Termination and Confluence in Infinitary Term Rewriting

The basic notions of the theory of term rewriting are defined for terms that may involve function letters of infinite arity. A sufficient condition for completeness is derived, and its use demonstrated by the example of abstract clones over infinitary signatures. Introduction In general, to see that two terms of some equational theory A are not equivalent requires a model of A in which their interpretations are distinct. The construction of such models is often bothersome. It may be avoided when the axioms of A can be oriented in such a way that they form a semi-complete term rewriting system. The normal forms then constitute a canonical model of A , which is sufficient for all demonstrations of nonequivalence (see [2]). A theory of rewriting for infinite terms, that is, terms constructed with function symbols of infinite arity, does not have the straightforward calculational motivation of the well-known finitary term rewriting (cf. [2] and [4]). It would still have its use, however, in demonstrating nonequivalence. For example, the theory of categories with certain infinite products may be axiomatized by an infinite set of infinitary equations. (Cf. [5] for the finite case.) One way of showing terms of this theory to be nonequivalent would be to interpret them as distinct arrows in some category of sets. As an application of the results of this note, we shall derive another method: the theory in question may be so oriented as to constitute a complete term rewriting system, and we can decide equivalence, for sufficiently conceivable terms, by normalizing and comparing the normal forms. The subject of the present note is essentially different from the infinitary term rewriting of Kennaway c.s. [3]. The infinite reduction sequences that they study will here be considered simply divergent. Terms By a vocabulary let us understand a set F oּf function letters, each associated with a fixed ordinal which we call its arity. If F is a vocabulary, then the set TF of terms over F is the least set T such that for any α-ary F ∈F and any αINFINITARY TERM REWRITING 2 termed sequence 〈tξ |ξ<α 〉 of elements of T , F (tξ |ξ<α)∈T. We picture F(tξ|ξ<α) as an oriented tree with a root labeled F from which the tξ grow, from left to right as ξ increases. Variables are special nullary function symbols. If X is a set of variables, then TF,X , the set of terms over F in variables from X, is TF∪X. Let t be a term over F (in variables from X), and ξ = 〈ξi| i<n〉 a finite sequence of ordinals. The label Lab(t, ξ ) of t at ξ , and the subterm Sub(t, ξ ) of t at ξ , are defined inductively as follows: (i) Sub(t, ∅) = t (∅ is the empty sequence), and if t ∈ X, Lab(t, ∅) = t, while if t = F(tξ|ξ<α ) with F ∈F, Lab(t, ∅) = F; (ii) Lab(F(tξ|ξ<α ), 〈γ , δ1,..., δn 〉) = Lab(tγ , 〈δ1,..., δn 〉) and Sub(F(tξ|ξ<α ), 〈γ , δ1,..., δn 〉) = Sub(tγ , 〈δ1,..., δn 〉). A sequence ξ for which Lab(t, ξ ) is defined will be called a position in t. For any ξ and F ∈F we define U(F, ξ ) := {t ∈ TF | Lab(t, ξ ) = F}. We topologize TF by taking the U(F, ξ ) as subbasic opens. Then t is the limit of a sequence 〈tξ|ξ<α〉 of terms if for every position η in t, there exists δ < α such that for all ξ with δ ≤ ξ < α, Lab(tξ, η ) = Lab(t, η ). Rewrite rules and reductions Let F be a vocabulary and X a set of variables. A rewrite rule over F in X is an ordered pair of terms s, t over F in variables from X such that s ∉ X and t contains only variables that also appear in s. We denote such a pair by s → t. A pair 〈TF , R 〉 consisting of the set of all terms over some vocabulary F and a set R of rewrite rules over F we call a term rewriting system, abbreviated trs. A substitution, with respect to F and X, is a function σ : TF,X a TF with the property that for every F ∈F and every suitable sequence 〈tξ|ξ<α〉 of arguments, σ (F(tξ|ξ<α)) = F(σ (tξ)|ξ<α) . We say σ (t) is an instance of t. Now suppose we also have a set R of rewrite rules over F. An instance of a rewrite rule s → t is a pair 〈σ (s), σ (t)〉 for some substitution σ. With reference to the trs 〈TF , R 〉, we call such a pair a contraction or head reduction; and we write σ (s)→ σ (t). Sometimes it is useful to make the variables in a term a little more explicit. If t is a term in variables from a set {xξ|ξ<α}, we write t as t(xξ |ξ<α). The result of substituting sξ for xξ, for all ξ<α, is then denoted by t(sξ |ξ<α). A reduction is either a contraction, or an internal reduction, i.e. a pair 〈F(sξ |ξ<α) , F(tξ|ξ<α)〉