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TermRew110701 - 1 LECTURE NOTES ON TERM REWRITING AND...

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1 LECTURE NOTES ON TERM REWRITING AND COMPUTATIONAL COMPLEXITY by Harvey M. Friedman Ohio State University [email protected] http://www.math.ohio-state.edu/~friedman/ November 7, 2001 Abstract. The main powerful method for establishing termination of term rewriting systems was discovered by Nachum Dershowitz through the introduction of certain natural well founded orderings (lexicographic path orderings). This leads to natural decision problems which may be of the highest computational complexity of any decidable problems appearing in a natural established computer science context. 1. TERM REWRITING. A signature S is a finite set of function symbols (arities 0). V is the set of variables x 1 ,x 2 ,... . T( S ,V) is the set of all terms using elements of S V. T( S ) is the restriction to closed terms (i.e., with no variables). A rewrite rule in T( S ,V) is an expression l r where l,r T( S ,V), l is not a variable, and every variable in r is a variable in l. These two restrictions are from [BN], p. 61. Only the second restriction is important for us. We write s t by l r iff s,t T( S ,V) and there is a substitution of variables in l by terms in T( S ,V) which converts l to s and r to t. A term rewriting system (trs) is a pair R = (R, S ), where R is a finite set of rewrite rules in T( S ,V). Term rewriting systems are implemented as follows.
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2 We write s t by R iff s t by some l r in R. An R-derivation is a nonempty sequence t 1 ,t 2 ,... of length 1 £ n £ such that for all 1 £ i £ n-1, t i t i+1 by R. THEOREM 1.1. Let R be a trs. All variables occurring in any R-derivation already occur in its first term. For all n 1 and s T( S ,V), there are finitely many R-derivations from s of length £ n. This depends heavily on the convention that, in each rewrite rule, all right variables are left variables. 2. ORDERED TERM REWRITING. Ordered term rewriting is discussed in [BN], 267-268. An ordered term rewriting system (otrs) is a triple (R, S ,<), where (R, S ) is a term rewriting system and < is a strict ordering on T( S ,V). An (R, S ,<)-derivation is an R-derivation which is strictly decreasing under <. Note that the presence of < only affects the allowable derivations. It does not have any affect on R, which can be any finite set of rewrite rules. A well founded term rewriting system (wftrs) is an otrs whose < is well founded (no infinite strictly decreasing sequences). As a consequence, every (R,<)-derivation is finite. In fact, more is true. THEOREM 2.1. In any wftrs there are finitely many derivations starting with any given term. Proof: Apply the fundamental fact that an infinite finitely branching tree has an infinite path. QED 3. TERMINATION FUNCTIONS. DERIVATION PROBLEMS.
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3 N = the set of all nonnegative integers. Z + = the set of positive integers. The size of a term, #(t), is the total number of occurrences of functions and variables.
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