Confluence and termination of fuzzy relations

Rewriting systems over similarity and generalized pseudometric spaces and their properties

“…The same applies for reductions in graded setting as they were introduced in [7], where * is a reflexive and ⊗-transitive closure of . This is quite natural since we are interested in reachability (subsitutability) in multiple steps.…”

“…Note that Definition 1 generalizes the notion of reducibility * that has been introduced in [7]. Namely, if one considers a trivial L-similarity space X, ≈ where ≈ is the crisp equality (i.e., the identity) then obviously x * ≈ y = x * y, where * is a reflexive and transitive closure of .…”

“…It is therefore appealing to consider and explore properties related to graded substitutability which may be important for the rewriting process, e.g., graded confluence. The initial study of graded substitutability and related notions is presented in [7].…”

“…Substitutability and similarity spaces In the present paper, we elaborate on the results from [7] and extend the utilized notions by equipping the universe of discourse by a similarity relation (i.e., a fuzzy relation which is reflexive, symmetric, and transitive) prescribing degrees to which elements are similar. Therefore, the definition of a degree to which "y is substitutable for x" that has been introduced in the earlier paper [7] can be extended so that the similarity is taken into account.…”

“…Therefore, the definition of a degree to which "y is substitutable for x" that has been introduced in the earlier paper [7] can be extended so that the similarity is taken into account. Namely, we can define a degree to which "there are z 1 and z 2 such that x is similar to z 1 and z 2 is substitutable for z 1 and z 2 is similar to y" as a more general degree of substitutability with respect to the underlying similarity.…”