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

Zero-one laws with respect to models of provability logic and two Grzegorczyk logics

It has been shown in the late 1960s that each formula of first-order logic without constants and function symbols obeys a zero-one law: As the number of elements of finite models increases, every formula holds either in almost all or in almost no models of that size. Therefore, many properties of models, such as having an even number of elements, cannot be expressed in the language of first-order logic. Halpern and Kapron proved zero-one laws for classes of models corresponding to the modal logics K, T, S4, and S5 and for frames corresponding to S4 and S5. In this paper, we prove zero-one laws for provability logic and its two siblings Grzegorczyk logic and weak Grzegorczyk logic, with respect to model validity. Moreover, we axiomatize validity in almost all relevant finite models, leading to three different axiom systems

Conjuntos construibles en modelos valuados en retículos

We investigate different set-theoretic constructions in Residuated Logic based on Fitting’s work on Intuitionistic Kripke models of Set Theory. Firstly, we consider constructable sets within valued models of Set Theory. We present two distinct constructions of the constructable universe: L B and L B , and prove that the they are isomorphic to V (von Neumann universe) and L (Gödel’s constructible universe), respectively. Secondly, we generalize Fitting’s work on Intuitionistic Kripke models of Set Theory using Ono and Komori’s Residuated Kripke models. Based on these models, we provide a general- ization of the von Neumann hierarchy in the context of Modal Residuated Logic and prove a translation of formulas between it and a suited Heyting valued model. We also propose a notion of universe of constructable sets in Modal Residuated Logic and discuss some aspects of it.Investigamos diferentes construcciones de la teoría de conjuntos en Lógica Residual basados en el trabajo de Fitting sobre los modelos intuicionistas de Kripke de la Teoría de Conjuntos. En primer lugar, consideramos conjuntos construibles dentro de modelos valuados de la Teoría de Conjuntos. Presentamos dos construcciones distintas del universo construible: L B y L B , y demostramos que son isomorfos a V (universo von Neumann) y L (universo construible de Gödel), respectivamente. En segundo lugar, generalizamos el trabajo de Fitting sobre los modelos intuicionistas de Kripke de la teoría de conjuntos utilizando los modelos residuados de Kripke de Ono y Komori. Con base en estos modelos, proporcionamos una generalización de la jerarquía de von Neumann en el contexto de la Lógica Modal Residuada y demostramos una traducción de fórmulas entre ella y un modelo Heyting valuado adecuado. También proponemos una noción de universo de conjuntos construibles en Lógica Modal Residuada y discutimos algunos aspectos de la misma. (Texto tomado de la fuente)MaestríaMagíster en Ciencias - MatemáticasLógica matemática, teoría de conjunto

Sequences of refinements of rough sets: logical and algebraic aspects

In this thesis, a generalization of the classical Rough set theory is developed considering the so-called sequences of orthopairs that we define as special sequences of rough sets. Mainly, our aim is to introduce some operations between sequences of orthopairs, and to discover how to generate them starting from the operations concerning standard rough sets. Also, we prove several representation theorems representing the class of finite centered Kleene algebras with the interpolation property, and some classes of finite residuated lattices (more precisely, we consider Nelson algebras, Nelson lattices, IUML-algebras and Kleene lattice with implication) as sequences of orthopairs. Moreover, as an application, we show that a sequence of orthopairs can be used to represent an examiner's opinion on a number of candidates applying for a job, and we show that opinions of two or more examiners can be combined using operations between sequences of orthopairs in order to get a final decision on each candidate. Finally, we provide the original modal logic SOn with semantics based on sequences of orthopairs, and we employ it to describe the knowledge of an agent that increases over time, as new information is provided. Modal logic Son is characterized by the sequences (\u25a11,\u2026, \u25a1n) and (O1,\u2026, On) of n modal operators corresponding to a sequence (t1,\u2026, tn) of consecutive times. Furthermore, the operator \u25a1i of (\u25a11,\u2026, \u25a1n) represents the knowledge of an agent at time ti, and it coincides with the necessity modal operator of S5 logic. On the other hand, the main innovative aspect of modal logic SOn is the presence of the sequence (O1,\u2026, On), since Oi establishes whether an agent is interested in knowing a given fact at time ti

Sequences of refinements of rough sets: logical and algebraic aspects

In this thesis, a generalization of the classical Rough set theory is developed considering the so-called sequences of orthopairs that we define as special sequences of rough sets. Mainly, our aim is to introduce some operations between sequences of orthopairs, and to discover how to generate them starting from the operations concerning standard rough sets. Also, we prove several representation theorems representing the class of finite centered Kleene algebras with the interpolation property, and some classes of finite residuated lattices (more precisely, we consider Nelson algebras, Nelson lattices, IUML-algebras and Kleene lattice with implication) as sequences of orthopairs. Moreover, as an application, we show that a sequence of orthopairs can be used to represent an examiner's opinion on a number of candidates applying for a job, and we show that opinions of two or more examiners can be combined using operations between sequences of orthopairs in order to get a final decision on each candidate. Finally, we provide the original modal logic SOn with semantics based on sequences of orthopairs, and we employ it to describe the knowledge of an agent that increases over time, as new information is provided. Modal logic Son is characterized by the sequences (□1,…, □n) and (O1,…, On) of n modal operators corresponding to a sequence (t1,…, tn) of consecutive times. Furthermore, the operator □i of (□1,…, □n) represents the knowledge of an agent at time ti, and it coincides with the necessity modal operator of S5 logic. On the other hand, the main innovative aspect of modal logic SOn is the presence of the sequence (O1,…, On), since Oi establishes whether an agent is interested in knowing a given fact at time ti