Some properties of substitutions in the framework of similarity relations
Pascual Julián-Iranzo, Ginés Moreno, Jose Antonio Riaza · Fuzzy Sets and Systems · 2023
A substitution is a mapping from variables to terms. There is a consensus in the Logic Programming community that the work with substitutions is a source of errors due to their poor algebraic properties. This problem increases when we manipulate substitutions in the framework of similarity relations, where some classical concepts have more complex definitions and the intuition of the result of the operations involving them is easily lost. In this work we analyze some algebraic properties of substitutions in the framework of similarity relations. Specifically, we study the lattice structure of the set of idempotent substitutions with the definition of the weak parallel composition and weak parallel factorization operations, acting as the join and meet of the lattice. Also we relate these operations with the classical operation of composition of substitutions. The aim is to provide fuzzy logic programming researchers with reasoning tools with richer properties that facilitate formal proofs and, even, able to define compositional and parallel semantics.