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Dimension in team semantics Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2024-03-12 Lauri Hella, Kerkko Luosto, Jouko Väänänen
We introduce three measures of complexity for families of sets. Each of the three measures, which we call dimensions, is defined in terms of the minimal number of convex subfamilies that are needed for covering the given family. For upper dimension, the subfamilies are required to contain a unique maximal set, for dual upper dimension a unique minimal set, and for cylindrical dimension both a unique
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Implicational Kleene algebra with domain and the substructural logic of partial correctness Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2024-03-04 Igor Sedlár
We show that Kozen and Tiuryn’s substructural logic of partial correctness $\mathsf{S}$ embeds into the equational theory of Kleene algebra with domain, $\mathsf{KAD}$ . We provide an implicational formulation of $\mathsf{KAD}$ which sets $\mathsf{S}$ in the context of implicational extensions of Kleene algebra.
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Parameterized complexity of weighted team definability Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2024-02-20 Juha Kontinen, Yasir Mahmood, Arne Meier, Heribert Vollmer
In this article, we study the complexity of weighted team definability for logics with team semantics. This problem is a natural analog of one of the most studied problems in parameterized complexity, the notion of weighted Fagin-definability, which is formulated in terms of satisfaction of first-order formulas with free relation variables. We focus on the parameterized complexity of weighted team
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Discrete equational theories Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2024-01-22 J. Rosický
On a locally $\lambda$-presentable symmetric monoidal closed category $\mathcal {V}$, $\lambda$-ary enriched equational theories correspond to enriched monads preserving $\lambda$-filtered colimits. We introduce discrete $\lambda$-ary enriched equational theories where operations are induced by those having discrete arities (equations are not required to have discrete arities) and show that they correspond
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Approximation Algorithm and FPT Algorithm for Connected-k-Subgraph Cover on Minor-Free Graphs Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2024-01-10 Pengcheng Liu, Zhao Zhang, Yingli Ran, Xiaohui Huang
Given a graph G, the minimum Connected-k-Subgraph Cover problem (MinCkSC) is to find a minimum vertex subset C of G such that every connected subgraph of G on k vertices has at least one vertex in C. If furthermore the subgraph of G induced by C is connected, then the problem is denoted as MinCkSC $_{con}$ . In this paper, we first present a PTAS for MinCkSC on an H-minor-free graph, where H is a graph
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A construction of free dcpo-cones Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2024-01-10 Yuxu Chen, Hui Kou, Zhenchao Lyu, Xiaolin Xie
We give a construction of the free dcpo-cone over any dcpo. There are two steps for getting this result. Firstly, we extend the notion of power domain to directed spaces which are equivalent to $T_0$ monotone-determined spaces introduced by Erné, and we construct the probabilistic powerspace of the monotone determined space, which is defined as a free monotone determined cone. Secondly, we take D-completion
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Game characterizations for the number of quantifiers Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2024-01-10 Lauri Hella, Kerkko Luosto
A game that characterizes equivalence of structures with respect to all first-order sentences containing a given number of quantifiers was introduced by Immerman in 1981. We define three other games and prove that they are all equivalent to the Immerman game, and hence also give a characterization for the number of quantifiers needed for separating structures. In the Immerman game, Duplicator has a
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The order-K-ification monads Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-12-21 Huijun Hou, Hualin Miao, Qingguo Li
Monads prove to be useful mathematical tools in theoretical computer science, notably in denoting different effects of programming languages. In this paper, we investigate a type of monads which arise naturally from Keimel and Lawson’s $\mathbf{K}$-ification. A subcategory of $\mathbf{TOP}_{\mathbf{0}}$ is called of type $\mathrm{K}^{*}$ if it consists of monotone convergence spaces and is of type
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Visibility and exploitation in social networks Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-12-19 Rustam Galimullin, Mina Young Pedersen
Social media is not a neutral channel. How visible information posted online is depends on many factors such as the network structure, the emotional volatility of the content, and the design of the social media platform. In this paper, we use formal methods to study the visibility of agents and information in a social network, as well as how vulnerable the network is to exploitation. We introduce a
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Fixed point logics and definable topological properties Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-12-13 David Fernández-Duque, Quentin Gougeon
Modal logic enjoys topological semantics that may be traced back to McKinsey and Tarski, and the classification of topological spaces via modal axioms is a lively area of research. In the past two decades, there has been interest in extending topological modal logic to the language of the mu-calculus, but previously no class of topological spaces was known to be mu-calculus definable that was not already
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A point-free perspective on lax extensions and predicate liftings Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-12-01 Sergey Goncharov, Dirk Hofmann, Pedro Nora, Lutz Schröder, Paul Wild
Lax extensions of set functors play a key role in various areas, including topology, concurrent systems, and modal logic, while predicate liftings provide a generic semantics of modal operators. We take a fresh look at the connection between lax extensions and predicate liftings from the point of view of quantale-enriched relations. Using this perspective, we show in particular that various fundamental
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Learning quantum finite automata with queries Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-11-30 Daowen Qiu
Learning finite automata (termed as model learning) has become an important field in machine learning and has been useful realistic applications. Quantum finite automata (QFA) are simple models of quantum computers with finite memory. Due to their simplicity, QFA have well physical realizability, but one-way QFA still have essential advantages over classical finite automata with regard to state complexity
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A concrete model for a typed linear algebraic lambda calculus Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-11-21 Alejandro Díaz-Caro, Octavio Malherbe
We give an adequate, concrete, categorical-based model for Lambda-${\mathcal S}$, which is a typed version of a linear-algebraic lambda calculus, extended with measurements. Lambda-${\mathcal S}$ is an extension to first-order lambda calculus unifying two approaches of non-cloning in quantum lambda-calculi: to forbid duplication of variables and to consider all lambda-terms as algebraic linear functions
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Material dialogues for first-order logic in constructive type theory: extended version Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-11-03 Dominik Wehr, Dominik Kirst
Dialogues are turn-taking games which model debates about the satisfaction of logical formulas. A novel variant played over first-order structures gives rise to a notion of first-order satisfaction. We study the induced notion of validity for classical and intuitionistic first-order logic in the constructive setting of the calculus of inductive constructions. We prove that such material dialogue semantics
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Dilations and information flow axioms in categorical probability Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-10-25 Tobias Fritz, Tomáš Gonda, Nicholas Gauguin Houghton-Larsen, Antonio Lorenzin, Paolo Perrone, Dario Stein
We study the positivity and causality axioms for Markov categories as properties of dilations and information flow and also develop variations thereof for arbitrary semicartesian monoidal categories. These help us show that being a positive Markov category is merely an additional property of a symmetric monoidal category (rather than extra structure). We also characterize the positivity of representable
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Bicategorical type theory: semantics and syntax Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-10-17 Benedikt Ahrens, Paige Randall North, Niels van der Weide
We develop semantics and syntax for bicategorical type theory. Bicategorical type theory features contexts, types, terms, and directed reductions between terms. This type theory is naturally interpreted in a class of structured bicategories. We start by developing the semantics, in the form of comprehension bicategories. Examples of comprehension bicategories are plentiful; we study both specific examples
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Computable soft separation axioms Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-10-12 S. M. Elsayed, Keng Meng Ng
Soft sets were introduced as a means to study objects that are not defined in an absolute way and have found applications in numerous areas of mathematics, decision theory, and in statistical applications. Soft topological spaces were first considered in Shabir and Naz ((2011). Computers & Mathematics with Applications 61 (7) 1786–1799) and soft separation axioms for soft topological spaces were studied
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Constrained read-once refutations in UTVPI constraint systems: A parallel perspective Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-09-11 K. Subramani, Piotr Wojciechowski
In this paper, we analyze two types of refutations for Unit Two Variable Per Inequality (UTVPI) constraints. A UTVPI constraint is a linear inequality of the form: $a_{i}\cdot x_{i}+a_{j} \cdot x_{j} \le b_{k}$ , where $a_{i},a_{j}\in \{0,1,-1\}$ and $b_{k} \in \mathbb{Z}$ . A conjunction of such constraints is called a UTVPI constraint system (UCS) and can be represented in matrix form as: ${\bf A
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The long exact sequence of homotopy n-groups Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-09-07 Ulrik Buchholtz, Egbert Rijke
Working in homotopy type theory, we introduce the notion of n-exactness for a short sequence $F\to E\to B$ of pointed types and show that any fiber sequence $F\hookrightarrow E \twoheadrightarrow B$ of arbitrary types induces a short sequence that is n-exact at $\| E\|_{n-1}$. We explain how the indexing makes sense when interpreted in terms of n-groups, and we compare our definition to the existing
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Substitution Principle and semidirect products Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-08-15 Célia Borlido, Mai Gehrke
In the classical theory of regular languages, the concept of recognition by profinite monoids is an important tool. Beyond regularity, Boolean spaces with internal monoids (BiMs) were recently proposed as a generalization. On the other hand, fragments of logic defining regular languages can be studied inductively via the so-called “Substitution Principle.” In this paper, we make the logical underpinnings
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Robustness, Scott continuity, and computability Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-08-04 Amin Farjudian, Eugenio Moggi
Robustness is a property of system analyses, namely monotonic maps from the complete lattice of subsets of a (system’s state) space to the two-point lattice. The definition of robustness requires the space to be a metric space. Robust analyses cannot discriminate between a subset of the metric space and its closure; therefore, one can restrict to the complete lattice of closed subsets. When the metric
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Apartness, sharp elements, and the Scott topology of domains Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-08-02 Tom de Jong
Working constructively, we study continuous directed complete posets (dcpos) and the Scott topology. Our two primary novelties are a notion of intrinsic apartness and a notion of sharp elements. Being apart is a positive formulation of being unequal, similar to how inhabitedness is a positive formulation of nonemptiness. To exemplify sharpness, we note that a lower real is sharp if and only if it is
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Game semantics of Martin-Löf type theory Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-07-31 Norihiro Yamada
This work presents game semantics of Martin-Löf type theory (MLTT) equipped with the One, the Zero, the N, Pi, Sigma and Id types. Game semantics interprets a wide range of logic and computation, even the polymorphic $\lambda$-calculus; however, it has remained a well-known challenge in the past 25 years to achieve game semantics of dependent type theories such as MLTT, and past attempts lack directness
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Divergences on monads for relational program logics Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-07-31 Tetsuya Sato, Shin-ya Katsumata
Several relational program logics have been introduced for integrating reasoning about relational properties of programs and measurement of quantitative difference between computational effects. Toward a general framework for such logics, in this paper, we formalize the concept of quantitative difference between computational effects as divergences on monads, then develop a relational program logic
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Not every countable complete distributive lattice is sober Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-07-28 Hualin Miao, Xiaoyong Xi, Qingguo Li, Dongsheng Zhao
The study of the sobriety of Scott spaces has got a relatively long history in domain theory. Lawson and Hoffmann independently proved that the Scott space of every continuous directed complete poset (usually called domain) is sober. Johnstone constructed the first directed complete poset whose Scott space is non-sober. Soon after, Isbell gave a complete lattice with a non-sober Scott space. Based
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Scott topology on Smyth power posets Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-07-27 Xiaoquan Xu, Xinpeng Wen, Xiaoyong Xi
For a $T_0$ space X, let $\mathsf{K}(X)$ be the poset of all nonempty compact saturated subsets of X endowed with the Smyth order $\sqsubseteq$. $(\mathsf{K}(X), \sqsubseteq)$ (shortly $\mathsf{K}(X)$) is called the Smyth power poset of X. In this paper, we mainly discuss some basic properties of the Scott topology on Smyth power posets. It is proved that for a well-filtered space X, its Smyth power
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A general framework for the semantics of type theory Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-07-24 Taichi Uemura
We propose an abstract notion of a type theory to unify the semantics of various type theories including Martin–Löf type theory, two-level type theory, and cubical type theory. We establish basic results in the semantics of type theory: every type theory has a bi-initial model; every model of a type theory has its internal language; the category of theories over a type theory is bi-equivalent to a
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CHAD for expressive total languages Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-07-14 Fernando Lucatelli Nunes, Matthijs Vákár
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;
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Two-level type theory and applications - ERRATUM Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-07-10 Danil Annenkov, Paolo Capriotti, Nicolai Kraus, Christian Sattler
We define and develop two-level type theory (2LTT), a version of Martin-Löf type theory which combines two different type theories. We refer to them as the ‘inner’ and the ‘outer’ type theory. In our case of interest, the inner theory is homotopy type theory (HoTT) which may include univalent universes and higher inductive types. The outer theory is a traditional form of type theory validating uniqueness
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Up-to techniques for behavioural metrics via fibrations Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-07-10 Filippo Bonchi, Barbara König, Daniela Petrişan
Up-to techniques are a well-known method for enhancing coinductive proofs of behavioural equivalences. We introduce up-to techniques for behavioural metrics between systems modelled as coalgebras, and we provide abstract results to prove their soundness in a compositional way. In order to obtain a general framework, we need a systematic way to lift functors: we show that the Wasserstein lifting of
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A (machine-oriented) logic based on pattern matching Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-07-05 Tim Lethen
Robinson’s unification algorithm can be identified as the underlying machinery of both C. Meredith’s rule D (condensed detachment) in propositional logic and of the construction of principal types in lambda calculus and combinatory logic. In combinatory logic, it also plays a crucial role in the construction of Meyer, Bunder & Powers’ Fool’s model. This paper now considers pattern matching, the unidirectional
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Univalent categories of modules Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-06-23 Jarl G. Taxerås Flaten
We show that categories of modules over a ring in homotopy type theory (HoTT) satisfy the internal versions of the AB axioms from homological algebra. The main subtlety lies in proving AB4, which is that coproducts indexed by arbitrary sets are left-exact. To prove this, we replace a set X with the strict category of lists of elements in X. From showing that the latter is filtered, we deduce left-exactness
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Two-level type theory and applications Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-05-30 Danil Annenkov, Paolo Capriotti, Nicolai Kraus, Christian Sattler
We define and develop two-level type theory (2LTT), a version of Martin-Löf type theory which combines two different type theories. We refer to them as the ‘inner’ and the ‘outer’ type theory. In our case of interest, the inner theory is homotopy type theory (HoTT) which may include univalent universes and higher inductive types. The outer theory is a traditional form of type theory validating uniqueness
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A domain-theoretic framework for robustness analysis of neural networks Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-05-23 Can Zhou, Razin A. Shaikh, Yiran Li, Amin Farjudian
A domain-theoretic framework is presented for validated robustness analysis of neural networks. First, global robustness of a general class of networks is analyzed. Then, using the fact that Edalat’s domain-theoretic L-derivative coincides with Clarke’s generalized gradient, the framework is extended for attack-agnostic local robustness analysis. The proposed framework is ideal for designing algorithms
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Coherent differentiation Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-04-28 Thomas Ehrhard
The categorical models of differential linear logic (LL) are additive categories and those of the differential lambda-calculus are left-additive categories because of the Leibniz rule which requires the summation of two expressions. This means that, as far as the differential lambda-calculus and differential LL are concerned, these models feature finite nondeterminism and indeed these languages are
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What should a generic object be? Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-04-25 Jonathan Sterling
Jacobs has proposed definitions for (weak, strong, split) generic objects for a fibered category; building on his definition of (split) generic objects, Jacobs develops a menagerie of important fibrational structures with applications to categorical logic and computer science, including higher order fibrations, polymorphic fibrations, $\lambda2$ -fibrations, triposes, and others. We observe that a
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An intuitionistic set-theoretical model of fully dependent CC Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-04-17 Masahiro Sato, Jacques Garrigue
Werner’s set-theoretical model is one of the simplest models of CIC. It combines a functional view of predicative universes with a collapsed view of the impredicative sort “ ${\tt Prop}$ ”. However, this model of ${\tt Prop}$ is so coarse that the principle of excluded middle $P \lor \neg P$ holds. Following our previous work, we interpret ${\tt Prop}$ into a topological space (a special case of Heyting
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From type theory to setoids and back Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-04-12 Erik Palmgren
A model of Martin-Löf extensional type theory with universes is formalized in Agda, an interactive proof system based on Martin-Löf intensional type theory. This may be understood, we claim, as a solution to the old problem of modeling the full extensional theory in the intensional theory. Types are interpreted as setoids, and the model is therefore a setoid model.We solve the problem of interpreting
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Behavioural equivalences for continuous-time Markov processes Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-03-30 Linan Chen, Florence Clerc, Prakash Panangaden
Bisimulation is a concept that captures behavioural equivalence of states in a variety of types of transition systems. It has been widely studied in a discrete-time setting. The core of this work is to generalise the discrete-time picture to continuous time by providing a notion of behavioural equivalence for continuous-time Markov processes. In Chen et al. [(2019). Electronic Notes in Theoretical
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Local Yoneda completions of quasi-metric spaces Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-03-24 Jing Lu, Bin Zhao
In this paper, we study quasi-metric spaces using domain theory. Given a quasi-metric space (X,d), we use $({\bf B}(X,d),\leq^{d^{+}}\!)$ to denote the poset of formal balls of the associated quasi-metric space (X,d). We introduce the notion of local Yoneda-complete quasi-metric spaces in terms of domain-theoretic properties of $({\bf B}(X,d),\leq^{d^{+}}\!)$ . The manner in which this definition is
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Stochastic linearized generalized alternating direction method of multipliers: Expected convergence rates and large deviation properties Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-03-14 Jia Hu, Tiande Guo, Congying Han
Alternating direction method of multipliers (ADMM) receives much attention in the field of optimization and computer science, etc. The generalized ADMM (G-ADMM) proposed by Eckstein and Bertsekas incorporates an acceleration factor and is more efficient than the original ADMM. However, G-ADMM is not applicable in some models where the objective function value (or its gradient) is computationally costly
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Monoidal reverse differential categories Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-02-20 Geoff Cruttwell, Jonathan Gallagher, Jean-Simon Pacaud Lemay, Dorette Pronk
Cartesian reverse differential categories (CRDCs) are a recently defined structure which categorically model the reverse differentiation operations used in supervised learning. Here, we define a related structure called a monoidal reverse differential category, prove important results about its relationship to CRDCs, and provide examples of both structures, including examples coming from models of
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On notions of compactness, object classifiers, and weak Tarski universes Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-02-20 Raffael Stenzel
We prove a correspondence between $\kappa$-small fibrations in simplicial presheaf categories equipped with the injective or projective model structure (and left Bousfield localizations thereof) and relatively $\kappa$-compact maps in their underlying quasi-categories for suitably large regular cardinals $\kappa$. We thus obtain a transition result between weakly universal small fibrations in the (type-theoretic)
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A rewriting coherence theorem with applications in homotopy type theory Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-02-06 Nicolai Kraus, Jakob von Raumer
Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by finding a homotopy basis for the rewriting system. We show that the basic notions of confluence and wellfoundedness are sufficient to recursively build such a homotopy
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On function spaces equipped with Isbell topology and Scott topology Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-02-05 Xiaoquan Xu, Meng Bao, Xiaoyuan Zhang
In this paper, we mainly study the function spaces related to H-sober spaces. For an irreducible subset system H and $T_{0}$ spaces X and Y, it is proved that the following three conditions are equivalent: (1) the Scott space $\Sigma \mathcal O(X)$ of the lattice of all open sets of X is H-sober; (2) for every H-sober space Y, the function space $\mathbb{C}(X, Y)$ of all continuous mappings from X
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Weighted synchronous automata Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-01-25 Leandro Gomes, Alexandre Madeira, Luis Soares Barbosa
This paper introduces a class of automata and associated languages, suitable to model a computational paradigm of fuzzy systems, in which both vagueness and simultaneity are taken as first-class citizens. This requires a weighted semantics for transitions and a precise notion of a synchronous product to enforce the simultaneous occurrence of actions. The usual relationships between automata and languages
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On reduction and normalization in the computational core Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-01-16 Claudia Faggian, Giulio Guerrieri, Ugo de’ Liguoro, Riccardo Treglia
We study the reduction in a $\lambda$ -calculus derived from Moggi’s computational one, which we call the computational core. The reduction relation consists of rules obtained by orienting three monadic laws. Such laws, in particular associativity and identity, introduce intricacies in the operational analysis. We investigate the central notions of returning a value versus having a normal form and
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Normalization in the simply typed -calculus Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-01-12 Péter Battyányi, Karim Nour
In this paper, in connection with the program of extending the Curry–Howard isomorphism to classical logic, we study the $\lambda \mu$ -calculus of Parigot emphasizing the difference between the original version of Parigot and the version of de Groote in terms of normalization properties. In order to talk about a satisfactory representation of the integers, besides the usual $\beta$ -, $\mu$ -, and
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Z property for the shuffling calculus Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2023-01-09 Koji Nakazawa, Ken-etsu Fujita, Yuta Imagawa
This paper gives a new proof of confluence for Carraro and Guerrieri’s call-by-value lambda calculus λvσ with permutation rules. We adapt the compositional Z theorem to λvσ.
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A dual-context sequent calculus for the constructive modal logic S4 Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2022-11-23 Favio Ezequiel Miranda-Perea, Lourdes del Carmen González Huesca, Pilar Selene Linares Arévalo
The proof theory of the constructive modal logic S4 (hereafter $\mathsf{CS4}$ ) has been settled since the beginning of this century by means of either standard natural deduction and sequent calculi or by the reconstruction of modal logic through hypothetical and categorical judgments à la Martin-Löf, an approach carried out by using a special kind of sequents, which keeps two separated contexts representing
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A linear logic framework for multimodal logics Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2022-11-22 Bruno Xavier, Carlos Olarte, Elaine Pimentel
One of the most fundamental properties of a proof system is analyticity, expressing the fact that a proof of a given formula F only uses subformulas of F. In sequent calculus, this property is usually proved by showing that the $\mathsf{cut}$ rule is admissible, i.e., the introduction of the auxiliary lemma H in the reasoning “if H follows from G and F follows from H, then F follows from G” can be
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Rewriting in Gray categories with applications to coherence Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2022-11-22 Simon Forest, Samuel Mimram
Over the recent years, the theory of rewriting has been used and extended in order to provide systematic techniques to show coherence results for strict higher categories. Here, we investigate a further generalization to Gray categories, which are known to be equivalent to tricategories. This requires us to develop the theory of rewriting in the setting of precategories, which are adapted to mechanized
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Semantic analysis of normalisation by evaluation for typed lambda calculus Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2022-11-22 Marcelo Fiore
This paper studies normalisation by evaluation for typed lambda calculus from a categorical and algebraic viewpoint. The first part of the paper analyses the lambda definability result of Jung and Tiuryn via Kripke logical relations and shows how it can be adapted to unify definability and normalisation, yielding an extensional normalisation result. In the second part of the paper, the analysis is
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Weakest preconditions in fibrations Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2022-10-28 Alejandro Aguirre, Shin-ya Katsumata, Satoshi Kura
Weakest precondition transformers are useful tools in program verification. One of their key properties is composability, that is, the weakest precondition predicate transformer (wppt for short) associated to program $f;\;g$ should be equal to the composition of the wppts associated to f and g. In this paper, we study the categorical structure behind wppts from a fibrational point of view. We characterize
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Preserving consistency in geometric modeling with graph transformations Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2022-10-18 Agnès Arnould, Hakim Belhaouari, Thomas Bellet, Pascale Le Gall, Romain Pascual
Labeled graphs are particularly well adapted to represent objects in the context of topology-based geometric modeling. Thus, graph transformation theory is used to implement modeling operations and check their consistency. This article defines a class of graph transformation rules dedicated to embedding computations. Objects are here defined as a particular subclass of labeled graphs in which arc labels
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An algebraic representation of the fixed-point closure of *-continuous Kleene algebras – A categorical Chomsky–Schützenberger theorem Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2022-10-14 Hans Leiß
The family ${\mathcal{R}} X^*$ of regular subsets of the free monoid $X^*$ generated by a finite set X is the standard example of a ${}^*$ -continuous Kleene algebra. Likewise, the family ${\mathcal{C}} X^*$ of context-free subsets of $X^*$ is the standard example of a $\mu$ -continuous Chomsky algebra, i.e. an idempotent semiring that is closed under a well-behaved least fixed-point operator $\mu$
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Complete algebraic semantics for second-order rewriting systems based on abstract syntax with variable binding Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2022-10-14 Makoto Hamana
By using algebraic structures in a presheaf category over finite sets, following Fiore, Plotkin and Turi, we develop sound and complete models of second-order rewriting systems called second-order computation systems (CSs). Restricting the algebraic structures to those equipped with well-founded relations, we obtain a complete characterisation of terminating CSs. We also extend the characterisation
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Some representations of real numbers using integer sequences Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2022-10-14 Loïc Mazo, Marie-Andrée Da Col-Jacob, Laurent Fuchs, Nicolas Magaud, Gaëlle Skapin
The paper describes three models of the real field based on subsets of the integer sequences. The three models are compared to the Harthong–Reeb line. Two of the new models, contrary to the Harthong–Reeb line, provide accurate integer “views” on real numbers at a sequence of growing scales $B^n$ ( $B\ge2$ ).
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The combinator M and the Mockingbird lattice Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2022-10-13 Samuele Giraudo
We study combinatorial and order theoretic structures arising from the fragment of combinatory logic spanned by the basic combinator ${{\mathbf{M}}}$ . This basic combinator, named as the Mockingbird by Smullyan, is defined by the rewrite rule ${{\mathbf{M}}} \mathsf{x}_1 \to \mathsf{x}_1 \mathsf{x}_1$ . We prove that the reflexive and transitive closure of this rewrite relation is a partial order
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String diagram rewrite theory II: Rewriting with symmetric monoidal structure Math. Struct. Comput. Sci. (IF 0.5) Pub Date : 2022-09-29 Filippo Bonchi, Fabio Gadducci, Aleks Kissinger, Pawel Sobocinski, Fabio Zanasi
Symmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are replaced by string diagrams, topological entities that can be intuitively thought of as diagrams of wires and boxes. Recently, string diagrams have become increasingly