Electronic Notes in Theoretical Computer Science最新文献_第3页

Bifibrations of Polycategories and Classical Linear Logic 多范畴的振动与经典线性逻辑

Electronic Notes in Theoretical Computer Science Pub Date : 2020-10-01 DOI: 10.1016/j.entcs.2020.09.003

Nicolas Blanco, Noam Zeilberger

{"title":"Bifibrations of Polycategories and Classical Linear Logic","authors":"Nicolas Blanco, Noam Zeilberger","doi":"10.1016/j.entcs.2020.09.003","DOIUrl":"10.1016/j.entcs.2020.09.003","url":null,"abstract":"<div><p>The main goal of this article is to expose and relate different ways of interpreting the multiplicative fragment of classical linear logic in polycategories. Polycategories are known to give rise to models of classical linear logic in so-called representable polycategories with duals, which ask for the existence of various polymaps satisfying the different universal properties needed to define tensor, par, and negation. We begin by explaining how these different universal properties can all be seen as instances of a single notion of universality of a polymap parameterised by an input or output object, which also generalises the classical notion of universal multimap in a multicategory. We then proceed to introduce a definition of in-cartesian and out-cartesian polymaps relative to a refinement system (= strict functor) of polycategories, in such a way that universal polymaps can be understood as a special case. In particular, we obtain that a polycategory is a representable polycategory with duals if and only if it is bifibred over the terminal polycategory <span><math><mn>1</mn></math></span>. Finally, we present a Grothendieck correspondence between bifibrations of polycategories and pseudofunctors into <strong>MAdj</strong>, the (weak) 2-polycategory of multivariable adjunctions. When restricted to bifibrations over <span><math><mn>1</mn></math></span> we get back the correspondence between *-autonomous categories and Frobenius pseudomonoids in <strong>MAdj</strong> that was recently observed by Shulman.</p></div>","PeriodicalId":38770,"journal":{"name":"Electronic Notes in Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.entcs.2020.09.003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117079878","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 2

Parametrized Fixed Points and Their Applications to Session Types 参数化不动点及其在会话类型中的应用

Electronic Notes in Theoretical Computer Science Pub Date : 2020-10-01 DOI: 10.1016/j.entcs.2020.09.008

Ryan Kavanagh

{"title":"Parametrized Fixed Points and Their Applications to Session Types","authors":"Ryan Kavanagh","doi":"10.1016/j.entcs.2020.09.008","DOIUrl":"10.1016/j.entcs.2020.09.008","url":null,"abstract":"<div><p>Parametrized fixed points are of particular interest to denotational semantics and are often given by “dagger operations” [Stephen L. Bloom and Zoltán Ésik, Fixed-Point Operations on ccc's. Part I, Theoretical Computer Science (ISSN 0304-3975) 155 (1996), 1–38, <span>https://doi.org/10.1016/0304-3975(95)00010-0</span><svg><path></path></svg>; Stephen L. Bloom and Zoltán Ésik, Iteration Theories. The Equational Logic of Iterative Processes, in: EATCS Monographs on Theoretical Computer Science, Springer-Verlag Berlin Heidelberg, ISBN 978-3-642-78034-9, 1993, xv+630 pp., <span>https://doi.org/10.1007/978-3-642-78034-9</span><svg><path></path></svg>; Stephen L. Bloom and Zoltán Ésik, Some Equational Laws of Initiality in 2CCC's, International Journal of Foundations of Computer Science 6 (1995) 95–118, <span>https://doi.org/10.1142/S0129054195000081</span><svg><path></path></svg>.]. Dagger operations that satisfy the Conway identities [Stephen L. Bloom and Zoltán Ésik, Fixed-Point Operations on ccc's. Part I, Theoretical Computer Science (ISSN 0304-3975) 155 (1996), 1–38, doi: <span>https://doi.org/10.1016/0304-3975(95)00010-0</span><svg><path></path></svg>.] are particularly useful, because these identities imply a large class of identities used in semantic reasoning. We generalize existing techniques to define dagger operations on <em>ω</em>-categories and on <strong>O</strong>-categories. These operations enjoy a 2-categorical structure that implies the Conway identities. We illustrate these operators by considering applications to the semantics of session-typed languages.</p></div>","PeriodicalId":38770,"journal":{"name":"Electronic Notes in Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.entcs.2020.09.008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125710648","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Continuous Monads 的连续单体

Electronic Notes in Theoretical Computer Science Pub Date : 2020-10-01 DOI: 10.1016/j.entcs.2020.09.009

Ernie Manes

引用次数: 0

Weakest Preconditions in Fibrations 纤颤的最弱前提条件

Electronic Notes in Theoretical Computer Science Pub Date : 2020-10-01 DOI: 10.1016/j.entcs.2020.09.002

Alejandro Aguirre , Shin-ya Katsumata

引用次数: 0

A Pebbling Comonad for Finite Rank and Variable Logic, and an Application to the Equirank-variable Homomorphism Preservation Theorem 有限秩变量逻辑的一个泛泡公，及其在等秩-变量同态保持定理中的应用

Electronic Notes in Theoretical Computer Science Pub Date : 2020-10-01 DOI: 10.1016/j.entcs.2020.09.010

Thomas Paine

引用次数: 10

Monads, Partial Evaluations, and Rewriting 单子、部分求值和重写

Electronic Notes in Theoretical Computer Science Pub Date : 2020-10-01 DOI: 10.1016/j.entcs.2020.09.007

Tobias Fritz, Paolo Perrone

{"title":"Monads, Partial Evaluations, and Rewriting","authors":"Tobias Fritz, Paolo Perrone","doi":"10.1016/j.entcs.2020.09.007","DOIUrl":"10.1016/j.entcs.2020.09.007","url":null,"abstract":"<div><p>Monads can be interpreted as encoding formal expressions, or formal operations in the sense of universal algebra. We give a construction which formalizes the idea of “evaluating an expression partially”: for example, “2+3” can be obtained as a partial evaluation of “2+2+1”. This construction can be given for any monad, and it is linked to the famous <em>bar construction</em> [Saunders Mac Lane, Categories for the Working Mathematician, Springer, 2000, VII.6], of which it gives an operational interpretation: the bar construction is a simplicial set, and its 1-cells are partial evaluations.</p><p>We study the properties of partial evaluations for general monads. We prove that whenever the monad is weakly cartesian, partial evaluations can be composed via the usual Kan filler property of simplicial sets, of which we give an interpretation in terms of substitution of terms.</p><p>For the case of probability monads, partial evaluations correspond to what probabilists call <em>conditional expectation</em> of random variables, and partial evaluation relation is known as <em>second-order stochastic dominance</em>.</p><p>In terms of rewritings, partial evaluations give an abstract reduction system which is reflexive, confluent, and transitive whenever the monad is weakly cartesian. This manuscript is part of a work in progress on a general rewriting interpretation of the bar construction.</p></div>","PeriodicalId":38770,"journal":{"name":"Electronic Notes in Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.entcs.2020.09.007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114191785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 7

Domain Theoretic Second-Order Euler's Method for Solving Initial Value Problems 初值问题的域论二阶欧拉解法

Electronic Notes in Theoretical Computer Science Pub Date : 2020-10-01 DOI: 10.1016/j.entcs.2020.09.006

Abbas Edalat , Amin Farjudian , Mina Mohammadian , Dirk Pattinson

{"title":"Domain Theoretic Second-Order Euler's Method for Solving Initial Value Problems","authors":"Abbas Edalat , Amin Farjudian , Mina Mohammadian , Dirk Pattinson","doi":"10.1016/j.entcs.2020.09.006","DOIUrl":"10.1016/j.entcs.2020.09.006","url":null,"abstract":"<div><p>A domain-theoretic method for solving initial value problems (IVPs) is presented, together with proofs of soundness, completeness, and some results on the algebraic complexity of the method. While the common fixed-precision interval arithmetic methods are restricted by the precision of the underlying machine architecture, domain-theoretic methods may be complete, i.e., the result may be obtained to any degree of accuracy. Furthermore, unlike methods based on interval arithmetic which require access to the syntactic representation of the vector field, domain-theoretic methods only deal with the semantics of the field, in the sense that the field is assumed to be given via finitely-representable approximations, to within any required accuracy.</p><p>In contrast to the domain-theoretic first-order Euler method, the second-order method uses the local Lipschitz properties of the field. This is achieved by using a domain for Lipschitz functions, whose elements are consistent pairs that provide approximations of the field and its local Lipschitz properties. In the special case where the field is differentiable, the local Lipschitz properties are exactly the local differential properties of the field. In solving IVPs, Lipschitz continuity of the field is a common assumption, as a sufficient condition for uniqueness of the solution. While the validated methods for solving IVPs commonly impose further restrictions on the vector field, the second-order Euler method requires no further condition. In this sense, the method may be seen as the most general of its kind.</p><p>To avoid complicated notations and lengthy arguments, the results of the paper are stated for the second-order Euler method. Nonetheless, the framework, and the results, may be extended to any higher-order Euler method, in a straightforward way.</p></div>","PeriodicalId":38770,"journal":{"name":"Electronic Notes in Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.entcs.2020.09.006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134102167","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 3

A Complete Equational Axiomatisation of Partial Differentiation 偏微分的完全等式公理化

Electronic Notes in Theoretical Computer Science Pub Date : 2020-10-01 DOI: 10.1016/j.entcs.2020.09.011

Gordon D. Plotkin

引用次数: 1

Towards a Classification of Behavioural Equivalences in Continuous-time Markov Processes 连续时间马尔可夫过程中行为等价的分类

Electronic Notes in Theoretical Computer Science Pub Date : 2020-10-01 DOI: 10.1016/j.entcs.2020.09.004

Linan Chen , Florence Clerc , Prakash Panangaden

引用次数: 1

Combining Algebraic Effect Descriptions Using the Tensor of Complete Lattices 用完全格张量组合代数效应描述

Electronic Notes in Theoretical Computer Science Pub Date : 2020-10-01 DOI: 10.1016/j.entcs.2020.09.013

Niels Voorneveld

{"title":"Combining Algebraic Effect Descriptions Using the Tensor of Complete Lattices","authors":"Niels Voorneveld","doi":"10.1016/j.entcs.2020.09.013","DOIUrl":"10.1016/j.entcs.2020.09.013","url":null,"abstract":"<div><p>Algebras can be used to interpret the behaviour of effectful programs. In particular, we use Eilenberg-Moore algebras given over a complete lattices of truth values, which specify answers to queries about programs. The algebras can be used to formulate a quantitative logic of behavioural properties, specifying a congruent notion of program equivalence coinciding with a notion of applicative bisimilarity. Many combinations of effects can be interpreted using these algebras. In this paper, we specify a method of generically combining effects and the algebras used to interpret them. At the core of this method is the tensor of complete lattices, which combines the carrier sets of the algebras. We show that this tensor preserves complete distributivity of complete lattices. Moreover, the universal properties of this tensor can then be used to properly combine the Eilenberg-Moore algebras. We will apply this method to combine the effects of probability, global store, cost, nondeterminism, and error effects. We will then compare this method of combining effects with the more traditional method of combining equational theories using interaction laws.</p></div>","PeriodicalId":38770,"journal":{"name":"Electronic Notes in Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.entcs.2020.09.013","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129773535","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0