72 research outputs found

    A Finite Axiomatisation of Finite-State Automata Using String Diagrams

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    We develop a fully diagrammatic approach to finite-state automata, based on reinterpreting their usual state-transition graphical representation as a two-dimensional syntax of string diagrams. In this setting, we are able to provide a complete equational theory for language equivalence, with two notable features. First, the proposed axiomatisation is finite. Second, the Kleene star is a derived concept, as it can be decomposed into more primitive algebraic blocks.Comment: arXiv admin note: text overlap with arXiv:2009.1457

    A FINITE AXIOMATISATION OF FINITE-STATE AUTOMATA USING STRING DIAGRAMS

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    We develop a fully diagrammatic approach to finite-state automata, based on reinterpreting their usual state-transition graphical representation as a two-dimensional syntax of string diagrams. In this setting, we are able to provide a complete equational theory for language equivalence, with two notable features. First, the proposed axiomatisation is finite. Second, the Kleene star is a derived concept, as it can be decomposed into more primitive algebraic blocks

    Categorical Modelling of Logic Programming: Coalgebra, Functorial Semantics, String Diagrams

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    Logic programming (LP) is driven by the idea that logic subsumes computation. Over the past 50 years, along with the emergence of numerous logic systems, LP has also grown into a large family, the members of which are designed to deal with various computation scenarios. Among them, we focus on two of the most influential quantitative variants are probabilistic logic programming (PLP) and weighted logic programming (WLP). In this thesis, we investigate a uniform understanding of logic programming and its quan- titative variants from the perspective of category theory. In particular, we explore both a coalgebraic and an algebraic understanding of LP, PLP and WLP. On the coalgebraic side, we propose a goal-directed strategy for calculating the probabilities and weights of atoms in PLP and WLP programs, respectively. We then develop a coalgebraic semantics for PLP and WLP, built on existing coalgebraic semantics for LP. By choosing the appropriate functors representing probabilistic and weighted computation, such coalgeraic semantics characterise exactly the goal-directed behaviour of PLP and WLP programs. On the algebraic side, we define a functorial semantics of LP, PLP, and WLP, such that they three share the same syntactic categories of string diagrams, and differ regarding to the semantic categories according to their data/computation type. This allows for a uniform diagrammatic expression for certain semantic constructs. Moreover, based on similar approaches to Bayesian networks, this provides a framework to formalise the connection between PLP and Bayesian networks. Furthermore, we prove a sound and complete aximatization of the semantic category for LP, in terms of string diagrams. Together with the diagrammatic presentation of the fixed point semantics, one obtain a decidable calculus for proving the equivalence between propositional definite logic programs

    The Formal Theory of Monads, Univalently

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    We develop the formal theory of monads, as established by Street, in univalent foundations. This allows us to formally reason about various kinds of monads on the right level of abstraction. In particular, we define the bicategory of monads internal to a bicategory, and prove that it is univalent. We also define Eilenberg-Moore objects, and we show that both Eilenberg-Moore categories and Kleisli categories give rise to Eilenberg-Moore objects. Finally, we relate monads and adjunctions in arbitrary bicategories. Our work is formalized in Coq using the https://github.com/UniMath/UniMath library

    Tape diagrams for rig categories with finite biproducts

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    Rig categories with finite biproducts are categories with two monoidal products, where one is a biproduct and the other distributes over it. In this report we present tape diagrams, a sound and complete diagrammatic language for rig categories with finite biproducts, which can be thought intuitively as string diagrams of string diagrams

    Localisable Monads

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    Monads govern computational side-effects in programming semantics. A collection of monads can be combined together in a local-to-global way to handle several instances of such effects. Indexed monads and graded monads do this in a modular way. Here, instead, we start with a single monad and equip it with a fine-grained structure by using techniques from tensor topology. This provides an intrinsic theory of local computational effects without needing to know how constituent effects interact beforehand. Specifically, any monoidal category decomposes as a sheaf of local categories over a base space. We identify a notion of localisable monads which characterises when a monad decomposes as a sheaf of monads. Equivalently, localisable monads are formal monads in an appropriate presheaf 2-category, whose algebras we characterise. Three extended examples demonstrate how localisable monads can interpret the base space as locations in a computer memory, as sites in a network of interacting agents acting concurrently, and as time in stochastic processes

    Foundations of Software Science and Computation Structures

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    This open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems

    Formalizing of Category Theory in Agda

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    The generality and pervasiness of category theory in modern mathematics makes it a frequent and useful target of formalization. It is however quite challenging to formalize, for a variety of reasons. Agda currently (i.e. in 2020) does not have a standard, working formalization of category theory. We document our work on solving this dilemma. The formalization revealed a number of potential design choices, and we present, motivate and explain the ones we picked. In particular, we find that alternative definitions or alternative proofs from those found in standard textbooks can be advantageous, as well as "fit" Agda's type theory more smoothly. Some definitions regarded as equivalent in standard textbooks turn out to make different "universe level" assumptions, with some being more polymorphic than others. We also pay close attention to engineering issues so that the library integrates well with Agda's own standard library, as well as being compatible with as many of supported type theories in Agda as possible

    Topos and Stacks of Deep Neural Networks

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    Every known artificial deep neural network (DNN) corresponds to an object in a canonical Grothendieck's topos; its learning dynamic corresponds to a flow of morphisms in this topos. Invariance structures in the layers (like CNNs or LSTMs) correspond to Giraud's stacks. This invariance is supposed to be responsible of the generalization property, that is extrapolation from learning data under constraints. The fibers represent pre-semantic categories (Culioli, Thom), over which artificial languages are defined, with internal logics, intuitionist, classical or linear (Girard). Semantic functioning of a network is its ability to express theories in such a language for answering questions in output about input data. Quantities and spaces of semantic information are defined by analogy with the homological interpretation of Shannon's entropy (P.Baudot and D.B. 2015). They generalize the measures found by Carnap and Bar-Hillel (1952). Amazingly, the above semantical structures are classified by geometric fibrant objects in a closed model category of Quillen, then they give rise to homotopical invariants of DNNs and of their semantic functioning. Intentional type theories (Martin-Loef) organize these objects and fibrations between them. Information contents and exchanges are analyzed by Grothendieck's derivators
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