On Soundness Conditions for Unraveling Deterministic Conditional Rewrite Systems

Confluence of Conditional Term Rewrite Systems via Transformations

“…Soundness and confluence of the transformed system implies confluence of the original CTRS (see e.g. [11]), yet there is not yet a positive or a negative result whether soundness is essential (although confluence of the transformed CTRS does not imply soundness which was shown in [9]). This paper will answer this question by first showing that innermost derivations are always sound and then show that this in fact implies confluence if the transformed TRS is terminating.…”

“…This was first shown by Marchiori in [13] using a normal 1-CTRS that consists of multiple non-linear rules. For DCTRSs we presented another example in [9].…”

“…The CTRSs of Example 4 and the counterexample of [13] are syntactically very complex. Based on this observation it was shown that many syntactic properties imply soundness: Left-linearity (normal 1-CTRSs [13]/ [19], DCTRSs [16]), weak left-linearity, right-linearity (normal 1-CTRSs [8], DCTRSs [9]), non-erasingness (normal 1-CTRSs [8], 2-DCTRSs [9], counterexample for 3-DCTRSs [9]) and weak right-linearity (DCTRSs [7]).…”

“…Example 5 (unsoundness for confluence [9]). Let R be the CTRS of Example 4 and R be the CTRS consisting of the unconditional rules…”

“…Although the previous example shows that confluence of the transformed TRS is not sufficient for soundness, it also shows that (in contrast to Example 4) the last term of the unsound derivation can be further reduced. In fact, all normalizing derivations in confluent DCTRSs are sound [9].…”