Koki Nishizawa

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.

Relational and multirelational representation theorems for complete idempotent left semirings

Brown and Gurr have shown a relational representation theorem for quantales. Complete idempotent left semirings are a relaxation of quantales by giving up strictness and distributivity of composition over arbitrary joins from the left. We show a relational representation theorem for them. Multirelations are generalisation of relations. We also show a multirelational representation theorem for complete idempotent left semirings.

A Coalgebraic Representation of Reduction by Cone of Influence

Abstract The Cone of Influence Reduction is a fundamental abstraction technique for reducing the size of models used in symbolic model checking. We develop coalgebraic representations of systems as composites of state transition maps and connectors. These representations include synchronous systems, asynchronous systems, asynchronous systems with synchronization by channels, and those with shared variables, probabilistic synchronous systems and so on. We schematically show the cone of influence reduction using these coalgebraic representations, which give a unified framework for providing the technique for various kinds of systems.

Relational representation theorem for powerset quantales

The paper gives a sufficient condition for a quantale to be isomorphic to a sub-quantale of the quantale whose elements are binary relations on a set and whose order and monoid structure are respectively given by inclusion and relational composition and the identity relation. A quantale has such a relational representation, if its underlying lattice is a powerset of some set. We also show some other equivalent conditions of the sufficient condition.

Composition of Different-Type Relations via the Kleisli Category for the Continuation Monad

We give the way of composing different types of relational notions under certain condition, for example, ordinary binary relations, up-closed multirelations, ordinary (possibly non-up-closed) multirelations, quantale-valued relations, and probabilistic relations. Our key idea is to represent a relational notion as a generalized predicate transformer based on some truth value in some category and to represent it as a Kleisli arrow for some continuation monad. The way of composing those relational notions is given via identity-on-object faithful functors between different Kleisli categories. We give a necessary and sufficient condition to have such identity-on-object faithful functor.

Multirelational representation theorems for complete idempotent left semirings

Abstract Complete idempotent left semirings are a relaxation of quantales by giving up strictness and distributivity of composition over arbitrary joins from the left. It is known that the set of up-closed multirelations over a set forms a complete idempotent left semiring together with union, multirelational composition, the empty multirelation, and the membership relation. This paper provides a sufficient condition for a complete idempotent left semiring to be isomorphic to a complete idempotent left semiring consisting of up-closed multirelations, in which all joins, the least element, multiplication, and the unit element are respectively given by unions, empty multirelations, the multirelational composition, and the membership relation. Some equivalent conditions of the sufficient condition are also provided.

A Sufficient Condition for Liftable Adjunctions between Eilenberg-Moore Categories

This paper gives a sufficient condition for monads P, P′ and T to have an adjunction between the category of P-algebras over T-algebras and the category of P′-algebras over T-algebras. The leading example is an adjunction between the category of idempotent semirings and the category of quantales, where P is the finite powerset monad, P′ is the powerset monad, and T is the free monoid monad. The left adjoint of this leading example is given by ideal completion. Applying our result, we show that ideal completion also gives an adjunction between the category of join semilattices over T-algebras and the category of complete join semilattices over T-algebras for a general monad T satisfying certain distributive law.

Implementation of high-definition lecture recording system for daily use

The recording of lecture videos and disseminating them is becoming popular. However, recording and post-processing daily lectures still demands much human effort and financial resources. Our goal is to develop a cost-effective and labor-saving video processing system that can record lectures in high-definition and process them for additional analysis and publishing. Our main idea is that high-definition recording itself is labor-saving. A stationary high-definition camcorder can record a lecturer and an entire classroom clearly so that notes can be read on a whiteboard and projector screen. Therefore, no camera operator is necessary during recording. We succeeded in implementing a cost-effective system without expensive hardware components for capturing a high-definition video signal. In this contribution, we describe the design and operational results of our system, which has recorded about 20 lectures every school week since 2010.

Multi-valued Modal Fixed Point Logics for Model Checking

In this paper, I will show how multi-valued logics are used for model checking. Model checking is an automatic technique to analyze correctness of hardware and software systems. A model checker is based on a temporal logic or a modal fixed point logic. That is to say, a system to be checked is formalized as a Kripke model, a property to be satisfied by the system is formalized as a temporal formula or a modal formula, and the model checker checks that the Kripke model satisfies the formula. Although most existing model checkers are based on 2-valued logics, recently new attempts have been made to extend the underlying logics of model checkers to multi-valued logics. I will show these new results.