Pub Date : 2022-06-27DOI: 10.1017/s0960129522000172
Thibaut Benjamin
Weak $omega$ -categories are notoriously difficult to define because of the very intricate nature of their axioms. Various approaches have been explored based on different shapes given to the cells. Interestingly, homotopy type theory encompasses a definition of weak $omega$ -groupoid in a globular setting, since every type carries such a structure. Starting from this remark, Brunerie could extract this definition of globular weak $omega$ -groupoids, formulated as a type theory. By refining its rules, Finster and Mimram have then defined a type theory called $mathsf{CaTT}$ , whose models are weak $omega$ -categories. Here, we generalize this approach to monoidal weak $omega$ -categories. Based on the principle that they should be equivalent to weak $omega$ -categories with only one 0-cell, we are able to derive a type theory $mathsf{MCaTT}$ whose models are monoidal weak $omega$ -categories. This requires changing the rules of the theory in order to encode the information carried by the unique 0-cell. The correctness of the resulting type theory is shown by defining a pair of translations between our type theory $mathsf{MCaTT}$ and the type theory $mathsf{CaTT}$ . Our main contribution is to show that these translations relate the models of our type theory to the models of the type theory $mathsf{CaTT}$ consisting of $omega$ -categories with only one 0-cell by analyzing in details how the notion of models interact with the structural rules of both type theories.
{"title":"Monoidal weak ω-categories as models of a type theory","authors":"Thibaut Benjamin","doi":"10.1017/s0960129522000172","DOIUrl":"https://doi.org/10.1017/s0960129522000172","url":null,"abstract":"\u0000 Weak \u0000 \u0000 \u0000 \u0000$omega$\u0000\u0000 \u0000 -categories are notoriously difficult to define because of the very intricate nature of their axioms. Various approaches have been explored based on different shapes given to the cells. Interestingly, homotopy type theory encompasses a definition of weak \u0000 \u0000 \u0000 \u0000$omega$\u0000\u0000 \u0000 -groupoid in a globular setting, since every type carries such a structure. Starting from this remark, Brunerie could extract this definition of globular weak \u0000 \u0000 \u0000 \u0000$omega$\u0000\u0000 \u0000 -groupoids, formulated as a type theory. By refining its rules, Finster and Mimram have then defined a type theory called \u0000 \u0000 \u0000 \u0000$mathsf{CaTT}$\u0000\u0000 \u0000 , whose models are weak \u0000 \u0000 \u0000 \u0000$omega$\u0000\u0000 \u0000 -categories. Here, we generalize this approach to monoidal weak \u0000 \u0000 \u0000 \u0000$omega$\u0000\u0000 \u0000 -categories. Based on the principle that they should be equivalent to weak \u0000 \u0000 \u0000 \u0000$omega$\u0000\u0000 \u0000 -categories with only one 0-cell, we are able to derive a type theory \u0000 \u0000 \u0000 \u0000$mathsf{MCaTT}$\u0000\u0000 \u0000 whose models are monoidal weak \u0000 \u0000 \u0000 \u0000$omega$\u0000\u0000 \u0000 -categories. This requires changing the rules of the theory in order to encode the information carried by the unique 0-cell. The correctness of the resulting type theory is shown by defining a pair of translations between our type theory \u0000 \u0000 \u0000 \u0000$mathsf{MCaTT}$\u0000\u0000 \u0000 and the type theory \u0000 \u0000 \u0000 \u0000$mathsf{CaTT}$\u0000\u0000 \u0000 . Our main contribution is to show that these translations relate the models of our type theory to the models of the type theory \u0000 \u0000 \u0000 \u0000$mathsf{CaTT}$\u0000\u0000 \u0000 consisting of \u0000 \u0000 \u0000 \u0000$omega$\u0000\u0000 \u0000 -categories with only one 0-cell by analyzing in details how the notion of models interact with the structural rules of both type theories.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42372414","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 : 2022-06-01DOI: 10.1017/s0960129522000147
Jan Hoffmann, D. Sannella, Ulrich Schöpp
This is the second part of a two-part special issue dedicated to the memory of our friend and colleague, Martin Hofmann. The first part was published asMathematical Structures in Computer Science (2021), 31(9). On 21 January 2018, Martin Hofmann died in a tragic mountain hiking accident in Japan. He was there to attend a workshop at NII Shonan and arrived early for the workshop in order to spend a day climbing Mount Nikkō-Shirane. On his way down from the 2578 m summit, he was caught in a severe snowstorm and lost his way back to safety. Martin Hofmann studied for a Diplom in Informatics at Universität Erlangen-Nürnberg from November 1984 until August 1991. During an exchange visit at the Université de Nice from October 1987 to June 1988 he obtained in addition the “Maitrise de Mathematiques.” In 1991, he joined the Laboratory for Foundations of Computer Science at the University of Edinburgh. He
这是纪念我们的朋友和同事马丁·霍夫曼的两期特刊的第二部分。第一部分发表在《计算机科学中的数学结构》(2021)，31(9)。2018年1月21日，马丁·霍夫曼在日本的一次登山事故中不幸去世。他是为了参加NII Shonan的研讨会而来的，为了花一天时间攀登Nikkō-Shirane山，他提前到了研讨会现场。在从海拔2578米的山顶下山的途中，他遇到了一场严重的暴风雪，失去了返回安全地带的路。从1984年11月到1991年8月，Martin Hofmann在Universität erlangen - n rnberg大学攻读信息学文凭。1987年10月至1988年6月在尼斯大学交换访问期间，他还获得了“Maitrise de Mathematiques”学位。1991年，他加入了爱丁堡大学计算机科学基础实验室。他
{"title":"Preface for the special issue in homage to Martin Hofmann Part 2","authors":"Jan Hoffmann, D. Sannella, Ulrich Schöpp","doi":"10.1017/s0960129522000147","DOIUrl":"https://doi.org/10.1017/s0960129522000147","url":null,"abstract":"This is the second part of a two-part special issue dedicated to the memory of our friend and colleague, Martin Hofmann. The first part was published asMathematical Structures in Computer Science (2021), 31(9). On 21 January 2018, Martin Hofmann died in a tragic mountain hiking accident in Japan. He was there to attend a workshop at NII Shonan and arrived early for the workshop in order to spend a day climbing Mount Nikkō-Shirane. On his way down from the 2578 m summit, he was caught in a severe snowstorm and lost his way back to safety. Martin Hofmann studied for a Diplom in Informatics at Universität Erlangen-Nürnberg from November 1984 until August 1991. During an exchange visit at the Université de Nice from October 1987 to June 1988 he obtained in addition the “Maitrise de Mathematiques.” In 1991, he joined the Laboratory for Foundations of Computer Science at the University of Edinburgh. He","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"682 - 684"},"PeriodicalIF":0.5,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41688985","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 : 2022-06-01DOI: 10.1017/S0960129522000329
Hans Leiß
Abstract The family ${mathcal{R}} X^*$ of regular subsets of the free monoid $X^*$ generated by a finite set X is the standard example of a ${}^*$ -continuous Kleene algebra. Likewise, the family ${mathcal{C}} X^*$ of context-free subsets of $X^*$ is the standard example of a $mu$ -continuous Chomsky algebra, i.e. an idempotent semiring that is closed under a well-behaved least fixed-point operator $mu$ . For arbitrary monoids M, ${mathcal{C}} M$ is the closure of ${mathcal{R}}M$ as a $mu$ -continuous Chomsky algebra, more briefly, the fixed-point closure of ${mathcal{R}} M$ . We provide an algebraic representation of ${mathcal{C}} M$ in a suitable product of ${mathcal{R}} M$ with $C_2'$ , a quotient of the regular sets over an alphabet $Delta_2$ of two pairs of bracket symbols. Namely, ${mathcal{C}}M$ is isomorphic to the centralizer of $C_2'$ in the product of ${mathcal{R}} M$ with $C_2'$ , i.e. the set of those elements that commute with all elements of $C_2'$ . This generalizes a well-known result of Chomsky and Schützenberger (1963, Computer Programming and Formal Systems, 118–161) and admits us to denote all context-free languages over finite sets $Xsubseteq M$ by regular expressions over $XcupDelta_2$ interpreted in the product of ${mathcal{R}} M$ and $C_2'$ . More generally, for any ${}^*$ -continuous Kleene algebra K the fixed-point closure of K can be represented algebraically as the centralizer of $C_2'$ in the product of K with $C_2'$ .
{"title":"An algebraic representation of the fixed-point closure of *-continuous Kleene algebras – A categorical Chomsky–Schützenberger theorem","authors":"Hans Leiß","doi":"10.1017/S0960129522000329","DOIUrl":"https://doi.org/10.1017/S0960129522000329","url":null,"abstract":"Abstract The family \u0000${mathcal{R}} X^*$\u0000 of regular subsets of the free monoid \u0000$X^*$\u0000 generated by a finite set X is the standard example of a \u0000${}^*$\u0000 -continuous Kleene algebra. Likewise, the family \u0000${mathcal{C}} X^*$\u0000 of context-free subsets of \u0000$X^*$\u0000 is the standard example of a \u0000$mu$\u0000 -continuous Chomsky algebra, i.e. an idempotent semiring that is closed under a well-behaved least fixed-point operator \u0000$mu$\u0000 . For arbitrary monoids M, \u0000${mathcal{C}} M$\u0000 is the closure of \u0000${mathcal{R}}M$\u0000 as a \u0000$mu$\u0000 -continuous Chomsky algebra, more briefly, the fixed-point closure of \u0000${mathcal{R}} M$\u0000 . We provide an algebraic representation of \u0000${mathcal{C}} M$\u0000 in a suitable product of \u0000${mathcal{R}} M$\u0000 with \u0000$C_2'$\u0000 , a quotient of the regular sets over an alphabet \u0000$Delta_2$\u0000 of two pairs of bracket symbols. Namely, \u0000${mathcal{C}}M$\u0000 is isomorphic to the centralizer of \u0000$C_2'$\u0000 in the product of \u0000${mathcal{R}} M$\u0000 with \u0000$C_2'$\u0000 , i.e. the set of those elements that commute with all elements of \u0000$C_2'$\u0000 . This generalizes a well-known result of Chomsky and Schützenberger (1963, Computer Programming and Formal Systems, 118–161) and admits us to denote all context-free languages over finite sets \u0000$Xsubseteq M$\u0000 by regular expressions over \u0000$XcupDelta_2$\u0000 interpreted in the product of \u0000${mathcal{R}} M$\u0000 and \u0000$C_2'$\u0000 . More generally, for any \u0000${}^*$\u0000 -continuous Kleene algebra K the fixed-point closure of K can be represented algebraically as the centralizer of \u0000$C_2'$\u0000 in the product of K with \u0000$C_2'$\u0000 .","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"685 - 728"},"PeriodicalIF":0.5,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41848301","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 : 2022-05-17DOI: 10.1017/S096012952200010X
Sandro Preto, M. Finger
Abstract This work concerns the representation of a class of continuous functions into Logic, so that one may automatically reason about properties of these functions using logical tools. Rational McNaughton functions may be implicitly represented by logical formulas in Łukasiewicz Infinitely-valued Logic by constraining the set of allowed valuations; such a restriction contemplates only those valuations that satisfy specific formulas. This work investigates two approaches to such depiction, called representation modulo satisfiability. Furthermore, a polynomial-time algorithm that builds this representation is presented, producing a pair of formulas consisting of the representative formula and the constraining one, given as input a rational McNaughton function in a suitable encoding. An implementation of the algorithm is discussed.
{"title":"Efficient representation of piecewise linear functions into Łukasiewicz logic modulo satisfiability","authors":"Sandro Preto, M. Finger","doi":"10.1017/S096012952200010X","DOIUrl":"https://doi.org/10.1017/S096012952200010X","url":null,"abstract":"Abstract This work concerns the representation of a class of continuous functions into Logic, so that one may automatically reason about properties of these functions using logical tools. Rational McNaughton functions may be implicitly represented by logical formulas in Łukasiewicz Infinitely-valued Logic by constraining the set of allowed valuations; such a restriction contemplates only those valuations that satisfy specific formulas. This work investigates two approaches to such depiction, called representation modulo satisfiability. Furthermore, a polynomial-time algorithm that builds this representation is presented, producing a pair of formulas consisting of the representative formula and the constraining one, given as input a rational McNaughton function in a suitable encoding. An implementation of the algorithm is discussed.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"1119 - 1144"},"PeriodicalIF":0.5,"publicationDate":"2022-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48680853","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 : 2022-05-01DOI: 10.1017/S0960129522000342
L. Mazo, Marie-Andrée Da Col-Jacob, Laurent Fuchs, Nicolas Magaud, Gaëlle Skapin
Abstract The paper describes three models of the real field based on subsets of the integer sequences. The three models are compared to the Harthong–Reeb line. Two of the new models, contrary to the Harthong–Reeb line, provide accurate integer “views” on real numbers at a sequence of growing scales $B^n$ ( $Bge2$ ).
{"title":"Some representations of real numbers using integer sequences","authors":"L. Mazo, Marie-Andrée Da Col-Jacob, Laurent Fuchs, Nicolas Magaud, Gaëlle Skapin","doi":"10.1017/S0960129522000342","DOIUrl":"https://doi.org/10.1017/S0960129522000342","url":null,"abstract":"Abstract The paper describes three models of the real field based on subsets of the integer sequences. The three models are compared to the Harthong–Reeb line. Two of the new models, contrary to the Harthong–Reeb line, provide accurate integer “views” on real numbers at a sequence of growing scales \u0000$B^n$\u0000 ( \u0000$Bge2$\u0000 ).","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"648 - 681"},"PeriodicalIF":0.5,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44200374","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 : 2022-04-01DOI: 10.1017/s0960129522000391
Masahito Hasegawa, Stephen Lack, G. McCusker
This volume is the second part of the Festschrift in honour of John Power, who turned 60 in December 2019. For a brief history of John and his work, we refer to the preface of the first volume of the Festschrift (Mathematical Structures in Computer Science (2021), 31(7)). Reflecting John’s influence on category theory and its applications to computer science throughout his career, we received a number of submissions on a wide range of topics. All the submitted papers have been fully peer-reviewed to the usual standards of the journal, and we finally accepted 11 papers. Among them, four papers already appeared in the first part. This volume consists of the remaining seven papers. We are very grateful to the authors and reviewers whose hard work was essential in preparing this special issue. We would like to thank all of them for their help and patience.
{"title":"A special issue on categorical algebras and computation in celebration of John Power’s 60th birthday, part II","authors":"Masahito Hasegawa, Stephen Lack, G. McCusker","doi":"10.1017/s0960129522000391","DOIUrl":"https://doi.org/10.1017/s0960129522000391","url":null,"abstract":"This volume is the second part of the Festschrift in honour of John Power, who turned 60 in December 2019. For a brief history of John and his work, we refer to the preface of the first volume of the Festschrift (Mathematical Structures in Computer Science (2021), 31(7)). Reflecting John’s influence on category theory and its applications to computer science throughout his career, we received a number of submissions on a wide range of topics. All the submitted papers have been fully peer-reviewed to the usual standards of the journal, and we finally accepted 11 papers. Among them, four papers already appeared in the first part. This volume consists of the remaining seven papers. We are very grateful to the authors and reviewers whose hard work was essential in preparing this special issue. We would like to thank all of them for their help and patience.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"348 - 348"},"PeriodicalIF":0.5,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48996064","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 : 2022-04-01DOI: 10.1017/S0960129522000287
M. Hamana
Abstract By using algebraic structures in a presheaf category over finite sets, following Fiore, Plotkin and Turi, we develop sound and complete models of second-order rewriting systems called second-order computation systems (CSs). Restricting the algebraic structures to those equipped with well-founded relations, we obtain a complete characterisation of terminating CSs. We also extend the characterisation to rewriting on meta-terms using the notion of $Sigma$ -monoid.
{"title":"Complete algebraic semantics for second-order rewriting systems based on abstract syntax with variable binding","authors":"M. Hamana","doi":"10.1017/S0960129522000287","DOIUrl":"https://doi.org/10.1017/S0960129522000287","url":null,"abstract":"Abstract By using algebraic structures in a presheaf category over finite sets, following Fiore, Plotkin and Turi, we develop sound and complete models of second-order rewriting systems called second-order computation systems (CSs). Restricting the algebraic structures to those equipped with well-founded relations, we obtain a complete characterisation of terminating CSs. We also extend the characterisation to rewriting on meta-terms using the notion of \u0000$Sigma$\u0000 -monoid.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"542 - 573"},"PeriodicalIF":0.5,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45084756","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 : 2022-04-01DOI: 10.1017/S0960129522000111
Thomas Cottrell, Soichiro Fujii
Abstract Classical definitions of weak higher-dimensional categories are given inductively, for example, a bicategory has a set of objects and hom categories, and a tricategory has a set of objects and hom bicategories. However, more recent definitions of weak n-categories for all natural numbers n, or of weak $omega$ -categories, take more sophisticated approaches, and the nature of the ‘hom is often not immediate from the definitions’. In this paper, we focus on Leinster’s definition of weak $omega$ -category based on an earlier definition by Batanin and construct, for each weak $omega$ -category $mathcal{A}$ , an underlying (weak $omega$ -category)-enriched graph consisting of the same objects and for each pair of objects x and y, a hom weak $omega$ -category $mathcal{A}(x,y)$ . We also show that our construction is functorial with respect to weak $omega$ -functors introduced by Garner.
{"title":"Hom weak ω-categories of a weak ω-category","authors":"Thomas Cottrell, Soichiro Fujii","doi":"10.1017/S0960129522000111","DOIUrl":"https://doi.org/10.1017/S0960129522000111","url":null,"abstract":"Abstract Classical definitions of weak higher-dimensional categories are given inductively, for example, a bicategory has a set of objects and hom categories, and a tricategory has a set of objects and hom bicategories. However, more recent definitions of weak n-categories for all natural numbers n, or of weak \u0000$omega$\u0000 -categories, take more sophisticated approaches, and the nature of the ‘hom is often not immediate from the definitions’. In this paper, we focus on Leinster’s definition of weak \u0000$omega$\u0000 -category based on an earlier definition by Batanin and construct, for each weak \u0000$omega$\u0000 -category \u0000$mathcal{A}$\u0000 , an underlying (weak \u0000$omega$\u0000 -category)-enriched graph consisting of the same objects and for each pair of objects x and y, a hom weak \u0000$omega$\u0000 -category \u0000$mathcal{A}(x,y)$\u0000 . We also show that our construction is functorial with respect to weak \u0000$omega$\u0000 -functors introduced by Garner.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"420 - 441"},"PeriodicalIF":0.5,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45614640","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 : 2022-03-24DOI: 10.1017/s0960129522000019
P. Szabó, J. Siekmann
Ordering is a well-established concept in mathematics and also plays an important role in many areas of computer science, where quasi-orderings, most notably well-founded quasi-orderings and well-quasi-orderings, are of particular interest. This paper deals with quasi-orderings on first-order terms and introduces a new notion of unification based on a special quasi-order, known as homeomorphic tree embedding. Historically, the development of unification theory began with the central notion of a most general unifier based on the subsumption order. A unifier $sigma$ is most general, if it subsumes any other unifier $tau$ , that is, if there is a substitution $lambda$ with $tau=_{E}sigmalambda$ , where E is an equational theory and $=_{E}$ denotes equality under E. Since there is in general more than one most general unifier for unification problems under equational theories E, called E-Unification, we have the notion of a complete and minimal set of unifiers under E for a unification problem $varGamma$ , denoted as $mumathcal{U}Sigma_{E}(Gamma)$ . This set is still the basic notion in unification theory today. But, unfortunately, the subsumption quasi-order is not a well-founded quasi-order, which is the reason why for certain equational theories there are solvable E-unification problems, but the set $mumathcal{U}Sigma_{E}(Gamma)$ does not exist. They are called type nullary in the unification hierarchy. In order to overcome this problem and also to substantially reduce the number of most general unifiers, we extended the well-known encompassment order on terms to an encompassment order on substitutions (modulo E). Unification under the encompassment order is called essential unification and if $mumathcal{U}Sigma_{E}(Gamma)$ exists, then the complete set of essential unifiers $emathcal{U}Sigma_{E}(Gamma)$ is a subset of $mumathcal{U}Sigma_{E}(Gamma)$ . An interesting effect is that many E-unification problems with an infinite set of most general unifiers (under the subsumption order) reduce to a problem with only finitely many essential unifiers. Moreover, there are cases of an equational theory E, for which the complete set of most general unifiers does not exist, the minimal and complete set of essential unifiers however does exist. Unfortunately again, the encompassment order is not a well-founded quasi-ordering either, that is, there are still theories with a solvable unification problem, for which a minimal and complete set of essential unifiers does not exist. This paper deals with a third approach, namely the extension of the well-known homeomorphic embedding of terms to a homeomorphic embedding of substitutions (modulo E). We examine the set of most general, minimal, and complete E-unifiers under the quasi-order of homeomorphic embedment modulo an equational theory E, called $varphi USigma_{E}(Gamma)$ , and p
{"title":"E-Unification based on Generalized Embedding","authors":"P. Szabó, J. Siekmann","doi":"10.1017/s0960129522000019","DOIUrl":"https://doi.org/10.1017/s0960129522000019","url":null,"abstract":"\u0000 Ordering is a well-established concept in mathematics and also plays an important role in many areas of computer science, where quasi-orderings, most notably well-founded quasi-orderings and well-quasi-orderings, are of particular interest. This paper deals with quasi-orderings on first-order terms and introduces a new notion of unification based on a special quasi-order, known as homeomorphic tree embedding. Historically, the development of unification theory began with the central notion of a most general unifier based on the subsumption order. A unifier \u0000 \u0000 \u0000 \u0000$sigma$\u0000\u0000 \u0000 is most general, if it subsumes any other unifier \u0000 \u0000 \u0000 \u0000$tau$\u0000\u0000 \u0000 , that is, if there is a substitution \u0000 \u0000 \u0000 \u0000$lambda$\u0000\u0000 \u0000 with \u0000 \u0000 \u0000 \u0000$tau=_{E}sigmalambda$\u0000\u0000 \u0000 , where E is an equational theory and \u0000 \u0000 \u0000 \u0000$=_{E}$\u0000\u0000 \u0000 denotes equality under E. Since there is in general more than one most general unifier for unification problems under equational theories E, called E-Unification, we have the notion of a complete and minimal set of unifiers under E for a unification problem \u0000 \u0000 \u0000 \u0000$varGamma$\u0000\u0000 \u0000 , denoted as \u0000 \u0000 \u0000 \u0000$mumathcal{U}Sigma_{E}(Gamma)$\u0000\u0000 \u0000 . This set is still the basic notion in unification theory today. But, unfortunately, the subsumption quasi-order is not a well-founded quasi-order, which is the reason why for certain equational theories there are solvable E-unification problems, but the set \u0000 \u0000 \u0000 \u0000$mumathcal{U}Sigma_{E}(Gamma)$\u0000\u0000 \u0000 does not exist. They are called type nullary in the unification hierarchy. In order to overcome this problem and also to substantially reduce the number of most general unifiers, we extended the well-known encompassment order on terms to an encompassment order on substitutions (modulo E). Unification under the encompassment order is called essential unification and if \u0000 \u0000 \u0000 \u0000$mumathcal{U}Sigma_{E}(Gamma)$\u0000\u0000 \u0000 exists, then the complete set of essential unifiers \u0000 \u0000 \u0000 \u0000$emathcal{U}Sigma_{E}(Gamma)$\u0000\u0000 \u0000 is a subset of \u0000 \u0000 \u0000 \u0000$mumathcal{U}Sigma_{E}(Gamma)$\u0000\u0000 \u0000 . An interesting effect is that many E-unification problems with an infinite set of most general unifiers (under the subsumption order) reduce to a problem with only finitely many essential unifiers. Moreover, there are cases of an equational theory E, for which the complete set of most general unifiers does not exist, the minimal and complete set of essential unifiers however does exist. Unfortunately again, the encompassment order is not a well-founded quasi-ordering either, that is, there are still theories with a solvable unification problem, for which a minimal and complete set of essential unifiers does not exist. This paper deals with a third approach, namely the extension of the well-known homeomorphic embedding of terms to a homeomorphic embedding of substitutions (modulo E). We examine the set of most general, minimal, and complete E-unifiers under the quasi-order of homeomorphic embedment modulo an equational theory E, called \u0000 \u0000 \u0000 \u0000$varphi USigma_{E}(Gamma)$\u0000\u0000 \u0000 , and p","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"1 1","pages":"898-917"},"PeriodicalIF":0.5,"publicationDate":"2022-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88921772","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 : 2022-03-23DOI: 10.1017/S096012952200038X
G. Cruttwell, J. Gallagher, J. Lemay, D. Pronk
Abstract Cartesian reverse differential categories (CRDCs) are a recently defined structure which categorically model the reverse differentiation operations used in supervised learning. Here, we define a related structure called a monoidal reverse differential category, prove important results about its relationship to CRDCs, and provide examples of both structures, including examples coming from models of quantum computation.
{"title":"Monoidal reverse differential categories","authors":"G. Cruttwell, J. Gallagher, J. Lemay, D. Pronk","doi":"10.1017/S096012952200038X","DOIUrl":"https://doi.org/10.1017/S096012952200038X","url":null,"abstract":"Abstract Cartesian reverse differential categories (CRDCs) are a recently defined structure which categorically model the reverse differentiation operations used in supervised learning. Here, we define a related structure called a monoidal reverse differential category, prove important results about its relationship to CRDCs, and provide examples of both structures, including examples coming from models of quantum computation.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"1313 - 1363"},"PeriodicalIF":0.5,"publicationDate":"2022-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47371244","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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