R. A. Shaikh, Li-Hao Yeh, Benjamin Rodatz, B. Coecke
Negation in natural language does not follow Boolean logic and is therefore inherently difficult to model. In particular, a broader understanding of what is being negated must be taken into account. In previous work, we proposed a framework for the negation of words that accounts for ‘worldly context’. This paper extends that proposal to now account for the compositional structure inherent in language within the DisCoCirc framework. We compose the negations of single words to capture the negation of sentences. We also describe how to model the negation of words whose meanings evolve in the text.
{"title":"Composing Conversational Negation","authors":"R. A. Shaikh, Li-Hao Yeh, Benjamin Rodatz, B. Coecke","doi":"10.4204/EPTCS.372.25","DOIUrl":"https://doi.org/10.4204/EPTCS.372.25","url":null,"abstract":"Negation in natural language does not follow Boolean logic and is therefore inherently difficult to model. In particular, a broader understanding of what is being negated must be taken into account. In previous work, we proposed a framework for the negation of words that accounts for ‘worldly context’. This paper extends that proposal to now account for the compositional structure inherent in language within the DisCoCirc framework. We compose the negations of single words to capture the negation of sentences. We also describe how to model the negation of words whose meanings evolve in the text.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"28 1","pages":"352-367"},"PeriodicalIF":0.0,"publicationDate":"2021-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80334279","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The constraint satisfaction problem (CSP) is a computational problem that includes a range of important problems in computer science. We point out that fundamental concepts of the CSP, such as the solution set of an instance and polymorphisms, can be formulated abstractly inside the 2-category P FinSet of finite sets and sets of functions between them. The 2-category P FinSet is a quantaloid, and the formulation relies mainly on structure available in any quantaloid. This observation suggests a formal development of generalisations of the CSP and concomitant notions of polymorphism in a large class of quantaloids. We extract a class of optimisation problems as a special case, and show that their computational complexity can be classified by the associated notion of polymorphism.
{"title":"Quantaloidal approach to constraint satisfaction","authors":"Soichiro Fujii, Yuni Iwamasa, Kei Kimura","doi":"10.4204/EPTCS.372.21","DOIUrl":"https://doi.org/10.4204/EPTCS.372.21","url":null,"abstract":"The constraint satisfaction problem (CSP) is a computational problem that includes a range of important problems in computer science. We point out that fundamental concepts of the CSP, such as the solution set of an instance and polymorphisms, can be formulated abstractly inside the 2-category P FinSet of finite sets and sets of functions between them. The 2-category P FinSet is a quantaloid, and the formulation relies mainly on structure available in any quantaloid. This observation suggests a formal development of generalisations of the CSP and concomitant notions of polymorphism in a large class of quantaloids. We extract a class of optimisation problems as a special case, and show that their computational complexity can be classified by the associated notion of polymorphism.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"54 1","pages":"289-305"},"PeriodicalIF":0.0,"publicationDate":"2021-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90980452","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop a comprehensive string diagrammatic treatment of electrical circuits. Building on previous, limited case studies, we introduce controlled sources and meters as elements, and the impedance calculus , a powerful toolbox for diagrammatic reasoning on circuit diagrams. We demonstrate the power of our approach by giving comprehensive proofs of several textbook results, including the superposition theorem and Th´evenin’s theorem.
{"title":"String Diagrammatic Electrical Circuit Theory","authors":"G. Boisseau, Pawel Soboci'nski","doi":"10.4204/EPTCS.372.13","DOIUrl":"https://doi.org/10.4204/EPTCS.372.13","url":null,"abstract":"We develop a comprehensive string diagrammatic treatment of electrical circuits. Building on previous, limited case studies, we introduce controlled sources and meters as elements, and the impedance calculus , a powerful toolbox for diagrammatic reasoning on circuit diagrams. We demonstrate the power of our approach by giving comprehensive proofs of several textbook results, including the superposition theorem and Th´evenin’s theorem.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"4 1","pages":"178-191"},"PeriodicalIF":0.0,"publicationDate":"2021-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88979073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
String diagrams are an increasingly popular algebraic language for the analysis of graphical models of computations across different research fields. Whereas string diagrams have been thoroughly studied as semantic structures, much less attention has been given to their algorithmic properties, and efficient implementations of diagrammatic reasoning are almost an unexplored subject. This work intends to be a contribution in such a direction. We introduce a data structure representing string diagrams in terms of adjacency matrices. This encoding has the key advantage of providing simple and efficient algorithms for composition and tensor product of diagrams. We demonstrate its effectiveness by showing that the complexity of the two operations is linear in the size of string diagrams. Also, as our approach is based on basic linear algebraic operations, we can take advantage of heavily optimised implementations, which we use to measure performances of string diagrammatic operations via several benchmarks.
{"title":"The Cost of Compositionality: A High-Performance Implementation of String Diagram Composition","authors":"Paul W. Wilson, F. Zanasi","doi":"10.4204/EPTCS.372.19","DOIUrl":"https://doi.org/10.4204/EPTCS.372.19","url":null,"abstract":"String diagrams are an increasingly popular algebraic language for the analysis of graphical models of computations across different research fields. Whereas string diagrams have been thoroughly studied as semantic structures, much less attention has been given to their algorithmic properties, and efficient implementations of diagrammatic reasoning are almost an unexplored subject. This work intends to be a contribution in such a direction. We introduce a data structure representing string diagrams in terms of adjacency matrices. This encoding has the key advantage of providing simple and efficient algorithms for composition and tensor product of diagrams. We demonstrate its effectiveness by showing that the complexity of the two operations is linear in the size of string diagrams. Also, as our approach is based on basic linear algebraic operations, we can take advantage of heavily optimised implementations, which we use to measure performances of string diagrammatic operations via several benchmarks.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"91 1","pages":"262-275"},"PeriodicalIF":0.0,"publicationDate":"2021-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84002002","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Matteo Capucci, Neil Ghani, J. Ledent, F. Forsberg
We show open games cover extensive form games with both perfect and imperfect information. Doing so forces us to address two current weaknesses in open games: the lack of a notion of player and their agency within open games, and the lack of choice operators. Using the former we construct the latter, and these choice operators subsume previous proposed operators for open games, thereby making progress towards a core, canonical and ergonomic calculus of game operators. Collectively these innovations increase the level of compositionality of open games, and demonstrate their expressiveness. The game on the left has three players, each of them making one decision. The strategy profile ( L , L , L ) is a Nash equilibrium of this game, which yields the utility ( 1 , 3 , 1 ) . The game on the right has only two players, with p 1 making two decisions. In this second game, (( L , L ) , L ) is not a Nash equilibrium because p 1 can change strategy to (( R , R ) , L ) and get a better reward. In the first game, even though p 1 and p 3 always get the same reward, they are different players and so cannot similarly coordinate changes to their strategies.
{"title":"Translating Extensive Form Games to Open Games with Agency","authors":"Matteo Capucci, Neil Ghani, J. Ledent, F. Forsberg","doi":"10.4204/EPTCS.372.16","DOIUrl":"https://doi.org/10.4204/EPTCS.372.16","url":null,"abstract":"We show open games cover extensive form games with both perfect and imperfect information. Doing so forces us to address two current weaknesses in open games: the lack of a notion of player and their agency within open games, and the lack of choice operators. Using the former we construct the latter, and these choice operators subsume previous proposed operators for open games, thereby making progress towards a core, canonical and ergonomic calculus of game operators. Collectively these innovations increase the level of compositionality of open games, and demonstrate their expressiveness. The game on the left has three players, each of them making one decision. The strategy profile ( L , L , L ) is a Nash equilibrium of this game, which yields the utility ( 1 , 3 , 1 ) . The game on the right has only two players, with p 1 making two decisions. In this second game, (( L , L ) , L ) is not a Nash equilibrium because p 1 can change strategy to (( R , R ) , L ) and get a better reward. In the first game, even though p 1 and p 3 always get the same reward, they are different players and so cannot similarly coordinate changes to their strategies.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"112 1","pages":"221-234"},"PeriodicalIF":0.0,"publicationDate":"2021-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75383946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tracelets are the intrinsic carriers of causal information in categorical rewriting systems. In this work, we assemble tracelets into a symmetric monoidal decomposition space, inducing a cocommutative Hopf algebra of tracelets. This Hopf algebra captures important combinatorial and algebraic aspects of rewriting theory, and is motivated by applications of its representation theory to stochastic rewriting systems such as chemical reaction networks.
{"title":"Tracelet Hopf algebras and decomposition spaces","authors":"Nicolas Behr, Joachim Kock","doi":"10.4204/EPTCS.372.23","DOIUrl":"https://doi.org/10.4204/EPTCS.372.23","url":null,"abstract":"Tracelets are the intrinsic carriers of causal information in categorical rewriting systems. In this work, we assemble tracelets into a symmetric monoidal decomposition space, inducing a cocommutative Hopf algebra of tracelets. This Hopf algebra captures important combinatorial and algebraic aspects of rewriting theory, and is motivated by applications of its representation theory to stochastic rewriting systems such as chemical reaction networks.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"10 1","pages":"323-337"},"PeriodicalIF":0.0,"publicationDate":"2021-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73505904","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Symplectic vector spaces are the phase spaces of linear mechanical systems. The symplectic form describes, for example, the relation between position and momentum as well as current and voltage. The category of linear Lagrangian relations between symplectic vector spaces is a symmetric monoidal subcategory of relations which gives a semantics for the evolution -- and more generally linear constraints on the evolution -- of various physical systems. We give a new presentation of the category of Lagrangian relations over an arbitrary field as a `doubled' category of linear relations. More precisely, we show that it arises as a variation of Selinger's CPM construction applied to linear relations, where the covariant orthogonal complement functor plays the role of conjugation. Furthermore, for linear relations over prime fields, this corresponds exactly to the CPM construction for a suitable choice of dagger. We can furthermore extend this construction by a single affine shift operator to obtain a category of affine Lagrangian relations. Using this new presentation, we prove the equivalence of the prop of affine Lagrangian relations with the prop of qudit stabilizer theory in odd prime dimensions. We hence obtain a unified graphical language for several disparate process theories, including electrical circuits, Spekkens' toy theory, and odd-prime-dimensional stabilizer quantum circuits.
{"title":"A Graphical Calculus for Lagrangian Relations","authors":"Cole Comfort, A. Kissinger","doi":"10.4204/EPTCS.372.24","DOIUrl":"https://doi.org/10.4204/EPTCS.372.24","url":null,"abstract":"Symplectic vector spaces are the phase spaces of linear mechanical systems. The symplectic form describes, for example, the relation between position and momentum as well as current and voltage. The category of linear Lagrangian relations between symplectic vector spaces is a symmetric monoidal subcategory of relations which gives a semantics for the evolution -- and more generally linear constraints on the evolution -- of various physical systems. We give a new presentation of the category of Lagrangian relations over an arbitrary field as a `doubled' category of linear relations. More precisely, we show that it arises as a variation of Selinger's CPM construction applied to linear relations, where the covariant orthogonal complement functor plays the role of conjugation. Furthermore, for linear relations over prime fields, this corresponds exactly to the CPM construction for a suitable choice of dagger. We can furthermore extend this construction by a single affine shift operator to obtain a category of affine Lagrangian relations. Using this new presentation, we prove the equivalence of the prop of affine Lagrangian relations with the prop of qudit stabilizer theory in odd prime dimensions. We hence obtain a unified graphical language for several disparate process theories, including electrical circuits, Spekkens' toy theory, and odd-prime-dimensional stabilizer quantum circuits.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"1 1","pages":"338-351"},"PeriodicalIF":0.0,"publicationDate":"2021-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74809189","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An old theorem of Adámek constructs initial algebras for sufficiently cocontinuous endofunctors via transfinite iteration over ordinals in classical set theory. We prove a new version that works in constructive logic, using “inflationary” iteration over a notion of size that abstracts from limit ordinals just their transitive, directed and well-founded properties. Borrowing from Taylor’s constructive treatment of ordinals, we show that sizes exist with upper bounds for any given signature of indexes. From this it follows that there is a rich class of endofunctors to which the new theorem applies, provided one admits a weak form of choice (WISC) due to Streicher, Moerdijk, van den Berg and Palmgren, and which is known to hold in the internal constructive logic of many kinds of topos.
在经典集合论中，Adámek的一个老定理通过序上的超限迭代构造了充分共连续内函子的初始代数。我们证明了一个新的版本，它在构造逻辑中工作，使用“膨胀”迭代的大小概念，从极限序数中抽象出它们的传递性，有向性和有根据的性质。利用泰勒对序数的构造处理，我们证明了对于任何给定的指标签名，大小都存在上界。由此可以得出，如果承认Streicher, Moerdijk, van den Berg和Palmgren提出的弱选择形式(WISC)，并且已知在许多拓扑的内部构造逻辑中成立，则存在一类丰富的内函子可以应用新定理。
{"title":"Constructing Initial Algebras Using Inflationary Iteration","authors":"A. Pitts, S. Steenkamp","doi":"10.4204/EPTCS.372.7","DOIUrl":"https://doi.org/10.4204/EPTCS.372.7","url":null,"abstract":"An old theorem of Adámek constructs initial algebras for sufficiently cocontinuous endofunctors via transfinite iteration over ordinals in classical set theory. We prove a new version that works in constructive logic, using “inflationary” iteration over a notion of size that abstracts from limit ordinals just their transitive, directed and well-founded properties. Borrowing from Taylor’s constructive treatment of ordinals, we show that sizes exist with upper bounds for any given signature of indexes. From this it follows that there is a rich class of endofunctors to which the new theorem applies, provided one admits a weak form of choice (WISC) due to Streicher, Moerdijk, van den Berg and Palmgren, and which is known to hold in the internal constructive logic of many kinds of topos.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"42 1","pages":"88-102"},"PeriodicalIF":0.0,"publicationDate":"2021-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84287193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In"Backprop as functor", the authors show that the fundamental elements of deep learning -- gradient descent and backpropagation -- can be conceptualized as a strong monoidal functor Para(Euc)$to$Learn from the category of parameterized Euclidean spaces to that of learners, a category developed explicitly to capture parameter update and backpropagation. It was soon realized that there is an isomorphism Learn$cong$Para(Slens), where Slens is the symmetric monoidal category of simple lenses as used in functional programming. In this note, we observe that Slens is a full subcategory of Poly, the category of polynomial functors in one variable, via the functor $Amapsto Ay^A$. Using the fact that (Poly,$otimes$) is monoidal closed, we show that a map $Ato B$ in Para(Slens) has a natural interpretation in terms of dynamical systems (more precisely, generalized Moore machines) whose interface is the internal-hom type $[Ay^A,By^B]$. Finally, we review the fact that the category p-Coalg of dynamical systems on any $p in$ Poly forms a topos, and consider the logical propositions that can be stated in its internal language. We give gradient descent as an example, and we conclude by discussing some directions for future work.
{"title":"Learners' languages","authors":"David I. Spivak","doi":"10.4204/EPTCS.372.2","DOIUrl":"https://doi.org/10.4204/EPTCS.372.2","url":null,"abstract":"In\"Backprop as functor\", the authors show that the fundamental elements of deep learning -- gradient descent and backpropagation -- can be conceptualized as a strong monoidal functor Para(Euc)$to$Learn from the category of parameterized Euclidean spaces to that of learners, a category developed explicitly to capture parameter update and backpropagation. It was soon realized that there is an isomorphism Learn$cong$Para(Slens), where Slens is the symmetric monoidal category of simple lenses as used in functional programming. In this note, we observe that Slens is a full subcategory of Poly, the category of polynomial functors in one variable, via the functor $Amapsto Ay^A$. Using the fact that (Poly,$otimes$) is monoidal closed, we show that a map $Ato B$ in Para(Slens) has a natural interpretation in terms of dynamical systems (more precisely, generalized Moore machines) whose interface is the internal-hom type $[Ay^A,By^B]$. Finally, we review the fact that the category p-Coalg of dynamical systems on any $p in$ Poly forms a topos, and consider the logical propositions that can be stated in its internal language. We give gradient descent as an example, and we conclude by discussing some directions for future work.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"37 1","pages":"14-28"},"PeriodicalIF":0.0,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86455238","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Native type systems are those in which type constructors are derived from term constructors, as well as the constructors of predicate logic and intuitionistic type theory. We present a method to construct native type systems for a broad class of languages, λ -theories with equality, by embedding such a theory into the internal language of its topos of presheaves. Native types provide total specification of the structure of terms; and by internalizing transition systems, native type systems serve to reason about structure and behavior simultaneously. The construction is functorial, thereby providing a shared framework of higher-order reasoning for many languages, including programming languages.
{"title":"Native Type Theory","authors":"Christian Williams, Michael Stay","doi":"10.4204/EPTCS.372.9","DOIUrl":"https://doi.org/10.4204/EPTCS.372.9","url":null,"abstract":"Native type systems are those in which type constructors are derived from term constructors, as well as the constructors of predicate logic and intuitionistic type theory. We present a method to construct native type systems for a broad class of languages, λ -theories with equality, by embedding such a theory into the internal language of its topos of presheaves. Native types provide total specification of the structure of terms; and by internalizing transition systems, native type systems serve to reason about structure and behavior simultaneously. The construction is functorial, thereby providing a shared framework of higher-order reasoning for many languages, including programming languages.","PeriodicalId":11810,"journal":{"name":"essentia law Merchant Shipping Act 1995","volume":"10 1","pages":"116-132"},"PeriodicalIF":0.0,"publicationDate":"2021-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80830451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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