# term rewriting system

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term rewriting system (TRS) A formal system for manipulating terms over a signature by means of rules. A set R of rules (each a rewrite rule) creates an abstract reduction systemR on the algebra T(Σ,X) of all terms over signature Σ and variables X. Usually, the rules are a set E of equations that determine a reduction system →E using rewrites based on equational logic.

Let E be a set of equations such that, for each t = t′ ∈ E, the left-hand side t is not a variable. The pair (Σ,E) is called an equational TRS. The equations of E are used in derivations of terms where the reduction tE t′ requires substitutions to be made in some equation eE and the left-hand side of e is replaced by the right-hand side of e in t to obtain t′.

The first set of properties of a term rewriting system (Σ,E) is now obtained from the properties of abstract reduction systems. The following are examples. (1) The term rewriting system (Σ,E) is complete if the reduction system →E on T(Σ) is Church–Rosser and strongly terminating.(2) Let ≡E be the smallest congruence containing →E and T(Σ,E) be the factor algebra T(Σ)|≡E. Then T(Σ,E) is the initial algebra of Alg(Σ,E). If (Σ,E) is a finite equational TRS specification that is complete, then T(Σ,E) is a computable algebra.