The programs, we consider are written in a restricted form of the language introduced by Dijkstra (1968). A program is said to be conservative when each of its loops restores all the resources it consumes. We define the geometric model of such a program and prove that the collection of directed paths on it is a reasonable over-approximation of its set of execution traces. In particular, two directed paths that are close enough with respect to the uniform distance result in the same action on the memory states of the system. The same holds for weakly dihomotopic directed paths. As a by-product, we obtain a notion of independence, which is favourably compared to more common ones. The geometric models actually belong to a handy class of local pospaces whose elements are called isothetic regions. The local pospaces we use differ from the original ones, we carefully explain why the alternative notion should be preferred. The title intentionally echoes the article by Carson and Reynolds (1987).

In the framework of effectively enumerable topological spaces, we introduce the notion of a partial computable function. We show that the class of partial computable functions is closed under composition, and the real-valued partial computable functions defined on a computable Polish space have a principal computable numbering. With respect to the principal computable numbering of the real-valued partial computable functions, we investigate complexity of important problems such as totality and root verification. It turns out that for some problems the corresponding complexity does not depend on the choice of a computable Polish space, whereas for other ones the corresponding choice plays a crucial role.

We prove a strictification theorem for cartesian closed bicategories. First, we adapt Power’s proof of coherence for bicategories with finite bilimits to show that every bicategory with bicategorical cartesian closed structure is biequivalent to a 2-category with 2-categorical cartesian closed structure. Then we show how to extend this result to a Mac Lane-style “all pasting diagrams commute” coherence theorem: precisely, we show that in the free cartesian closed bicategory on a graph, there is at most one 2-cell between any parallel pair of 1-cells. The argument we employ is reminiscent of that used by Čubrić, Dybjer, and Scott to show normalisation for the simply-typed lambda calculus (Čubrić et al., 1998). The main results first appeared in a conference paper (Fiore and Saville, 2020) but for reasons of space many details are omitted there; here we provide the full development.

Type theory allows us to extract from a constructive proof that a specification is satisfiable a program that satisfies the specification. Algorithms for optimization of such programs are currently the object of research.

In this paper we consider one such algorithm, which was described in Beeson (1985) and which we will call ‘Harrop’. This algorithm greatly simplifies programs extracted from proofs in the Pure Construction Calculus. We use a Partial Equivalence Relation model for higher order lambda calculus, to check that t and Harrop(t) return the same outputs from the same inputs, i.e. that they are extensionally equal.

As a corollary, we show that it is correct (and, of course, useful) to replace a program t with Harrop(t). Such a correctness result has already been proved by Möhring (Möhring 1989a, 1989b) using realizability semantics, but we obtain it as a corollary of a new result, the extensional equality between t and Harrop(t). Also the semantic method we use is interesting in its own right.

For any locally cartesian closed category ℰ, we prove that a local fibred right adjoint between slices of ℰ is given by a polynomial. The slices in question are taken in a well-known fibred sense.

Herbrand’s theorem is one of the most fundamental insights in logic. From the syntactic point of view, it suggests a compact representation of proofs in classical first- and higher-order logics by recording the information of which instances have been chosen for which quantifiers. This compact representation is known in the literature as Miller’s expansion tree proof. It is inherently analytic and hence corresponds to a cut-free sequent calculus proof. Recently several extensions of such proof representations to proofs with cuts have been proposed. These extensions are based on graphical formalisms similar to proof nets and are limited to prenex formulas.

In this paper, we present a new syntactic approach that directly extends Miller’s expansion trees by cuts and also covers non-prenex formulas. We describe a cut-elimination procedure for our expansion trees with cut that is based on the natural reduction steps and shows that it is weakly normalizing.

The original software reliability demonstration test (SRDT) does not take adequate account of prior knowledge or the prior distribution, which can lead to an expensive use of many resources. In the current paper, we propose a new improved Bayesian based SRDT method. We begin by constructing a framework for the SRDT scheme, then we use decreasing functions to construct the prior distribution density functions for both discrete and continuous safety-critical software, and then present schemes for both discrete and continuous Bayesian software demonstration functions (which we call DBSDF and CBSDF, respectively). We have carried out a set of experiments comparing our new schemes with the classic demonstration testing scheme on several published data sets. The results reveal that the DBSDF and CBSDF schemes are both more efficient and more applicable, and this is especially the case for safety-critical software with high reliability requirements.

We study the longstanding problem of semantics for input/output (I/O) expressed using side-effects. Our vehicle is a small higher-order imperative language, with operations for interactive character I/O and based on ML syntax. Unlike previous theories, we present both operational and denotational semantics for I/O effects. We use a novel labelled transition system that uniformly expresses both applicative and imperative computation. We make a standard definition of bisimilarity and prove bisimilarity is a congruence using Howe's method.

Next, we define a metalanguage [Mscr ] in which we may give a denotational semantics to [Oscr ]. [Mscr ] generalises Crole and Pitts' FIX-logic by adding in a parameterised recursive datatype, which is used to model I/O. [Mscr ] comes equipped both with an operational semantics and a domain-theoretic semantics in the category [Cscr ][Pscr ][Pscr ][Oscr ] of cppos (bottom-pointed posets with joins of ω-chains) and Scott continuous functions. We use the [Cscr ][Pscr ][Pscr ][Oscr ] semantics to prove that [Mscr ] is computationally adequate for the operational semantics using formal approximation relations. The existence of such relations is based on recent work of Pitts (Pitts 1994b) for untyped languages, and uses the idea of minimal invariant objects due to Freyd.

A monadic-style textual translation into [Mscr ] induces a denotational semantics on [Oscr ]. Our final result validates the denotational semantics: if the denotations of two [Oscr ] programs are equal, then the [Oscr ] programs are in fact operationally equivalent.

This paper introduces an endofunctor VT on the category of frames that is parametrised by an endofunctor T on the category Set that satisfies certain constraints. This generalises Johnstone's construction of the Vietoris powerlocale in the sense that his construction is obtained by taking for T the finite covariant power set functor. Our construction of the T-powerlocale VT out of a frame is based on ideas from coalgebraic logic and makes explicit the connection between the Vietoris construction and Moss's coalgebraic cover modality.

We show how to extend certain natural transformations between set functors to natural transformations between T-powerlocale functors. Finally, we prove that the operation VT preserves some properties of frames, such as regularity, zero-dimensionality and the combination of zero-dimensionality and compactness.

We examine a construction due to Fouché in which a Brownian motion is constructed from an algorithmically random infinite binary sequence. We show that although the construction is provably not computable in the sense of computable analysis, a lower bound for the rate of convergence is computable in any upper bound for the compressibilty of the sequence, making the construction layerwise computable.

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 prove the conservation theorem for differential nets – the graph-theoretical syntax of the differential extension of Linear Logic (Ehrhard and Regnier's DiLL). The conservation theorem states that the property of having infinite reductions (here infinite chains of cut elimination steps) is preserved by non-erasing steps. This turns the quest for strong normalisation (SN) into one for non-erasing weak normalisation (WN), and indeed we use this result to prove SN of simply typed DiLL (with promotion). Along the way to the theorem we achieve a number of additional results having their own interest, such as a standardisation theorem and a slightly modified system of nets, DiLL∂ϱ.

Every accessible category is proved to be sketchable by a sketch with finite colimits. In contrast, a finitely accessible category is presented that cannot be sketched by a finitary sketch, i.e., a sketch with finite limits and finite colimits. Also, a category sketchable by a finitary sketch is found that is not finitely accessible.

Process algebraic techniques for distributed systems are increasingly being targeted at identifying abstractions that are adequate for both high-level programming and specification and security analysis and verification. Drawing on our earlier work in Bugliesi and Focardi, (2008), we investigate the expressive power of a core set of security and network abstractions that provide high-level primitives for specifying the honest principals in a network, while at the same time enabling an analysis of the network-level adversarial attacks that may be mounted by an intruder.

We analyse various bisimulation equivalences for security that arise from endowing the intruder with:

1. (i) different adversarial capabilities; and

2. (ii) increasingly powerful control over the interaction among the distributed principals of a network.

The statistical mechanical interpretation of algorithmic information theory (AIT for short) was introduced and developed in our previous papers Tadaki (2008; 2012), where we introduced into AIT the notion of thermodynamic quantities, such as the partition function Z(T), free energy F(T), energy E(T) and statistical mechanical entropy S(T). We then discovered that in the interpretation, the temperature T is equal to the partial randomness of the values of all these thermodynamic quantities, where the notion of partial randomness is a stronger representation of the compression rate by means of program-size complexity. Furthermore, we showed that this situation holds for the temperature itself as a thermodynamic quantity, namely, for each of the thermodynamic quantities above, the computability of its value at temperature T gives a sufficient condition for T ∈ (0, 1) to be a fixed point on partial randomness. In this paper, we develop the statistical mechanical interpretation of AIT further and pursue its formal correspondence to normal statistical mechanics. The thermodynamic quantities in AIT are defined on the basis of the halting set of an optimal prefix-free machine, which is a universal decoding algorithm used to define the notion of program-size complexity. We show that there are infinitely many optimal prefix-free machines that give completely different sufficient conditions for each of the thermodynamic quantities in AIT. We do this by introducing the notion of composition of prefix-free machines into AIT, which corresponds to the notion of the composition of systems in normal statistical mechanics.

In this paper we interpret (fragments of) intuitionistic logic in categories with weak closure properties, such as quasi left exact categories and locally cartesian closed categories (LCCC) with sums. We also interpret the full choice scheme in an LCCC. The interpretation can be seen as a categorical form of the usual Brouwer–Heyting– Kolmogorov (BHK) interpretation. The standard interpretation of geometric logic in a pretopos is obtained by applying the image functor to the BHK-interpretation.

Differential categories axiomatize the basics of differentiation and provide categorical models of differential linear logic. A differential category is said to have antiderivatives if a natural transformation , which all differential categories have, is a natural isomorphism. Differential categories with antiderivatives come equipped with a canonical integration operator such that generalizations of the Fundamental Theorems of Calculus hold. In this paper, we show that Blute, Ehrhard, and Tasson's differential category of convenient vector spaces has antiderivatives. To help prove this result, we show that a differential linear category – which is a differential category with a monoidal coalgebra modality – has antiderivatives if and only if one can integrate over the monoidal unit and such that the Fundamental Theorems of Calculus hold. We also show that generalizations of the relational model (which are biproduct completions of complete semirings) are also differential linear categories with antiderivatives.

We describe some formal topological results, formalised in Matita 1/2, presented in predicative intuitionistic logic and in terms of Overlap Algebras.

Overlap Algebras are new algebraic structures designed to ease reasoning about subsets in an algebraic way within intuitionistic logic. We find that they also ease the formalisation of formal topological results in an interactive theorem prover.

Our main result is the existence of a functor between two categories of ‘generalised topological spaces’, one with points (Basic Pairs) and the other point-free (Basic Topologies). This formalisation is part of a wider scientific collaboration with the inventor of the theory, Giovanni Sambin. His goal is to verify in what sense his theory is ‘implementable’, and to discover what problems may arise in the process. We check that all intermediate constructions respect the stringent size requirements imposed by predicative logic. The formalisation is quite unusual, since it has to make explicit size information that is often hidden.

We found that the version of Matita used for the formalisation was largely inappropriate. The formalisation drove several major improvements of Matita that will be integrated in the next major release (Matita 1.0). We show some motivating examples, taken directly from the formalisation, for these improvements. We also describe a possibly sub-optimal solution in Matita 1/2, which is exploitable in other similar systems. We briefly discuss a better solution available in Matita 1.0.

We present a formalism called addressed term rewriting systems, which can be used to model implementations of theorem proving, symbolic computation and programming languages, especially aspects of sharing, recursive computations and cyclic data structures. Addressed Term Rewriting Systems are therefore well suited to describing object-based languages, and as an example we present a language called $\lambda{\cal O}bj^{a}$, incorporating both functional and object-based features. As a case study in how reasoning about languages is supported in the ATRS formalism, we define a type system for $\lambda{\cal O}bj^{a}$ and prove a type soundness result.

In this paper we study partial equivalence relations (PERs) over graph models of the λcalculus. We define categories of PERs that behave like predomains, and like domains. These categories are small and complete; so we can solve domain equations and construct polymorphic types inside them. Upper, lower and convex powerdomain constructions are also available, as well as interpretations of subtyping and bounded quantification. Rather than performing explicit calculations with PERs, we work inside the appropriate realizability topos: this is a model of constructive set theory in which PERs, can be regarded simply as special kinds of sets. In this framework, most of the definitions and proofs become quite smple and attractives. They illustrative some general technicques in ‘synthetic domain theory’ that rely heavily on category theory; using these methods, we can obtain quite powerful results about classes of PERs, even when we know very little about their internal structure.