On conditional rewrite systems with extra variables and deterministic logic programs

Certifying Confluence of Quasi-Decreasing Strongly Deterministic Conditional Term Rewrite Systems

“…This property is used in the proof of [4,Theorem 4.2]. By a small change to the definition of unfeasibility we avoid this requirement for our extension to quasi-decreasingness.…”

“…4). Both unfeasibility and context-joinability rely on the notion of contextual rewriting, which we formalize together with the crucial lemma that contextual rewriting implies conditional rewriting for satisfying substitutions, a result that was stated without proof by Avenhaus and Loría-Sáenz [4,Lemma 4.2]. Unfeasibility further employs strong irreducibility, which like strong determinism is an undecidable property.…”

“…Along the way, we identify and fix some problems in proofs and definitions (of absolute irreducibility, contextual rewriting, and unfeasibility) and provide a (not entirely obvious) proof for [4,Lemma 4.2]. We further adapt the original proof of AL to the new definitions and extend it by infeasibility.…”

“…Alas, even in the presence of termination, confluence is in general still undecidable for conditional term rewrite systems (CTRSs). While Avenhaus and Loría-Sáenz [4] gave a critical pair criterion for quasi-reductive and strongly deterministic CTRSs: joinability of all conditional critical pairs (CCPs) implies confluence; joinability of CCPs is undecidable in general, due to the inherent complexities of conditional rewriting. This lead to the development of sufficient criteria that are implemented in confluence tools for CTRSs like ConCon [27].…”

“…= x min (x:xs) | x < y = x | otherwise = y where y = min xs min(cons(x, nil)) → x (1) min(cons(x, xs)) → x ⇐ min(xs) ≈ y, x < y ≈ true (2) min(cons(x, xs)) → y ⇐ min(xs) ≈ y, x < y ≈ false (3) x < 0 → false (4) 0 < s(y) → true (5) s(x) < s(y) → x< y (6) Issue. Alas, even in the presence of termination, confluence is in general still undecidable for conditional term rewrite systems (CTRSs).…”