Behavioural equivalences are a means of establishing whether computing systems possess the same properties. The specific set of properties that are preserved by a specific behavioural equivalence clearly depends on how the system behaviour is observed and can usually be characterised by means of a modal logic. In this paper we consider three different approaches to the definition of behavioural equivalences – bisimulation, testing and trace – applied to three different classes of systems – non-deterministic, probabilistic and Markovian – and we survey the nine resulting modal logic characterisations, each of which stems from the Hennessy–Milner logic. We then compare the nine characterisations with respect to the logical operators, in order to emphasise the differences between the three approaches in the definition of behavioural equivalences and the regularities within each of them. In the probabilistic and Markovian cases we also address the issue of whether the probabilistic and temporal aspects should be treated in a local or global way and consequently whether the modal logic interpretation should be qualitative or quantitative.

We study a type system with a notion of subtyping that involves a largest type ⊤, a smallest type ⊥, atomic coercions between base types, and the usual ordering of function types. We prove that any λ-term typable in this system is strongly normalizing, which solves an open problem of Thatte. We also prove that the fragment without ⊥ types has strictly fewer terms. This demonstrates that ⊥ adds power to a type system.

The goal of this paper is to report on a formalization of the p-adic numbers in the setting of the second author's univalent foundations program. This formalization, which has been verified in the Coq proof assistant, provides an approach to the p-adic numbers in constructive algebra and analysis.

This paper presents a novel method for comparing computational properties of λ-terms that are typeable with intersection types, with respect to terms that are typeable with Curry types. We introduce a translation from intersection typing derivations to Curry typeable terms that is preserved by β-reduction: this allows the simulation of a computation starting from a term typeable in the intersection discipline by means of a computation starting from a simply typeable term. Our approach proves strong normalisation for the intersection system naturally by means of purely syntactical techniques. The paper extends the results presented in Bucciarelli et al. (1999) to the whole intersection type system of Barendregt, Coppo and Dezani, thus providing a complete proof of the conjecture, proposed in Leivant (1990), that all functions uniformly definable using intersection types are already definable using Curry types.

The importance of an abstract approach to a computation theory over general data types has been stressed by Tucker in many of his papers. Berger and Seisenberger recently elaborated the idea for extraction out of proofs involving (only) abstract reals. They considered a proof involving coinduction of the proposition that any two reals in [−1, 1] have their average in the same interval, and informally extract a Haskell program from this proof, which works with stream representations of reals. Here we formalize the proof, and machine extract its computational content using the Minlog proof assistant. This required an extension of this system to also take coinduction into account.

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 investigate the expressiveness of two classical distributed paradigms by defining the first encoding of the pure mobile ambient calculus into the synchronous π-calculus. Our encoding, whose correctness has been proved by relying on the notion of operational correspondence, shows how the hierarchical ambient structure can be reformulated within a flat channel interconnection amongst independent processes, without centralised control. To easily handle the computation for simulating a capability, we introduce the notions of simulating trace (representing the computation that a π-calculus process has to execute to mimic a capability) and of aborting trace (representing the computation that a π-calculus process executes when the simulation of a capability cannot succeed). Thus, the encoding may introduce loops, but, as it will be shown, the number of steps of any trace, therefore of any aborting trace, is limited, and the number of states of the transition system of the encoding processes still remains finite. In particular, an aborting trace makes a sort of backtracking, leaving the involved sub-processes in the same starting configurations. We also discuss two run-time support methods to make these loops harmless at execution time. Our work defines a relatively simple, direct, and precise translation that reproduces the ambient structure by means of channel links, and keeps track of the dissolving of an ambient.

Haplotypes are more useful in complex disease gene mapping than single-nucleotide polymorphisms (SNPs). However, haplotypes are difficult to obtain directly using biological experiments, which has prompted research into efficient computational methods for determining haplotypes. The individual haplotyping problem called Minimum Letter Flip (MLF) is a computational problem that, given a set of aligned DNA sequence fragment data of an individual, induces the corresponding haplotypes by flipping minimum SNPs. There has been no practical exact algorithm for solving the problem. Due to technical limits in DNA sequencing experiments, the maximum length of a fragment sequenced directly is about 1kb. In consequence, with a genome-average SNP density of 1.84 SNPs per 1 kb of DNA sequence, the maximum number k1 of SNP sites that a fragment covers is usually small. Moreover, in order to save time and money, the maximum number k2 of fragments that cover an SNP site is usually no more than 19. Building on these fragment data properties, the current paper introduces a new parameterised algorithm with running time O(nk22k2 + mlogm + mk1), where m is the number of fragments and n is the number of SNP sites. In practical biological applications, the algorithm solves the MLF problem efficiently even if m and n are large.

We study the asymptotic behaviour of (a) information leakage and (b) adversary's error probability in information hiding systems modelled as noisy channels. Specifically, we assume the attacker can make a single guess after observing n independent executions of the system, throughout which the secret information is kept fixed. We show that the asymptotic behaviour of quantities (a) and (b) can be determined in a simple way from the channel matrix. Moreover, simple and tight bounds on them as functions of n show that the convergence is exponential. We also discuss feasible methods to evaluate the rate of convergence. Our results cover both the Bayesian case, where an a priori probability distribution on the secrets is assumed known to the attacker, and the maximum-likelihood case, where the attacker does not know such distribution. In the Bayesian case, we identify the distributions that maximize leakage. We consider both the min-entropy setting studied by Smith and the additive form recently proposed by Braun et al. and show the two forms do agree asymptotically. Next, we extend these results to a more sophisticated eavesdropping scenario, where the attacker can perform a (noisy) observation at each state of the computation and the systems are modelled as hidden Markov models.

A longstanding open problem is whether there exists a non-syntactical model of the untyped λ-calculus whose theory is exactly the least λ-theory λβ. In this paper we investigate the more general question of whether the equational/order theory of a model of the untyped λ-calculus can be recursively enumerable (r.e. for short). We introduce a notion of effective model of λ-calculus, which covers, in particular, all the models individually introduced in the literature. We prove that the order theory of an effective model is never r.e.; from this it follows that its equational theory cannot be λβ or λβη. We then show that no effective model living in the stable or strongly stable semantics has an r.e. equational theory. For Scott's semantics, we investigate the class of graph models and prove that no order theory of a graph model can be r.e., and that there exists an effective graph model whose equational/order theory is the minimum among the theories of graph models. Finally, we show that the class of graph models enjoys a kind of downwards Löwenheim–Skolem theorem.

By the Final Coalgebra Theorem of Aczel and Mendler, every endofunctor of the category of sets has a final coalgebra, which, however, may be a proper class. We generalise this to all ‘well-behaved’ categories ${\frak K}$. The role of the category of classes is played by a free cocompletion ${\frak K}^\infty$ of ${\frak K}$ under transfinite colimits, that is, colimits of ordinal-indexed chains. Every endofunctor $F$ of ${\frak K}$ has a canonical extension to an endofunctor $F^\infty$ of ${\frak K}^\infty$, which is proved to have a final coalgebra (and an initial algebra). Based on this, we prove a general solution theorem: for every endofunctor of a locally presentable category ${\frak K}$ all guarded equation-morphisms have unique solutions. The last result does not need the extension ${\frak K}^\infty$: the solutions are always found within the category ${\frak K}$.

We show how to apply forward and reverse mode Combinatory Homomorphic Automatic Differentiation (CHAD) (Vákár (2021). ESOP, 607–634; Vákár and Smeding (2022). ACM Transactions on Programming Languages and Systems 44 (3) 20:1–20:49.) to total functional programming languages with expressive type systems featuring the combination of

• • tuple types;

• • sum types;

• • inductive types;

• • coinductive types;

• • function types.

We achieve this by analyzing the categorical semantics of such types in $\Sigma$-types (Grothendieck constructions) of suitable categories. Using a novel categorical logical relations technique for such expressive type systems, we give a correctness proof of CHAD in this setting by showing that it computes the usual mathematical derivative of the function that the original program implements. The result is a principled, purely functional and provably correct method for performing forward- and reverse-mode automatic differentiation (AD) on total functional programming languages with expressive type systems.

This work extends a rule-based specification of nominal C-unification formalised in Coq to include ‘protected variables’ that cannot be instantiated during the unification process. By introducing protected variables, we are able to reuse the C-unification simplification rules to solve nominal C-matching (as well as equality check) problems. From the algorithmic point of view, this extension is sufficient to obtain a generalised C-unification procedure; however, it cannot be formally checked by simple reuse of the original formalisation. This paper describes the additional effort necessary in order to adapt the specification of the inference rules and reuse previous formalisations. We also generalise a functional recursive nominal C-unification algorithm specified in PVS with protected variables, effectively adapting this algorithm to the tasks of nominal C-matching and nominal equality check. The PVS formalisation is applied to test the correctness of a Python manual implementation of the algorithm.

We propose a framework for reasoning about program security building on language-theoretic and coalgebraic concepts. The behaviour of a system is viewed as a mapping from traces of high (unobservable) events to low (observable) events: the less the degree of dependency of low events on high traces, the more secure the system. We take the abstract view that low events are drawn from a generic semiring, where they can be combined using product and sum operations; throughout the paper, we provide instances of this framework, obtained by concrete instantiations of the underlying semiring. We specify systems via a simple process calculus, whose semantics is given as the unique homomorphism from the calculus into the set of behaviours, i.e. formal power series, seen as a final coalgebra. We provide a compositional semantics for the calculus in terms of rational operators on formal power series and show that the final and the compositional semantics coincide. This compositional, syntax-driven framework lays a foundation for automation and abstraction of a quantified approach to flow security of system specifications.

The representation of the inductively defined abstract data type for lists was left incomplete in Seldin (1997, Section 9). Here that representation is completed, and it is proved that all extra axioms needed are consistent. Among the innovations of this paper is a definition of cdr, whose definition was left for future work in Seldin (1997, Section 9). The results are then extended to other abstract data types – those of Berardi (1993). The method used to define cdr for lists is extended to obtain the definition of an inverse for each argument of each constructor of an abstract data type. These inverses are used to prove the injective property for the constructors. Also, Dedekind's method of defining the natural numbers is used to define a predicate associated with each abstract data type, and the use of this predicate makes it unnecessary to postulate the induction principle. The only axioms left to be proved are those asserting the disjointness of the co-domains of different constructors, and it is shown that those axioms can be proved consistent.

The present paper uses spectral theory of linear operators to construct approximatelyminimal realizations of weighted languages. Our new contributions are: (i) a new algorithm for the singular value decomposition (SVD) decomposition of finite-rank infinite Hankel matrices based on their representation in terms of weighted automata, (ii) a new canonical form for weighted automata arising from the SVD of its corresponding Hankelmatrix, and (iii) an algorithmto construct approximateminimizations of given weighted automata by truncating the canonical form.We give bounds on the quality of our approximation.

This paper presents a dependently-typed construction of semi-simplicial sets in a type theory where sets are taken to be types. This addresses an open question raised on the wiki of the special year on Univalent Foundations at the Institute of Advanced Study (2012–2013).

In this paper we study (interpret) the precise composability guarantee of the generalised universal composability (GUC) feasibility with global setups that was proposed in the recent paper Canetti et al. (2007) from the point of view of full universal composability (FUC), that is, composability with arbitrary protocols, which was the original security goal and motivation for UC. By observing a counter-intuitive phenomenon, we note that the GUC feasibility implicitly assumes that the adversary has limited access to arbitrary external protocols. We then clarify a general principle for achieving FUC security, and propose some approaches for fixing the GUC feasibility under the general principle. Finally, we discuss the relationship between GUC and FUC from both technical and philosophical points of view. This should be helpful in gaining a precise understanding of the GUC feasibility, and for preventing potential misinterpretations and/or misuses in practice.

It is well known that weakening and contraction cause naive categorical models of the classical sequent calculus to collapse to Boolean lattices. In previous work, summarised briefly herein, we have provided a class of models called classical categories that is sound and complete and avoids this collapse by interpreting cut reduction by a poset enrichment. Examples of classical categories include boolean lattices and the category of sets and relations, where both conjunction and disjunction are modelled by the set-theoretic product. In this article, which is self-contained, we present an improved axiomatisation of classical categories, together with a deep exploration of their structural theory. Observing that the collapse already happens in the absence of negation, we start with negation-free models called Dummett categories. Examples of these include, besides the classical categories mentioned above, the category of sets and relations, where both conjunction and disjunction are modelled by the disjoint union. We prove that Dummett categories are MIX, and that the partial order can be derived from hom-semilattices, which have a straightforward proof-theoretic definition. Moreover, we show that the Geometry-of-Interaction construction can be extended from multiplicative linear logic to classical logic by applying it to obtain a classical category from a Dummett category.

Along the way, we gain detailed insights into the changes that proofs undergo during cut elimination in the presence of weakening and contraction.

We prove two results for the sequential topology on countable products of sequential topological spaces. First we show that a countable product of topological quotients yields a quotient map between the product spaces. Then we show that the reflection from sequential spaces to its subcategory of monotone ω-convergence spaces preserves countable products. These results are motivated by applications to the modelling of computation on non-discrete spaces.