Conditions for confluence of innermost terminating term rewriting systems

Abstract

This paper presents a counterexample for the open conjecture whether innermost joinability of all critical pairs ensures confluence of innermost terminating term rewriting systems. We then show that innermost joinability of all normalized instances of the critical pairs is a necessary and sufficient condition. Using this condition, we give a decidable sufficient condition for confluence of innermost terminating systems. Finally, we enrich the condition by introducing the notion of left-stable rules. As a corollary, confluence of innermost terminating left-weakly-shallow TRSs is shown to be decidable.

A Priority rewriting system

We show that there exists a rewrite relation satisfying that , , and by using priority term rewriting systems (PRS) [3].

Definition 5

A priority term rewriting system (PRS) [3] is a pair $(\mathcal {R}, \blacktriangleright )$ of a TRS $\mathcal {R}$ and a partial order $\blacktriangleright$ on the rules of $\mathcal {R}$. A rule $l_1 \rightarrow r_1$ has a higher priority than a rule $l_2 \rightarrow r_2$ iff $l_1 \rightarrow r_1 \blacktriangleright l_2 \rightarrow r_2$. A priority rewrite step is defined as: iff for a substitution $\sigma$, a position $p \in \mathrm {Pos}(s)$ and a rule $l \rightarrow r \in \mathcal {R}$,

• and

• $l \rightarrow r$ is maximal with respect to $\blacktriangleright$ among rules that reduce $l\sigma$, i.e. $l' \rightarrow r' \blacktriangleright l \rightarrow r$ for any different rule $l' \rightarrow r' \in \mathcal {R}$ such that $l\sigma = l'\sigma '$ for some $\sigma '$.

It is clear that .

Example 7

For a PRS

$$\begin{aligned} \begin{array}{lll} \begin{aligned} &{}f(g(x)) \rightarrow b \ \ &{}(1)\\ &{}g(a) \rightarrow c \ \ \ \ \ &{}(2)\\ &{}g(x) \rightarrow x \ \ \ \ \ &{}(3) \end{aligned} &{} \text {and} &{} (1) \blacktriangleright (2) \blacktriangleright (3), \end{array} \end{aligned}$$

, but is not a priority rewrite step.

Lemma 13

Proof

Since , we will show . For , it is obvious from .

Now, we prove by contradiction. We assume that for . Then there exist a rule $l \rightarrow r \in \mathcal {R}$, normalized substitution $\sigma$ and position $p \in \mathrm {Pos}(t)$ such that $t = t[l\sigma ]_p$ and $t' = t[r\sigma ]_p$. Since , is not possible. This means that there must exist $l' \rightarrow r' \in \mathcal {R}$ such that $l' \rightarrow r' \blacktriangleright l \rightarrow r$ and $l\sigma = l'\sigma '$ for some substitution $\sigma '$. Note that since every proper subterm of $l\sigma ({}=l'\sigma ')$ is irreducible. This means that t can be reduced by . This is a contradiction. $\square$

Lemma 14

If $\blacktriangleright$ is total then .

Proof

We first show that implies $u=v$ or . If the rewriting steps to u and v occur at the same position, $u = v$ since the same rule is used for the rewriting from the totality of $\blacktriangleright$. Otherwise, the rewriting steps to u and v occur at parallel positions, and hence there exists a term t such that . Therefore the lemma is easily obtained. $\square$