### Bi-rewrite Systems

Jordi Levy, Jaume Agustí
1996 Journal of symbolic computation
1 . An inclusion is a pair of terms s, t ∈ T (F, X ) written s ⊆ t. Given a finite set of inclusions Ax and a pair of terms s and t, we say that s ⊆ Ax t iff Ax POL s ⊆ t, where POL stands for Pre-Order Logic and POL is the entailment relation defined by the following inference rules where σ is a substitution, p a position in u, i.e. u[·] p is a context, and ∆ is a finite set of inclusions. Meseguer (1990, 1992) has studied the logic of conditional inequalities widely, which he names rewriting
more » ... ogic, and its models. † As we will see later, in most cases we also require the finiteness of F . We suppose that Fn are disjoint sets. The set T (F, X ) is defined as the smallest set containing X such that if f ∈ Fn and t i ∈ T (F, X ) for i = 1, . . . , n then f (t 1 , . . . , tn) ∈ T (F, X ). ‡ We write p 1 ≺ p 2 when there exists a sequence q such that p 2 = p 1 · q, and p 1 |p 2 when p 1 ≺ p 2 and p 2 ≺ p 1 . If p is an empty sequence then t|p is defined by t| were well-founded orderings. This is a weaker condition and clearly it is not enough to prove the equivalence between the Church-Rosser and the local bi-confluence properties. This error was communicated to the authors by Professor Harald Ganzinger. We can prove the following results for the decision procedure based on a bi-rewrite system, and the Ax POL t ⊆ u deduction problem of its corresponding inclusion theory. Lemma 2.5. If the bi-rewrite system R ⊆ , R ⊇ is Church-Rosser then the decision procedure based on it is sound and complete, i.e. Ax POL t ⊆ u holds if, and only if, the procedure terminates and answers true. If the bi-rewrite system is Church-Rosser and quasi-terminating then the decision procedure is sound, complete and terminates, therefore the satisfiability problem is decidable. We only need to require the quasi-termination property of the bi-rewrite system-which is (strictly) weaker than the termination property-in order to prove the termination of the procedure; whereas in the equational case, the termination property of the rewrite u t v t