Pub Date : 2022-03-17DOI: 10.1017/S0960129522000044
Gilles Dowek, Gaspard Férey, J. Jouannaud, Jiaxiang Liu
Abstract User-defined higher-order rewrite rules are becoming a standard in proof assistants based on intuitionistic type theory. This raises the question of proving that they preserve the properties of beta-reductions for the corresponding type systems. In a series of papers, we develop techniques based on van Oostrom’s decreasing diagrams that reduce confluence proofs to the checking of various forms of critical pairs for higher-order rewrite rules extending beta-reduction on pure lambda-terms. As shown in a previous paper of the two middle authors, confluence of a terminating set of left-linear rewrite rules is obtained when their critical pairs are joinable, beta-rewrite steps being disallowed. The present paper concentrates on the case where arbitrary beta-rewrite steps are allowed for joining critical pairs. The rewrite relation used for analyzing confluence may rewrite arbitrarily many non-overlapping redexes in a single step. This relation gives rise to critical pairs that overlap both horizontally, as with parallel rewriting, but also vertically, forming chains of successive overlaps. Practical examples of use of this technique are analyzed.
{"title":"Confluence of left-linear higher-order rewrite theories by checking their nested critical pairs","authors":"Gilles Dowek, Gaspard Férey, J. Jouannaud, Jiaxiang Liu","doi":"10.1017/S0960129522000044","DOIUrl":"https://doi.org/10.1017/S0960129522000044","url":null,"abstract":"Abstract User-defined higher-order rewrite rules are becoming a standard in proof assistants based on intuitionistic type theory. This raises the question of proving that they preserve the properties of beta-reductions for the corresponding type systems. In a series of papers, we develop techniques based on van Oostrom’s decreasing diagrams that reduce confluence proofs to the checking of various forms of critical pairs for higher-order rewrite rules extending beta-reduction on pure lambda-terms. As shown in a previous paper of the two middle authors, confluence of a terminating set of left-linear rewrite rules is obtained when their critical pairs are joinable, beta-rewrite steps being disallowed. The present paper concentrates on the case where arbitrary beta-rewrite steps are allowed for joining critical pairs. The rewrite relation used for analyzing confluence may rewrite arbitrarily many non-overlapping redexes in a single step. This relation gives rise to critical pairs that overlap both horizontally, as with parallel rewriting, but also vertically, forming chains of successive overlaps. Practical examples of use of this technique are analyzed.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"898 - 933"},"PeriodicalIF":0.5,"publicationDate":"2022-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44418770","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 : 2022-03-16DOI: 10.1017/S0960129521000487
Jan Hoffmann, Steffen Jost
Abstract This article gives an overview of automatic amortized resource analysis (AARA), a technique for inferring symbolic resource bounds for programs at compile time. AARA has been introduced by Hofmann and Jost in 2003 as a type system for deriving linear worst-case bounds on the heap-space consumption of first-order functional programs with eager evaluation strategy. Since then AARA has been the subject of dozens of research articles, which extended the analysis to different resource metrics, other evaluation strategies, non-linear bounds, and additional language features. All these works preserved the defining characteristics of the original paper: local inference rules, which reduce bound inference to numeric (usually linear) optimization; a soundness proof with respect to an operational cost semantics; and the support of amortized analysis with the potential method.
{"title":"Two decades of automatic amortized resource analysis","authors":"Jan Hoffmann, Steffen Jost","doi":"10.1017/S0960129521000487","DOIUrl":"https://doi.org/10.1017/S0960129521000487","url":null,"abstract":"Abstract This article gives an overview of automatic amortized resource analysis (AARA), a technique for inferring symbolic resource bounds for programs at compile time. AARA has been introduced by Hofmann and Jost in 2003 as a type system for deriving linear worst-case bounds on the heap-space consumption of first-order functional programs with eager evaluation strategy. Since then AARA has been the subject of dozens of research articles, which extended the analysis to different resource metrics, other evaluation strategies, non-linear bounds, and additional language features. All these works preserved the defining characteristics of the original paper: local inference rules, which reduce bound inference to numeric (usually linear) optimization; a soundness proof with respect to an operational cost semantics; and the support of amortized analysis with the potential method.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"729 - 759"},"PeriodicalIF":0.5,"publicationDate":"2022-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48548709","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 : 2022-02-01DOI: 10.1017/S0960129522000238
Xin He, Huaming Zhang, Yijie Han
Abstract Given a plane graph $G=(V,E)$ , a Petrie tour of G is a tour P of G that alternately turns left and right at each step. A Petrie tour partition of G is a collection ${mathscr P}={P_1,ldots,P_q}$ of Petrie tours so that each edge of G is in exactly one tour $P_i in {mathscr P}$ . A Petrie tour P is called a Petrie cycle if all its vertices are distinct. A Petrie cycle partition of G is a collection ${mathscr C}={C_1,ldots,C_p}$ of Petrie cycles so that each vertex of G is in exactly one cycle $C_i in {mathscr C}$ . In this paper, we study the properties of 3-regular plane graphs that have Petrie cycle partitions and 4-regular plane multi-graphs that have Petrie tour partitions. Given a 4-regular plane multi-graph $G=(V,E)$ , a 3-regularization of G is a 3-regular plane graph $G_3$ obtained from G by splitting every vertex $vin V$ into two degree-3 vertices. G is called Petrie partitionable if it has a 3-regularization that has a Petrie cycle partition. The general version of this problem is motivated by a data compression method, tristrip, used in computer graphics. In this paper, we present a simple characterization of Petrie partitionable graphs and show that the problem of determining if G is Petrie partitionable is NP-complete.
摘要给定平面图$G=(V,E)$， G的Petrie巡回是每一步交替向左和向右转的G的巡回P。G的Petrie tour分区是Petrie tour的集合${mathscr P}={P_1，ldots,P_q}$，使得G的每条边都恰好在一个tour $P_i in {mathscr P}$中。如果一个皮特里环的顶点都是不同的，那么它就叫做皮特里环。G的Petrie环划分是一个Petrie环的集合${mathscr C}={C_1，ldots,C_p}$，使得G的每个顶点恰好在一个循环$C_i in {mathscr C}$中。本文研究了具有Petrie循环划分的3正则平面图和具有Petrie循环划分的4正则平面多图的性质。给定一个4-正则平面多图$G=(V,E)$， G的3-正则化是将V$中的每个顶点$ V 拆分为两个3次顶点，得到一个由G得到的3-正则平面图$G_3$。G被称为皮特里可分的如果它有一个3正则化并且有一个皮特里循环划分。这个问题的一般版本是由计算机图形学中使用的数据压缩方法tritrip引起的。本文给出了Petrie可分图的一个简单刻划，并证明了判定G是否Petrie可分的问题是np完全的。
{"title":"On Petrie cycle and Petrie tour partitions of 3- and 4-regular plane graphs","authors":"Xin He, Huaming Zhang, Yijie Han","doi":"10.1017/S0960129522000238","DOIUrl":"https://doi.org/10.1017/S0960129522000238","url":null,"abstract":"Abstract Given a plane graph \u0000$G=(V,E)$\u0000 , a Petrie tour of G is a tour P of G that alternately turns left and right at each step. A Petrie tour partition of G is a collection \u0000${mathscr P}={P_1,ldots,P_q}$\u0000 of Petrie tours so that each edge of G is in exactly one tour \u0000$P_i in {mathscr P}$\u0000 . A Petrie tour P is called a Petrie cycle if all its vertices are distinct. A Petrie cycle partition of G is a collection \u0000${mathscr C}={C_1,ldots,C_p}$\u0000 of Petrie cycles so that each vertex of G is in exactly one cycle \u0000$C_i in {mathscr C}$\u0000 . In this paper, we study the properties of 3-regular plane graphs that have Petrie cycle partitions and 4-regular plane multi-graphs that have Petrie tour partitions. Given a 4-regular plane multi-graph \u0000$G=(V,E)$\u0000 , a 3-regularization of G is a 3-regular plane graph \u0000$G_3$\u0000 obtained from G by splitting every vertex \u0000$vin V$\u0000 into two degree-3 vertices. G is called Petrie partitionable if it has a 3-regularization that has a Petrie cycle partition. The general version of this problem is motivated by a data compression method, tristrip, used in computer graphics. In this paper, we present a simple characterization of Petrie partitionable graphs and show that the problem of determining if G is Petrie partitionable is NP-complete.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"240 - 256"},"PeriodicalIF":0.5,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46159547","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 : 2022-02-01DOI: 10.1017/S0960129522000184
H. Fernau, Jens Bruchertseifer
Abstract The concept of a synchronizing word is a very important notion in the theory of finite automata. We consider the associated decision problem to decide if a given DFA possesses a synchronizing word of length at most k, where k is the standard parameter. We show that this problem DFA-SW is equivalent to the problem Monoid Factorization introduced by Cai, Chen, Downey, and Fellows. Apart from the known $textsf{W}[2]$ -hardness results, we show that these problems belong to $textsf{A}[2]$ , $textsf{W}[textsf{P}],$ and $textsf{WNL}$ . This indicates that DFA-SW is not complete for any of these classes, and hence, we suggest a new parameterized complexity class $textsf{W}[textsf{Sync}]$ as a proper home for these (and more) problems. We present quite a number of problems that belong to $textsf{W}[textsf{Sync}]$ or are hard or complete for this new class.
{"title":"Synchronizing words and monoid factorization, yielding a new parameterized complexity class?","authors":"H. Fernau, Jens Bruchertseifer","doi":"10.1017/S0960129522000184","DOIUrl":"https://doi.org/10.1017/S0960129522000184","url":null,"abstract":"Abstract The concept of a synchronizing word is a very important notion in the theory of finite automata. We consider the associated decision problem to decide if a given DFA possesses a synchronizing word of length at most k, where k is the standard parameter. We show that this problem DFA-SW is equivalent to the problem Monoid Factorization introduced by Cai, Chen, Downey, and Fellows. Apart from the known \u0000$textsf{W}[2]$\u0000 -hardness results, we show that these problems belong to \u0000$textsf{A}[2]$\u0000 , \u0000$textsf{W}[textsf{P}],$\u0000 and \u0000$textsf{WNL}$\u0000 . This indicates that DFA-SW is not complete for any of these classes, and hence, we suggest a new parameterized complexity class \u0000$textsf{W}[textsf{Sync}]$\u0000 as a proper home for these (and more) problems. We present quite a number of problems that belong to \u0000$textsf{W}[textsf{Sync}]$\u0000 or are hard or complete for this new class.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"189 - 215"},"PeriodicalIF":0.5,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45026659","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 : 2022-02-01DOI: 10.1017/S0960129522000093
N. Bazhenov, M. Mustafa, S. Ospichev
Abstract The paper works within the framework of punctual computability, which is focused on eliminating unbounded search from constructions in algebra and infinite combinatorics. We study punctual numberings, that is, uniform computations for families S of primitive recursive functions. The punctual reducibility between numberings is induced by primitive recursive functions. This approach gives rise to upper semilattices of degrees, which are called Rogers pr-semilattices. We show that any infinite, uniformly primitive recursive family S induces an infinite Rogers pr-semilattice R. We prove that the semilattice R does not have minimal elements, and every nontrivial interval inside R contains an infinite antichain. In addition, every non-greatest element from R is a part of an infinite antichain. We show that the $Sigma_1$ -fragment of the theory Th(R) is decidable.
{"title":"Rogers semilattices of punctual numberings","authors":"N. Bazhenov, M. Mustafa, S. Ospichev","doi":"10.1017/S0960129522000093","DOIUrl":"https://doi.org/10.1017/S0960129522000093","url":null,"abstract":"Abstract The paper works within the framework of punctual computability, which is focused on eliminating unbounded search from constructions in algebra and infinite combinatorics. We study punctual numberings, that is, uniform computations for families S of primitive recursive functions. The punctual reducibility between numberings is induced by primitive recursive functions. This approach gives rise to upper semilattices of degrees, which are called Rogers pr-semilattices. We show that any infinite, uniformly primitive recursive family S induces an infinite Rogers pr-semilattice R. We prove that the semilattice R does not have minimal elements, and every nontrivial interval inside R contains an infinite antichain. In addition, every non-greatest element from R is a part of an infinite antichain. We show that the \u0000$Sigma_1$\u0000 -fragment of the theory Th(R) is decidable.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"164 - 188"},"PeriodicalIF":0.5,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44059358","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}
Abstract One of the most important results in domain theory is the Hofmann-Mislove Theorem, which reveals a very distinct characterization for the sober spaces via open filters. In this paper, we extend this result to the d-spaces and well-filtered spaces. We do this by introducing the notions of Hofmann-Mislove-system (HM-system for short) and $Psi$ -well-filtered space, which provide a new unified approach to sober spaces, well-filtered spaces, and d-spaces. In addition, a characterization for $Psi$ -well-filtered spaces is provided via $Psi$ -sets. We also discuss the relationship between $Psi$ -well-filtered spaces and H-sober spaces considered by Xu. We show that the category of complete $Psi$ -well-filtered spaces is a full reflective subcategory of the category of $T_0$ spaces with continuous mappings. For each HM-system $Psi$ that has a designated property, we show that a $T_0$ space X is $Psi$ -well-filtered if and only if its Smyth power space $P_s(X)$ is $Psi$ -well-filtered.
{"title":"Hofmann-Mislove type definitions of non-Hausdorff spaces","authors":"Chong Shen, Xiaoyong Xi, Xiaoquan Xu, Dongsheng Zhao","doi":"10.1017/S0960129522000196","DOIUrl":"https://doi.org/10.1017/S0960129522000196","url":null,"abstract":"Abstract One of the most important results in domain theory is the Hofmann-Mislove Theorem, which reveals a very distinct characterization for the sober spaces via open filters. In this paper, we extend this result to the d-spaces and well-filtered spaces. We do this by introducing the notions of Hofmann-Mislove-system (HM-system for short) and \u0000$Psi$\u0000 -well-filtered space, which provide a new unified approach to sober spaces, well-filtered spaces, and d-spaces. In addition, a characterization for \u0000$Psi$\u0000 -well-filtered spaces is provided via \u0000$Psi$\u0000 -sets. We also discuss the relationship between \u0000$Psi$\u0000 -well-filtered spaces and H-sober spaces considered by Xu. We show that the category of complete \u0000$Psi$\u0000 -well-filtered spaces is a full reflective subcategory of the category of \u0000$T_0$\u0000 spaces with continuous mappings. For each HM-system \u0000$Psi$\u0000 that has a designated property, we show that a \u0000$T_0$\u0000 space X is \u0000$Psi$\u0000 -well-filtered if and only if its Smyth power space \u0000$P_s(X)$\u0000 is \u0000$Psi$\u0000 -well-filtered.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"111 - 124"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49542231","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 : 2022-01-01DOI: 10.1017/S0960129522000159
I. Petrakis
Abstract Bishop’s presentation of his informal system of constructive mathematics BISH was on purpose closer to the proof-irrelevance of classical mathematics, although a form of proof-relevance was evident in the use of several notions of moduli (of convergence, of uniform continuity, of uniform differentiability, etc.). Focusing on membership and equality conditions for sets given by appropriate existential formulas, we define certain families of proof sets that provide a BHK-interpretation of formulas that correspond to the standard atomic formulas of a first-order theory, within Bishop set theory $(mathrm{BST})$ , our minimal extension of Bishop’s theory of sets. With the machinery of the general theory of families of sets, this BHK-interpretation within BST is extended to complex formulas. Consequently, we can associate to many formulas $phi$ of BISH a set ${texttt{Prf}}(phi)$ of “proofs” or witnesses of $phi$ . Abstracting from several examples of totalities in BISH, we define the notion of a set with a proof-relevant equality, and of a Martin-Löf set, a special case of the former, the equality of which corresponds to the identity type of a type in intensional Martin-Löf type theory $(mathrm{MLTT})$ . Through the concepts and results of BST notions and facts of MLTT and its extensions (either with the axiom of function extensionality or with Vooevodsky’s axiom of univalence) can be translated into BISH. While Bishop’s theory of sets is standardly understood through its translation to MLTT, our development of BST offers a partial translation in the converse direction.
{"title":"Proof-relevance in Bishop-style constructive mathematics","authors":"I. Petrakis","doi":"10.1017/S0960129522000159","DOIUrl":"https://doi.org/10.1017/S0960129522000159","url":null,"abstract":"Abstract Bishop’s presentation of his informal system of constructive mathematics BISH was on purpose closer to the proof-irrelevance of classical mathematics, although a form of proof-relevance was evident in the use of several notions of moduli (of convergence, of uniform continuity, of uniform differentiability, etc.). Focusing on membership and equality conditions for sets given by appropriate existential formulas, we define certain families of proof sets that provide a BHK-interpretation of formulas that correspond to the standard atomic formulas of a first-order theory, within Bishop set theory \u0000$(mathrm{BST})$\u0000 , our minimal extension of Bishop’s theory of sets. With the machinery of the general theory of families of sets, this BHK-interpretation within BST is extended to complex formulas. Consequently, we can associate to many formulas \u0000$phi$\u0000 of BISH a set \u0000${texttt{Prf}}(phi)$\u0000 of “proofs” or witnesses of \u0000$phi$\u0000 . Abstracting from several examples of totalities in BISH, we define the notion of a set with a proof-relevant equality, and of a Martin-Löf set, a special case of the former, the equality of which corresponds to the identity type of a type in intensional Martin-Löf type theory \u0000$(mathrm{MLTT})$\u0000 . Through the concepts and results of BST notions and facts of MLTT and its extensions (either with the axiom of function extensionality or with Vooevodsky’s axiom of univalence) can be translated into BISH. While Bishop’s theory of sets is standardly understood through its translation to MLTT, our development of BST offers a partial translation in the converse direction.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"1 - 43"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46111842","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 : 2022-01-01DOI: 10.1017/S0960129521000505
Ugo Dal Lago
{"title":"Implicit computation complexity in higher-order programming languages: A Survey in Memory of Martin Hofmann","authors":"Ugo Dal Lago","doi":"10.1017/S0960129521000505","DOIUrl":"https://doi.org/10.1017/S0960129521000505","url":null,"abstract":"","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"760-776"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"57259158","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 : 2022-01-01DOI: 10.1017/S0960129522000160
G. Amato, M. Meo, F. Scozzari
Abstract Sharing analysis is used to statically discover data structures which may overlap in object-oriented programs. Using the abstract interpretation framework, we show that sharing analysis greatly benefits from linearity information. A variable is linear in a program state when different field paths starting from it always reach different objects. We propose a graph-based abstract domain which can represent aliasing, linearity, and sharing information and define all the necessary abstract operators for the analysis of a Java-like language.
{"title":"The role of linearity in sharing analysis","authors":"G. Amato, M. Meo, F. Scozzari","doi":"10.1017/S0960129522000160","DOIUrl":"https://doi.org/10.1017/S0960129522000160","url":null,"abstract":"Abstract Sharing analysis is used to statically discover data structures which may overlap in object-oriented programs. Using the abstract interpretation framework, we show that sharing analysis greatly benefits from linearity information. A variable is linear in a program state when different field paths starting from it always reach different objects. We propose a graph-based abstract domain which can represent aliasing, linearity, and sharing information and define all the necessary abstract operators for the analysis of a Java-like language.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"44 - 110"},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42850724","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 : 2021-12-10DOI: 10.1017/s0960129521000311
Anders Mörtberg
Cubical methods have played an important role in the development of Homotopy Type Theory and Univalent Foundations (HoTT/UF) in recent years. The original motivation behind these developments was to give constructive meaning to Voevodsky’s univalence axiom, but they have since then led to a range of new results. Among the achievements of these methods is the design of new type theories and proof assistants with native support for notions from HoTT/UF, syntactic and semantic consistency results for HoTT/UF, as well as a variety of independence results and establishing that the univalence axiom does not increase the proof theoretic strength of type theory. This paper is based on lecture notes that were written for the 2019 Homotopy Type Theory Summer School at Carnegie Mellon University. The goal of these lectures was to give an introduction to cubical methods and provide sufficient background in order to make the current research in this very active area of HoTT/UF more accessible to newcomers. The focus of these notes is hence on both the syntactic and semantic aspects of these methods, in particular on cubical type theory and the various cubical set categories that give meaning to these theories.
{"title":"Cubical methods in homotopy type theory and univalent foundations","authors":"Anders Mörtberg","doi":"10.1017/s0960129521000311","DOIUrl":"https://doi.org/10.1017/s0960129521000311","url":null,"abstract":"\u0000 Cubical methods have played an important role in the development of Homotopy Type Theory and Univalent Foundations (HoTT/UF) in recent years. The original motivation behind these developments was to give constructive meaning to Voevodsky’s univalence axiom, but they have since then led to a range of new results. Among the achievements of these methods is the design of new type theories and proof assistants with native support for notions from HoTT/UF, syntactic and semantic consistency results for HoTT/UF, as well as a variety of independence results and establishing that the univalence axiom does not increase the proof theoretic strength of type theory. This paper is based on lecture notes that were written for the 2019 Homotopy Type Theory Summer School at Carnegie Mellon University. The goal of these lectures was to give an introduction to cubical methods and provide sufficient background in order to make the current research in this very active area of HoTT/UF more accessible to newcomers. The focus of these notes is hence on both the syntactic and semantic aspects of these methods, in particular on cubical type theory and the various cubical set categories that give meaning to these theories.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"15 1","pages":"1147-1184"},"PeriodicalIF":0.5,"publicationDate":"2021-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88929140","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}
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