Advanced Topics in Term Rewriting by Enno Ohlebusch

By Enno Ohlebusch

Term rewriting options are acceptable in numerous fields of laptop sci­ ence: in software program engineering (e.g., equationally unique summary info types), in programming languages (e.g., functional-logic programming), in desktop algebra (e.g., symbolic computations, Grabner bases), in seasoned­ gram verification (e.g., instantly proving termination of programs), in automatic theorem proving (e.g., equational unification), and in algebra (e.g., Boolean algebra, staff theory). In different phrases, time period rewriting has functions in sensible laptop technology, theoretical computing device technology, and arithmetic. approximately talking, time period rewriting recommendations 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 sessions of rewrite structures that experience a fascinating estate like confluence or termination. If a time period rewriting procedure is conflu­ ent, then the conventional kind of a given time period is exclusive. A terminating rewrite procedure doesn't enable limitless computations, that's, each computation ranging from a time period needs to result in a standard shape. accordingly, in a method that's either terminating and confluent each computation ends up in a end result that's certain, whatever the order during which the rewrite principles are utilized. This ebook offers a entire examine of termination and confluence in addition to similar properties.

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8 lack WN and SCOR",. In the following, we will use a special measure based on multisets. Let A = (A, {-+aJaEI, H) be an ARS. The label of a finite reduction sequence is the string of labels of its constituent reduction steps. For example, if I = {I, 2, 3, 4}, then the label of the reduction sequence a -+2 b -+4 C -+2 d is the string 242. The Greek letters a, T, /-L, v etc. will be used to denote strings. The concatenation of two strings a and T is denoted by aT. Moreover, e denotes the empty string.

3. D We have seen that critical pairs arise from "overlaps" between rewrite rules. Let us specify this notion. 3 A term s overlaps a term t if s is unifiable with a nonvariable subterm of t. sand t are called nonoverlapping if neither s overlaps t nor t overlaps s. A term s overlays a term t if sand t are unifiable. We say that a rewrite rule 12 -+ r2 overlaps (overlays) a rewrite rule 11 -+ rl if there are renamed versions l~ -+ r~ and l~ -+ r~ of 12 -+ r2 and h -+ rl, respectively, which have no variables in common and satisfy l~ overlaps (overlays) l~.

Is a wellfounded partial ordering. We iteratively construct trees To, T I , T 2 , ... with the following properties: 1. The nodes of~, except the root, are labeled with elements from AJ.. 2. 1 of the leaves of Ti coincides with Mi. 3. t. >-. The initial tree To has a nonlabeled root and, for every element a of M o , a leaflabeled with a. Given T i , we construct Ti+1 as follows. Because Mi >-mul M HI , there are finite multisets X, Y E FM(A) such that 0 =I- X s::;; M i , MHI = (Mi \ X) EB Y, and, for all y E Y, there exists an x E X such that x >- y.

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