Z. Khasidashvili, Perpetual reductions and strong normalization in orthogonal term rewriting systems, CWI Report CS-R9345, Amsterdam, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....to have been explored. As far as we know, only in [11, ch. II.6.2] a sufficient condition for termination of regular CRSs is given. With this condition, stated in terms of redexes and descendants, a termination proof for CRSs remains a syntactical matter. Other work on this line is done in [10]. We also refer to [9] where a recursion scheme for higher order rules is given that guarantees termination. Termination of first order Term Rewriting is already an undecidable problem. But as the termination of TRSs is an interesting question, many semi algorithms and characterisations of ....

Z. Khasidashvili. Perpetual reductions and strong normalization in orthogonal term rewriting systems. Technical Report CS-R9345, CWI, Amsterdam, July 1993.

.... rewriting can skip the next section, and refer to it only for the notation (Notation 3) for an informal description of the descendant concept [8,27] and for the concept of similarity of redexes (Definition 6) The results on the longest perpetual reductions have been reported previously in [29,31,32,34]. Here we simplify and correct several concepts and some proofs. Some of the termination proofs employing the memory method and establishing exact upper bounds were obtained much earlier in [22,24] 2 Preliminaries: Orthogonal Expression Reduction Systems Klop introduced Combinatory Reduction ....

....starting from t, and s is an R normal form of t, then jQj = jQ j (by Corollary 12) ksk (by confluence of R ) jP j (by Corollary 29) Thus of all reductions of t to normal form, P has the maximal length. Remark 31 A direct proof that limit reductions are the longest can be found in [29]. The idea is that the residual of a reduction P : t of length m under a limit reduction Q : t s of length n yields a reduction P=Q [19] of length at least m Gamma n. In case m = 1, m Gamma n = 1, hence the limit strategy is perpetual. And if P and Q are normalizing, then 0 m Gamma n, ....

Z. Khasidashvili, Perpetual reductions and strong normalization in orthogonal term rewriting systems, CWI Report CS-R9345, Amsterdam, 1993.

....decidability of strong normalization if decidability of weak normalization is known. For example, all the above classes of OTRSs are closed under extension. We describe some applications of our results in section 3. The main results are obtained in section 2. Complete proofs can be found in [7]. 2 Perpetual strategies in OTRSs We recall some basic notions of TRS theory; one can find comprehensive introductions to the subject in [3] and [10] A TRS is a pair ( Sigma; R) where the alphabet Sigma consists of variables and function symbols and R is a set of rewrite rules r of the form t ....

....jQ j = ksn k , i.e. each step of Q increases the number of occurrences exactly by 1. This is achieved by contracting the limit redexes only. Indeed, in this case the old occurrences do not duplicate, and the only new symbol created in each step is the head symbol of the contractum. In [7], we give a simple direct proof of the fact that limit reductions are the longest. Definition 2.1 The extension ( Sigma ; R ) of an OTRS ( Sigma; R) is defined as follows: 1. Sigma = Sigma [ f n j n = 0; 1; g, where n is a fresh n ary function symbol. For any subterm s = ....

Khasidashvili Z. Perpetual reductions and strong normalization in orthogonal term rewriting systems. CWI Report CS-R9345, Amsterdam, July 1993.

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