Programming Languages

Advanced Topics in Term Rewriting by Enno Ohlebusch

By Enno Ohlebusch

Term rewriting concepts are appropriate in quite a few fields of computing device sci­ ence: in software program engineering (e.g., equationally specific summary information types), in programming languages (e.g., functional-logic programming), in computing device algebra (e.g., symbolic computations, Grabner bases), in professional­ gram verification (e.g., immediately proving termination of programs), in computerized theorem proving (e.g., equational unification), and in algebra (e.g., Boolean algebra, crew theory). In different phrases, time period rewriting has purposes in useful laptop technology, theoretical desktop technological know-how, and arithmetic. approximately conversing, time period rewriting thoughts can suc­ cessfully be utilized in parts that call for effective tools for reasoning with equations. one of many significant difficulties one encounters within the conception of time period rewriting is the characterization of periods of rewrite structures that experience a fascinating estate like confluence or termination. If a time period rewriting method is conflu­ ent, then the conventional kind of a given time period is exclusive. A terminating rewrite procedure doesn't enable countless computations, that's, each computation ranging from a time period needs to result in a standard shape. hence, in a approach that's either terminating and confluent each computation ends up in a end result that's particular, whatever the order during which the rewrite ideas are utilized. This publication offers a complete examine of termination and confluence in addition to comparable properties.

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6. o We conclude this chapter by showing how every string rewriting system S over an alphabet L: can be viewed as a term rewriting system. To this end, let the signature F contain a constant c and a unary function symbol for every letter a E L:. , o}, then let F = {b, w, c}, where b is associated with. and w with o. Then a word over L: can be represented by a term over F. For example, w(w(w(b(c)))) represents the word 0 0 0 •. Furthermore, a string rewrite rule like . 0 --+ 0 0 o. can be represented by the term rewrite rule b(w(x)) --+ w(w(w(b(x)))).

First, we recall PCP, which can be stated as follows: Given a finite alphabet r and a finite set P c r+ x r+, is there some natural number n > 0 and (ai,f3i) E P for i E {l, ... ,n} such that a1 a2 ... an = f31f32 ... f3n? This problem is known to be undecidable even in the case of a two-letter alphabet; see Post [Pos46]. The set P is called an instance of PCP, the string a1 a2 ... an = f31f32'" f3n is a solution for P. Matiyasevich and Senizergues [MS96] showed that PCP is undecidable even when restricted to instances consisting of seven pairs.

We distinguish between the following cases: Case q < p: If u is a proper subterm of v, then root(tl q) is underlined and tl q is the descendant of v. Because R is nonoverlapping, u must be a subterm of xa' for some x E Var(l'). It follows that tl q is still a redex in t because the rule l' ~ r' is left-linear. It should be stressed that tl q need not necessarily be a redex if R is not orthogonal. Case q II p: If u and v occur at independent positions, then root(tl q) is underlined and tl q, the descendant of v, is unaltered in t.

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