Some uniform categorical theoretical treatment of automata and lattice-valued fuzzy automata using quantale theory is studied in this paper. First, L-relational sheaves on a monoidM andQ-enriched categories are introduced for quantales L andQ, the equivalence of the corresponding categories are proved next. Then lattice-valued (fuzzy) automata are described by Q-enriched categories. In fact, la...

A principled approach to the design of program verification and construction tools is applied to separation logic. The control flow is modelled by power series with convolution as separating conjunction. A generic construction lifts resource monoids to assertion and predicate transformer quantales. The data flow is captured by concrete store/heap models. These are linked to the separation algeb...

We provide a new characterization of the notion of sheaf on an étale groupoid G, in terms of a particular kind of Hilbert module on the quantale O(G) of the groupoid. All the theory is developed in the context of the more general class of quantales known as stable quantal frames, of which examples are easy to construct because their category is algebraic. The homomorphisms of our Hilbert module...

Dynamic epistemic algebra establishes Galois connections and quantales as a basis for reasoning about knowledge in multi-agent systems. To date, these algebraic-axiomatic methods have been restricted to a positive fragment of dynamic epistemic logic with communication events only. This paper proposes Boolean algebraic extensions to overcome these limitations by generalizing dynamic epistemic al...

An algebraic approach to the design of program construction and verification tools is applied to separation logic. The control-flow level is modelled by power series with convolution as separating conjunction. A generic construction lifts resource monoids to assertion and predicate transformer quantales. The data domain is captured by concrete store-heap models. These are linked to the separati...

A notion of convolution is presented in the context of formal power series together with lifting constructions characterising algebras of such series, which usually are quantales. A number of examples underpin the universality of these constructions, the most prominent ones being separation logics, where convolution is separating conjunction in an assertion quantale; interval logics, where conv...

Variants of Kleene algebra support program construction and verification by algebraic reasoning. This entry provides a verification component for Hoare logic based on Kleene algebra with tests, verification components for weakest preconditions and strongest postconditions based on Kleene algebra with domain and a component for step-wise refinement based on refinement Kleene algebra with tests. ...

Equivalences, partitions and (bi)simulations are usually tackled using concrete relations. There are only few treatments by abstract relation algebra or category theory. We give an approach based on the theory of semirings and quantales. This allows applying the results directly to structures such as path and tree algebras which is not as straightforward in the other approaches mentioned. Also,...

We present an algebraic semantics for Duration Calculus based on semirings and quantales. Duration Calculus was originally introduced in 1991 as a powerful logic for specifying the safety of real-time systems. We embed the Duration Calculus into the theory of Boolean semirings and extend them to Kleene algebras and ω-algebras, respectively, to express finite and infinite iteration. This allows ...