Basic Logic, SMT solvers and finitely generated varieties of GBL-algebras

Basic Logic was introduced by Petr Hájek to provide a uni ed approach to fuzzy logics, and judging by its rapid adoption in the research community, it has enjoyed considerable success in this regard. One of the reasons is that while it is a very general logic, it has elegant semantics with respect to the real unit interval, which allow for practical applications and use of standard software tools. Here we show how to use these semantics to encode propositional Basic Logic into the Satis ability Modulo Theories (SMT) framework, based on an interpretation of Lukasiewicz logic, Gödel logic and product logic into SMT [4]. Ultimately these ideas go back to Mundici's result [13] that satis ablity for Lukasiewicz logic is NP-complete, and Hähnle's translation from Lukasiewicz logic to integer linear programming [8, 9]. In the current setting the translation to SMT is very simple, and since there are several e cient SMT-solvers available, this is probably one of the most e ective and exible ways of implementing a decision procedure for propositional basic logic. Basic logic algebras (or BL-algebras for short) are bounded integral commutative residuated lattices that satisfy divisibility and prelinearity. The latter property implies that subdirectly irreducible BL-algebras are linearly ordered. Generalized BL-algebras (or GBL-algebras) are just divisible residuated lattices, but still retain many of the properties of BL-algebras. For example they have distributive lattice reducts, the fusion operation distributes over the meet operation, and in the n-potent case they are integral and commutative [10]. The subdirectly irreducible GBL-algebras are no longer linearly ordered, but in the nite case they have a well-understood structure theory based on the so-called poset product construction [10, 11]. In [3] it is shown that the category of nite BL-algebras is dually equivalent to a category of nite rooted forests, labeled by positive integers. This result is extended here to nite GBL-algebras, with rooted forests replaced by posets. Since subdirectly irreducible GBL-algebras correspond to labeled rooted posets under this duality, it becomes feasible to calculate the HS-poset of subdirectly irreducible GBL-algebras up to a xed size, giving a view of the lattice of nitely generated GBL varieties. The duality also implies that GBL-algebras with n join-irreducibles are in one-one correspondence with preorders on n-elements, which simpli es the enumeration of nite GBL-algebras. The Kripke models of the modal logic S4 are determined by preorders, so the duality can be extended to the category of nite closure algebras (the algebraic models of S4). Since the larger category of all modal algebras has a well-developed duality theory using descriptive frames, these results may be useful for obtaining a duality for all n-potent commutative GBL-algebras.