Pub Date : 2023-10-17DOI: 10.1017/s0960129523000312
Benedikt Ahrens, Paige Randall North, Niels van der Weide
Abstract 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 as well as classes of examples constructed from other data. From the notion of comprehension bicategory, we extract the syntax of bicategorical type theory, that is, judgment forms and structural inference rules. We prove soundness of the rules by giving an interpretation in any comprehension bicategory. The semantic aspects of our work are fully checked in the Coq proof assistant, based on the UniMath library.
{"title":"Bicategorical type theory: semantics and syntax","authors":"Benedikt Ahrens, Paige Randall North, Niels van der Weide","doi":"10.1017/s0960129523000312","DOIUrl":"https://doi.org/10.1017/s0960129523000312","url":null,"abstract":"Abstract 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 as well as classes of examples constructed from other data. From the notion of comprehension bicategory, we extract the syntax of bicategorical type theory, that is, judgment forms and structural inference rules. We prove soundness of the rules by giving an interpretation in any comprehension bicategory. The semantic aspects of our work are fully checked in the Coq proof assistant, based on the UniMath library.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135994931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-12DOI: 10.1017/s0960129523000336
S. M. Elsayed, Keng Meng Ng
Abstract 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 in El-Shafei et al. ((2018). Filomat 32 (13) 4755–4771), El-Shafei and Al-Shami ((2020). Computational and Applied Mathematics 39 (3) 1–17), Al-shami ((2021). Mathematical Problems in Engineering 2021 ). In this paper, we introduce the effective versions of soft separation axioms. Specifically, we focus our attention on computable u-soft and computable p-soft separation axioms and investigate various relations between them. We also compare the effective and classical versions of these soft separation axioms.
软集作为一种研究非绝对定义对象的方法被引入，并在数学、决策理论和统计应用的许多领域中得到了应用。软拓扑空间首先在Shabir和Naz(2011)中被考虑。电脑,El-Shafei et al.(2018)研究了软拓扑空间的软分离公理和应用数学61(7)1786-1799。Filomat 32 (13) 4755-4771)， El-Shafei和Al-Shami(2020)。[3]杨建军，杨建军，李建军。计算与应用数学39(3):1-17)。工程数学问题(2021)。本文介绍了软分离公理的有效版本。具体来说，我们关注可计算u-soft和可计算p-soft分离公理，并研究它们之间的各种关系。我们还比较了这些软分离公理的有效版本和经典版本。
{"title":"Computable soft separation axioms","authors":"S. M. Elsayed, Keng Meng Ng","doi":"10.1017/s0960129523000336","DOIUrl":"https://doi.org/10.1017/s0960129523000336","url":null,"abstract":"Abstract 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 in El-Shafei et al. ((2018). Filomat 32 (13) 4755–4771), El-Shafei and Al-Shami ((2020). Computational and Applied Mathematics 39 (3) 1–17), Al-shami ((2021). Mathematical Problems in Engineering 2021 ). In this paper, we introduce the effective versions of soft separation axioms. Specifically, we focus our attention on computable u-soft and computable p-soft separation axioms and investigate various relations between them. We also compare the effective and classical versions of these soft separation axioms.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135969273","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-11DOI: 10.1017/s0960129523000300
K. Subramani, Piotr Wojciechowski
Abstract 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: ${bf A cdot x le b}$ . UTVPI constraints are used in many domains including operations research and program verification. We focus on two variants of read-once refutation (ROR). An ROR is a refutation in which each constraint is used at most once. A literal-once refutation (LOR), a more restrictive form of ROR, is a refutation in which each literal ( $x_i$ or $-x_i$ ) is used at most once. First, we examine the constraint-required read-once refutation (CROR) problem and the constraint-required literal-once refutation (CLOR) problem. In both of these problems, we are given a set of constraints that must be used in the refutation. RORs and LORs are incomplete since not every system of linear constraints is guaranteed to have such a refutation. This is still true even when we restrict ourselves to UCSs. In this paper, we provide NC reductions between the CROR and CLOR problems in UCSs and the minimum weight perfect matching problem. The reductions used in this paper assume a CREW PRAM model of parallel computation. As a result, the reductions establish that, from the perspective of parallel algorithms, the CROR and CLOR problems in UCSs are equivalent to matching. In particular, if an NC algorithm exists for either of these problems, then there is an NC algorithm for matching.
摘要本文分析了Unit two Variable Per Inequality (UTVPI)约束的两类反驳。UTVPI约束是如下形式的线性不等式:$a_{i}cdot x_{i}+a_{j} cdot x_{j} le b_{k}$，其中$a_{i}，a_{j}in {0,1，-1}$和$b_{k} in mathbb{Z}$。这些约束的结合称为UTVPI约束系统(UCS)，可以用矩阵形式表示为:${bf A cdot x le b}$。UTVPI约束被广泛应用于运筹学和程序验证等领域。我们重点讨论了两次读一次驳斥(ROR)的变体。ROR是一种驳斥，其中每个约束最多使用一次。一次字面量反驳(LOR)是一种更严格的ROR形式，它是一种反驳，其中每个字面量($x_i$或$-x_i$)最多使用一次。首先，我们研究了约束要求读一次反驳(CROR)问题和约束要求字面一次反驳(CLOR)问题。在这两个问题中，我们都给定了一组必须在反驳中使用的约束。error和lor是不完全的，因为不是每个线性约束系统都保证有这样的反驳。即使我们将自己限制为ucs，这仍然是正确的。在本文中，我们提供了ucs中CROR和CLOR问题之间的NC约简以及最小权值完美匹配问题。本文所使用的缩减假设了并行计算的CREW PRAM模型。因此，约简表明，从并行算法的角度来看，ucs中的CROR和CLOR问题等价于匹配。特别地，如果对于这两个问题中的任何一个存在NC算法，那么就存在匹配的NC算法。
{"title":"Constrained read-once refutations in UTVPI constraint systems: A parallel perspective","authors":"K. Subramani, Piotr Wojciechowski","doi":"10.1017/s0960129523000300","DOIUrl":"https://doi.org/10.1017/s0960129523000300","url":null,"abstract":"Abstract 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: ${bf A cdot x le b}$ . UTVPI constraints are used in many domains including operations research and program verification. We focus on two variants of read-once refutation (ROR). An ROR is a refutation in which each constraint is used at most once. A literal-once refutation (LOR), a more restrictive form of ROR, is a refutation in which each literal ( $x_i$ or $-x_i$ ) is used at most once. First, we examine the constraint-required read-once refutation (CROR) problem and the constraint-required literal-once refutation (CLOR) problem. In both of these problems, we are given a set of constraints that must be used in the refutation. RORs and LORs are incomplete since not every system of linear constraints is guaranteed to have such a refutation. This is still true even when we restrict ourselves to UCSs. In this paper, we provide NC reductions between the CROR and CLOR problems in UCSs and the minimum weight perfect matching problem. The reductions used in this paper assume a CREW PRAM model of parallel computation. As a result, the reductions establish that, from the perspective of parallel algorithms, the CROR and CLOR problems in UCSs are equivalent to matching. In particular, if an NC algorithm exists for either of these problems, then there is an NC algorithm for matching.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135981574","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-01DOI: 10.1017/s0960129523000221
Dan Licata, Peter LeFanu Lumsdaine
{"title":"Special issue on homotopy type theory 2019 vol. 2","authors":"Dan Licata, Peter LeFanu Lumsdaine","doi":"10.1017/s0960129523000221","DOIUrl":"https://doi.org/10.1017/s0960129523000221","url":null,"abstract":"","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135254529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-08-07DOI: 10.1017/s0960129523000270
U. Dal Lago, Francesco Gavazzo, Paolo Pistone
{"title":"Preface to the special issue on metric and differential semantics","authors":"U. Dal Lago, Francesco Gavazzo, Paolo Pistone","doi":"10.1017/s0960129523000270","DOIUrl":"https://doi.org/10.1017/s0960129523000270","url":null,"abstract":"","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57258874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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 on Isbell’s example, Xu, Xi, and Zhao showed that there is even a complete Heyting algebra whose Scott space is non-sober. Achim Jung then asked whether every countable complete lattice has a sober Scott space. The main aim of this paper is to answer Jung’s problem by constructing a countable complete lattice whose Scott space is non-sober. This lattice is then modified to obtain a countable distributive complete lattice with a non-sober Scott space. In addition, we prove that the topology of the product space $Sigma Ptimes Sigma Q$ coincides with the Scott topology of the product poset $Ptimes Q$ if the set Id(P) and Id(Q) of all incremental ideals of posets P and Q are both countable. Based on this, it is deduced that a directed complete poset P has a sober Scott space, if Id(P) is countable and $Sigma P$ is coherent and well filtered. In particular, every complete lattice L with Id(L) countable has a sober Scott space.
{"title":"Not every countable complete distributive lattice is sober","authors":"Hualin Miao, Xiaoyong Xi, Qingguo Li, Dongsheng Zhao","doi":"10.1017/s0960129523000269","DOIUrl":"https://doi.org/10.1017/s0960129523000269","url":null,"abstract":"\u0000 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 on Isbell’s example, Xu, Xi, and Zhao showed that there is even a complete Heyting algebra whose Scott space is non-sober. Achim Jung then asked whether every countable complete lattice has a sober Scott space. The main aim of this paper is to answer Jung’s problem by constructing a countable complete lattice whose Scott space is non-sober. This lattice is then modified to obtain a countable distributive complete lattice with a non-sober Scott space. In addition, we prove that the topology of the product space \u0000 \u0000 \u0000 \u0000$Sigma Ptimes Sigma Q$\u0000\u0000 \u0000 coincides with the Scott topology of the product poset \u0000 \u0000 \u0000 \u0000$Ptimes Q$\u0000\u0000 \u0000 if the set Id(P) and Id(Q) of all incremental ideals of posets P and Q are both countable. Based on this, it is deduced that a directed complete poset P has a sober Scott space, if Id(P) is countable and \u0000 \u0000 \u0000 \u0000$Sigma P$\u0000\u0000 \u0000 is coherent and well filtered. In particular, every complete lattice L with Id(L) countable has a sober Scott space.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47701224","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-27DOI: 10.1017/s0960129523000257
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 poset $mathsf{K}(X)$ with the Scott topology is still well-filtered, and a $T_0$ space Y is well-filtered iff the Smyth power poset $mathsf{K}(Y)$ with the Scott topology is well-filtered and the upper Vietoris topology is coarser than the Scott topology on
{"title":"Scott topology on Smyth power posets","authors":"Xiaoquan Xu, Xinpeng Wen, Xiaoyong Xi","doi":"10.1017/s0960129523000257","DOIUrl":"https://doi.org/10.1017/s0960129523000257","url":null,"abstract":"\u0000\t <jats:p>For a <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000257_inline1.png\" />\u0000\t\t<jats:tex-math>\u0000$T_0$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> space <jats:italic>X</jats:italic>, let <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000257_inline2.png\" />\u0000\t\t<jats:tex-math>\u0000$mathsf{K}(X)$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> be the poset of all nonempty compact saturated subsets of <jats:italic>X</jats:italic> endowed with the Smyth order <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000257_inline3.png\" />\u0000\t\t<jats:tex-math>\u0000$sqsubseteq$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>. <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000257_inline4.png\" />\u0000\t\t<jats:tex-math>\u0000$(mathsf{K}(X), sqsubseteq)$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> (shortly <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000257_inline5.png\" />\u0000\t\t<jats:tex-math>\u0000$mathsf{K}(X)$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula>) is called the Smyth power poset of <jats:italic>X</jats:italic>. 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 <jats:italic>X</jats:italic>, its Smyth power poset <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000257_inline6.png\" />\u0000\t\t<jats:tex-math>\u0000$mathsf{K}(X)$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> with the Scott topology is still well-filtered, and a <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000257_inline7.png\" />\u0000\t\t<jats:tex-math>\u0000$T_0$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> space <jats:italic>Y</jats:italic> is well-filtered iff the Smyth power poset <jats:inline-formula>\u0000\t <jats:alternatives>\u0000\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129523000257_inline8.png\" />\u0000\t\t<jats:tex-math>\u0000$mathsf{K}(Y)$\u0000</jats:tex-math>\u0000\t </jats:alternatives>\u0000\t </jats:inline-formula> with the Scott topology is well-filtered and the upper Vietoris topology is coarser than the Scott topology on <jats:inline-formula>\u0000\t ","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-07-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47692020","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-10DOI: 10.1017/s096012952300021x
D. 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 of identity proofs (UIP). One point of view on it is as internalised meta-theory of the inner type theory. There are two motivations for 2LTT. Firstly, there are certain results about HoTT which are of meta-theoretic nature, such as the statement that semisimplicial types up to level n can be constructed in HoTT for any externally fixed natural number n. Such results cannot be expressed in HoTT itself, but they can be formalised and proved in 2LTT, where n will be a variable in the outer theory. This point of view is inspired by observations about conservativity of presheaf models. Secondly, 2LTT is a framework which is suitable for formulating additional axioms that one might want to add to HoTT. This idea is heavily inspired by Voevodsky’s Homotopy Type System (HTS), which constitutes one specific instance of a 2LTT. HTS has an axiom ensuring that the type of natural numbers behaves like the external natural numbers, which allows the construction of a universe of semisimplicial types. In 2LTT, this axiom can be assumed by postulating that the inner and outer natural numbers types are isomorphic. After defining 2LTT, we set up a collection of tools with the goal of making 2LTT a convenient language for future developments. As a first such application, we develop the theory of Reedy fibrant diagrams in the style of Shulman. Continuing this line of thought, we suggest a definition of $(infty,1)$ -category and give some examples.
{"title":"Two-level type theory and applications - ERRATUM","authors":"D. Annenkov, Paolo Capriotti, Nicolai Kraus, Christian Sattler","doi":"10.1017/s096012952300021x","DOIUrl":"https://doi.org/10.1017/s096012952300021x","url":null,"abstract":"\u0000 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 of identity proofs (UIP). One point of view on it is as internalised meta-theory of the inner type theory. There are two motivations for 2LTT. Firstly, there are certain results about HoTT which are of meta-theoretic nature, such as the statement that semisimplicial types up to level n can be constructed in HoTT for any externally fixed natural number n. Such results cannot be expressed in HoTT itself, but they can be formalised and proved in 2LTT, where n will be a variable in the outer theory. This point of view is inspired by observations about conservativity of presheaf models. Secondly, 2LTT is a framework which is suitable for formulating additional axioms that one might want to add to HoTT. This idea is heavily inspired by Voevodsky’s Homotopy Type System (HTS), which constitutes one specific instance of a 2LTT. HTS has an axiom ensuring that the type of natural numbers behaves like the external natural numbers, which allows the construction of a universe of semisimplicial types. In 2LTT, this axiom can be assumed by postulating that the inner and outer natural numbers types are isomorphic. After defining 2LTT, we set up a collection of tools with the goal of making 2LTT a convenient language for future developments. As a first such application, we develop the theory of Reedy fibrant diagrams in the style of Shulman. Continuing this line of thought, we suggest a definition of \u0000 \u0000 \u0000 \u0000$(infty,1)$\u0000\u0000 \u0000 -category and give some examples.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43600185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-07-05DOI: 10.1017/s0960129523000191
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 variant of unification, as a basis for logical inference, typing, and a very simple and natural model for untyped combinatory logic. An analysis of the new typing scheme will enable us to characterize a large class of terms of combinatory logic which do not change their principal type when being weakly reduced. We also consider the question whether the major or the minor premisse should be used as the fixed pattern.
{"title":"A (machine-oriented) logic based on pattern matching","authors":"Tim Lethen","doi":"10.1017/s0960129523000191","DOIUrl":"https://doi.org/10.1017/s0960129523000191","url":null,"abstract":"\u0000 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 variant of unification, as a basis for logical inference, typing, and a very simple and natural model for untyped combinatory logic. An analysis of the new typing scheme will enable us to characterize a large class of terms of combinatory logic which do not change their principal type when being weakly reduced. We also consider the question whether the major or the minor premisse should be used as the fixed pattern.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42973210","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-01DOI: 10.1017/s0960129523000233
Amin Farjudian, Eugenio Moggi
Abstract 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 space is compact, the complete lattice of closed subsets ordered by reverse inclusion is $omega$ -continuous, and robust analyses are exactly the Scott-continuous maps. Thus, one can also ask whether a robust analysis is computable (with respect to a countable base). The main result of this paper establishes a relation between robustness and Scott continuity when the metric space is not compact. The key idea is to replace the metric space with a compact Hausdorff space, and relate robustness and Scott continuity by an adjunction between the complete lattice of closed subsets of the metric space and the $omega$ -continuous lattice of closed subsets of the compact Hausdorff space. We demonstrate the applicability of this result with several examples involving Banach spaces.
{"title":"Robustness, Scott continuity, and computability","authors":"Amin Farjudian, Eugenio Moggi","doi":"10.1017/s0960129523000233","DOIUrl":"https://doi.org/10.1017/s0960129523000233","url":null,"abstract":"Abstract 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 space is compact, the complete lattice of closed subsets ordered by reverse inclusion is $omega$ -continuous, and robust analyses are exactly the Scott-continuous maps. Thus, one can also ask whether a robust analysis is computable (with respect to a countable base). The main result of this paper establishes a relation between robustness and Scott continuity when the metric space is not compact. The key idea is to replace the metric space with a compact Hausdorff space, and relate robustness and Scott continuity by an adjunction between the complete lattice of closed subsets of the metric space and the $omega$ -continuous lattice of closed subsets of the compact Hausdorff space. We demonstrate the applicability of this result with several examples involving Banach spaces.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135046364","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：