IA Scholar Query: Positive fragments of coalgebraic logics.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgWed, 23 Nov 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Data-Codata Symmetry and its Interaction with Evaluation Order
https://scholar.archive.org/work/5grx7ackmnb3dp7urxwykt3yui
Data types and codata types are, as the names suggest, often seen as duals of each other. However, most programming languages do not support both of them in their full generality, or if they do, they are still seen as distinct constructs with separately defined type-checking, compilation, etc. Rendel et al. were the first to propose variants of two standard program transformations, de- and refunctionalization, as a test to gauge and improve the symmetry between data and codata types. However, in previous works, codata and data were still seen as separately defined language constructs, with de- and refunctionalization being defined as similar but separate algorithms. These works also glossed over interactions between the aforementioned transformations and evaluation order, which leads to a loss of desirable η expansion equalities. We argue that the failure of complete symmetry is due to the inherent asymmetry of natural deduction as the logical foundation of the language design. Natural deduction is asymmetric in that its focus is on producers (proofs) of types, whereas consumers (contexts, continuations, refutations) have a second-class status. Inspired by existing sequent-calculus-based language designs, we present the first language design that is fully symmetric in that the issues of polarity (data type vs codata types) and evaluation order (call-by-value vs call-by-name) are untangled and become independent attributes of a single form of type declaration. Both attributes, polarity and evaluation order, can be changed independently by one algorithm each. In particular, defunctionalization and refunctionalization are now one algorithm. Evaluation order can be defined and changed individually for each type, independently from polarity. By allowing only certain combinations of evaluation order and polarity, the aforementioned η laws can be restored.David Binder, Julian Jabs, Ingo Skupin, Klaus Ostermannwork_5grx7ackmnb3dp7urxwykt3yuiWed, 23 Nov 2022 00:00:00 GMTTaylor Expansion Finitely Simulates Infinitary β-Reduction
https://scholar.archive.org/work/qgcm27ykoff4dmx3b4vqhmbxhq
Originating in Girard's Linear logic, Ehrhard and Regnier's Taylor expansion of λ-terms has been broadly used as a tool to approximate the terms of several variants of the λ-calculus. Many results arise from a Commutation theorem relating the normal form of the Taylor expansion of a term to its Böhm tree. This led us to consider extending this formalism to the infinitary λ-calculus, since the Λ_∞^001 version of this calculus has Böhm trees as normal forms and seems to be the ideal framework to reformulate the Commutation theorem. We give a (co-)inductive presentation of Λ_∞^001. We define a Taylor expansion on this calculus, and state that the infinitary β-reduction can be simulated through this Taylor expansion. The target language is the usual resource calculus, and in particular the resource reduction remains finite, confluent and terminating. Finally, we state the generalised Commutation theorem and use our results to provide simple proofs of some normalisation and confluence properties in the infinitary λ-calculus.Rémy Cerda, Lionel Vaux Auclairwork_qgcm27ykoff4dmx3b4vqhmbxhqSun, 13 Nov 2022 00:00:00 GMTCategorical Semantics for Model Comparison Games for Description Logics
https://scholar.archive.org/work/3kxg4n5dnrg4np4max5g6uzn4a
A categorical approach to study model comparison games in terms of comonads was recently initiated by Abramsky et al. In this work, we analyse games that appear naturally in the context of description logics and supplement them with suitable game comonads. More precisely, we consider expressive sublogics of ALCSelfIbO, namely, the logics that extend ALC with any combination of inverses, nominals, safe boolean roles combinations, and Self operator. Our construction augments and modifies the so-called modal comonad by Abramsky and Shah. The approach that we took heavily relies on the use of relative comonads, which we leverage to encapsulate additional capabilities within the bisimulation games in a compositional manner.Mateusz Urbańczykwork_3kxg4n5dnrg4np4max5g6uzn4aFri, 11 Nov 2022 00:00:00 GMTCoalgebraic Geometric Logic: Basic Theory
https://scholar.archive.org/work/m3b5u6annvakflhapirmxinuf4
Using the theory of coalgebra, we introduce a uniform framework for adding modalities to the language of propositional geometric logic. Models for this logic are based on coalgebras for an endofunctor on some full subcategory of the category of topological spaces and continuous functions. We investigate derivation systems, soundness and completeness for such geometric modal logics, and we specify a method of lifting an endofunctor on Set, accompanied by a collection of predicate liftings, to an endofunctor on the category of topological spaces, again accompanied by a collection of (open) predicate liftings. Furthermore, we compare the notions of modal equivalence, behavioural equivalence and bisimulation on the resulting class of models, and we provide a final object for the corresponding category.Nick Bezhanishvili, Jim de Groot, Yde Venemawork_m3b5u6annvakflhapirmxinuf4Mon, 31 Oct 2022 00:00:00 GMTThe Philosophy of Nature of the Natural Realism. The Operator Algebra from Physics to Logic
https://scholar.archive.org/work/g4txsavu7jgzxpwjslf2zlsq2e
This contribution is an essay of formal philosophy—and more specifically of formal ontology and formal epistemology—applied, respectively, to the philosophy of nature and to the philosophy of sciences, interpreted the former as the ontology and the latter as the epistemology of the modern mathematical, natural, and artificial sciences, the theoretical computer science included. I present the formal philosophy in the framework of the category theory (CT) as an axiomatic metalanguage—in many senses "wider" than set theory (ST)—of mathematics and logic, both of the "extensional" logics of the pure and applied mathematical sciences (=mathematical logic), and the "intensional" modal logics of the philosophical disciplines (=philosophical logic). It is particularly significant in this categorical framework the possibility of extending the operator algebra formalism from (quantum and classical) physics to logic, via the so-called "Boolean algebras with operators" (BAOs), with this extension being the core of our formal ontology. In this context, I discuss the relevance of the algebraic Hopf coproduct and colimit operations, and then of the category of coalgebras in the computations over lattices of quantum numbers in the quantum field theory (QFT), interpreted as the fundamental physics. This coalgebraic formalism is particularly relevant for modeling the notion of the "quantum vacuum foliation" in QFT of dissipative systems, as a foundation of the notion of "complexity" in physics, and "memory" in biological and neural systems, using the powerful "colimit" operators. Finally, I suggest that in the CT logic, the relational semantics of BAOs, applied to the modal coalgebraic relational logic of the "possible worlds" in Kripke's model theory, is the proper logic of the formal ontology and epistemology of the natural realism, as a formalized philosophy of nature and sciences.Gianfranco Bastiwork_g4txsavu7jgzxpwjslf2zlsq2eWed, 26 Oct 2022 00:00:00 GMTA Linear Exponential Comonad in s-finite Transition Kernels and Probabilistic Coherent Spaces
https://scholar.archive.org/work/dqzaeui3gvd2ngp7qyv2esobzq
This paper concerns a stochastic construction of probabilistic coherent spaces by employing novel ingredients (i) linear exponential comonads arising properly in the measure-theory (ii) continuous orthogonality between measures and measurable functions. A linear exponential comonad is constructed over a symmetric monoidal category of transition kernels, relaxing Markov kernels of Panangaden's stochastic relations into s-finite kernels. The model supports an orthogonality in terms of an integral between measures and measurable functions, which can be seen as a continuous extension of Girard-Danos-Ehrhard's linear duality for probabilistic coherent spaces. The orthogonality is formulated by a Hyland-Schalk double glueing construction, into which our measure theoretic monoidal comonad structure is accommodated. As an application to countable measurable spaces, a dagger compact closed category is obtained, whose double glueing gives rise to the familiar category of probabilistic coherent spaces.Masahiro Hamanowork_dqzaeui3gvd2ngp7qyv2esobzqSun, 16 Oct 2022 00:00:00 GMTThe lattice of varieties of monoids
https://scholar.archive.org/work/ltiqaqr3vjbntexhehrmwwusdm
We survey results devoted to the lattice of varieties of monoids. Along with known results, some unpublished results are given with proofs. A number of open questions and problems are also formulated.Sergey V. Gusev, Edmond W. H. Lee, Boris M. Vernikovwork_ltiqaqr3vjbntexhehrmwwusdmFri, 14 Oct 2022 00:00:00 GMTStructure and Power: an emerging landscape
https://scholar.archive.org/work/puc7r5bhnjcaxdwmmowibb7bva
In this paper, we give an overview of some recent work on applying tools from category theory in finite model theory, descriptive complexity, constraint satisfaction, and combinatorics. The motivations for this work come from Computer Science, but there may also be something of interest for model theorists and other logicians. The basic setting involves studying the category of relational structures via a resource-indexed family of adjunctions with some process category - which unfolds relational structures into treelike forms, allowing natural resource parameters to be assigned to these unfoldings. One basic instance of this scheme allows us to recover, in a purely structural, syntax-free way: the Ehrenfeucht-Fraisse~game; the quantifier rank fragments of first-order logic; the equivalences on structures induced by (i) the quantifier rank fragments, (ii) the restriction of this fragment to the existential positive part, and (iii) the extension with counting quantifiers; and the combinatorial parameter of tree-depth (Nesetril and Ossona de Mendez). Another instance recovers the k-pebble game, the finite-variable fragments, the corresponding equivalences, and the combinatorial parameter of treewidth. Other instances cover modal, guarded and hybrid fragments, generalized quantifiers, and a wide range of combinatorial parameters. This whole scheme has been axiomatized in a very general setting, of arboreal categories and arboreal covers. Beyond this basic level, a landscape is beginning to emerge, in which structural features of the resource categories, adjunctions and comonads are reflected in degrees of logical and computational tractability of the corresponding languages. Examples include semantic characterisation and preservation theorems, and Lovasz-type results on counting homomorphisms.Samson Abramskywork_puc7r5bhnjcaxdwmmowibb7bvaMon, 03 Oct 2022 00:00:00 GMTNested Session Types
https://scholar.archive.org/work/cdzjx4x355eyjn7slpugmdj6di
Session types statically describe communication protocols between concurrent message-passing processes. Unfortunately, parametric polymorphism even in its restricted prenex form is not fully understood in the context of session types. In this article, we present the metatheory of session types extended with prenex polymorphism and, as a result, nested recursive datatypes. Remarkably, we prove that type equality is decidable by exhibiting a reduction to trace equivalence of deterministic first-order grammars. Recognizing the high theoretical complexity of the latter, we also propose a novel type equality algorithm and prove its soundness. We observe that the algorithm is surprisingly efficient and, despite its incompleteness, sufficient for all our examples. We have implemented our ideas by extending the Rast programming language with nested session types. We conclude with several examples illustrating the expressivity of our enhanced type system.Ankush Das, Henry Deyoung, Andreia Mordido, Frank Pfenningwork_cdzjx4x355eyjn7slpugmdj6diFri, 30 Sep 2022 00:00:00 GMTLinear-time logics – a coalgebraic perspective
https://scholar.archive.org/work/a2zudo3mevdg5doimugga7cq2m
We describe a general approach to deriving linear-time logics for a wide variety of state-based, quantitative systems, by modelling the latter as coalgebras whose type incorporates both branching and linear behaviour. Concretely, we define logics whose syntax is determined by the type of linear behaviour, and whose domain of truth values is determined by the type of branching behaviour, and we provide two equivalent semantics for them: a step-wise semantics akin to that of standard coalgebraic logics, and a path-based semantics akin to that of standard linear-time logics. We also provide a semantic characterisation of a notion of logical distance induced by these logics. Instances of our logics support reasoning about the possibility, likelihood or minimal cost of exhibiting a given linear-time property.Corina Cirsteawork_a2zudo3mevdg5doimugga7cq2mWed, 21 Sep 2022 00:00:00 GMTSpan(Graph): a Canonical Feedback Algebra of Open Transition Systems
https://scholar.archive.org/work/iklphrfi3zechgg3cc7h3rlmgq
We show that Span(Graph)*, an algebra for open transition systems introduced by Katis, Sabadini and Walters, satisfies a universal property. By itself, this is a justification of the canonicity of this model of concurrency. However, the universal property is itself of interest, being a formal demonstration of the relationship between feedback and state. Indeed, feedback categories, also originally proposed by Katis, Sabadini and Walters, are a weakening of traced monoidal categories, with various applications in computer science. A state bootstrapping technique, which has appeared in several different contexts, yields free such categories. We show that Span(Graph)* arises in this way, being the free feedback category over Span(Set). Given that the latter can be seen as an algebra of predicates, the algebra of open transition systems thus arises - roughly speaking - as the result of bootstrapping state to that algebra. Finally, we generalize feedback categories endowing state spaces with extra structure: this extends the framework from mere transition systems to automata with initial and final states.Elena Di Lavore, Alessandro Gianola, Mario Román, Nicoletta Sabadini, Paweł Sobocińskiwork_iklphrfi3zechgg3cc7h3rlmgqFri, 16 Sep 2022 00:00:00 GMTOn Feller continuity and full abstraction
https://scholar.archive.org/work/3mxaqxc32zdhxoo4vapqzpe3da
We study the nature of applicative bisimilarity in λ-calculi endowed with operators for sampling from contin- uous distributions. On the one hand, we show that bisimilarity, logical equivalence, and testing equivalence all coincide with contextual equivalence when real numbers can be manipulated through continuous functions only. The key ingredient towards this result is a notion of Feller-continuity for labelled Markov processes, which we believe of independent interest, giving rise a broad class of LMPs for which coinductive and logically inspired equivalences coincide. On the other hand, we show that if no constraint is put on the way real numbers are manipulated, characterizing contextual equivalence turns out to be hard, and most of the aforementioned notions of equivalence are even unsound.Gilles Barthe, Raphaëlle Crubillé, Ugo Dal Lago, Francesco Gavazzowork_3mxaqxc32zdhxoo4vapqzpe3daMon, 29 Aug 2022 00:00:00 GMTThe Univalence Principle
https://scholar.archive.org/work/5cturxqkjncxrgkbwqyq47iaka
The Univalence Principle is the statement that equivalent mathematical structures are indistinguishable. We prove a general version of this principle that applies to all set-based, categorical, and higher-categorical structures defined in a non-algebraic and space-based style, as well as models of higher-order theories such as topological spaces. In particular, we formulate a general definition of indiscernibility for objects of any such structure, and a corresponding univalence condition that generalizes Rezk's completeness condition for Segal spaces and ensures that all equivalences of structures are levelwise equivalences. Our work builds on Makkai's First-Order Logic with Dependent Sorts, but is expressed in Voevodsky's Univalent Foundations (UF), extending previous work on the Structure Identity Principle and univalent categories in UF. This enables indistinguishability to be expressed simply as identification, and yields a formal theory that is interpretable in classical homotopy theory, but also in other higher topos models. It follows that Univalent Foundations is a fully equivalence-invariant foundation for higher-categorical mathematics, as intended by Voevodsky.Benedikt Ahrens and Paige Randall North and Michael Shulman and Dimitris Tsementziswork_5cturxqkjncxrgkbwqyq47iakaMon, 29 Aug 2022 00:00:00 GMTComonadic semantics for hybrid logic
https://scholar.archive.org/work/p6h6tcb6yjbgloyc3jjawkbrnq
Hybrid logic is a widely-studied extension of basic modal logic, which corresponds to the bounded fragment of first-order logic. We study it from two novel perspectives: (1) We apply the recently introduced paradigm of comonadic semantics, which provides a new set of tools drawing on ideas from categorical semantics which can be applied to finite model theory, descriptive complexity and combinatorics. (2) We give a novel semantic characterization of hybrid logic in terms of invariance under disjoint extensions, a minimal form of locality. A notable feature of this result is that we give a uniform proof, valid for both the finite and infinite cases.Samson Abramsky, Dan Marsden, Stefan Szeider, Robert Ganian, Alexandra Silvawork_p6h6tcb6yjbgloyc3jjawkbrnqMon, 22 Aug 2022 00:00:00 GMTOn the Reinhardt Conjecture and Formal Foundations of Optimal Control
https://scholar.archive.org/work/rxdljdx5hnf7jmix4rh3hxse2u
We describe a reformulation (following Hales (2017)) of a 1934 conjecture of Reinhardt on pessimal packings of convex domains in the plane as a problem in optimal control theory. Several structural results of this problem including its Hamiltonian structure and Lax pair formalism are presented. General solutions of this problem for constant control are presented and are used to prove that the Pontryagin extremals of the control problem are constrained to lie in a compact domain of the state space. We further describe the structure of the control problem near its singular locus, and prove that we recover the Pontryagin system of the multi-dimensional Fuller optimal control problem (with two dimensional control) in this case. We show how this system admits logarithmic spiral trajectories when the control set is the circumscribing disk of the 2-simplex with the associated control performing an infinite number of rotations on the boundary of the disk in finite time. We also describe formalization projects in foundational optimal control viz., model-based and model-free Reinforcement Learning theory. Key ingredients which make these formalization novel viz., the Giry monad and contraction coinduction are considered and some applications are discussed.Koundinya Vajjhawork_rxdljdx5hnf7jmix4rh3hxse2uMon, 08 Aug 2022 00:00:00 GMTDeciding All Behavioral Equivalences at Once: A Game for Linear-Time–Branching-Time Spectroscopy
https://scholar.archive.org/work/jjeatgzwwfhehd6uawfx5uer4y
We introduce a generalization of the bisimulation game that finds distinguishing Hennessy-Milner logic formulas from every finitary, subformula-closed language in van Glabbeek's linear-time--branching-time spectrum between two finite-state processes. We identify the relevant dimensions that measure expressive power to yield formulas belonging to the coarsest distinguishing behavioral preorders and equivalences; the compared processes are equivalent in each coarser behavioral equivalence from the spectrum. We prove that the induced algorithm can determine the best fit of (in)equivalences for a pair of processes.Benjamin Bisping, David N. Jansen, Uwe Nestmannwork_jjeatgzwwfhehd6uawfx5uer4yMon, 08 Aug 2022 00:00:00 GMTMoss' logic for ordered coalgebras
https://scholar.archive.org/work/424byatauvagzo7pjd3zjkv2pi
We present a finitary version of Moss' coalgebraic logic for T-coalgebras, where T is a locally monotone endofunctor of the category of posets and monotone maps. The logic uses a single cover modality whose arity is given by the least finitary subfunctor of the dual of the coalgebra functor T_ω^∂, and the semantics of the modality is given by relation lifting. For the semantics to work, T is required to preserve exact squares. For the finitary setting to work, T_ω^∂ is required to preserve finite intersections. We develop a notion of a base for subobjects of T_ω X. This in particular allows us to talk about the finite poset of subformulas for a given formula. The notion of a base is introduced generally for a category equipped with a suitable factorisation system. We prove that the resulting logic has the Hennessy-Milner property for the notion of similarity based on the notion of relation lifting. We define a sequent proof system for the logic, and prove its completeness.Marta Bílková, Matěj Dostálwork_424byatauvagzo7pjd3zjkv2piSat, 06 Aug 2022 00:00:00 GMTGraded Monads and Behavioural Equivalence Games
https://scholar.archive.org/work/5vvsxffo3nelfjcksilzdw2xha
The framework of graded semantics uses graded monads to capture behavioural equivalences of varying granularity, for example as found in the linear-time / branching-time spectrum, over general system types. We describe a generic Spoiler-Duplicator game for graded semantics that is extracted from the given graded monad, and may be seen as playing out an equational proof; instances include standard pebble games for simulation and bisimulation as well as games for trace-like equivalences and coalgebraic behavioural equivalence. Considerations on an infinite variant of such games lead to a novel notion of infinite-depth graded semantics. Under reasonable restrictions, the infinite-depth graded semantics associated to a given graded equivalence can be characterized in terms of a determinization construction for coalgebras under the equivalence at hand.Chase Ford, Stefan Milius, Lutz Schröder, Harsh Beohar, Barbara Königwork_5vvsxffo3nelfjcksilzdw2xhaTue, 02 Aug 2022 00:00:00 GMTArboreal Categories: An Axiomatic Theory of Resources
https://scholar.archive.org/work/k7qttq5sqrbf5ngnhilquyduaq
Game comonads provide a categorical syntax-free approach to finite model theory, and their Eilenberg-Moore coalgebras typically encode important combinatorial parameters of structures. In this paper, we develop a framework whereby the essential properties of these categories of coalgebras are captured in a purely axiomatic fashion. To this end, we introduce arboreal categories, which have an intrinsic process structure, allowing dynamic notions such as bisimulation and back-and-forth games, and resource notions such as number of rounds of a game, to be defined. These are related to extensional or "static" structures via arboreal covers, which are resource-indexed comonadic adjunctions. These ideas are developed in a general, axiomatic setting, and applied to relational structures, where the comonadic constructions for pebbling, Ehrenfeucht-Fra\"iss\'e and modal bisimulation games recently introduced by Abramsky et al. are recovered, showing that many of the fundamental notions of finite model theory and descriptive complexity arise from instances of arboreal covers.Samson Abramsky, Luca Reggiowork_k7qttq5sqrbf5ngnhilquyduaqTue, 26 Jul 2022 00:00:00 GMTOn Feller Continuity and Full Abstraction (Long Version)
https://scholar.archive.org/work/2zjvqmwnezcxvbowgjchaw3opm
We study the nature of applicative bisimilarity in λ-calculi endowed with operators for sampling from continuous distributions. On the one hand, we show that bisimilarity, logical equivalence, and testing equivalence all coincide with contextual equivalence when real numbers can be manipulated only through continuous functions. The key ingredient towards this result is a novel notion of Feller-continuity for labelled Markov processes, which we believe of independent interest, being a broad class of LMPs for which coinductive and logically inspired equivalences coincide. On the other hand, we show that if no constraint is put on the way real numbers are manipulated, characterizing contextual equivalence turns out to be hard, and most of the aforementioned notions of equivalence are even unsound.Gilles Barthe, Raphaëlle Crubillé, Ugo Dal Lago, Francesco Gavazzowork_2zjvqmwnezcxvbowgjchaw3opmThu, 21 Jul 2022 00:00:00 GMT