On (Un)Soundness of Unravelings

Confluence of Conditional Term Rewrite Systems via Transformations

“…The CTRS of Example 4 is not confluent and in [8] it is shown that the simultaneous unraveling is sound for confluent normal 1-CTRSs. Yet, this result does not hold for DCTRSs:…”

“…Since U seq preserves all variable bindings of the left-hand side of a conditional rule it is possible to extract these bindings and insert them into the corresponding left-hand side, thus obtaining the backtranslation tb (defined in [8], similar mappings are used in the proofs in [13] and [19]).…”

“…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]).…”

Sound Structure-Preserving Transformation for Weakly-Left-Linear Deterministic Conditional Term Rewriting Systems

“…Neither any unraveling nor the SR transformation is sound for all CTRSs. Since soundness is one of the most important properties for transformations of CTRSs, sufficient conditions for soundness have been well investigated, especially for unravelings (see, e.g., [5,11,6]). For example, the simultaneous unraveling that has been proposed by Marchiori [9] (and then has been improved by Ohlebusch [13]) is sound for weakly-left-linear (WLL, for short), confluent, non-erasing, or ground conditional normal CTRSs [5], and for DCTRSs that are confluent and right-stable, WLL, or ultra-right-linear [6].…”

“…Since soundness is one of the most important properties for transformations of CTRSs, sufficient conditions for soundness have been well investigated, especially for unravelings (see, e.g., [5,11,6]). For example, the simultaneous unraveling that has been proposed by Marchiori [9] (and then has been improved by Ohlebusch [13]) is sound for weakly-left-linear (WLL, for short), confluent, non-erasing, or ground conditional normal CTRSs [5], and for DCTRSs that are confluent and right-stable, WLL, or ultra-right-linear [6]. Normal CTRSs admit a rewrite rule to have conditions to test terms received via variables in the left-hand side, e.g., whether a term with such variables can reach a ground normal form specified by the rule.…”