Constructive Fuzzy Logics

We generalise Kripke’s semantics for Intuitionistic logic to Hajek’s BL and consider the constructive subsystems of GBLewf and Intuitionistic Affine logic or ALi. The genesis of our semantics is the Poset Product construction for GBL-algebras elucidated in a series of papers by Peter Jipsen, Simone Bova, and Franco Montagna. We present natural deduction systems for all of these systems and corresponding deduction theorems for these same. We present the algebraic semantics for each of the logics under consideration, demonstrate their soundness and completeness with respect to these algebraic semantics. We also show how the classical Kripke semantics for Intuitionistic logic can be recast in terms of Poset Products. We then proceed to the main results, showing how a very natural generalisation of the Kripke semantics holds for each of GBLewf , ALi and Hajek’s BL based on the embedding results of Jipsen and Montagna and the decidability results of Bova and Montagna. We demonstrate soundness and completeness of the logics under our semantics in each case, with the exception of ALi, whose robust completeness with respect to the intended models (relational models with frames valued in involutive pocrims) we leave as an open problem for the ambitious reader

Dual Logic Concepts based on Mathematical Morphology in Stratified Institutions: Applications to Spatial Reasoning

Several logical operators are defined as dual pairs, in different types of logics. Such dual pairs of operators also occur in other algebraic theories, such as mathematical morphology. Based on this observation, this paper proposes to define, at the abstract level of institutions, a pair of abstract dual and logical operators as morphological erosion and dilation. Standard quantifiers and modalities are then derived from these two abstract logical operators. These operators are studied both on sets of states and sets of models. To cope with the lack of explicit set of states in institutions, the proposed abstract logical dual operators are defined in an extension of institutions, the stratified institutions, which take into account the notion of open sentences, the satisfaction of which is parametrized by sets of states. A hint on the potential interest of the proposed framework for spatial reasoning is also provided.Comment: 36 page

Distributive Residuated Frames and Generalized Bunched Implication Algebras

We show that all extensions of the (non-associative) Gentzen system for distributive full Lambek calculus by simple structural rules have the cut elimination property. Also, extensions by such rules that do not increase complexity have the finite model property, hence many subvarieties of the variety of distributive residuated lattices have decidable equational theories. For some other extensions, we prove the finite embeddability property, which implies the decidability of the universal theory, and we show that our results also apply to generalized bunched implication algebras. Our analysis is conducted in the general setting of residuated frames

Bisimulations for Kripke models of Fuzzy Multimodal Logics

The main objective of the dissertation is to provide a detailed study of several different types of simulations and bisimulations for Kripke models of fuzzy multimodal logics. Two types of simulations (forward and backward) and five types of bisimulations (forward, backward, forward-backward, backward-forward and regular) are presented hereby. For each type of simulation and bisimulation, an algorithm is created to test the existence of the simulation or bisimulation and, if it exists, the algorithm computes the greatest one. The dissertation presents the application of bisimulations in the state reduction of fuzzy Kripke models, while preserving their semantic properties. Next, weak simulations and bisimulations were considered and the Hennessy-Milner property was examined. Finally, an algorithm was created to compute weak simulations and bisimulations for fuzzy Kripke models over locally finite algebras