Pub Date : 2021-11-29DOI: 10.1017/S0960129521000256
M. Hofmann, J. Ledent
Abstract We use a simplified version of the framework of resource monoids, introduced by Dal Lago and Hofmann, to interpret simply typed λ-calculus with constants zero and successor. We then use this model to prove a simple quantitative result about bounding the size of the normal form of λ-terms. While the bound itself is already known, this is to our knowledge the first semantic proof of this fact. Our use of resource monoids differs from the other instances found in the literature, in that it measures the size of λ-terms rather than time complexity.
{"title":"A quantitative model for simply typed λ-calculus","authors":"M. Hofmann, J. Ledent","doi":"10.1017/S0960129521000256","DOIUrl":"https://doi.org/10.1017/S0960129521000256","url":null,"abstract":"Abstract We use a simplified version of the framework of resource monoids, introduced by Dal Lago and Hofmann, to interpret simply typed λ-calculus with constants zero and successor. We then use this model to prove a simple quantitative result about bounding the size of the normal form of λ-terms. While the bound itself is already known, this is to our knowledge the first semantic proof of this fact. Our use of resource monoids differs from the other instances found in the literature, in that it measures the size of λ-terms rather than time complexity.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"777 - 793"},"PeriodicalIF":0.5,"publicationDate":"2021-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42699072","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-16DOI: 10.1017/s0960129521000372
Zhicheng Liu, Hong Chang, Ran Ma, D. Du, Xiaoyan Zhang
We consider a two-stage submodular maximization problem subject to a cardinality constraint and k matroid constraints, where the objective function is the expected difference of a nonnegative monotone submodular function and a nonnegative monotone modular function. We give two bi-factor approximation algorithms for this problem. The first is a deterministic $left( {{1 over {k + 1}}left( {1 - {1 over {{e^{k + 1}}}}} right),1} right)$ -approximation algorithm, and the second is a randomized $left( {{1 over {k + 1}}left( {1 - {1 over {{e^{k + 1}}}}} right) - varepsilon ,1} right)$ -approximation algorithm with improved time efficiency.
{"title":"Two-stage submodular maximization problem beyond nonnegative and monotone","authors":"Zhicheng Liu, Hong Chang, Ran Ma, D. Du, Xiaoyan Zhang","doi":"10.1017/s0960129521000372","DOIUrl":"https://doi.org/10.1017/s0960129521000372","url":null,"abstract":"\u0000 We consider a two-stage submodular maximization problem subject to a cardinality constraint and k matroid constraints, where the objective function is the expected difference of a nonnegative monotone submodular function and a nonnegative monotone modular function. We give two bi-factor approximation algorithms for this problem. The first is a deterministic \u0000 \u0000 \u0000 $left( {{1 over {k + 1}}left( {1 - {1 over {{e^{k + 1}}}}} right),1} right)$\u0000 \u0000 -approximation algorithm, and the second is a randomized \u0000 \u0000 \u0000 $left( {{1 over {k + 1}}left( {1 - {1 over {{e^{k + 1}}}}} right) - varepsilon ,1} right)$\u0000 \u0000 -approximation algorithm with improved time efficiency.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2021-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48621947","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-16DOI: 10.1017/s096012952100027x
B. Jacobs, A. Kissinger, F. Zanasi
Extracting causal relationships from observed correlations is a growing area in probabilistic reasoning, originating with the seminal work of Pearl and others from the early 1990s. This paper develops a new, categorically oriented view based on a clear distinction between syntax (string diagrams) and semantics (stochastic matrices), connected via interpretations as structure-preserving functors. A key notion in the identification of causal effects is that of an intervention, whereby a variable is forcefully set to a particular value independent of any prior propensities. We represent the effect of such an intervention as an endo-functor which performs ‘string diagram surgery’ within the syntactic category of string diagrams. This diagram surgery in turn yields a new, interventional distribution via the interpretation functor. While in general there is no way to compute interventional distributions purely from observed data, we show that this is possible in certain special cases using a calculational tool called comb disintegration. We demonstrate the use of this technique on two well-known toy examples: one where we predict the causal effect of smoking on cancer in the presence of a confounding common cause and where we show that this technique provides simple sufficient conditions for computing interventions which apply to a wide variety of situations considered in the causal inference literature; the other one is an illustration of counterfactual reasoning where the same interventional techniques are used, but now in a ‘twinned’ set-up, with two version of the world – one factual and one counterfactual – joined together via exogenous variables that capture the uncertainties at hand.
{"title":"Causal inference via string diagram surgery: A diagrammatic approach to interventions and counterfactuals","authors":"B. Jacobs, A. Kissinger, F. Zanasi","doi":"10.1017/s096012952100027x","DOIUrl":"https://doi.org/10.1017/s096012952100027x","url":null,"abstract":"\u0000 Extracting causal relationships from observed correlations is a growing area in probabilistic reasoning, originating with the seminal work of Pearl and others from the early 1990s. This paper develops a new, categorically oriented view based on a clear distinction between syntax (string diagrams) and semantics (stochastic matrices), connected via interpretations as structure-preserving functors. A key notion in the identification of causal effects is that of an intervention, whereby a variable is forcefully set to a particular value independent of any prior propensities. We represent the effect of such an intervention as an endo-functor which performs ‘string diagram surgery’ within the syntactic category of string diagrams. This diagram surgery in turn yields a new, interventional distribution via the interpretation functor. While in general there is no way to compute interventional distributions purely from observed data, we show that this is possible in certain special cases using a calculational tool called comb disintegration. We demonstrate the use of this technique on two well-known toy examples: one where we predict the causal effect of smoking on cancer in the presence of a confounding common cause and where we show that this technique provides simple sufficient conditions for computing interventions which apply to a wide variety of situations considered in the causal inference literature; the other one is an illustration of counterfactual reasoning where the same interventional techniques are used, but now in a ‘twinned’ set-up, with two version of the world – one factual and one counterfactual – joined together via exogenous variables that capture the uncertainties at hand.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"484 1","pages":"553-574"},"PeriodicalIF":0.5,"publicationDate":"2021-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77783997","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-01DOI: 10.1017/s0960129522000202
Daniel R. Licata, P. Lumsdaine
{"title":"Special issue on homotopy type theory 2019","authors":"Daniel R. Licata, P. Lumsdaine","doi":"10.1017/s0960129522000202","DOIUrl":"https://doi.org/10.1017/s0960129522000202","url":null,"abstract":"","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"75 1","pages":"1145-1146"},"PeriodicalIF":0.5,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74271748","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-12DOI: 10.1017/s0960129522000305
Marc Bataille
We describe quantum circuits generating four-qubit maximally entangled states, the amount of entanglement being quantified by using the absolute value of the Cayley hyperdeterminant as an entanglement monotone. More precisely we show that this type of four-qubit entangled states can be obtained by the action of a family of $mathtt{CNOT}$ circuits on some special states of the LU orbit of the state $|0000rangle$ .
{"title":"Quantum circuits generating four-qubit maximally entangled states","authors":"Marc Bataille","doi":"10.1017/s0960129522000305","DOIUrl":"https://doi.org/10.1017/s0960129522000305","url":null,"abstract":"\u0000 We describe quantum circuits generating four-qubit maximally entangled states, the amount of entanglement being quantified by using the absolute value of the Cayley hyperdeterminant as an entanglement monotone. More precisely we show that this type of four-qubit entangled states can be obtained by the action of a family of \u0000 \u0000 \u0000 \u0000$mathtt{CNOT}$\u0000\u0000 \u0000 circuits on some special states of the LU orbit of the state \u0000 \u0000 \u0000 \u0000$|0000rangle$\u0000\u0000 \u0000 .","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"14 1","pages":"257-270"},"PeriodicalIF":0.5,"publicationDate":"2021-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88422306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-01DOI: 10.1017/s0960129522000135
Jan Hoffmann, D. Sannella, Ulrich Schöpp
This is the first part of a two-part special issue of Mathematical Structures in Computer Science dedicated to the memory of our friend and colleague, Martin Hofmann. 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 obtained his PhD in June 1995 with a dissertation entitled ‘Extensional Concepts in Intensional
{"title":"Preface for the special issue in homage to Martin Hofmann Part 1","authors":"Jan Hoffmann, D. Sannella, Ulrich Schöpp","doi":"10.1017/s0960129522000135","DOIUrl":"https://doi.org/10.1017/s0960129522000135","url":null,"abstract":"This is the first part of a two-part special issue of Mathematical Structures in Computer Science dedicated to the memory of our friend and colleague, Martin Hofmann. 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 obtained his PhD in June 1995 with a dissertation entitled ‘Extensional Concepts in Intensional","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"31 1","pages":"950 - 952"},"PeriodicalIF":0.5,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47005668","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-01DOI: 10.1017/S0960129521000475
Markus Latte
Abstract Emerson and Halpern (1986, Journal of the Association for Computing Machinery 33, 151–178) prove that the Computation Tree Logic (CTL) cannot express the existence of a path on which a proposition holds infinitely often (fairness for short). The scope is widened from CTL to a general branching-time logic. A path quantifier is followed by a language with temporal descriptions. In this extended setting, the said inexpressiveness is strengthened in two aspects. First, universal path quantifiers are unrestricted. In this way, they are relieved of any temporal quantifiers such as of those in $mathtt{AU}$ and $mathtt{AR}$ from CTL. Second, existential path quantifiers are allowed with any countable language. Instances are the temporal quantifiers in $mathtt{EU}$ and $mathtt{ER}$ from CTL. By contrast, the fairness statement is an existential path quantifier with an uncountable language. Both aspects indicate that this inexpressiveness is optimal with respect to the polarity of path quantifiers and to the cardinality of their languages.
摘要Emerson和Halpern (1986, Journal of the Association for Computing Machinery, 33,151 - 178)证明了计算树逻辑(CTL)不能表达一个命题在其上无限常(简称公平)持有的路径的存在性。范围从CTL扩展到一般的分支时间逻辑。路径量词后跟带有时间描述的语言。在这个扩展的环境中，上述的无表达性在两个方面得到加强。首先，通用路径量词是不受限制的。通过这种方式，它们可以从CTL中免去任何时间量词，例如$mathtt{AU}$和$mathtt{AR}$中的时间量词。其次，任何可数语言都允许使用存在路径量词。实例是CTL中的$mathtt{EU}$和$mathtt{ER}$中的时间量词。相比之下，公平语句是一个存在路径量词，带有不可数语言。这两个方面都表明，就路径量词的极性和语言的基数性而言，这种非表达性是最佳的。
{"title":"Branching-time logics and fairness, revisited","authors":"Markus Latte","doi":"10.1017/S0960129521000475","DOIUrl":"https://doi.org/10.1017/S0960129521000475","url":null,"abstract":"Abstract Emerson and Halpern (1986, Journal of the Association for Computing Machinery 33, 151–178) prove that the Computation Tree Logic (CTL) cannot express the existence of a path on which a proposition holds infinitely often (fairness for short). The scope is widened from CTL to a general branching-time logic. A path quantifier is followed by a language with temporal descriptions. In this extended setting, the said inexpressiveness is strengthened in two aspects. First, universal path quantifiers are unrestricted. In this way, they are relieved of any temporal quantifiers such as of those in \u0000$mathtt{AU}$\u0000 and \u0000$mathtt{AR}$\u0000 from CTL. Second, existential path quantifiers are allowed with any countable language. Instances are the temporal quantifiers in \u0000$mathtt{EU}$\u0000 and \u0000$mathtt{ER}$\u0000 from CTL. By contrast, the fairness statement is an existential path quantifier with an uncountable language. Both aspects indicate that this inexpressiveness is optimal with respect to the polarity of path quantifiers and to the cardinality of their languages.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"31 1","pages":"1135 - 1144"},"PeriodicalIF":0.5,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42129369","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-01DOI: 10.1017/S096012952100030X
Jacopo Emmenegger, Fabio Pasquali, G. Rosolini
Abstract The present paper aims at stressing the importance of the Hofmann–Streicher groupoid model for Martin Löf Type Theory as a link with the first-order equality and its semantics via adjunctions. The groupoid model was introduced by Martin Hofmann in his Ph.D. thesis and later analysed in collaboration with Thomas Streicher. In this paper, after describing an algebraic weak factorisation system $$mathsf {L, R}$$ on the category $${cal C}-{cal Gpd}$$ of $${cal C}$$ -enriched groupoids, we prove that its fibration of algebras is elementary (in the sense of Lawvere) and use this fact to produce the factorisation of diagonals for $$mathsf {L, R}$$ needed to interpret identity types.
{"title":"Elementary fibrations of enriched groupoids","authors":"Jacopo Emmenegger, Fabio Pasquali, G. Rosolini","doi":"10.1017/S096012952100030X","DOIUrl":"https://doi.org/10.1017/S096012952100030X","url":null,"abstract":"Abstract The present paper aims at stressing the importance of the Hofmann–Streicher groupoid model for Martin Löf Type Theory as a link with the first-order equality and its semantics via adjunctions. The groupoid model was introduced by Martin Hofmann in his Ph.D. thesis and later analysed in collaboration with Thomas Streicher. In this paper, after describing an algebraic weak factorisation system \u0000$$mathsf {L, R}$$\u0000 on the category \u0000$${cal C}-{cal Gpd}$$\u0000 of \u0000$${cal C}$$\u0000 -enriched groupoids, we prove that its fibration of algebras is elementary (in the sense of Lawvere) and use this fact to produce the factorisation of diagonals for \u0000$$mathsf {L, R}$$\u0000 needed to interpret identity types.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"31 1","pages":"958 - 978"},"PeriodicalIF":0.5,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48613754","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-13DOI: 10.1017/S0960129522000123
F. Bonchi, F. Gadducci, A. Kissinger, Pawel Soboci'nski, F. Zanasi
Abstract In this paper, we address the problem of proving confluence for string diagram rewriting, which was previously shown to be characterised combinatorially as double-pushout rewriting with interfaces (DPOI) on (labelled) hypergraphs. For standard DPO rewriting without interfaces, confluence for terminating rewriting systems is, in general, undecidable. Nevertheless, we show here that confluence for DPOI, and hence string diagram rewriting, is decidable. We apply this result to give effective procedures for deciding local confluence of symmetric monoidal theories with and without Frobenius structure by critical pair analysis. For the latter, we introduce the new notion of path joinability for critical pairs, which enables finitely many joins of a critical pair to be lifted to an arbitrary context in spite of the strong non-local constraints placed on rewriting in a generic symmetric monoidal theory.
{"title":"String diagram rewrite theory III: Confluence with and without Frobenius","authors":"F. Bonchi, F. Gadducci, A. Kissinger, Pawel Soboci'nski, F. Zanasi","doi":"10.1017/S0960129522000123","DOIUrl":"https://doi.org/10.1017/S0960129522000123","url":null,"abstract":"Abstract In this paper, we address the problem of proving confluence for string diagram rewriting, which was previously shown to be characterised combinatorially as double-pushout rewriting with interfaces (DPOI) on (labelled) hypergraphs. For standard DPO rewriting without interfaces, confluence for terminating rewriting systems is, in general, undecidable. Nevertheless, we show here that confluence for DPOI, and hence string diagram rewriting, is decidable. We apply this result to give effective procedures for deciding local confluence of symmetric monoidal theories with and without Frobenius structure by critical pair analysis. For the latter, we introduce the new notion of path joinability for critical pairs, which enables finitely many joins of a critical pair to be lifted to an arbitrary context in spite of the strong non-local constraints placed on rewriting in a generic symmetric monoidal theory.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"829 - 869"},"PeriodicalIF":0.5,"publicationDate":"2021-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46239939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-09-11DOI: 10.1017/S0960129522000299
Simon Forest, S. Mimram
Abstract Over the recent years, the theory of rewriting has been used and extended in order to provide systematic techniques to show coherence results for strict higher categories. Here, we investigate a further generalization to Gray categories, which are known to be equivalent to tricategories. This requires us to develop the theory of rewriting in the setting of precategories, which are adapted to mechanized computations and include Gray categories as particular cases. We show that a finite rewriting system in precategories admits a finite number of critical pairs, which can be efficiently computed. We also extend Squier’s theorem to our context, showing that a convergent rewriting system is coherent, which means that any two parallel 3-cells are necessarily equal. This allows us to prove coherence results for several well-known structures in the context of Gray categories: monoids, adjunctions, and Frobenius monoids.
{"title":"Rewriting in Gray categories with applications to coherence","authors":"Simon Forest, S. Mimram","doi":"10.1017/S0960129522000299","DOIUrl":"https://doi.org/10.1017/S0960129522000299","url":null,"abstract":"Abstract Over the recent years, the theory of rewriting has been used and extended in order to provide systematic techniques to show coherence results for strict higher categories. Here, we investigate a further generalization to Gray categories, which are known to be equivalent to tricategories. This requires us to develop the theory of rewriting in the setting of precategories, which are adapted to mechanized computations and include Gray categories as particular cases. We show that a finite rewriting system in precategories admits a finite number of critical pairs, which can be efficiently computed. We also extend Squier’s theorem to our context, showing that a convergent rewriting system is coherent, which means that any two parallel 3-cells are necessarily equal. This allows us to prove coherence results for several well-known structures in the context of Gray categories: monoids, adjunctions, and Frobenius monoids.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"574 - 647"},"PeriodicalIF":0.5,"publicationDate":"2021-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42833880","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：