Formal Properties of Needed Narrowing with Similarity Relations

• Authors: Ginés Moreno;Vicente Pascual
• Venue: Electronic Notes in Theoretical Computer Science (ENTCS)
• Year: 2007

The integration of declarative paradigms such as functional-logic and fuzzy-logic programming seems to be an interesting challenge in the design of highly expressive declarative languages where mathematical functions cohabit with fuzzy logic features. Starting with two representative languages from both settings, namely Curry and Likelog, we have recently proposed an hybrid dialect where a set of (Curry) rewrite rules together with a set of (Likelog) similarity equations can be safely executed by means of a fuzzy variant of needed narrowing. This paper is devoted to show some important properties enjoyed by the new narrowing strategy. Firstly, we prove the termination of the process which determines the set of tuples that enable new narrowing steps for a given term. Termination in the fuzzy context is not trivial since, apart that it is required to compute the transitive closure of the initial set of similarity equations, it is also mandatory to ensure the finiteness of the set of recursive calls performed by the narrowing strategy, as well as that each call never falls into an infinite loop. From here, we prove two important (somehow complementary) properties, that we call crispness and fuzziness, respectively. The first one implies that the new strategy is (at least) conservative with respect to the original one of needed narrowing, in the sense that each tuple obtained in the crisp case, is also replicated (with the maximum truth degree) in the fuzzy case. On the other hand, fuzziness means the maximality of the new strategy when exploiting (as much as possible) the similarity relations collected in a given program.