We present an algorithm for deciding polarised higher-order subtyping without bounded quantification. Constructors are identified not only modulo β, but also η. We give a direct proof of completeness, without constructing a model or establishing a strong normalisation theorem. Inductive and coinductive types are enriched with a notion of size and the subtyping calculus is extended to account for the inclusions arising between the sized types.

We present a novel coalgebraic formulation of infinite extensive games. We define both the game trees and the strategy profiles by possibly infinite systems of corecursive equations. Certain strategy profiles are proved to be subgame-perfect equilibria using a novel proof principle of predicate coinduction which is shown to be sound. We characterize all subgame-perfect equilibria for the dollar auction game. The economically interesting feature is that in order to prove these results we do not need to rely on continuity assumptions on the pay-offs which amount to discounting the future. In particular, we prove a form of one-deviation principle without any such assumptions. This suggests that coalgebra supports a more adequate treatment of infinite-horizon models in game theory and economics.

We present a result of C.C. Squier relating two topics:

— the canonical rewriting systems, which have been widely studied by computer scientists as a tool for solving word problems,

— the homology of groups (more generally of monoids), which belongs to the core of pure mathematics, on the borders of algebra and topology.

Noether classes of posets arise in a natural way from the constructively meaningful variants of the notion of a Noetherian ring. Using an axiomatic characterisation of a Noether class, we prove that if a poset belongs to a Noether class, then so does the poset of the finite descending chains. When applied to the poset of finitely generated ideals of a ring, this helps towards a unified constructive proof of the Hilbert basis theorem for all Noether classes.

One natural way to generalise domain theory is to replace partially ordered sets by categories. This kind of generalisation has recently found application in the study of concurrency. An outline is given of the elegant mathematical foundations that have been developed. This is specialised to give a construction of cartesian closed categories of domains, which throws light on standard presentations of domain theory.

The correct and efficient implementation of general real-time applications remains very much an open problem. A key issue is meeting timing constraints whose satisfaction depends on features of the execution platform, in particular its speed. Existing rigorous implementation techniques are applicable to specific classes of systems, for example, with periodic tasks or time-deterministic systems.

We present a general model-based implementation method for real-time systems based on the use of two models:

• • An abstract model representing the behaviour of real-time software as a timed automaton, which describes user-defined platform-independent timing constraints. Its transitions are timeless and correspond to the execution of statements of the real-time software.

• • A physical model representing the behaviour of the real-time software running on a given platform. It is obtained by assigning execution times to the transitions of the abstract model.

We show that, as a rule, physical models are not time-robust, and that time-determinism is a sufficient condition for time-robustness. For a given piece of real-time software and an execution platform corresponding to a time-robust model, we define an execution engine that coordinates the execution of the application software so that it meets its timing constraints. Furthermore, in the case of non-robustness, the execution engine can detect violations of time-safety and stop execution.

We have implemented the execution engine for BIP programs with real-time constraints and validated the implementation method for two case studies. The experimental results for a module of a robotic application show that the CPU utilisation and the size of the model are reduced compared with existing implementations. The experimental results for an adaptive video encoder also show that a lack of time-robustness may seriously degrade the performance for increasing platform execution speed.

We give an algorithm allowing the construction of bases of local unitary invariants of pure k-qubit states from a knowledge of the polynomial covariants of the group of invertible local filtering operations. The simplest invariants obtained in this way are made explicit and compared with various known entanglement measures. Complete sets of generators are obtained for up to four qubits, and the structure of the invariant algebras is discussed in detail.

This paper surveys what we have learned during the last ten years about the lattice $\lambda \mathcal{T}$ of all $\lambda$-theories (= equational extensions of untyped $\lambda$-calculus), via the sets $\lambda \mathcal{C}$ consisting of the $\lambda$-theories that are representable in a uniform class $\mathcal{C}$ of $\lambda$-models. This includes positive answers to several questions raised in Berline (2000), as well as several independent results, the state of the art on the long-standing open questions concerning the representability of $\lambda _{\beta},\lambda _{\beta\eta}$, $H$ as theories of models, and 22 open problems.

We will focus on the class $\mathcal{G}$ of graph models, since almost all the existing semantic proofs on $\lambda \mathcal{T}$ have been, or could be, more easily, obtained via graph models, or slight variations of them. But in this paper we will also give some evidence that, for all uniform classes $\mathcal{C},\mathcal{C}^{\prime}$ of proper $\lambda$-models living in functional semantics, $\lambda \mathcal{C}-\lambda \mathcal{C}^{\prime}$ should have cardinality $2^{\omega }$, provided $ \mathcal{C}$ is not included in $\mathcal{C}^{\prime}.$

This paper presents an extension of the simply typed λ-calculus that allows iteration and case reasoning over terms of functional types that arise when using higher order abstract syntax. This calculus aims at being the kernel for a type theory in which the user will be able to formalize logics or formal systems using the LF methodology, while taking advantage of new induction and recursion principles, extending the principles available in a calculus such as the Calculus of Inductive Constructions. The key idea of our system is the use of modal logic S4. We present here the system, its typing rules and reduction rules. The system enjoys the decidability of typability, soundness of typed reduction with respect to the typing rules, the Church–Rosser and strong normalization properties and it is a conservative extension over the simply typed λ-calculus. These properties entail the preservation of the adequacy of encodings.

Recent developments in the categorical foundations of universal algebra have given an impetus to an understanding of the lambda calculus coming from categorical logic: an interpretation is a semi-closed algebraic theory. Scott's representation theorem is then completely natural and leads to a precise Fundamental Theorem showing the essential equivalence between the categorical and more familiar notions.

We extend the work of A. Ciaffaglione and P. di Gianantonio on the mechanical verification of algorithms for exact computation on real numbers, using infinite streams of digits implemented as a co-inductive type. Four aspects are studied. The first concerns the proof that digit streams correspond to axiomatised real numbers when they are already present in the proof system. The second re-visits the definition of an addition function, looking at techniques to let the proof search engine perform the effective construction of an algorithm that is correct by construction. The third concerns the definition of a function to compute affine formulas with positive rational coefficients. This is an example where we need to combine co-recursion and recursion. Finally, the fourth aspect concerns the definition of a function to compute series, with an application on the series that is used to compute Euler's number e. All these experiments should be reproducible in any proof system that supports co-inductive types, co-recursion and general forms of terminating recursion; we used the COQ system (Dowek et al. 1993; Bertot and Castéran 2004; Giménez 1994).

The local solver TD is a generic fixpoint engine which explores a given system of equations on demand. It has been successfully applied to the interprocedural analysis of procedural languages. The solver TD gains efficiency by detecting dependencies between unknowns on the fly. This algorithm has been recently extended to deal with widening and narrowing as well. In particular, it has been equipped with an automatic detection of widening and narrowing points. That version, however, is only guaranteed to terminate under two conditions: only finitely many unknowns are encountered, and all right-hand sides are monotonic. While the first condition is unavoidable, the second limits the applicability of the solver. Another limitation is that the solver maintains the current abstract values of all encountered unknowns instead of a minimal set sufficient for performing the iteration. By consuming unnecessarily much space, interprocedural analyses may not succeed on seemingly small programs. In the present paper, we therefore extend the top-down solver TD in three ways. First, we indicate how the restriction to monotonic right-hand sides can be lifted without compromising termination. We then show how the solver can be tuned to store abstract values only when their preservation is inevitable. Finally, we also show how the solver can be extended to side-effecting equation systems. Right-hand sides of these may not only provide values for the corresponding left-hand side unknowns but at the same time produce contributions to other unknowns. This practical extension has successfully been used for a seamless combination of context-sensitive analyses (e.g., of local states) with flow-insensitive analyses (e.g., of globals).

This paper is about a categorical approach for modelling the pure (i.e., without constants) call-by-value λ-calculus, defined by Plotkin as a restriction of the call-by-name λ-calculus. In particular, we give the properties that a category Cbv must enjoy to describe a model of call-by-value λ-calculus. The category Cbv is general enough to catch models in Scott Domains and Coherence Spaces.

The concurrency theory literature offers a wealth of examples of characteristic-formula constructions for various behavioural relations over finite labelled transition systems and Kripke structures that are defined in terms of fixed points of suitable functions. Such constructions and their proofs of correctness have been developed independently, but have a common underlying structure. This paper provides a general view of characteristic formulae that are expressed in terms of logics that have a facility for the recursive definition of formulae. We show how several examples of characteristic-formula constructions in the literature can be recovered as instances of the proposed general framework, and how the framework can be used to yield novel constructions. The paper also offers general results pertaining to the definition of co-characteristic formulae and of characteristic formulae expressed in terms of infinitary modal logics.

Results of Anick (1986), Squier (1987), Kobayashi (1990), Brown (1992b), and others, show that a monoid with a finite convergent rewriting system satisfies a homological condition known as FP∞.

In this paper we give a simplified version of Brown's proof, which is conceptual, in contrast with the other proofs, which are computational.

We also collect together a large number of results and examples of monoids and groups that satisfy FP∞ and others that do not. These may provide techniques for showing that various monoids do not have finite convergent rewriting systems, as well as explicit examples with which methods can be tested.

Session types describe and constrain the input/output behaviour of systems. Existing session typing systems have limited support for polymorphism. For example, existing systems cannot provide the most general type for a generic proxy process that forwards messages between two channels. We provide a polymorphic session typing system for the π calculus, and demonstrate the utility of session-type-level functions in combination with polymorphic session typing. The type system guarantees subject reduction and safety properties, but not deadlock freedom. We describe a formalization of the type system in Coq. The proofs of subject reduction and safety properties, as well as typing of example processes, have been mechanically verified.

Synthetic domain theory (SDT) is a version of Domain Theory where ‘all functions are continuous’. Following the original suggestion of Dana Scott, several approaches to SDT have been developed that are logical or categorical, axiomatic or model-oriented in character and that are either specialised towards Scott domains or aim at providing a general theory axiomatising the structure common to the various notions of domains studied so far.

In Reus and Streicher (1993), Reus (1995) and Reus (1998), we have developed a logical and axiomatic version of SDT, which is special in the sense that it captures the essence of Domain Theory à la Scott but rules out, for example, Stable Domain Theory, as it requires order on function spaces to be pointwise. In this article we will give a logical and axiomatic account of a general SDT with the aim of grasping the structure common to all notions of domains.

As in loc.cit., the underlying logic is a sufficiently expressive version of constructive type theory. We start with a few basic axioms giving rise to a core theory on top of which we study various notions of predomains (such as, for example, complete and well-complete S-spaces (Longley and Simpson 1997)), define the appropriate notion of domain and verify the usual induction principles of domain theory.

Although each domain carries a logically definable ‘specialization order’, we avoid order-theoretic notions as much as possible in the formulation of axioms and theorems. The reason is that the order on function spaces cannot be required to be pointwise, as this would rule out the model of stable domains à la Berry.

The consequent use of logical language – understood as the internal language of some categorical model of type theory – avoids the irritating coexistence of the internal and the external view pervading purely categorical approaches. Therefore, the paper is aimed at providing an elementary introduction to synthetic domain theory, albeit requiring some knowledge of basic type theory.

We generalise the construction of the formal ball model for metric spaces due to A. Edalat and R. Heckmann in order to obtain computational models for separated -categories. We fully describe -categories that are

1. (a) Yoneda complete

2. (b) continuous Yoneda complete

1. (a) a quasi-metric space X is Yoneda complete if and only if its formal ball model is a dcpo, and

2. (b) a quasi-metric space X is continuous and Yoneda complete if and only if its formal ball model BX is a domain that admits a simple characterisation of approximation.

We investigate several geometric models of networks that simultaneously have some nice global properties, including the small-diameter property, the small-community phenomenon, which is defined to capture the common experience that (almost) everyone in society also belongs to some meaningful small communities, and the power law degree distribution, for which our result significantly strengthens those given in van den Esker (2008) and Jordan (2010). These results, together with our previous work in Li and Peng (2011), build a mathematical foundation for the study of both communities and the small-community phenomenon in various networks.

In the proof of the power law degree distribution, we develop the method of alternating concentration analysis to build a concentration inequality by alternately and iteratively applying both the sub- and super-martingale inequalities, which seems to be a powerful technique with further potential applications.

Previous work has demonstrated that categories are useful and expressive models for databases. In the current paper we build on that model, showing that certain queries and constraints correspond to lifting problems, as found in modern approaches to algebraic topology. In our formulation, each SPARQL graph pattern query corresponds to a category-theoretic lifting problem, whereby the set of solutions to the query is precisely the set of lifts. We interpret constraints within the same formalism, and then investigate some basic properties of queries and constraints. In particular, to any database π, we can associate a certain derived database Qry(π) of queries on π. As an application, we explain how giving users access to certain parts of Qry(π), rather than direct access to π, improves the ability to manage the impact of schema evolution.