On Obtaining Knuth, Morris, and Pratt's String Matcher by Partial Evaluation

Abstract

We present the first formal proof that partial evaluation of a quadratic string matcher can yield the precise behaviour of Knuth, Morris, and Pratt's linear string matcher.Obtaining a KMP-like string matcher is a canonical example of partial evaluation: starting from the naive, quadratic program checking whether a pattern occurs in a text, one ensures that backtracking can be performed at partial-evaluation time (a binding-time shift that yields a staged string matcher); specializing the resulting staged program yields residual programs that do not back up on the text,à la KMP. We are not aware, however, of any formal proof that partial evaluation of a staged string matcher precisely yields the KMP string matcher, or in fact any other specific string matcher.In this article, we present a staged string matcher and we formally prove that it performs the same sequence of comparisons between pattern and text as the KMP string matcher. To this end, we operationally specify each of the programming languages in which the matchers are written, and we formalize each sequence of comparisons with a trace semantics. We also state the (mild) conditions under which specializing the staged string matcher with respect to a pattern string provably yields a specialized string matcher whose size is proportional to the length of this pattern string and whose time complexity is proportional to the length of the text string. Finally, we show how tabulating one of the functions in this staged string matcher gives rise to the 'next' table of the original KMP algorithm.The method scales for obtaining other linear string matchers, be they known or new. * Extended version of an article

Fast partial evaluation of pattern matching in strings

“…As already shown in the literature [Ager et al 2002;Grobauer and Lawall 2002], each specialized version of a staged matcher such as that of Figure 1 has size linear in the length of the pattern. For two reasons, however, specialization does not proceed in time linear in the length of the pattern: -The first reason is that for every position in the pattern, the specializer blindly performs the backtracking steps of the staged quadratic matcher.…”

“…In this article, the starting point is a staged quadratic-time matcher and a simple memoizing partial evaluator such as Similix, where specialization points are dynamic conditional expressions [Bondorf and Danvy 1991]. Figure 1 displays a staged matcher similar to the ones developed in the literature [Ager et al 2002;Amtoft et al 2002;Consel and Danvy 1989;Jones et al 1993] (see Appendix A for a more complete picture). Matching is done naively from left to right.…”

On obtaining the Boyer–Moore string-matching algorithm by partial evaluation

“…We would like to thank Mads Sig Ager for our pleasant joint initial study of partial evaluation of string matchers [1,2]. We are also grateful to him, Julia Lawall, and the anonymous referees for their insightful comments.…”

“…As in our earlier work [1], we characterize a string matcher by a notion of trace: the sequence of character comparisons between the pattern and the text in the course of a run. Using a large test suite (several hundreds of runs), we have verified the correctness of each of our programs by automatically comparing its traces and the corresponding traces of a reference implementation [6].…”

“…Using a large test suite (several hundreds of runs), we have verified the correctness of each of our programs by automatically comparing its traces and the corresponding traces of a reference implementation [6]. A formal alternative would be to give a trace semantics to our programs and to the reference programs and to prove that they operate in lock step [1].…”