Symmetric Heyting relation algebras with applications to hypergraphs

A relation on a hypergraph is a binary relation on the set consisting of all the nodes and the edges, and which satisfies a constraint involving the incidence structure of the hypergraph. These relations correspond to join preserving mappings on the lattice of sub-hypergraphs. This paper introduces a generalization of a relation algebra in which the Boolean algebra part is replaced by a Heyting algebra that supports an order-reversing involution. A general construction for these symmetric Heyting relation algebras is given which includes as a special case the algebra of relations on a hypergraph. A particular feature of symmetric Heyting relation algebras is that instead of an involutory converse operation they possess both a left converse and a right converse which form an adjoint pair of operations. Properties of the converses are established and used to derive a generalization of the well-known connection between converse, complement, erosion and dilation in mathematical morphology. This provides part of the foundation necessary to develop mathematical morphology on hypergraphs based on relations on hypergraphs

Weakening Relation Algebras and FL\u3csup\u3e2\u3c/sup\u3e-algebras

FL2-algebras are lattice-ordered algebras with two sets of residuated operators. The classes RA of relation algebras and GBI of generalized bunched implication algebras are subvarieties of FL2-algebras. We prove that the congruences of FL2-algebras are determined by the congruence class of the respective identity elements, and we characterize the subsets that correspond to this congruence class. For involutive GBI-algebras the characterization simplifies to a form similar to relation algebras. For a positive idempotent element p in a relation algebra A, the double division conucleus image p/A/p is an (abstract) weakening relation algebra, and all representable weakening relation algebras (RWkRAs) are obtained in this way from representable relation algebras (RRAs). The class S(dRA) of subalgebras of {p/A/p∶ A ϵ RA; 1 ≤ p2 = p ϵ A} is a discriminator variety of cyclic involutive GBI-algebras that includes RA. We investigate S(dRA) to find additional identities that are valid in all RWkRAs. A representable weakening relation algebra is determined by a chain if and only if it satisfies 0 ≤ 1, and we prove that the identity 1 ≤ 0 holds only in trivial members of S(dRA).https://digitalcommons.chapman.edu/scs_books/1050/thumbnail.jp

Dualities and Dual Pairs in Heyting Algebras

We extract the abstract core of finite homomorphism dualities using the techniques of Heyting algebras and (combinatorial) categorie

On the Mereological Structure of Complex States of Affairs

The aim of this paper is to elucidate the mereological structure of complex states of affairs without relying on the problematic notion of structural universals. For this task tools from graph theory, lattice theory, and the theory of relational systems are employed. Our starting point is the mereology of similarity structures. Since similarity structures are structured sets, their mereology can be considered as a generalization of the mereology of sets.

Ordered Information Systems and Graph Granulation

The concept of an Information System, as used in Rough Set theory, is extended to the case of a partially ordered universe equipped with a set of order preserving attributes. These information systems give rise to partitions of the universe where the set of equivalence classes is partially ordered. Such ordered partitions correspond to relations on the universe which are reflexive and transitive. This correspondence allows the definition of approximation operators for an ordered information system by using the concepts of opening and closing from mathematical morphology. A special case of partial orders are graphs and hypergraphs and these provide motivation for the need to consider approximations on partial orders

Strong Completeness and the Finite Model Property for Bi-Intuitionistic Stable Tense Logics

Bi-Intuitionistic Stable Tense Logics (BIST Logics) are tense logics with a Kripke semantics where worlds in a frame are equipped with a pre-order as well as with an accessibility relation which is ‘stable’ with respect to this pre-order. BIST logics are extensions of a logic, BiSKt, which arose in the semantic context of hypergraphs, since a special case of the pre-order can represent the incidence structure of a hypergraph. In this paper we provide, for the first time, a Hilbert-style axiomatisation of BISKt and prove the strong completeness of BiSKt. We go on to prove strong completeness of a class of BIST logics obtained by extending BiSKt by formulas of a certain form. Moreover we show that the finite model property and the decidability hold for a class of BIST logics

A Bi-Intuitionistic Modal Logic: Foundations and Automation

The paper introduces a bi-intuitionistic modal logic, called BISKT, with two adjoint pairs of tense operators. The semantics of BISKT is defined using Kripke models in which the set of worlds carries a pre-order relation as well as an accessibility relation, and the two relations are linked by a stability condition. A special case of these models arises from graphs in which the worlds are interpreted as nodes and edges of graphs, and formulae represent subgraphs. The pre-order is the incidence structure of the graphs. We present a comprehensive study of the logic, giving decidability, complexity and correspondence results. We also show the logic has the effective finite model property. We present a sound, complete and terminating tableau calculus for the logic and use the MetTeL system to explore implementations of different versions of the calculus. An experimental evaluation gave good results for satisfiable problems using predecessor blocking

Graph Granularity through Bi-intuitionistic Modal Logic

This thesis concerns the use of a bi-intuitionistic modal logic, UBiSKt, in the field of Knowledge Representation and Reasoning. The logic is shown to be able to represent qualitative spatial relations between subgraphs at different levels of detail, or granularity. The level of detail is provided by the modal accessibility relation R defined on the set of nodes and edges. The connection between modal logic and mathematical morphology is exploited to study notions of granulation on subgraphs, namely the process of changing granularity, and to define qualitative spatial relations between these “granular” regions. In addition, a special case of graph and hypergraph granularity is analysed, namely when the accessibility relation gives rise to a partition of the underlying set of nodes and edges. Different S5 extensions of intuitionistic modal logic are considered and compared in the thesis. It is shown that these logics, and their associated semantics, provide different ways of partitioning a graph, a hypergraph, or, more generally, a partially ordered set