Hitoshi Furusawa

An algebraic formalization of fuzzy relations

Abstract This paper provides an algebraic formalization of mathematical structures formed by fuzzy relations with sup-min composition. A simple proof of a representation theorem for Boolean relation algebras satisfying Tarski rule and point axiom has been given by Schmidt and Strohlein. Unlike Boolean relation algebras, fuzzy relation algebras are not Boolean but equipped with semi-scalar multiplication. First, we present a set of axioms for fuzzy relation algebras and improve the definition of point relations. Then by using relational calculus, a representation theorem for such relation algebras is deduced without Tarski rule.

Monodic tree kleene algebra

We propose a quasi-equational sound axiomatization of regular tree languages, called monodic tree Kleene algebra. The algebra is weaker than Kleene algebra introduced by Kozen. We find a subclass of regular tree languages, for which monodic tree Kleene algebra is complete. While regular tree expressions may have two or more kinds of place holders, the subclass can be equipped with only one kind of them. Along the lines of the original proof by Kozen, we prove the completeness theorem based on determinization and minimization of tree automata represented by matrices on monodic tree Kleene algebra.

Categorical representation theorems of fuzzy relations

Abstract This paper provides a notion of Zadeh categories as a categorical structure formed by fuzzy relations with sup-min composition, and proves two representation theorems for Dedekind categories (relation categories) with a unit object analogous to one-point set, and for Zadeh categories without unit objects.

Dedekind categories with cutoff operators

Dedekind categories provide a suitable categorical framework for lattice-valued binary relations. It is known that the notion of crispness cannot be described by the basic tools of this theory only. In this paper we will study Dedekind categories with a cutoff operator in order to circumvent this shortage. We will introduce and investigate the properties of the operator and its relationship with other tools previously used for the same purpose. The main result of this paper is a representation theorem for Dedekind categories with a cutoff operator satisfying the point axiom.

The Cube of Kleene Algebras and the Triangular Prism of Multirelations

We refine and extend the known results that the set of ordinary binary relations forms a Kleene algebra, the set of up-closed multirelations forms a lazy Kleene algebra, the set of up-closed finite multirelations forms a monodic tree Kleene algebra, and the set of total up-closed finite multirelations forms a probabilistic Kleene algebra. For the refinement, we introduce a notion of type of multirelations. For each of eight classes of relaxation of Kleene algebra, we give a sufficient condition on type T so that the set of up-closed multirelations of T belongs to the class. Some of the conditions are not only sufficient, but also necessary.

A non-probabilistic relational model of probabilistic Kleene algebras

This paper studies basic properties of up-closed multirelations, and then shows that the set of finitary total up-closed multirelations over a set forms a probabilistic Kleene algebra. In Kleene algebras, the star operator is very essential. We investigate the reflexive transitive closure of a finitary up-closed multirelation and show that the closure operator plays a role of the star operator of a probabilistic Kleene algebra consisting of the set of finitary total up-closed multirelations as in the case of a Kozens Kleene algebra consisting of the set of (usual) binary relations.

Concurrent Dynamic Algebra

We reconstruct Peleg’s concurrent dynamic logic in the context of modal Kleene algebras. We explore the algebraic structure of its multirelational semantics and develop an axiomatization of concurrent dynamic algebras from that basis. In this context, sequential composition is not associative. It interacts with parallel composition through a weak distributivity law. The modal operators of concurrent dynamic algebra are obtained from abstract axioms for domain and antidomain operators; the Kleene star is modelled as a least fixpoint. Algebraic variants of Peleg’s axioms are shown to be derivable in these algebras, and their soundness is proved relative to the multirelational model. Additional results include iteration principles for the Kleene star and a refutation of variants of Segerberg’s axiom in the multirelational setting. The most important results have been verified formally with Isabelle/HOL.

Continuous relations and richardson's theorem

The paper intends to seek a definition of continuous relations with relational methods and gives another proof of Richardsons theorem on nondeterministic cellular automata.

Taming Multirelations

Binary multirelations generalise binary relations by associating elements of a set to its subsets. We study the structure and algebra of multirelations under the operations of union, intersection, sequential, and parallel composition, as well as finite and infinite iteration. Starting from a set-theoretic investigation, we propose axiom systems for multirelations in contexts ranging from bi-monoids to bi-quantales.

Point axioms in dedekind categories

A Dedekind category is a convenient algebraic framework to treat relations. Concepts of points and some axioms such as the point axiom, the axiom of totality, the axiom of subobject, the axiom of complement, and the relational axiom of choice are introduced in Dedekind categories to connect functional ideas to set-theoretical intuition. This paper summarises interrelations of these axioms.