BL-Functions and Free BL-Algebra

Fuzzy logics are designed to support logical inferences on vague or uncertain premises, and they are useful in several theoretical and applicative areas of computer science. A central paradigm in mathematical fuzzy logic, popularized by Hájek [Háj98], is based on the idea of weakening Boolean logic starting from a suitable generalization of Boolean conjunction, namely, a class of [0, 1]-valued binary functions known as (continuous) triangular norms. Any continuous triangular norm gives raise to a propositional logic, and Hájek’s Basic fuzzy logic (for short, Basic logic) is the intersecting common fragment of all these logics. Despite the intensive research efforts devoted to Basic logic in the last decade, this logic still resists to a complete understanding, as it appears from the lack of a satisfactory proof theory [MOG]. The algebraic counterpart of Basic logic is given by a very natural subvariety of residuated bounded lattices, namely commutative, divisible and prelinear residuated lattices, or BL-algebras. A representation result of Aglianó and Montagna [AM03] establishes that the variety generated by all the n-generated BL-algebras is singly generated by the BL-chain (n + 1)[0, 1], given by the ordinal sum of n + 1 copies of the generic MV-chain [0, 1]. As a consequence, validity problems in Basic logic turn out to have the same computational complexity of their Boolean counterparts [BHMV02, BM08]. This fact provides further motivation for the investigation of Basic logic in the computer science setting. The aforementioned result of Aglianó and Montagna is the starting point of this thesis. By universal algebra, it gives an implicit functional representation of the free n-generated BL-algebra: the free n-generated BL-algebra is isomorphic to the clone of n-ary term operations of (n + 1)[0, 1], with the basic operations defined pointwise. Hence, to provide an explicit functional representation of the free n-generated BL-algebra, it is sufficient to describe exactly the subset of n-ary functions over the domain of (n + 1)[0, 1] that contains all projections and is closed under the basic operations of (n + 1)[0, 1]: we call these functions, n-ary BLfunctions. By algebraic logic, the Lindenbaum-Tarski algebra of the nvariate fragment of Basic logic is isomorphic to the free BL-algebra over n generators, thus n-ary BL-functions coincide with the truthfunctions of the n-variate fragment of Basic logic. The main contribution of this thesis is the explicit representation of the free n-generated BL-algebra in terms of n-ary BL-functions. Our result accounts as the BL-algebraic counterpart of Mundici’s constructive version of the McNaughton theorem for MV-algebras [Mun94], and improves the previous knowledge on the subject, that was limited to the case of one generator settled by Montagna [Mon00] and Aguzzoli and Gerla [AG05].