Given a synchronous system, we study the question whether – or, under which conditions – the behaviour of that system can be realized by a (non-trivially) distributed and hence asynchronous implementation. In this paper, we partially answer this question by examining the role of causality for the implementation of synchrony in two fundamental different formalisms of concurrency, Petri nets and the π-calculus. For both formalisms it turns out that each ‘good’ encoding of synchronous interactions using just asynchronous interactions introduces causal dependencies in the translation.

Setoids commonly take the place of sets when formalising mathematics inside type theory. In this note, the category of setoids is studied in type theory with universes that are as small as possible (and thus, the type theory is as weak as possible). In particular, we will consider epimorphisms and disjoint sums. We show that, given the minimal type universe, all epimorphisms are surjections, and disjoint sums exist. Further, without universes, there are countermodels for these statements, and if we use the Logical Framework formulation of type theory, these statements are provably non-derivable.

Extracting causal relationships from observed correlations is a growing area in probabilistic reasoning, originating with the seminal work of Pearl and others from the early 1990s. This paper develops a new, categorically oriented view based on a clear distinction between syntax (string diagrams) and semantics (stochastic matrices), connected via interpretations as structure-preserving functors. A key notion in the identification of causal effects is that of an intervention, whereby a variable is forcefully set to a particular value independent of any prior propensities. We represent the effect of such an intervention as an endo-functor which performs ‘string diagram surgery’ within the syntactic category of string diagrams. This diagram surgery in turn yields a new, interventional distribution via the interpretation functor. While in general there is no way to compute interventional distributions purely from observed data, we show that this is possible in certain special cases using a calculational tool called comb disintegration. We demonstrate the use of this technique on two well-known toy examples: one where we predict the causal effect of smoking on cancer in the presence of a confounding common cause and where we show that this technique provides simple sufficient conditions for computing interventions which apply to a wide variety of situations considered in the causal inference literature; the other one is an illustration of counterfactual reasoning where the same interventional techniques are used, but now in a ‘twinned’ set-up, with two version of the world – one factual and one counterfactual – joined together via exogenous variables that capture the uncertainties at hand.

User-defined higher-order rewrite rules are becoming a standard in proof assistants based on intuitionistic type theory. This raises the question of proving that they preserve the properties of beta-reductions for the corresponding type systems. In a series of papers, we develop techniques based on van Oostrom’s decreasing diagrams that reduce confluence proofs to the checking of various forms of critical pairs for higher-order rewrite rules extending beta-reduction on pure lambda-terms. As shown in a previous paper of the two middle authors, confluence of a terminating set of left-linear rewrite rules is obtained when their critical pairs are joinable, beta-rewrite steps being disallowed. The present paper concentrates on the case where arbitrary beta-rewrite steps are allowed for joining critical pairs. The rewrite relation used for analyzing confluence may rewrite arbitrarily many non-overlapping redexes in a single step. This relation gives rise to critical pairs that overlap both horizontally, as with parallel rewriting, but also vertically, forming chains of successive overlaps. Practical examples of use of this technique are analyzed.

Item recognition has become a hotspot in the field of computer vision research. SIFT has the advantage of requiring a low amount of information, a fast running speed and high precision, but it requires large data calculations and thus takes a long time to perform the item recognition. In this paper we propose a method of item recognition based on SIFT and SURF that provides a new way to solve the problem of item recognition, and has both feasibility and availability. This technique currently ignores colour information when dealing with colour images, but the evaluation method is capable of taking colour quality characteristics into account so it should be possible to improve the algorithm in the future. Experimental results show that this system of item recognition based on the SURF algorithm gives better matching recognition, is faster and has greater robustness.

Following Lutz's approach to effective (constructive) dimension, we define a notion of dimension for individual sequences based on Schnorr's concept(s) of randomness. In contrast to computable randomness and Schnorr randomness, the dimension concepts defined via computable martingales and Schnorr tests coincide, that is, the Schnorr Hausdorff dimension of a sequence always equals its computable Hausdorff dimension. Furthermore, we give a machine characterisation of the Schnorr dimension, based on prefix-free machines whose domain has computable measure. Finally, we show that there exist computably enumerable sets that are Schnorr (computably) irregular: while every c.e. set has Schnorr Hausdorff dimension 0, there are c.e. sets of computable packing dimension 1, which is, from Barzdiņš' Theorem, an impossible property for the case of effective (constructive) dimension. In fact, we prove that every hyperimmune Turing degree contains a set of computable packing dimension 1.

The object-calculus is an imperative and object-based programming language in which every object comes equipped with its own method suite. Consequently, methods need to reside in the store (‘higher-order store’), which complicates the semantics. Abadi and Leino defined a program logic for this language enriching object types by method specifications. We present a new soundness proof for their logic using denotational semantics. It turns out that denotations of store specifications are predicates defined by mixed-variant recursion. A benefit of our approach is that derivability and validity can be kept distinct. Moreover, it reveals which of the limitations of Abadi and Leino's logic are incidental design decisions and which follow inherently from the use of a higher-order store. We discuss the implications for the development of other, more expressive, program logics.

Let Σ be an arbitrary signature and ϒ be a non-empty set of operation symbols within it. A (partial) Σ-algebra is ϒ-total when all its operations in ϒ are total: these are the partly total algebras in the title, and they include total algebras and attributed graphs. In this paper we establish a necessary and sufficient condition on a pair of homomorphisms of ϒ-total Σ-algebras for the existence of a pushout complement of them. This solves the application problem for the double-pushout transformation of these kinds of structure.

In this paper we describe a method for proving the normalization property for a large variety of typed lambda calculi of first and second order, which is based on a proof of equivalence of two deduction systems. We first illustrate the method on the elementary example of simply typed lambda calculus, and then we show how to extend it to a more expressive dependent type system. Finally we use it to prove the normalization theorem for Girard's system F.

HyLL (Hybrid Linear Logic) is an extension of intuitionistic linear logic (ILL) that has been used as a framework for specifying systems that exhibit certain modalities. In HyLL, truth judgements are labelled by worlds (having a monoidal structure) and hybrid connectives (at and ↓) relate worlds with formulas. We start this work by showing that HyLL's axioms and rules can be adequately encoded in linear logic (LL), so that one focused step in LL will correspond to a step of derivation in HyLL. This shows that any proof in HyLL can be exactly mimicked by a LL focused derivation. Another extension of LL that has extensively been used for specifying systems with modalities is Subexponential Linear Logic (SELL). In SELL, the LL exponentials (!, ?) are decorated with labels representing locations, and a pre-order on such labels defines the provability relation. We propose an encoding of HyLL into SELL⋒ (SELL plus quantification over locations) that gives better insights about the meaning of worlds in HyLL. More precisely, we identify worlds as locations, and show that a flat subexponential structure is sufficient for representing any world structure in HyLL. This shows that HyLL's monoidal structure is not reflected in LL derivations, hence not increasing the expressiveness of LL, from a proof theoretical point of view. We conclude by proposing the notion of fixed points in multiplicative additive HyLL (μHyMALL), which can be encoded into multiplicative additive linear logic with fixed points (μMALL). As an application, we propose encodings of Computational Tree Logic (CTL) into both μMALL and μHyMALL. In the former, states are represented as atoms in the linear context, hence reflecting a more operational view of CTL connectives. In the latter, worlds represent states of the transition system, thus exhibiting a pleasant similarity with the semantics of CTL.

Given a set $\mathcal{D}$ of q documents, the Longest Common Substring (LCS) problem asks, for any integer 2 ⩽ k ⩽ q, the longest substring that appears in k documents. LCS is a well-studied problem having a wide range of applications in Bioinformatics: from microarrays to DNA sequences alignments and analysis. This problem has been solved by Hui (2000International Journal of Computer Science and Engineering15 73–76) by using a famous constant-time solution to the Lowest Common Ancestor (LCA) problem in trees coupled with the use of suffix trees.

In this article, we present a simple method for solving the LCS problem by using suffix trees (STs) and classical union-find data structures. In turn, we show how this simple algorithm can be adapted in order to work with other space efficient data structures such as the enhanced suffix arrays (ESA) and the compressed suffix tree.

We give a denotational semantics to a calculus λ[otimes ] with overloading and subtyping. In λ[otimes ], the interaction between overloading and subtyping causes self application, and non-normalizing terms exist for each type. Moreover, the semantics of a type depends not on that type alone, but also on infinitely many others. Thus, we need to consider infinitely many domains, which are related by an infinite number of mutually recursive equations. We solve this by considering a functor category from the poset of types modulo equivalence to a category in which each type is interpreted. We introduce a categorical constructor corresponding to overloading, and formalize the equations as a single equation in the functor category. A semantics of λ[otimes ] is then expressed in terms of the minimal solution of this equation. We prove the adequacy theorem for λ[otimes ] following the construction in Pitts (1994) and use it to derive some syntactic properties.

Frame and anti-frame rules have been proposed as proof rules for modular reasoning about programs. Frame rules allow the hiding of irrelevant parts of the state during verification, whereas the anti-frame rule allows the hiding of local state from the context.

We discuss the semantic foundations of frame and anti-frame rules, and present the first sound model for Charguéraud and Pottier's type and capability system including both of these rules. The model is a possible worlds model based on the operational semantics and step-indexed heap relations, and the worlds are given by a recursively defined metric space. We also extend the model to account for Pottier's generalised frame and anti-frame rules, where invariants are generalised to families of invariants indexed over preorders. This generalisation enables reasoning about some well-bracketed as well as (locally) monotone uses of local state.

We give a formulation of simple type theory in which types are just sets. This theory is obtained by relativizing quantifiers and replacing typing predicates by membership of some sets. It can easily be extended to include predicate subtyping.

Finitary monads on Pos are characterized as precisely the free-algebra monads of varieties of algebras. These are classes of ordered algebras specified by inequations in context. Analogously, finitary enriched monads on Pos are characterized: here we work with varieties of coherent algebras which means that their operations are monotone.

We show analogues of the Daniell–Kolmogorov and Prohorov theorems on the existence of projective limits of measures, in the setting of continuous valuations on T0 topological spaces.

In the concurrency theory, various semantic equivalences on transition systems are based on traces decorated with some additional observations, generally referred to as decorated traces. Using the generalized powerset construction, recently introduced by a subset of the authors (Silva et al.2010 FSTTCS. LIPIcs8 272–283), we give a coalgebraic presentation of decorated trace semantics. The latter include ready, failure, (complete) trace, possible futures, ready trace and failure trace semantics for labelled transition systems, and ready, (maximal) failure and (maximal) trace semantics for generative probabilistic systems. This yields a uniform notion of minimal representatives for the various decorated trace equivalences, in terms of final Moore automata. As a consequence, proofs of decorated trace equivalence can be given by coinduction, using different types of (Moore-) bisimulation (up-to context).

Any modality in homotopy type theory gives rise to an orthogonal factorization system of which the left class is stable under pullbacks. We show that there is a second orthogonal factorization system associated with any modality, of which the left class is the class of ○-equivalences and the right class is the class of ○-étale maps. This factorization system is called the modal reflective factorization system of a modality, and we give a precise characterization of the orthogonal factorization systems that arise as the modal reflective factorization system of a modality. In the special case of the n-truncation, the modal reflective factorization system has a simple description: we show that the n-étale maps are the maps that are right orthogonal to the map $${\rm{1}} \to {\rm{ }}{{\rm{S}}^{n + 1}}$$ . We use the ○-étale maps to prove a modal descent theorem: a map with modal fibers into ○X is the same thing as a ○-étale map into a type X. We conclude with an application to real-cohesive homotopy type theory and remark how ○-étale maps relate to the formally etale maps from algebraic geometry.

We investigate how various forms of bisimulation can be characterised using the technology of logical relations. The approach taken is that each form of bisimulation corresponds to an algebraic structure derived from a transition system, and the general result is that a relation R between two transition systems on state spaces S and T is a bisimulation if and only if the derived algebraic structures are in the logical relation automatically generated from R. We show that this approach works for the original Park–Milner bisimulation and that it extends to weak bisimulation, and branching and semi-branching bisimulation. The paper concludes with a discussion of probabilistic bisimulation, where the situation is slightly more complex, partly owing to the need to encompass bisimulations that are not just relations.

Bishop’s presentation of his informal system of constructive mathematics BISH was on purpose closer to the proof-irrelevance of classical mathematics, although a form of proof-relevance was evident in the use of several notions of moduli (of convergence, of uniform continuity, of uniform differentiability, etc.). Focusing on membership and equality conditions for sets given by appropriate existential formulas, we define certain families of proof sets that provide a BHK-interpretation of formulas that correspond to the standard atomic formulas of a first-order theory, within Bishop set theory $(\mathrm{BST})$ , our minimal extension of Bishop’s theory of sets. With the machinery of the general theory of families of sets, this BHK-interpretation within BST is extended to complex formulas. Consequently, we can associate to many formulas $\phi$ of BISH a set ${\texttt{Prf}}(\phi)$ of “proofs” or witnesses of $\phi$ . Abstracting from several examples of totalities in BISH, we define the notion of a set with a proof-relevant equality, and of a Martin-Löf set, a special case of the former, the equality of which corresponds to the identity type of a type in intensional Martin-Löf type theory $(\mathrm{MLTT})$ . Through the concepts and results of BST notions and facts of MLTT and its extensions (either with the axiom of function extensionality or with Vooevodsky’s axiom of univalence) can be translated into BISH. While Bishop’s theory of sets is standardly understood through its translation to MLTT, our development of BST offers a partial translation in the converse direction.