This paper is an extended version of Bílková et al. ((2023b). Logic, Language, Information, and Computation. WoLLIC 2023, Lecture Notes in Computer Science, vol. 13923, Cham, Springer Nature Switzerland, 101–117.). We discuss two-layered logics formalising reasoning with probabilities and belief functions that combine the Łukasiewicz $[0,1]$-valued logic with Baaz $\triangle$ operator and the Belnap–Dunn logic. We consider two probabilistic logics – $\mathsf {Pr}^{{\mathsf {\unicode {x0141}}}^2}_\triangle$ (introduced by Bílková et al. 2023d. Annals of Pure and Applied Logic, 103338.) and $\mathbf {4}\mathsf {Pr}^{{\mathsf {\unicode {x0141}}}_\triangle }$ (from Bílková et al. 2023b. Logic, Language, Information, and Computation. WoLLIC 2023, Lecture Notes in Computer Science, vol. 13923, Cham, Springer Nature Switzerland, 101–117.) – that present two perspectives on the probabilities in the Belnap–Dunn logic. In $\mathsf {Pr}^{{\mathsf {\unicode {x0141}}}^2}_\triangle$, every event $\phi$ has independent positive and negative measures that denote the likelihoods of $\phi$ and $\neg \phi$, respectively. In $\mathbf {4}\mathsf {Pr}^{{\mathsf {\unicode {x0141}}}_\triangle }$, the measures of the events are treated as partitions of the sample into four exhaustive and mutually exclusive parts corresponding to pure belief, pure disbelief, conflict and uncertainty of an agent in $\phi$. In addition to that, we discuss two logics for the paraconsistent reasoning with belief and plausibility functions from Bílková et al. ((2023d). Annals of Pure and Applied Logic, 103338.) – $\mathsf {Bel}^{{\mathsf {\unicode {x0141}}}^2}_\triangle$ and $\mathsf {Bel}^{\mathsf {N}{\mathsf {\unicode {x0141}}}}$. Both these logics equip events with two measures (positive and negative) with their main difference being that in $\mathsf {Bel}^{{\mathsf {\unicode {x0141}}}^2}_\triangle$, the negative measure of $\phi$ is defined as the belief in $\neg \phi$ while in $\mathsf {Bel}^{\mathsf {N}{\mathsf {\unicode {x0141}}}}$, it is treated independently as the plausibility of $\neg \phi$. We provide a sound and complete Hilbert-style axiomatisation of $\mathbf {4}\mathsf {Pr}^{{\mathsf {\unicode {x0141}}}_\triangle }$ and establish faithful translations between it and $\mathsf {Pr}^{\mathsf {\unicode {x0141}}^2}_\triangle$. We also show that the validity problem in all the logics is $\mathsf {coNP}$-complete.

We revisit the communication primitive in ambient calculi. Previously, such communication was confined to first-order (FO) mode (e.g., merely names or capabilities of ambients can be sent), local mode (e.g., the communication only occurs inside an ambient), or particular cross-hierarchy mode (e.g., parent-child communication). In this work, we explore further higher-order (HO) communication in ambient calculi. Specifically, such a communication mechanism allows sending a whole piece of a program across the borders of ambients and is the only form of communication that can happen exactly between ambients. Since ambients are basically of HO nature (i.e., those being moved may be ambients themselves), in a sense, it appears more natural to have HO communication than FO communication. We stipulate that communications merely occur between equally positioned ambients in a peer-to-peer fashion (e.g., between sibling ambients). Following this line, we drop the local or other forms of communication that violate this criterion. As the workbench, we work on a variant of Fair Ambients extended with HO communication, FAHO. This variant also strengthens the original version in that entirely real-identity interaction is guaranteed. We study the semantics, bisimulation, and expressiveness of FAHO. Particularly, we provide the operational semantics using a labeled transition system. Over the semantics, we define the bisimulation in line with the standard notion of bisimulation for ambients and prove that the bisimulation equivalence (i.e., bisimilarity) is a congruence. In addition, we demonstrate that bisimilarity coincides with observational congruence (i.e., barbed congruence). Moreover, we show that FAHO can encode a minimal Turing-complete HO calculus and thus is computationally complete.

We provide a fine classification of bisimilarities between states of possibly different labelled Markov processes (LMP). We show that a bisimilarity relation proposed by Panangaden that uses direct sums coincides with “event bisimilarity” from his joint work with Danos, Desharnais, and Laviolette. We also extend Giorgio Bacci’s notions of bisimilarity between two different processes to the case of nondeterministic LMP and generalize the game characterization of state bisimilarity by Clerc et al. for the latter.

In this paper, by means of upper approximation operators in rough set theory, we study representations for sL-domains and its special subclasses. We introduce the concepts of sL-approximation spaces, L-approximation spaces, and bc-approximation spaces, which are special types of CF-approximation spaces. We prove that the collection of CF-closed sets in an sL-approximation space (resp., an L-approximation space, a bc-approximation space) ordered by set-theoretic inclusion is an sL-domain (resp., an L-domain, a bc-domain); conversely, every sL-domain (resp., L-domain, bc-domain) is order-isomorphic to the collection of CF-closed sets of an sL-approximation space (resp., an L-approximation space, a bc-approximation space). Consequently, we establish an equivalence between the category of sL-domains (resp., L-domains) with Scott continuous mappings and that of sL-approximation spaces (resp., L-approximation spaces) with CF-approximable relations.

A topological space has a domain model if it is homeomorphic to the maximal point space $\mbox{Max}(P)$ of a domain $P$. Lawson proved that every Polish space $X$ has an $\omega$-domain model $P$ and for such a model $P$, $\mbox{Max}(P)$ is a $G_{\delta }$-set of the Scott space of $P$. Martin (2003) then asked whether it is true that for every $\omega$-domain $Q$, $\mbox{Max}(Q)$ is $G_{\delta }$-set of the Scott space of $Q$. In this paper, we give a negative answer to Martin’s long-standing open problem by constructing a counterexample. The counterexample here actually shows that the answer is no even for $\omega$-algebraic domains. In addition, we also construct an $\omega$-ideal domain $\widetilde{Q}$ for the constructed $Q$ such that their maximal point spaces are homeomorphic. Therefore, $\textrm{Max}(Q)$ is a $G_\delta$-set of the Scott space of the new model $\widetilde{Q}$ .

A categorical axiomatic theory of creation/annihilation operators on symmetric Fock space is introduced, and the combinatorial model that motivated it is presented. Commutation relations and coherent states are considered in both frameworks.

We present an axiom system for what we call Prior’s Ideal Language and prove its completeness and pure completeness with respect to general models. With this is done, we explain, with examples, why this system provides a useful setting for exploring Arthur Prior’s work.

We show that the principal types of the closed terms of the affine fragment of λ-calculus, with respect to a simple type discipline, are structurally isomorphic to their interpretations, as partial involutions, in a natural Geometry of Interaction model à la Abramsky. This permits to explain in elementary terms the somewhat awkward notion of linear application arising in Geometry of Interaction, simply as the resolution between principal types using an alternate unification algorithm. As a consequence, we provide an answer, for the purely affine fragment, to the open problem raised by Abramsky of characterizing those partial involutions which are denotations of combinatory terms.

We extend the recently introduced setting of coherent differentiation by taking into account not only differentiation but also Taylor expansion in categories which are not necessarily (left) additive. The main idea consists in extending summability into an infinitary functor which intuitively maps any object to the object of its countable summable families. This functor is endowed with a canonical structure of a bimonad. In a linear logical categorical setting, Taylor expansion is then axiomatized as a distributive law between this summability functor and the resource comonad (aka. exponential). This distributive law allows to extend the summability functor into a bimonad on the coKleisli category of the resource comonad: this extended functor computes the Taylor expansion of the (nonlinear) morphisms of the coKleisli category. We also show how this categorical axiomatization of Taylor expansion can be generalized to arbitrary cartesian categories, leading to a general theory of Taylor expansion formally similar to that of cartesian differential categories, although it does not require the underlying cartesian category to be left additive. We provide several examples of concrete categories that arise in denotational semantics and feature such analytic structures.

We investigate causal computations, which take sequences of inputs to sequences of outputs such that the $n$th output depends on the first $n$ inputs only. We model these in category theory via a construction taking a Cartesian category $\mathbb{C}$ to another category $\mathrm{St}(\mathbb{C})$ with a novel trace-like operation called “delayed trace,” which misses yanking and dinaturality axioms of the usual trace. The delayed trace operation provides a feedback mechanism in $\mathrm{St}(\mathbb{C})$ with an implicit guardedness guarantee. When $\mathbb{C}$ is equipped with a Cartesian differential operator, we construct a differential operator for $\mathrm{St}(\mathbb{C})$ using an abstract version of backpropagation through time (BPTT), a technique from machine learning based on unrolling of functions. This obtains a swath of properties for BPTT, including a chain rule and Schwartz theorem. Our differential operator is also able to compute the derivative of a stateful network without requiring the network to be unrolled.

Veltman semantics is the basic Kripke-like semantics for interpretability logic. Verbrugge semantics is a generalization of Veltman semantics. An appropriate notion of bisimulation between a Verbrugge model and a Veltman model is developed in this paper. We show that each Verbrugge model can be transformed to a bisimilar Veltman model.

In this paper, the notion of structural completeness is explored in the context of a generalized class of superintuitionistic logics that also involve systems that are not closed under uniform substitution. We just require that each logic must be closed under $D$-substitutions assigning to atomic formulas only $\vee$-free formulas. For these systems, we introduce four different notions of structural completeness and study how they are related. We focus on superintuitionistic inquisitive logics that validate a schema called Split and have the disjunction property. In these logics, disjunction can be interpreted in the sense of inquisitive semantics as a question-forming operator. It is shown that a logic is structurally complete with respect to $D$-substitutions if and only if it validates Split. Various consequences of this result are explored. For example, it is shown that every superintuitionistic inquisitive logic can be characterized by a Kripke model built from $D$-substitutions. We also formulate an algebraic counterpart of this result that says that the Lindenbaum–Tarski algebra ${\mathscr{H}}$ of any inquisitive logic can be embedded into the Heyting algebra formed from left ideals of endomorphisms on ${\mathscr{H}}$. Additionally, we resolve a conjecture concerning superintuitionistic inquisitive logics due to Miglioli et al. and show that a false conjecture about superintuitionistic logics due to Minari and Wroński becomes true in the broader space of regular generalized superintuitionistic logics.

Semi-simplicial and semi-cubical sets are commonly defined as presheaves over, respectively, the semi-simplex or semi-cube category. Homotopy type theory then popularized an alternative definition, where the set of $n$-simplices or $n$-cubes are instead regrouped into the families of the fibers over their faces, leading to a characterization we call indexed. Moreover, it is known that semi-simplicial and semi-cubical sets are related to iterated Reynolds parametricity, respectively, in their unary and binary variants. We exploit this correspondence to develop an original uniform indexed definition of both augmented semi-simplicial and semi-cubical sets, and fully formalize it in Coq.

Often in Software Engineering, a modeling formalism has to support scenarios of inconsistency in which several requirements either reinforce or contradict each other. Paraconsistent transition systems are proposed in this paper as one such formalism: states evolve through two accessibility relations capturing weighted evidence of a transition or its absence, respectively. Their weights come, parametrically, from a residuated lattice. This paper explores both i) a category of these systems, and the corresponding compositional operators and ii) a modal logic to reason upon them. Furthermore, two notions of crisp and graded simulation and bisimulation are introduced in order to relate two paraconsistent transition systems. Finally, results of modal invariance, for specific subsets of formulas, are discussed.

We discuss the logical principle of extensionality for set-valued operators and its relation to mathematical notions of continuity for these operators in the context of systems of finite types as used in proof mining. Concretely, we initially exhibit an issue that arises with treating full extensionality in the context of the prevalent intensional approach to set-valued operators in such systems. Motivated by these issues, we discuss a range of useful fragments of this full extensionality statement where these issues are avoided and discuss their interrelations. Further, we study the continuity principles associated with these fragments of extensionality and show how they can be introduced in the logical systems via a collection of axioms that do not contribute to the growth of extractable bounds from proofs. In particular, we place an emphasis on a variant of extensionality and continuity formulated using the Hausdorff-metric and, in the course of our discussion, we in particular employ a tame treatment of suprema over bounded sets developed by the author in previous work to provide the first proof-theoretically tame treatment of the Hausdorff metric in systems geared for proof mining. To illustrate the applicability of these treatments for the extraction of quantitative information from proofs, we provide an application of proof mining to the Mann iteration of set-valued mappings which are nonexpansive w.r.t. the Hausdorff metric and extract highly uniform and effective quantitative information on the convergence of that method.

This paper discusses the development of synthetic cohomology in Homotopy Type Theory (HoTT), as well as its computer formalisation. The objectives of this paper are (1) to generalise previous work on integral cohomology in HoTT by the current authors and Brunerie (2022) to cohomology with arbitrary coefficients and (2) to provide the mathematical details of, as well as extend, results underpinning the computer formalisation of cohomology rings by the current authors and Lamiaux (2023). With respect to objective (1), we provide new direct definitions of the cohomology group operations and of the cup product, which, just as in the previous work by the current authors and Brunerie (2022), enable significant simplifications of many earlier proofs in synthetic cohomology theory. In particular, the new definition of the cup product allows us to give the first complete formalisation of the axioms needed to turn the cohomology groups into a graded commutative ring. We also establish that this cohomology theory satisfies the HoTT formulation of the Eilenberg–Steenrod axioms for cohomology and study the classical Mayer–Vietoris and Gysin sequences. With respect to objective (2), we characterise the cohomology groups and rings of various spaces, including the spheres, torus, Klein bottle, real/complex projective planes, and infinite real projective space. All results have been formalised in Cubical Agda, and we obtain multiple new numbers, similar to the famous ‘Brunerie number’, which can be used as benchmarks for computational implementations of HoTT. Some of these numbers are infeasible to compute in Cubical Agda and hence provide new computational challenges and open problems which are much easier to define than the original Brunerie number.

Every directed graph $G$ induces a locally ordered metric space $\mathcal{X}_{(G)}$ together with a local order $\tilde {\mathcal{X}}_{(G)}$ that is locally dihomeomorphic to the standard pospace $\mathbb{R}$; both are related by a morphism ${\beta }_{(G)} G:\tilde {\mathcal{X}}_{(G)}\to {\mathcal{X}}_{(G)}$ satisfying a universal property. The underlying set of $\tilde {\mathcal{X}_{(G)}}$ admits a non-Hausdorff atlas $\mathcal{A}_{G}$ equipped with a non-vanishing vector field ${{f}}_{G}$; the latter is associated to $\tilde {\mathcal{X}}_{(G)}$ through the correspondence between local orders and cone fields on manifolds. The above constructions are compatible with Cartesian products, so the geometric model of a conservative program is lifted through ${{\beta }_{G_1}} \times \cdots \times {{\beta }}_{G_n}$ to a subset $M$ of the parallelized manifold $\mathcal{A}_{G_1} \times \cdots \times \mathcal{A}_{G_n}$. By assigning the suitable norm to each tangent space of $\mathcal{A}_{G_1} \times \cdots \times \mathcal{A}_{G_n}$, the length of every directed smooth path $\gamma$ on $M$, i.e. $\int {{|\gamma '(t)|}}_{\gamma (t)}dt$, corresponds to the execution time of the sequence of multi-instructions associated to $\gamma$. This induces a pseudometric ${{d}}_{\mathcal{A}}$ whose restrictions to sufficiently small open sets of $\mathcal{A}_{G_1} \times \cdots \times \mathcal{A}_{G_n}$ (we refer to the manifold topology, which is strictly finer than the pseudometric topology) are isometric to open subspaces of ${\mathbb{R}}^n$ with the $\alpha$-norm for some $\alpha \in [{{1}},{{\infty }}]$. The transition maps of $\mathcal{A}_{G}$ are translations, so the representation of a tangent vector does not depend on the chart of $\mathcal{A}_{G}$ in which it is represented; consequently, differentiable maps between open subsets of $\mathcal{A}_{G_{1}} \times \cdots \times \mathcal{A}_{G_{n}}$ are handled as if they were maps between open subsets of ${\mathbb{R}}^n$. For every directed path $\gamma$ on $M$ (possibly the representation of a sequence $\sigma$ of multi-instructions), there is a shorter directed smooth path on $M$ that is arbitrarily close to $\gamma$, and that can replace $\gamma$ as a representation of $\sigma$.

The Grothendieck construction establishes an equivalence between fibrations, a.k.a. fibred categories and indexed categories and is one of the fundamental results of category theory. Cockett and Cruttwell introduced the notion of fibrations into the context of tangent categories and proved that the fibres of a tangent fibration inherit a tangent structure from the total tangent category. The main goal of this paper is to provide a Grothendieck construction for tangent fibrations. Our first attempt will focus on providing a correspondence between tangent fibrations and indexed tangent categories, which are collections of tangent categories and tangent morphisms indexed by the objects and morphisms of a base tangent category. We will show that this construction inverts Cockett and Cruttwell’s result, but it does not provide a full equivalence between these two concepts. In order to understand how to define a genuine Grothendieck equivalence in the context of tangent categories, inspired by Street’s formal approach to monad theory we introduce a new concept: tangent objects. We show that tangent fibrations arise as tangent objects of a suitable $2$-category and we employ this characterisation to lift the Grothendieck construction between fibrations and indexed categories to a genuine Grothendieck equivalence between tangent fibrations and tangent indexed categories.

Stone locales together with continuous maps form a coreflective subcategory of spectral locales and perfect maps. A proof in the internal language of an elementary topos was previously given by the second-named author. This proof can be easily translated to univalent type theory using resizing axioms. In this work, we show how to achieve such a translation without resizing axioms, by working with large, locally small, and small-complete frames with small bases. This requires predicative reformulations of several fundamental concepts of locale theory in predicative HoTT/UF, which we investigate systematically.

We give a simple diagrammatic proof of the Frobenius property for generic fibrations that does not depend on any additional structure on the interval object such as connections.