Full-commutation and fair-termination in equational (and combined) term-rewriting systems

Twenty years of rewriting logic

“…However, under fairness assumptions about how each receiver object will apply the receive rule, and each sender object will receive acknowledgments and clear its cell, the fault-tolerant system is indeed fairly terminating. Proof techniques for termination of rewrite theories under fairness assumptions have been studied in [293], substantially extending prior work in [386,387]. Another way in which termination techniques need to be extended is to reason about termination of R when executed under a given strategy (see Section 3.5).…”

Termination of just/fair computations in term rewriting

“…The related question of finding general methods for proving fair termination of term rewriting systems has indeed been studied before, particularly by Porat and Francez [28,29]. However, their efforts followed Floyd's classical approach, which uses predicates on states (in our setting, ground terms) to achieve termination (see [10,Chapter 2] for a general description of this approach, and also [19]).…”

“…However, their efforts followed Floyd's classical approach, which uses predicates on states (in our setting, ground terms) to achieve termination (see [10,Chapter 2] for a general description of this approach, and also [19]). In particular, their characterization of fair-termination of a rewrite system in terms of the compatibility of a well-founded ordering with all possible full derivations [29,Definition 9] does not lend itself to mechanization. Some parallelism can be found with the use of Manna and Ness's versus Lankford's termination criteria in proofs of termination of rewriting (see, e.g., [6]).…”

“…In our setting, these notions are specially relevant because (as shown in the subsequent sections) we are able to characterize the corresponding termination notions as termination of (combinations of) more standard reduction relations. In the literature on fairness, though, labels are usually identified with rules, thus leading to the notion of rule justice/fairness, where each rule in R F is assumed to 'own' a different label (e.g., [29]). Furthermore, when R F = R this becomes the notion of fairness in term rewriting proposed in [28].…”

“…Once this theoretical stance is adopted, since fairness is a pervasive property of concurrent systems, needed to establish many properties of interest, the first thing required is to correctly express the fairness notion within the rewriting framework. In this regard, the early work of Porat and Francez [28,29], and the work of Tison for the ground fair termination case [32], complemented by the more recent "localized fairness" notion in [24] offer a good basis. In this setting, a subset R F of the rules of a Term Rewriting System (TRS) R, whose rules are conveniently labelled can be used to localize the desired fairness requirements over particular rules of R.…”