We consider computation with real numbers that arise through a process of physical measurement. We have developed a theory in which physical experiments that measure quantities can be used as oracles to algorithms and we have begun to classify the computational power of various forms of experiment using non-uniform complexity classes. Earlier, in Beggs et al. (2014 Reviews of Symbolic Logic7(4) 618–646), we observed that measurement can be viewed as a process of comparing a rational number z – a test quantity – with a real number y – an unknown quantity; each oracle call performs such a comparison. Experiments can then be classified into three categories, that correspond with being able to return test results

$$\begin{eqnarray*} z < y\text{ or }z > y\text{ or }\textit{timeout},\\ z < y\text{ or }\textit{timeout},\\ z \neq y\text{ or }\textit{timeout}. \end{eqnarray*} $$

Web of Linked Data introduces common format and principles for publishing and linking data on the Web. Such a network of linked data is publicly available and easily consumable. This paper introduces a calculus for modelling networks of linked data with encoded privacy preferences.

In that calculus, a network is a parallel composition of users, where each user is named and consists of data, representing the user's profile, and a process. Data is a parallel composition of triples with names (resources) as components. Associated with each name and each triple of names are their privacy protection policies, that are represented by queries. A data triple is accessible to a user if the user's data satisfies the query assigned to that triple.

The main contribution of this model lies in the type system which together with the introduced query order ensures that static type-checking prevents privacy violations. We say that a network is well behaved if

• — access to a triple is more restrictive than access to its components and less restrictive than access to the user name it is enclosed with,

• — each user can completely access their own profile,

• — each user can update or partly delete profiles that they own (can access the whole profiles), and

• — each user can update the privacy preference policy of data of another profile that they own or write data to another profile only if the newly obtained profile stays fully accessible to their owner.

We prove that any well-typed network is well behaved.

In this paper, we introduce the notion of consistent F-augmented contexts by adding a special family of finite subsets into the structure of a formal context, which essentially establishes the basis of the representation of general algebraic domains. In particular, we investigate the association rule systems which are derived from the consistent F-augmented contexts and propose the notion of formal association rule systems. By the notion of antecedent connections, we obtain the equivalence between the category of formal association rule systems and that of algebraic domains, which demonstrates that the proposed notion of formal association rule systems provides a concrete approach to representing algebraic domains.

Even though the multiplicative–additive fragment of linear logic forbids structural rules in general, is does admit a bounded form of exponential modalities enjoying a bounded form of structural rules. The approximation theorem, originally proved by Girard, states that if full linear logic proves a propositional formula, then the multiplicative–additive fragment proves every bounded approximation of it. This may be understood as the fact that multiplicative–additive linear logic is somehow dense in full linear logic. Our goal is to give a technical formulation of this informal remark. We introduce a Cauchy-complete space of infinitary affine term-proofs and we show that it yields a fully complete model of multiplicative exponential polarised linear logic, in the style of Girard's ludics. Moreover, the subspace of finite term-proofs, which is a model of multiplicative polarised linear logic, is dense in the space of all term-proofs.

The axioms of iteration theories, or iteration categories, capture the equational properties of fixed point operations in several computationally significant categories. Iteration categories may be axiomatized by the Conway identities and identities associated with finite automata. We show that the Conway identities and the identities associated with the members of a subclass $\mathcal{Q}$ of finite automata is complete for iteration categories iff for every finite simple group G there is an automaton Q ∈ $\mathcal{Q}$ such that G is a quotient of a group in the monoid M(Q) of the automaton Q. We also prove a stronger result that concerns identities associated with finite automata with a distinguished initial state.

We relate the so-called powercone models of mixed non-deterministic and probabilistic choice proposed by Tix, Keimel, Plotkin, Mislove, Ouaknine, Worrell, Morgan and McIver, to our own models of previsions. Under suitable topological assumptions, we show that they are isomorphic. We rely on Keimel's cone-theoretic variants of the classical Hahn–Banach separation theorems, using functional analytic methods, and on the Schröder–Simpson Theorem.

A natural question in the λ-calculus asks what is the possible number of fixed points of a combinator (closed term). A complete answer to this question is still missing (Problem 25 of TLCA Open Problems List) and we investigate the related question about the number of fixed points of a combinator in λ-theories. We show the existence of a recursively enumerable lambda theory where the number is always one or infinite. We also show that there are λ-theories such that some terms have only two fixed points. In a first example, this is obtained by means of a non-constructive (more precisely non-r.e.) λ-theory where the range property is violated. A second, more complex example of a non-r.e. λ-theory (with a higher unsolvability degree) shows that some terms can have only two fixed points while the range property holds for every term.

We characterise quasivarieties and varieties of ordered algebras categorically in terms of regularity, exactness and the existence of a suitable generator. The notions of regularity and exactness need to be understood in the sense of category theory enriched over posets. We also prove that finitary varieties of ordered algebras are cocompletions of their theories under sifted colimits (again, in the enriched sense).

We give a realizability interpretation of an intuitionistic version of Church's Simple Theory of Types (CST) which can be viewed as a formalization of intuitionistic higher-order logic. Although definable in CST we include operators for monotone induction and coinduction and provide simple realizers for them. Realizers are formally represented in an untyped lambda–calculus with pairing and case-construct. The purpose of this interpretation is to provide a foundation for the extraction of verified programs from formal proofs as an alternative to type-theoretic systems. The advantages of our approach are that (a) induction and coinduction are not restricted to the strictly positive case, (b) abstract mathematical structures and results may be imported, (c) the formalization is technically simpler than in other systems, for example, regarding the definition of realizability, which is a simple syntactical substitution, and the treatment of nested and simultaneous (co)inductive definitions.

This paper demonstrates the usefulness of a comonadic approach to give previously unknown characterisation of projective objects in certain categories over particular subclasses of epimorphisms. This approach is a simple adaptation of a powerful technique due to Escardó which has been used extensively to characterise injective spaces and locales over various kinds of embeddings, but never previously for projective structures. Using some examples, we advertise the versatility of this approach – in particular, highlighting its advantage over existing methods on characterisation of projectives, which is that the comonadic machinery forces upon us the structural properties of projectives without relying on extraneous characterisations of the underlying object of the coalgebra arising from the comonad.

Circular string matching is a problem which naturally arises in many contexts. It consists in finding all occurrences of the rotations of a pattern of length m in a text of length n. There exist optimal worst- and average-case algorithms for circular string matching. Here, we present a suboptimal average-case algorithm for circular string matching requiring time $\mathcal{O}$(n) and space $\mathcal{O}$(m). The importance of our contribution is underlined by the fact that the proposed algorithm can be easily adapted to deal with circular dictionary matching. In particular, we show how the circular dictionary-matching problem can be solved in average-case time $\mathcal{O}$(n + M) and space $\mathcal{O}$(M), where M is the total length of the dictionary patterns, assuming that the shortest pattern is sufficiently long. Moreover, the presented average-case algorithms and other worst-case approaches were also implemented. Experimental results, using real and synthetic data, demonstrate that the implementation of the presented algorithms can accelerate the computations by more than a factor of two compared to the corresponding implementation of other approaches.

This paper gives a complete characterisation of type isomorphism definable by terms of a λ-calculus with intersection and union types. Unfortunately, when union is considered the Subject Reduction property does not hold in general. However, it is well known that in the λ-calculus, independently of the considered type system, the isomorphism between two types can be realised only by invertible terms. Notably, all invertible terms are linear terms. In this paper, the isomorphism of intersection and union types is investigated using a relevant type system for linear terms enjoying the Subject Reduction property. To characterise type isomorphism, a similarity between types and a type reduction are introduced. Types have a unique normal form with respect to the reduction rules and two types are isomorphic if and only if their normal forms are similar.

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 $\mathcal{^KOCA}$.

In this paper, we treat the specification problem in Krivine classical realizability (Krivine 2009 Panoramas et synthèses 27), in the case of arithmetical formulæ. In the continuity of previous works from Miquel and the first author (Guillermo 2008 Jeux de réalisabilité en arithmétique classique, Ph.D. thesis, Université Paris 7; Guillermo and Miquel 2014 Mathematical Structures in Computer Science, Epub ahead of print), we characterize the universal realizers of a formula as being the winning strategies for a game (defined according to the formula). In the first sections, we recall the definition of classical realizability, as well as a few technical results. In Section 5, we introduce in more details the specification problem and the intuition of the game-theoretic point of view we adopt later. We first present a game 1, that we prove to be adequate and complete if the language contains no instructions ‘quote’ (Krivine 2003 Theoretical Computer Science 308 259–276), using interaction constants to do substitution over execution threads. We then show that as soon as the language contain ‘Quote,’ the game is no more complete, and present a second game 2 that is both adequate and complete in the general case. In the last Section, we draw attention to a model-theoretic point of view and use our specification result to show that arithmetical formulæ are absolute for realizability models.

Processes are often viewed as coalgebras, with the structure maps specifying the state transitions. In the simplest case, the state spaces are discrete, and the structure map simply takes each state to the next states. But the coalgebraic view is also quite effective for studying processes over structured state spaces, e.g. measurable, or continuous. In the present paper, we consider coalgebras over manifolds. This means that the captured processes evolve over state spaces that are not just continuous, but also locally homeomorphic to normed vector spaces, and thus carry a differential structure. Both dynamical systems and differential forms arise as coalgebras over such state spaces, for two different endofunctors over manifolds. A duality induced by these two endofunctors provides a formal underpinning for the informal geometric intuitions linking differential forms and dynamical systems in the various practical applications, e.g. in physics. This joint functorial reconstruction of tangent bundles and cotangent bundles uncovers the universal properties and a high-level view of these fundamental structures, which are implemented rather intricately in their standard form. The succinct coalgebraic presentation provides unexpected insights even about the situations as familiar as Newton's laws.

This article introduces an analogue of permutation classes in the context of polyominoes. For both permutation classes and polyomino classes, we present an original way of characterizing them by avoidance constraints (namely, with excluded submatrices) and we discuss how canonical such a description by submatrix-avoidance can be. We provide numerous examples of permutation and polyomino classes which may be defined and studied from the submatrix-avoidance point of view, and conclude with various directions for future research on this topic.

We revisit the definition of Cartesian differential categories, showing that a slightly more general version is useful for a number of reasons. As one application, we show that these general differential categories are comonadic over categories with finite products, so that every category with finite products has an associated cofree differential category. We also work out the corresponding results when the categories involved have restriction structure, and show that these categories are closed under splitting restriction idempotents.

We consider a uniform model of computation over algebraic structures resulting from a generalization of the Turing machine and the BSS model of computation. This model allows us to gain more insight into the reasons for unsolvability of algorithmic decision problems from different perspectives. For example, classes of undecidable problems can be introduced in several ways by analogy with the classical arithmetical hierarchy and, for many structures, the different definitions lead to different hierarchies of undecidable problems. Here, we will investigate some classes of a hierarchy that is defined semantically by our deterministic oracle machines and that can be syntactically characterized by formulas whose quantifiers range only over an enumerable set. Starting from machines over algebraic structures endowed with some relations and containing an infinite recursively enumerable sequence of individuals, we will also consider this hierarchy for BSS RAM's over the reals and some undecidable problems defined by algebraic properties of the real numbers.

A topological space X is called well-filtered if for any filtered family $\mathcal{F}$ of compact saturated sets and an open set U, ∩ $\mathcal{F}$ ⊆ U implies F ⊆ U for some F ∈ $\mathcal{F}$. Every sober space is well-filtered and the converse is not true. A dcpo (directed complete poset) is called well-filtered if its Scott space is well-filtered. In 1991, Heckmann asked whether every UK-admitting (the same as well-filtered) dcpo is sober. In 2001, Kou constructed a counterexample to give a negative answer. In this paper, for each T1 space X we consider a dcpo D(X) whose maximal point space is homeomorphic to X and prove that X is well-filtered if and only if D(X) is well-filtered. The main result proved here enables us to construct new well-filtered dcpos that are not sober (only one such example is known by now). A space will be called K-closed if the intersection of every filtered family of compact saturated sets is compact. Every well-filtered space is K-closed. Some similar results on K-closed spaces are also proved.

Intersection type assignment systems can be used as a general framework for building logical models of λ-calculus that allow to reason about the denotation of terms in a finitary way. We define essential models (a new class of logical models) through a parametric type assignment system using non-idempotent intersection types. Under an interpretation of terms based on typings instead than the usual one based on types, every suitable instance of the parameters induces a λ-model, whose theory is sensible. We prove that this type assignment system provides a logical description of a family of λ-models arising from a category of sets and relations.