Pub Date : 2024-09-19DOI: 10.1017/s0960129524000203
Steve Awodey
We take another look at the construction by Hofmann and Streicher of a universe $(U,{mathcal{E}l})$ for the interpretation of Martin-Löf type theory in a presheaf category $[{{{mathbb{C}}}^{textrm{op}}},textsf{Set}]$ . It turns out that $(U,{mathcal{E}l})$ can be described as the nerve of the classifier $dot{{textsf{Set}}}^{textsf{op}} rightarrow{{textsf{Set}}}^{textsf{op}}$ for discrete fibrations in $textsf{Cat}$ , where the nerve functor is right adjoint to the so-called “Grothendieck construction” taking a presheaf $P :{{{mathbb{C}}}^{textrm{op}}}rightarrow{textsf{Set}}$ to its category of elements $int _{mathbb{C}} P$ . We also consider change of base for such universes, as well as universes of structured families, such as fibrations.
{"title":"On Hofmann–Streicher universes","authors":"Steve Awodey","doi":"10.1017/s0960129524000203","DOIUrl":"https://doi.org/10.1017/s0960129524000203","url":null,"abstract":"We take another look at the construction by Hofmann and Streicher of a universe <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000203_inline1.png\"/> <jats:tex-math> $(U,{mathcal{E}l})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for the interpretation of Martin-Löf type theory in a presheaf category <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000203_inline2.png\"/> <jats:tex-math> $[{{{mathbb{C}}}^{textrm{op}}},textsf{Set}]$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. It turns out that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000203_inline3.png\"/> <jats:tex-math> $(U,{mathcal{E}l})$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> can be described as the <jats:italic>nerve</jats:italic> of the classifier <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000203_inline4.png\"/> <jats:tex-math> $dot{{textsf{Set}}}^{textsf{op}} rightarrow{{textsf{Set}}}^{textsf{op}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> for discrete fibrations in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000203_inline5.png\"/> <jats:tex-math> $textsf{Cat}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, where the nerve functor is right adjoint to the so-called “Grothendieck construction” taking a presheaf <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000203_inline6.png\"/> <jats:tex-math> $P :{{{mathbb{C}}}^{textrm{op}}}rightarrow{textsf{Set}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> to its category of elements <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000203_inline7.png\"/> <jats:tex-math> $int _{mathbb{C}} P$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. We also consider change of base for such universes, as well as universes of structured families, such as fibrations.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"29 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259193","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 : 2024-09-19DOI: 10.1017/s0960129524000240
Jimmie Lawson, Xiaoquan Xu
The authors’ primary goal in this paper is to enhance the study of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0960129524000240_inline1.png"/> <jats:tex-math> $T_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> topological spaces by using the order of specialization of a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0960129524000240_inline2.png"/> <jats:tex-math> $T_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-space to introduce the lower topology (with a subbasis of closed sets <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0960129524000240_inline3.png"/> <jats:tex-math> $mathord{uparrow } x$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>) and studying the interaction of the original topology and the lower topology. Using the lower topology, one can define and study new properties of the original space that provide deeper insight into its structure. One focus of study is the property R, which asserts that if the intersection of a family of finitely generated sets <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0960129524000240_inline4.png"/> <jats:tex-math> $mathord{uparrow } F$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0960129524000240_inline5.png"/> <jats:tex-math> $F$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> finite, is contained in an open set <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0960129524000240_inline6.png"/> <jats:tex-math> $U$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, then the same is true for finitely many of the family. We first show that property R is equivalent to several other interesting properties, for example, the property that all closed subsets of the original space are compact in the lower topology. We then find conditions under which these spaces are compact, well-filtered, and coherent, a weaker variant of stably compact spaces. We also investigate what have been called strong <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0960129524000240_inline7.png"/> <jats:tex-math> $d$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-spaces, develop some of their basic properties, and make connections with the earlier considerations involving spaces satisfying property R. Two key results we obtain are that if a dcpo <jats:inline-formula
{"title":"T0-spaces and the lower topology","authors":"Jimmie Lawson, Xiaoquan Xu","doi":"10.1017/s0960129524000240","DOIUrl":"https://doi.org/10.1017/s0960129524000240","url":null,"abstract":"The authors’ primary goal in this paper is to enhance the study of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000240_inline1.png\"/> <jats:tex-math> $T_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> topological spaces by using the order of specialization of a <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000240_inline2.png\"/> <jats:tex-math> $T_0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-space to introduce the lower topology (with a subbasis of closed sets <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000240_inline3.png\"/> <jats:tex-math> $mathord{uparrow } x$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>) and studying the interaction of the original topology and the lower topology. Using the lower topology, one can define and study new properties of the original space that provide deeper insight into its structure. One focus of study is the property R, which asserts that if the intersection of a family of finitely generated sets <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000240_inline4.png\"/> <jats:tex-math> $mathord{uparrow } F$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000240_inline5.png\"/> <jats:tex-math> $F$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> finite, is contained in an open set <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000240_inline6.png\"/> <jats:tex-math> $U$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, then the same is true for finitely many of the family. We first show that property R is equivalent to several other interesting properties, for example, the property that all closed subsets of the original space are compact in the lower topology. We then find conditions under which these spaces are compact, well-filtered, and coherent, a weaker variant of stably compact spaces. We also investigate what have been called strong <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000240_inline7.png\"/> <jats:tex-math> $d$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-spaces, develop some of their basic properties, and make connections with the earlier considerations involving spaces satisfying property R. Two key results we obtain are that if a dcpo <jats:inline-formula","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"20 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259194","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 : 2024-08-27DOI: 10.1017/s0960129524000161
Pierre Cagne, Enrico Ghiorzi, Patricia Johann
Generalized Algebraic Data Types (GADTs) are a syntactic generalization of the usual algebraic data types (ADTs), such as lists, trees, etc. ADTs’ standard initial algebra semantics (IAS) in the category $mathit{Set}$ of sets justify critical syntactic constructs – such as recursion, pattern matching, and fold – for programming with them. In this paper, we show that semantics for GADTs that specialize to the IAS for ADTs are necessarily unsatisfactory. First, we show that the functorial nature of such semantics for GADTs in $mathit{Set}$ introduces ghost elements, i.e., elements not writable in syntax. Next, we show how such ghost elements break parametricity. We observe that the situation for GADTs contrasts dramatically with that for ADTs, whose IAS coincides with the parametric model constructed via their Church encodings in System F. Our analysis reveals that the fundamental obstacle to giving a functorial IAS for GADTs is the inherently partial nature of their map functions. We show that this obstacle cannot be overcome by replacing $mathit{Set}$ with other categories that account for this partiality.
{"title":"GADTs are not (Even partial) functors","authors":"Pierre Cagne, Enrico Ghiorzi, Patricia Johann","doi":"10.1017/s0960129524000161","DOIUrl":"https://doi.org/10.1017/s0960129524000161","url":null,"abstract":"<jats:italic>Generalized Algebraic Data Types</jats:italic> (GADTs) are a syntactic generalization of the usual algebraic data types (ADTs), such as lists, trees, etc. ADTs’ standard initial algebra semantics (IAS) in the category <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000161_inline1.png\"/> <jats:tex-math> $mathit{Set}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> of sets justify critical syntactic constructs – such as recursion, pattern matching, and fold – for programming with them. In this paper, we show that semantics for GADTs that specialize to the IAS for ADTs are necessarily unsatisfactory. First, we show that the functorial nature of such semantics for GADTs in <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000161_inline2.png\"/> <jats:tex-math> $mathit{Set}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> introduces <jats:italic>ghost</jats:italic> elements, i.e., elements not writable in syntax. Next, we show how such ghost elements break parametricity. We observe that the situation for GADTs contrasts dramatically with that for ADTs, whose IAS coincides with the parametric model constructed via their Church encodings in System F. Our analysis reveals that the fundamental obstacle to giving a functorial IAS for GADTs is the inherently partial nature of their map functions. We show that this obstacle cannot be overcome by replacing <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000161_inline3.png\"/> <jats:tex-math> $mathit{Set}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> with other categories that account for this partiality.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"19 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142176661","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 : 2024-05-31DOI: 10.1017/s0960129524000197
Alejandro Díaz-Caro, Gilles Dowek
We present a linearity theorem for a proof language of intuitionistic multiplicative additive linear logic, incorporating addition and scalar multiplication. The proofs in this language are linear in the algebraic sense. This work is part of a broader research program aiming to define a logic with a proof language that forms a quantum programming language.
{"title":"A linear linear lambda-calculus","authors":"Alejandro Díaz-Caro, Gilles Dowek","doi":"10.1017/s0960129524000197","DOIUrl":"https://doi.org/10.1017/s0960129524000197","url":null,"abstract":"We present a linearity theorem for a proof language of intuitionistic multiplicative additive linear logic, incorporating addition and scalar multiplication. The proofs in this language are linear in the algebraic sense. This work is part of a broader research program aiming to define a logic with a proof language that forms a quantum programming language.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"74 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141190298","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 : 2024-05-30DOI: 10.1017/s0960129524000173
Pedro Hack, Daniel A. Braun, Sebastian Gottwald
Computability on uncountable sets has no standard formalization, unlike that on countable sets, which is given by Turing machines. Some of the approaches to define computability in these sets rely on order-theoretic structures to translate such notions from Turing machines to uncountable spaces. Since these machines are used as a baseline for computability in these approaches, countability restrictions on the ordered structures are fundamental. Here, we show several relations between the usual countability restrictions in order-theoretic theories of computability and some more common order-theoretic countability constraints, like order density properties and functional characterizations of the order structure in terms of multi-utilities. As a result, we show how computability can be introduced in some order structures via countability order density and multi-utility constraints.
{"title":"Countability constraints in order-theoretic approaches to computability","authors":"Pedro Hack, Daniel A. Braun, Sebastian Gottwald","doi":"10.1017/s0960129524000173","DOIUrl":"https://doi.org/10.1017/s0960129524000173","url":null,"abstract":"Computability on uncountable sets has no standard formalization, unlike that on countable sets, which is given by Turing machines. Some of the approaches to define computability in these sets rely on order-theoretic structures to translate such notions from Turing machines to uncountable spaces. Since these machines are used as a baseline for computability in these approaches, countability restrictions on the ordered structures are fundamental. Here, we show several relations between the usual countability restrictions in order-theoretic theories of computability and some more common order-theoretic countability constraints, like order density properties and functional characterizations of the order structure in terms of multi-utilities. As a result, we show how computability can be introduced in some order structures via countability order density and multi-utility constraints.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"43 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141190294","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 : 2024-05-13DOI: 10.1017/s0960129524000136
Timon Barlag, Florian Chudigiewitsch, Sabrina A. Gaube
We present an adapted construction of algebraic circuits over the reals introduced by Cucker and Meer to arbitrary infinite integral domains and generalize the $mathrm{AC}_{mathbb{R}}$ and $mathrm{NC}_{mathbb{R}}^{}$ classes for this setting. We give a theorem in the style of Immerman’s theorem which shows that for these adapted formalisms, sets decided by circuits of constant depth and polynomial size are the same as sets definable by a suitable adaptation of first-order logic. Additionally, we discuss a generalization of the guarded predicative logic by Durand, Haak and Vollmer, and we show characterizations for the $mathrm{AC}_{R}$ and $mathrm{NC}_R^{}$ hierarchy. Those generalizations apply to the Boolean $mathrm{AC}$ and $mathrm{NC}$ hierarchies as well. Furthermore, we introduce a formalism to be able to compare some of the aforementioned complexity classes with different underlying integral domains.
{"title":"Logical characterizations of algebraic circuit classes over integral domains","authors":"Timon Barlag, Florian Chudigiewitsch, Sabrina A. Gaube","doi":"10.1017/s0960129524000136","DOIUrl":"https://doi.org/10.1017/s0960129524000136","url":null,"abstract":"We present an adapted construction of algebraic circuits over the reals introduced by Cucker and Meer to arbitrary infinite integral domains and generalize the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000136_inline1.png\"/> <jats:tex-math> $mathrm{AC}_{mathbb{R}}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000136_inline2.png\"/> <jats:tex-math> $mathrm{NC}_{mathbb{R}}^{}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> classes for this setting. We give a theorem in the style of Immerman’s theorem which shows that for these adapted formalisms, sets decided by circuits of constant depth and polynomial size are the same as sets definable by a suitable adaptation of first-order logic. Additionally, we discuss a generalization of the guarded predicative logic by Durand, Haak and Vollmer, and we show characterizations for the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000136_inline3.png\"/> <jats:tex-math> $mathrm{AC}_{R}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000136_inline4.png\"/> <jats:tex-math> $mathrm{NC}_R^{}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> hierarchy. Those generalizations apply to the Boolean <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000136_inline5.png\"/> <jats:tex-math> $mathrm{AC}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000136_inline6.png\"/> <jats:tex-math> $mathrm{NC}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> hierarchies as well. Furthermore, we introduce a formalism to be able to compare some of the aforementioned complexity classes with different underlying integral domains.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"16 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140931685","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 : 2024-05-08DOI: 10.1017/s0960129524000100
Jonathan Prieto-Cubides, Håkon Robbestad Gylterud
In this paper, we present a constructive and proof-relevant development of graph theory, including the notion of maps, their faces and maps of graphs embedded in the sphere, in homotopy type theory (HoTT). This allows us to provide an elementary characterisation of planarity for locally directed finite and connected multigraphs that takes inspiration from topological graph theory, particularly from combinatorial embeddings of graphs into surfaces. A graph is planar if it has a map and an outer face with which any walk in the embedded graph is walk-homotopic to another. A result is that this type of planar maps forms a homotopy set for a graph. As a way to construct examples of planar graphs inductively, extensions of planar maps are introduced. We formalise the essential parts of this work in the proof assistant Agda with support for HoTT.
{"title":"On planarity of graphs in homotopy type theory","authors":"Jonathan Prieto-Cubides, Håkon Robbestad Gylterud","doi":"10.1017/s0960129524000100","DOIUrl":"https://doi.org/10.1017/s0960129524000100","url":null,"abstract":"In this paper, we present a constructive and proof-relevant development of graph theory, including the notion of maps, their faces and maps of graphs embedded in the sphere, in homotopy type theory (HoTT). This allows us to provide an elementary characterisation of planarity for locally directed finite and connected multigraphs that takes inspiration from topological graph theory, particularly from combinatorial embeddings of graphs into surfaces. A graph is planar if it has a map and an outer face with which any walk in the embedded graph is walk-homotopic to another. A result is that this type of planar maps forms a homotopy set for a graph. As a way to construct examples of planar graphs inductively, extensions of planar maps are introduced. We formalise the essential parts of this work in the proof assistant Agda with support for HoTT.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"43 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140932043","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 : 2024-04-19DOI: 10.1017/s0960129524000082
Hans van Ditmarsch, Malvin Gattinger
It is known that without synchronization via a global clock one cannot obtain common knowledge by communication. Moreover, it is folklore that without communicating higher-level information one cannot obtain arbitrary higher-order shared knowledge. Here, we make this result precise in the setting of gossip where agents make one-to-one telephone calls to share secrets: we prove that “everyone knows that everyone knows that everyone knows all secrets” is unsatisfiable in a logic of knowledge for gossiping. We also prove that, given n agents, $2n-3$ calls are optimal to reach “someone knows that everyone knows all secrets” and that $n - 2 + binom{n}{2}$ calls are optimal to reach “everyone knows that everyone knows all secrets.”
{"title":"You can only be lucky once: optimal gossip for epistemic goals","authors":"Hans van Ditmarsch, Malvin Gattinger","doi":"10.1017/s0960129524000082","DOIUrl":"https://doi.org/10.1017/s0960129524000082","url":null,"abstract":"It is known that without synchronization via a global clock one cannot obtain common knowledge by communication. Moreover, it is folklore that without communicating higher-level information one cannot obtain arbitrary higher-order shared knowledge. Here, we make this result precise in the setting of gossip where agents make one-to-one telephone calls to share secrets: we prove that “everyone knows that everyone knows that everyone knows all secrets” is unsatisfiable in a logic of knowledge for gossiping. We also prove that, given <jats:italic>n</jats:italic> agents, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000082_inline1.png\" /> <jats:tex-math> $2n-3$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> calls are optimal to reach “someone knows that everyone knows all secrets” and that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0960129524000082_inline2.png\" /> <jats:tex-math> $n - 2 + binom{n}{2}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> calls are optimal to reach “everyone knows that everyone knows all secrets.”","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"209 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140625889","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 : 2024-04-19DOI: 10.1017/s0960129524000070
Bahareh Afshari, Dominik Wehr
Cyclic proof systems permit derivations that are finite graphs in contrast to conventional derivation trees. The soundness of such proofs is ensured by imposing a soundness condition on derivations. The most common such condition is the global trace condition (GTC), a condition on the infinite paths through the derivation graph. To give a uniform treatment of such cyclic proof systems, Brotherston proposed an abstract notion of trace. We extend Brotherston’s approach into a category theoretical rendition of cyclic derivations, advancing the framework in two ways: first, we introduce activation algebras which allow for a more natural formalisation of trace conditions in extant cyclic proof systems. Second, accounting for the composition of trace information allows us to derive novel results about cyclic proofs, such as introducing a Ramsey-style trace condition. Furthermore, we connect our notion of trace to automata theory and prove that verifying the GTC for abstract cyclic proofs with certain trace conditions is PSPACE-complete.
{"title":"Abstract cyclic proofs","authors":"Bahareh Afshari, Dominik Wehr","doi":"10.1017/s0960129524000070","DOIUrl":"https://doi.org/10.1017/s0960129524000070","url":null,"abstract":"Cyclic proof systems permit derivations that are finite graphs in contrast to conventional derivation trees. The soundness of such proofs is ensured by imposing a soundness condition on derivations. The most common such condition is the <jats:italic>global trace condition</jats:italic> (GTC), a condition on the infinite paths through the derivation graph. To give a uniform treatment of such cyclic proof systems, Brotherston proposed an abstract notion of trace. We extend Brotherston’s approach into a category theoretical rendition of cyclic derivations, advancing the framework in two ways: first, we introduce <jats:italic>activation algebras</jats:italic> which allow for a more natural formalisation of trace conditions in extant cyclic proof systems. Second, accounting for the composition of trace information allows us to derive novel results about cyclic proofs, such as introducing a Ramsey-style trace condition. Furthermore, we connect our notion of trace to automata theory and prove that verifying the GTC for abstract cyclic proofs with certain trace conditions is PSPACE-complete.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"82 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140625588","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 : 2024-04-18DOI: 10.1017/s0960129524000057
Axel Muller, Metod Saniga, Alain Giorgetti, Henri de Boutray, Frédéric Holweck
We present algorithms and a C code to reveal quantum contextuality and evaluate the contextuality degree (a way to quantify contextuality) for a variety of point-line geometries located in binary symplectic polar spaces of small rank. With this code we were not only able to recover, in a more efficient way, all the results of a recent paper by de Boutray et al. [(2022). Journal of Physics A: Mathematical and Theoretical55 475301], but also arrived at a bunch of new noteworthy results. The paper first describes the algorithms and the C code. Then it illustrates its power on a number of subspaces of symplectic polar spaces whose rank ranges from 2 to 7. The most interesting new results include: (i) non-contextuality of configurations whose contexts are subspaces of dimension 2 and higher, (ii) non-existence of negative subspaces of dimension 3 and higher, (iii) considerably improved bounds for the contextuality degree of both elliptic and hyperbolic quadrics for rank 4, as well as for a particular subgeometry of the three-qubit space whose contexts are the lines of this space, (iv) proof for the non-contextuality of perpsets and, last but not least, (v) contextual nature of a distinguished subgeometry of a multi-qubit doily, called a two-spread, and computation of its contextuality degree. Finally, in the three-qubit polar space we correct and improve the contextuality degree of the full configuration and also describe finite geometric configurations formed by unsatisfiable/invalid constraints for both types of quadrics as well as for the geometry whose contexts are all 315 lines of the space.
我们提出了揭示量子情境性的算法和 C 代码，并评估了位于小秩二元交点极空间中的各种点线几何的情境性程度（量化情境性的一种方法）。有了这套代码，我们不仅能以更高效的方式恢复 de Boutray 等人最近发表的论文[（2022）. 物理期刊 A：数学与理论 55 475301]中的所有结果，而且还得出了一系列值得注意的新结果。论文首先介绍了算法和 C 代码。然后说明了它在秩为 2 到 7 的交错极空间的一些子空间上的威力。最有趣的新结果包括(i) 上下文为维数为 2 或更高的子空间的构型的非上下文性，(ii) 维数为 3 或更高的负子空间的不存在性，(iii) 极大地改进了阶为 4 的椭圆和双曲四边形的上下文性程度边界、(iv)证明了周集的非上下文性，最后但并非最不重要的是，(v)证明了多比特多面体的一个杰出子几何（称为双展宽）的上下文性质及其上下文性度的计算。最后，在三比特极坐标空间中，我们修正并改进了完整构型的上下文度，还描述了由两类四面体以及上下文均为空间 315 条线的几何体的不可满足/无效约束所形成的有限几何构型。
{"title":"New and improved bounds on the contextuality degree of multi-qubit configurations","authors":"Axel Muller, Metod Saniga, Alain Giorgetti, Henri de Boutray, Frédéric Holweck","doi":"10.1017/s0960129524000057","DOIUrl":"https://doi.org/10.1017/s0960129524000057","url":null,"abstract":"<p>We present algorithms and a C code to reveal quantum contextuality and evaluate the contextuality degree (a way to quantify contextuality) for a variety of point-line geometries located in binary symplectic polar spaces of small rank. With this code we were not only able to recover, in a more efficient way, all the results of a recent paper by de Boutray et al. [(2022). <span>Journal of Physics A: Mathematical and Theoretical</span> <span>55</span> 475301], but also arrived at a bunch of new noteworthy results. The paper first describes the algorithms and the C code. Then it illustrates its power on a number of subspaces of symplectic polar spaces whose rank ranges from 2 to 7. The most interesting new results include: (i) non-contextuality of configurations whose contexts are subspaces of dimension 2 and higher, (ii) non-existence of negative subspaces of dimension 3 and higher, (iii) considerably improved bounds for the contextuality degree of both elliptic and hyperbolic quadrics for rank 4, as well as for a particular subgeometry of the three-qubit space whose contexts are the lines of this space, (iv) proof for the non-contextuality of perpsets and, last but not least, (v) contextual nature of a distinguished subgeometry of a multi-qubit doily, called a two-spread, and computation of its contextuality degree. Finally, in the three-qubit polar space we correct and improve the contextuality degree of the full configuration and also describe finite geometric configurations formed by unsatisfiable/invalid constraints for both types of quadrics as well as for the geometry whose contexts are all 315 lines of the space.</p>","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"85 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140609687","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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