### Rewrite Systems [chapter]

Nachum DERSHOWITZ, Jean-Pierre JOUANNAUD
1990 Formal Models and Semantics
This is the same as saying that s = u l p and t = u r p , for some context u and position p in u. A subterm sj p at which a rewrite can take place is called a redex; w e s a y that s is irreducible, o r i n normal form, i f i t has no redex, i.e. if there is no t in T such t h a t s ! R t. Systems of rules are used to compute by rewriting repeatedly, u n til, perhaps, a normal form is reached. A derivation in R is any nite or in nite sequence t 0 ! R t 1 ! R ! R t i ! R of applications of
more » ... e rules in R. The reducibility, o r derivability, relation is the quasi-ordering ! R , i.e. the re exive-transitive closure of ! R . W e write s ! ! R t if s ! R t and t is irreducible, in which c a s e w e s a y t h a t t is a normal form of s. This normalizability relation ! ! R is not a rewrite relation, since normalizing a subterm does not mean that its superterm is in normal form. One says that a rewrite system is normalizing if every term has at least one normal form. A ground rewriting-system is one all the rules of which are ground i.e. elements of G G ; an important early paper on ground rewriting is Rosen, 1973 . A string-rewriting system, o r semi-Thue system, i s o n e that has monadic words ending in the same variable i.e. strings of elements of T F 1 ; fxg as left-and right-hand side terms; Book, 1987 is a survey of string rewriting. The rst three Co ee Can Games can Such relations have sometimes been called Noetherian in the term-rewriting literature|after the algebraicist, Emily Noether|though the adjective is ordinarily used to exclude in nite ascending chains. Termination is more than being normalizing, since the latter allows some derivations to be in nite. A partial irre exive ordering of a set T is well-founded if there exists no in nite descending chain t 1 t 2 of elements of T. T h us, a relation ! is terminating i its transitive closure ! + is a well-founded ordering. The importance of terminating relations lies in the possibility of inductive proofs in which t h e h ypothesis is assumed to hold for all elements t such that s ! + t when proving it for arbitrary s. Induction on terminating relations, sometimes called Noetherian induction," is essentially well-founded induction i.e. trans nite induction extended to partial orderings; see, for example, Cohn, 1981 . W e will have occasion to employ this technique in Sections 4.1 and 8.2. A rewrite system R is terminating for a set of terms T if the rewrite relation ! R over T is terminating, i.e. if there are no in nite derivations t 1 ! R t 2 ! R of terms in T . When a system is terminating,