Igor Sedlár (ICS – CAS; CZ)

**Title: Kleene Algebras for Weighted Programs**

Abstract: Weighted programs are a recent generalization of probabilistic programs which can also be used to represent optimization problems and, in general, programs whose execution traces carry some sort of weight. In this talk, I will discuss semantics for weighted programs, and a generalization of Kleene algebras with tests abstracting this semantics. In particular, I define a language model based on weighted sets of guarded strings, and a relational model based on weighted relations on a state space. Both kinds of semantics are special cases of a more general functional semantics based on functions from multimonoids to quantales. The proposed generalization of Kleene algebras with tests adds a third sort to programs and Boolean statements, corresponding to the algebra of weights. Several open problems will be discussed, including questions of completeness, complexity, and relation to weighted automata.

Amanda Vidal (IIIA – CSIC; ES)

**Title: Standard and general completeness of modal many-valued logics**

Abstract: In this talk we will introduce and explore the similarities and differences between the modal logics evaluated over standard algebras and those over arbitrary algebras of the corresponding varieties, for some well-known fuzzy logics. In particular, we will present results affecting both the local and the global logical entailments of the modal Gödel, Łukasiewicz and Product logics, understood both as the logics arising from their respective standard algebra (namely, over [0,1]) and from their generated varieties. These coincide at the propositional level, but we will see (might) have different behaviors when we move to their modal extensions. The semantics of the above modal many-valued logics is based on classical frames (i.e., where the accessibility relation is as in the classical modal logic). Henceforth, their adaptation to a multi-modal case with the usual axioms from dynamic logic would allow for the modeling of problems analogous to those of dynamic logic but offering the possibility of working with formulas valued on the corresponding many-valued algebras.