We show that, for any abstract complexity measure in the sense of Blum and for any computable function f (or computable operator F), the class of problems that are f-speedable (or F-speedable) does not have effective measure 0. On the other hand, for sufficiently fast growing f (or F), the class of non-speedable computable problems does not have effective measure 0. These results answer some questions raised by Calude and Zimand. We also give a quantitative analysis of Borodin and Trakhtenbrot's Gap Theorem, which corrects a claim by Calude and Zimand.

We consider the problem of the robust stability of a class of uncertain T–S fuzzy neutral systems with time delay under time-varying parametric uncertainties using the Lyapunov–Krasovskii approach, where the parametric uncertainty is assumed to be norm bounded. By choosing a new Lyapunov–Krasovskii function, we are able to propose less conservative robust stability criteria in terms of linear matrix inequalities (LMIs) such that the uncertain T–S fuzzy neutral system under consideration remains asymptotically stable. The reduced conservatism of the proposed stability criterion compared with recently reported results is attributed to our using the Jensen inequality. The obtained results can also reduce the computational complexity.

We present a new data structure, called a Decomposition Tree (DT), for analysing Boolean functions, and demonstrate a variety of applications. In each node of the DT, appropriate bit-string decomposition fragments are combined by a logical operator. The DT has 2k nodes in the worst case, which implies exponential complexity for problems where the whole tree has to be considered. However, it is important to note that many problems are simpler. We show that these can be handled in an efficient way using the DT. Nevertheless, many problems are of exponential complexity and cannot be made any simpler: for example, the calculation of prime implicants. Using our general DT structure, we present a new worst case algorithm to compute all prime implicants. This algorithm has a lower time complexity than the well-known Quine–McCluskey algorithm and is the fastest corresponding worst case algorithm so far.

We present the no-iteration version of a mixed distributive law of a monad over a comonad in a general 2-category. We present the simplifications that occur in the case of the 2-category Cat.

We show that the following instance of the principle of excluded middle holds: any function on the one-point compactification of the natural numbers with values on the natural numbers is either classically continuous or classically discontinuous. The proof does not require choice and can be understood in any of the usual varieties of constructive mathematics. Classical (dis)continuity is a weakening of the notion of (dis)continuity, where the existential quantifiers are replaced by negated universal quantifiers. We also show that the classical continuity of all functions is equivalent to the negation of the weak limited principle of omniscience. We use this to relate uniform continuity and searchability of the Cantor space.

We provide motivation for and then study the synthesis of Petri nets. Synthesis can avoid the state exploration problem by guaranteeing correctness for the Petri net. We propose conditions to be imposed on a synthesis shared pb-type subnet for systems specified in Petri nets that ensure the preservation of the liveness and boundedness structural properties. Specifically, we propose a group of sufficient conditions, or both sufficient and necessary conditions, for liveness preservation and boundedness preservation. Possible applications of this synthesis method are illustrated through an example in the form of a flexible manufacturing system. These results are useful for studying the static and dynamic properties of Petri nets for analysing the properties of large complex systems.

We prove a compactness theorem in the context of Hennessy–Milner logic and use it to derive a sufficient condition on modal characterisations for the approximation induction principle to be sound modulo the corresponding process equivalence. We show that this condition is necessary when the equivalence in question is compositional with respect to the projection operators. Furthermore, we derive different upper bounds for the constructive version of the approximation induction principle with respect to simulation and decorated trace semantics.

We introduce a category of vector spaces modelling full propositional linear logic, similar to probabilistic coherence spaces and to Koethe sequences spaces. Its objects are rigged sequence spaces, Banach spaces of sequences, with norms defined from pairing with finite sequences, and morphisms are bounded linear maps, continuous in a suitable topology. The main interest of the work is that our model gives a realization of the free linear logic exponentials construction.

This article continues the study of the genus of regular languages that the authors introduced in a 2013 paper (published in 2018). In order to understand further the genus g(L) of a regular language L, we introduce the genus size of |L|gen to be the minimal size of all finite deterministic automata of genus g(L) computing L.We show that the minimal finite deterministic automaton of a regular language can be arbitrarily far away from a finite deterministic automaton realizing the minimal genus and computing the same language, in terms of both the difference of genera and the difference in size. In particular, we show that the genus size |L|gen can grow at least exponentially in size |L|. We conjecture, however, the genus of every regular language to be computable. This conjecture implies in particular that the planarity of a regular language is decidable, a question asked in 1976 by R. V. Book and A. K. Chandra. We prove here the conjecture for a fairly generic class of regular languages having no short cycles. The methods developed for the proof are used to produce new genus-based hierarchies of regular languages and in particular, we show a new family of regular languages on a two-letter alphabet having arbitrary high genus.

We investigate cut elimination in multi-focused sequent calculi and the impact on the cut elimination proof of design choices in such calculi. The particular design we advocate is illustrated by a multi-focused calculus for full linear logic using an explicitly polarised syntax and incremental focus handling, for which we provide a syntactic cut elimination procedure. We discuss the effect of cut elimination on the structure of proofs, leading to a conceptually simple proof exploiting the strong structure of multi-focused proofs.

We characterize Yoneda completeness for non-symmetric distances by combinations of metric and directed completeness. One of these generalizes the Kostanek–Waszkiewicz theorem on formal balls.

We consider a variation of the Travelling Salesman Problem (TSP) in which the cities visited have non-zero spatial extent, in contrast with the classical TSP, which has destinations that are mathematical points. This new approach opens up both new analyses of the problem and new algorithms for solutions, while remaining an economic first approximation to the standard problem. We present one particular solution that, depending on the number and size of the cities, can improve existing algorithms solving the classical TSP.

The categorical models of differential linear logic (LL) are additive categories and those of the differential lambda-calculus are left-additive categories because of the Leibniz rule which requires the summation of two expressions. This means that, as far as the differential lambda-calculus and differential LL are concerned, these models feature finite nondeterminism and indeed these languages are essentially non-deterministic. We introduce a categorical framework for differentiation which does not require additivity and is compatible with deterministic models such as coherence spaces and probabilistic models such as probabilistic coherence spaces.

We present a linearity theorem for a proof language of intuitionistic multiplicative additive linear logic, incorporating addition and scalar multiplication. The proofs in this language are linear in the algebraic sense. This work is part of a broader research program aiming to define a logic with a proof language that forms a quantum programming language.

We provide a constructive version of the notion of sheaf models of univalent type theory. We start by relativizing existing constructive models of univalent type theory to presheaves over a base category. Any Grothendieck topology of the base category then gives rise to a family of left-exact modalities, and we recover a model of type theory by localizing the presheaf model with respect to this family of left-exact modalities. We provide then some examples.

We consider anti-unification for simply typed lambda terms in theories defined by associativity, commutativity, identity (unit element) axioms and their combinations and develop a sound and complete algorithm which takes two lambda terms and computes their equational generalizations in the form of higher-order patterns. The problem is finitary: the minimal complete set of such generalizations contains finitely many elements. We define the notion of optimal solution and investigate special restrictions of the problem for which the optimal solution can be computed in linear or polynomial time.

We develop universal algebra over an enriched category and relate it to finitary enriched monads over . Using it, we deduce recent results about ordered universal algebra where inequations are used instead of equations. Then we apply it to metric universal algebra where quantitative equations are used instead of equations. This contributes to understanding of finitary monads on the category of metric spaces.