term rewriting system
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 system →_{R} 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 lefthand 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 t →_{E} t′ requires substitutions to be made in some equation e ∈ E and the lefthand side of e is replaced by the righthand 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.
See also orthogonal term rewriting system.
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