of cardmahty [formula omitted], then the equivalence problem is NP-hard. The case when the field Q is finite but its cardinality is a funcuon of the size of the instance to the eqmvalence problem is also considered An example is shown for which a sharp boundary between the classes NP-hard and probabihsticaUy decidable exists (provided they are not identical classes). © 1983, ACM. All rights reserved.