Motivated by the need to reason about hybrid systems, we study limits in categories of coalgebras whose underlying functor is a Vietoris polynomial one – intuitively, the topological analogue of a Kripke polynomial functor. Among other results, we prove that every Vietoris polynomial functor admits a final coalgebra if it respects certain conditions concerning separation axioms and compactness. When the functor is restricted to some of the categories induced by these conditions, the resulting categories of coalgebras are even complete.

As a practical application, we use these developments in the specification and analysis of non-deterministic hybrid systems, in particular to obtain suitable notions of stability and behaviour.

It is an open problem whether weak bisimilarity is decidable for Basic Process Algebra (BPA) and Basic Parallel Processes (BPP). A PSPACE lower bound for BPA and NP lower bound for BPP were demonstrated by Stribrna. Mayr recently achieved a result, saying that weak bisimilarity for BPP is $\Pi_2^P$-hard. We improve this lower bound to PSPACE, and, moreover, prove this result for the restricted class of normed BPP. It is also not known whether weak regularity (finiteness) of BPA and BPP is decidable. In the case of BPP there is a $\Pi_2^P$-hardness result by Mayr, which we improve to PSPACE. No lower bound has previously been established for BPA. We demonstrate DP-hardness, which, in particular, implies both NP and co-NP-hardness. In each of the bisimulation/regularity problems we also consider the classes of normed processes. Finally, we show how the technique for proving co-NP lower bound for weak bisimilarity of BPA can be applied to strong bisimilarity of BPP.

This paper focuses on the use of linear logic as a specification language for the operational semantics of advanced concepts of programming such as concurrency and object-orientation. Our approach is based on a refinement of linear logic sequent calculi based on the proof-theoretic characterization of logic programming. A well-founded combination of higher-order logic programming and linear logic will be used to give an accurate encoding of the traditional features of concurrent object-oriented programming languages, whose corner-stone is the notion of encapsulation.

Frequently, mathematical structures of a certain type and their morphisms fail to form a category for lack of composability of the morphisms; one example of this problem is the class of probabilistic automata when equipped with morphisms that allow restriction as well as relabelling. The proper mathematical framework for this situation is provided by a generalisation of category theory in the shape of the so-called precategories, which are introduced and studied in this paper. In particular, notions of adjointness, weak adjointness and partial adjointness for precategories are presented and justified in detail. This makes it possible to use universal properties as characterisations of well-known basic constructions in the theory of (generative) probabilistic automata: we show that accessible automata and decision trees, respectively, form coreflective subprecategories of the precategory of probabilistic automata. Moreover, the aggregation of two automata is identified as a partial product, whereas restriction and interconnection of automata are recognised as Cartesian lifts.

It is well known that classical varieties of $\Sigma$ -algebras correspond bijectively to finitary monads on $\mathsf{Set}$ . We present an analogous result for varieties of ordered $\Sigma$ -algebras, that is, categories of algebras presented by inequations between $\Sigma$ -terms. We prove that they correspond bijectively to strongly finitary monads on $\mathsf{Pos}$ . That is, those finitary monads which preserve reflexive coinserters. We deduce that strongly finitary monads have a coinserter presentation, analogous to the coequalizer presentation of finitary monads due to Kelly and Power. We also show that these monads are liftings of finitary monads on $\mathsf{Set}$ . Finally, varieties presented by equations are proved to correspond to extensions of finitary monads on $\mathsf{Set}$ to strongly finitary monads on $\mathsf{Pos}$ .

This paper studies normalisation by evaluation for typed lambda calculus from a categorical and algebraic viewpoint. The first part of the paper analyses the lambda definability result of Jung and Tiuryn via Kripke logical relations and shows how it can be adapted to unify definability and normalisation, yielding an extensional normalisation result. In the second part of the paper, the analysis is refined further by considering intensional Kripke relations (in the form of Artin–Wraith glueing) and shown to provide a function for normalising terms, casting normalisation by evaluation in the context of categorical glueing. The technical development includes an algebraic treatment of the syntax and semantics of the typed lambda calculus that allows the definition of the normalisation function to be given within a simply typed metatheory. A normalisation-by-evaluation program in a dependently typed functional programming language is synthesised.

We propose the new concept of Krivine ordered combinatory algebra ($\mathcal{^KOCA}$) as foundation for the categorical study of Krivine's classical realizability, as initiated by Streicher (2013).

We show that $\mathcal{^KOCA}$'s are equivalent to Streicher's abstract Krivine structures for the purpose of modeling higher-order logic, in the precise sense that they give rise to the same class of triposes. The difference between the two representations is that the elements of a $\mathcal{^KOCA}$ play both the role of truth values and realizers, whereas truth values are sets of realizers in $\mathcal{AKS}$s.

To conclude, we give a direct presentation of the realizability interpretation of a higher order language in a $\mathcal{^KOCA}$, which showcases the dual role that is played by the elements of the $\mathcal{^KOCA}$.

This paper presents a simple and powerful diagrammatic framework for dealing with specifications in computer science. Following a classical line, we define diagrammatic specifications as a kind of generalised sketch. In addition, the specifications themselves are defined as the realisations of projective sketches. This meta level provides adjunction properties: this is due to a well-known result of Ehresmann. Moreover, we prove in this paper that this meta level also provides an efficient definition of deduction. This work results from a collaboration with Christian Lair.

In recent decades, qualitative approaches to probabilistic uncertainty have received more and more attention. We propose a characterisation of partial preference orders through a uniform axiomatic treatment of a variety of qualitative uncertainty notions. To this end, we prove a representation result that connects qualitative notions of partial uncertainty to their numerical counterparts. We describe an executable specification, in the declarative framework of Answer Set Programming, that constitutes the core engine for qualitative management of uncertainty. Some basic reasoning tasks are also identified.

This paper reviews recent attempts to describe the two- and three-qubit Hilbert space geometries. In the first part, this is done with the help of Hopf fibrations of hyperspheres. It is shown that the associated Hopf map is strongly sensitive to states’ entanglement content. In the two-qubit case, a generalisation of the celebrated one-qubit Bloch sphere representation is described. In the second part, we present Hilbert space discrete versions, which are comparable to polyhedral approximations of spheres in standard geometry.

A semigroup-based setting for developing Hoare logics and refinement calculi is introduced together with procedures for translating between verification and refinement proofs. A new Hoare logic for multirelations and two minimalist generic verification and refinement components, implemented in an interactive theorem prover, are presented as applications that benefit from this generalisation.

Interactive theorem provers based on dependent type theory have the flexibility to support both constructive and classical reasoning. Constructive reasoning is supported natively by dependent type theory, and classical reasoning is typically supported by adding additional non-constructive axioms. However, there is another perspective that views constructive logic as an extension of classical logic. This paper will illustrate how classical reasoning can be supported in a practical manner inside dependent type theory without additional axioms. We will show several examples of how classical results can be applied to constructive mathematics. Finally, we will show how to extend this perspective from logic to mathematics by representing classical function spaces using a weak value monad.

We propose an adaptive genetic algorithm (AGA) for the multi-objective optimisation design of a fuzzy PID controller and apply it to the control of an active magnetic bearing (AMB) system. Unlike PID controllers with fixed gains, a fuzzy PID controller is expressed in terms of fuzzy rules whose consequences employ analytical PID expressions. The PID gains are adaptive and the fuzzy PID controller has more flexibility and capability than conventional ones. Moreover, it can be easily used to develop a precise and fast control algorithm in an optimal design. An adaptive genetic algorithm is proposed to design the fuzzy PID controller. The centres of the triangular membership functions and the PID gains for all fuzzy control rules are selected as parameters to be determined. We also present a dynamic model of an AMB system for axial motion. The simulation results of this AMB system show that a fuzzy PID controller designed using the proposed AGA has good performance.

In this paper, we describe a denotational model of Intuitionist Linear Logic which is also a differential category. Formulas are interpreted as Mackey-complete topological vector space and linear proofs are interpreted as bounded linear functions. So as to interpret non-linear proofs of Linear Logic, we use a notion of power series between Mackey-complete spaces, generalizing entire functions in $\mathbb{C}$ . Finally, we get a quantitative model of Intuitionist Differential Linear Logic, with usual syntactic differentiation and where interpretations of proofs decompose as a Taylor expansion.

Quasi-interpretations are a technique for guaranteeing complexity bounds on first-order functional programs: in particular, with termination orderings, they give a sufficient condition for a program to be executable in polynomial time (Marion and Moyen 2000), which we call the P-criterion here. We study properties of the programs satisfying the P-criterion in order to improve the understanding of its intensional expressive power. Given a program, its blind abstraction is the non-deterministic program obtained by replacing all constructors with the same arity by a single one. A program is blindly polytime if its blind abstraction terminates in polynomial time. We show that all programs satisfying a variant of the P-criterion are in fact blindly polytime. Then we give two extensions of the P-criterion: one relaxing the termination ordering condition and the other (the bounded-value property) giving a necessary and sufficient condition for a program to be polynomial time executable, with memoisation.

Categorical compositional distributional semantics is a model of natural language; it combines the statistical vector space models of words with the compositional models of grammar. We formalise in this model the generalised quantifier theory of natural language, due to Barwise and Cooper. The underlying setting is a compact closed category with bialgebras. We start from a generative grammar formalisation and develop an abstract categorical compositional semantics for it, and then instantiate the abstract setting to sets and relations and to finite-dimensional vector spaces and linear maps. We prove the equivalence of the relational instantiation to the truth theoretic semantics of generalised quantifiers. The vector space instantiation formalises the statistical usages of words and enables us to, for the first time, reason about quantified phrases and sentences compositionally in distributional semantics.

A special final coalgebra theorem, in the style of Aczel (1988), is proved within standard Zermelo–Fraenkel set theory. Aczel's Anti-Foundation Axiom is replaced by a variant definition of function that admits non-well-founded constructions. Variant ordered pairs and tuples, of possibly infinite length, are special cases of variant functions. Analogues of Aczel's solution and substitution lemmas are proved in the style of Rutten and Turi (1993). The approach is less general than Aczel's, but the treatment of non-well-founded objects is simple and concrete. The final coalgebra of a functor is its greatest fixedpoint.

Compared with previous work (Paulson, 1995a), iterated substitutions and solutions are considered, as well as final coalgebras defined with respect to parameters. The disjoint sum construction is replaced by a smoother treatment of urelements that simplifies many of the derivations.

The theory facilitates machine implementation of recursive definitions by letting both inductive and coinductive definitions be represented as fixed points. It has already been applied to the theorem prover Isabelle (Paulson, 1994).

We study two well-known linear-time metrics on Markov chains (MCs), namely, the strong and strutter trace distances. Our interest in these metrics is motivated by their relation to the probabilistic linear temporal logic (LTL)-model checking problem: we prove that they correspond to the maximal differences in the probability of satisfying the same LTL and LTL−X (LTL without next operator) formulas, respectively.

The threshold problem for these distances (whether their value exceeds a given threshold) is NP-hard and not known to be decidable. Nevertheless, we provide an approximation schema where each lower and upper approximant is computable in polynomial time in the size of the MC.

The upper approximants are bisimilarity-like pseudometrics (hence, branching-time distances) that converge point-wise to the linear-time metrics. This convergence is interesting in itself, because it reveals a non-trivial relation between branching and linear-time metric-based semantics that does not hold in equivalence-based semantics.

Computer Science has witnessed the emergence of a plethora of different logics, models and paradigms for the description of computation. Yet, the classic Church–Turing thesis may be seen as indicating that all general models of computation are equivalent. Alan Perlis referred to this as the ‘Turing tarpit’, and argued that some of the most crucial distinctions in computing methodology, such as sequential versus parallel, deterministic versus non-deterministic, local versus distributed disappear if all one sees in computation is pure symbol pushing. How can we express formally the difference between these models of computation?

The theory ACUI of an associative, commutative, and idempotent binary function symbol + with unit 0 was one of the first equational theories for which the complexity of testing solvability of unification problems was investigated in detail. In this paper, we investigate two extensions of ACUI. On one hand, we consider approximate ACUI-unification, where we use appropriate measures to express how close a substitution is to being a unifier. On the other hand, we extend ACUI-unification to ACUIG-unification, that is, unification in equational theories that are obtained from ACUI by adding a finite set G of ground identities. Finally, we combine the two extensions, that is, consider approximate ACUI-unification. For all cases we are able to determine the exact worst-case complexity of the unification problem.