The main purpose of this paper is to study prime spectrums of EQ-algebras and to solve two open problems about $wedge $-prime spectrums of involutive and prelinear EQ-algebras, which were proposed by N. Akhlaghinia, R.A. Borzooei and M. A. Kologani. In order to do so, we first give some characterizations of preideals, prime preideals and maximal preideals on (good) EQ-algebras, respectively. Then we introduce the notion of quasi De Morgan EQ-algebras (MEQ-algebras for short) and obtain that $wedge $-prime preideals coincide with prime preideals for MEQ-algebras, and each involutive EQ-algebra is an MEQ-algebra. Following, we show that the prime spectrum space of a good EQ-algebra is a compact topological space and obtain that for any involutive EQ-algebra the prime spectrum space is connected if and only if its Boolean center is indeed 2-element. Also, we prove that the maximal spectrum space of a good and prelinear EQ-algebra (or an involutive and prelinear EQ-algebra) is a normal Hausdorff space. These results totally answer the above two open problems. Finally, we give some characterizations of the spectrum space of an MEQ-algebra by its reticulation.
本文的主要目的是研究EQ代数的素谱，并解决N.Akhlaghinia、R.A.Borzooei和M.A.Kologani提出的关于对合和预线性EQ代数$wedge$-素谱的两个开放问题。为了做到这一点，我们首先分别给出了（好）EQ代数上的预理想、素数预理想和极大预理想的一些刻画。然后我们引入了拟De Morgan EQ代数（简称MEQ代数）的概念，得到了$wedge$-素数预理想与MEQ代数的素数预理想一致，并且每个对合EQ代数都是一个MEQ代数。接下来，我们证明了一个好的EQ代数的素谱空间是一个紧致拓扑空间，并得到了对于任何对合EQ代数，素谱空间都是连通的，当且仅当其布尔中心确实是2-元。此外，我们还证明了一个好的预线性EQ代数（或对合的预线性EQ代数）的最大谱空间是一个正规的Hausdorff空间。这些结果完全回答了上述两个悬而未决的问题。最后，我们通过MEQ代数的网状给出了它的谱空间的一些特征。
{"title":"Prime spectrums of EQ-algebras","authors":"Binjie Zhao, Wei Wang","doi":"10.1093/logcom/exad045","DOIUrl":"https://doi.org/10.1093/logcom/exad045","url":null,"abstract":"\u0000 The main purpose of this paper is to study prime spectrums of EQ-algebras and to solve two open problems about $wedge $-prime spectrums of involutive and prelinear EQ-algebras, which were proposed by N. Akhlaghinia, R.A. Borzooei and M. A. Kologani. In order to do so, we first give some characterizations of preideals, prime preideals and maximal preideals on (good) EQ-algebras, respectively. Then we introduce the notion of quasi De Morgan EQ-algebras (MEQ-algebras for short) and obtain that $wedge $-prime preideals coincide with prime preideals for MEQ-algebras, and each involutive EQ-algebra is an MEQ-algebra. Following, we show that the prime spectrum space of a good EQ-algebra is a compact topological space and obtain that for any involutive EQ-algebra the prime spectrum space is connected if and only if its Boolean center is indeed 2-element. Also, we prove that the maximal spectrum space of a good and prelinear EQ-algebra (or an involutive and prelinear EQ-algebra) is a normal Hausdorff space. These results totally answer the above two open problems. Finally, we give some characterizations of the spectrum space of an MEQ-algebra by its reticulation.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44952639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Transition systems are often used to describe the behaviour of software systems. If viewed as a graph then, at their most basic level, vertices correspond to the states of a program and each edge represents a transition between states via the (atomic) action labelled. In this setting, systems are thought to be consistent and at each state formulas are evaluated as either true or false. On the other hand, when a structure of this sort—e.g. a map where states represent locations, some local properties are known and labelled transitions represent information available about different routes—is built resorting to multiple sources of information, it is common to find inconsistent or incomplete information regarding what holds at each state, both at the level of propositional variables and transitions. This paper aims at bringing together Belnap’s four values, Dynamic Logic and hybrid machinery such as nominals and the satisfaction operator, so that reasoning is still possible in face of contradicting evidence. Proof-theory for this new logic is explored by means of a terminating, sound and complete tableau system.
{"title":"4DL: a four-valued dynamic logic and its proof theory","authors":"Diana Costa","doi":"10.1093/logcom/exad044","DOIUrl":"https://doi.org/10.1093/logcom/exad044","url":null,"abstract":"Transition systems are often used to describe the behaviour of software systems. If viewed as a graph then, at their most basic level, vertices correspond to the states of a program and each edge represents a transition between states via the (atomic) action labelled. In this setting, systems are thought to be consistent and at each state formulas are evaluated as either true or false. On the other hand, when a structure of this sort—e.g. a map where states represent locations, some local properties are known and labelled transitions represent information available about different routes—is built resorting to multiple sources of information, it is common to find inconsistent or incomplete information regarding what holds at each state, both at the level of propositional variables and transitions. This paper aims at bringing together Belnap’s four values, Dynamic Logic and hybrid machinery such as nominals and the satisfaction operator, so that reasoning is still possible in face of contradicting evidence. Proof-theory for this new logic is explored by means of a terminating, sound and complete tableau system.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"1978 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138496230","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
FDE, LP and $mathbf {K_{3}}$ are closely related to each other and admit of an intuitive informational interpretation. However, all these logics are co-NP complete, and so idealized models of how an agent can think. We address this issue by shifting to signed formulae, where the signs express imprecise values associated with two bipartitions of the corresponding set of standard values. We present proof systems whose operational rules are all linear and have only two structural branching rules that express a generalized Principle of Bivalence. Each of these systems leads to defining an infinite hierarchy of tractable approximations to the respective logic, in terms of the maximum number of allowed nested applications of the two branching rules. Further, each resulting hierarchy admits of an intuitive 5-valued non-deterministic semantics.
{"title":"Tractable depth-bounded approximations to FDE and its satellites","authors":"Marcello D'Agostino, Alejandro Solares-Rojas","doi":"10.1093/logcom/exad040","DOIUrl":"https://doi.org/10.1093/logcom/exad040","url":null,"abstract":"\u0000 FDE, LP and $mathbf {K_{3}}$ are closely related to each other and admit of an intuitive informational interpretation. However, all these logics are co-NP complete, and so idealized models of how an agent can think. We address this issue by shifting to signed formulae, where the signs express imprecise values associated with two bipartitions of the corresponding set of standard values. We present proof systems whose operational rules are all linear and have only two structural branching rules that express a generalized Principle of Bivalence. Each of these systems leads to defining an infinite hierarchy of tractable approximations to the respective logic, in terms of the maximum number of allowed nested applications of the two branching rules. Further, each resulting hierarchy admits of an intuitive 5-valued non-deterministic semantics.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46150199","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper presents a novel concept of a polyatomic logic and initiates its systematic study. This approach, inspired by inquisitive semantics, is obtained by taking a variant of a given logic, obtained by looking at the fragment covered by a selector term. We introduce an algebraic semantics for these logics and prove algebraic completeness. These logics are then related to translations, through the introduction of a number of classes of translations involving selector terms, which are noted to be ubiquitous in algebraic logic. In this setting, we also introduce a generalized Blok–Esakia theory, which can be developed for special classes of translations. We conclude by showing some systematic connections between the theory of polyatomic logics and the general Blok–Esakia theory for a wide class of interesting translations.
{"title":"Polyatomic logics and generalized Blok–Esakia theory","authors":"Rodrigo Nicolau Almeida","doi":"10.1093/logcom/exad043","DOIUrl":"https://doi.org/10.1093/logcom/exad043","url":null,"abstract":"\u0000 This paper presents a novel concept of a polyatomic logic and initiates its systematic study. This approach, inspired by inquisitive semantics, is obtained by taking a variant of a given logic, obtained by looking at the fragment covered by a selector term. We introduce an algebraic semantics for these logics and prove algebraic completeness. These logics are then related to translations, through the introduction of a number of classes of translations involving selector terms, which are noted to be ubiquitous in algebraic logic. In this setting, we also introduce a generalized Blok–Esakia theory, which can be developed for special classes of translations. We conclude by showing some systematic connections between the theory of polyatomic logics and the general Blok–Esakia theory for a wide class of interesting translations.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46380410","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The problem of finding a Master Surgical Schedule (MSS) consists of scheduling different specialties to the operating rooms (ORs) of a hospital clinic. To produce a proper MSS, each specialty must be assigned to some ORs, where the number of assignments is different for each specialty and can also vary during the considered planning horizon. The problem is enriched by considering resource availability such as beds, surgical teams and nurses. Realizing a satisfying schedule is of upmost importance for a hospital clinic, since a poorly scheduled MSS may lead to unbalanced specialties availability and increase patients’ waiting list, thus negatively affecting both the administrative costs of the hospital and the patient satisfaction. In this paper, we present compact solutions based on Answer Set Programming (ASP) to the MSS problem. We tested our solutions on different scenarios: experiments show that our ASP solutions provide satisfying results in short time, also when compared to other logic-based formalisms. Finally, we describe a web application we have developed for easy usage of our solution.
{"title":"Master Surgical Scheduling via Answer Set Programming","authors":"Marco Mochi, G. Galatà, M. Maratea","doi":"10.1093/logcom/exad035","DOIUrl":"https://doi.org/10.1093/logcom/exad035","url":null,"abstract":"\u0000 The problem of finding a Master Surgical Schedule (MSS) consists of scheduling different specialties to the operating rooms (ORs) of a hospital clinic. To produce a proper MSS, each specialty must be assigned to some ORs, where the number of assignments is different for each specialty and can also vary during the considered planning horizon. The problem is enriched by considering resource availability such as beds, surgical teams and nurses. Realizing a satisfying schedule is of upmost importance for a hospital clinic, since a poorly scheduled MSS may lead to unbalanced specialties availability and increase patients’ waiting list, thus negatively affecting both the administrative costs of the hospital and the patient satisfaction. In this paper, we present compact solutions based on Answer Set Programming (ASP) to the MSS problem. We tested our solutions on different scenarios: experiments show that our ASP solutions provide satisfying results in short time, also when compared to other logic-based formalisms. Finally, we describe a web application we have developed for easy usage of our solution.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47182711","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Playwright Gerhard Gentzen","authors":"C. Smorynski","doi":"10.1093/logcom/exad041","DOIUrl":"https://doi.org/10.1093/logcom/exad041","url":null,"abstract":"","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48858091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We propose a novel method to inject symbolic knowledge in form of Datalog formulæ into neural networks (NN), called Knowledge Injection via Network Structuring (KINS). The idea behind our method is to extend NN internal structure with ad-hoc layers built out of the injected symbolic knowledge. KINS does not constrain NN to any specific architecture, neither requires logic formulæ to be ground. Moreover, it is robust w.r.t. both lack of data and imperfect/incomplete knowledge. Experiments are reported, involving multiple datasets and predictor types, to demonstrate how KINS can significantly improve the predictive performance of the neural networks it is applied to.
摘要提出了一种将符号知识以Datalog公式的形式注入神经网络的新方法——通过网络结构注入知识(knowledge Injection via Network Structuring, KINS)。我们的方法背后的思想是用注入的符号知识构建的特别层来扩展神经网络的内部结构。KINS不将神经网络约束到任何特定的架构，也不要求逻辑公式是接地的。此外，它在缺乏数据和不完善/不完整知识的情况下是稳健的。报告了涉及多个数据集和预测器类型的实验，以证明KINS如何显着提高其应用的神经网络的预测性能。
{"title":"Knowledge injection of Datalog rules via Neural Network Structuring with KINS","authors":"Matteo Magnini, Giovanni Ciatto, Andrea Omicini","doi":"10.1093/logcom/exad037","DOIUrl":"https://doi.org/10.1093/logcom/exad037","url":null,"abstract":"Abstract We propose a novel method to inject symbolic knowledge in form of Datalog formulæ into neural networks (NN), called Knowledge Injection via Network Structuring (KINS). The idea behind our method is to extend NN internal structure with ad-hoc layers built out of the injected symbolic knowledge. KINS does not constrain NN to any specific architecture, neither requires logic formulæ to be ground. Moreover, it is robust w.r.t. both lack of data and imperfect/incomplete knowledge. Experiments are reported, involving multiple datasets and predictor types, to demonstrate how KINS can significantly improve the predictive performance of the neural networks it is applied to.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135325055","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we continue the study of the enriched topological representation of $Q$-algebras where $Q$ is a unital quantale and give two kinds of enriched topological representations of $Q$-algebras. The first one is based on strong $M_3$-valued $Q$-algebra homomorphisms, and the second way is based on strong $M_6$-valued $Q$-algebra homomorphisms. For the first way, we first construct a spatial and semiunital $Q$-algebra $M_3$ containing three elements and show that prime elements of semiunital $Q$-algebras are identified with strong $Q$-algebra homomorphisms taking their values in $M_3$. Then we prove that a semiunital $Q$-algebra is spatial iff strong $Q$-algebra homomorphisms with values in $M_3$ separate elements. Based on this, we obtain that every spatial and semiunital $Q$-algebra can be identified with an $M_3$-enriched sober space. For the second enriched topological representation, we construct a spatial and semiunital $Q$-algebra $M_6$ containing exactly six elements and prove that every spatial and semiunital $Q$-algebra can be identified with an $M_6$-enriched sober space.
{"title":"Two kinds of enriched topological representations of <i>Q</i>-algebras","authors":"Xianglong Ruan","doi":"10.1093/logcom/exad029","DOIUrl":"https://doi.org/10.1093/logcom/exad029","url":null,"abstract":"Abstract In this paper, we continue the study of the enriched topological representation of $Q$-algebras where $Q$ is a unital quantale and give two kinds of enriched topological representations of $Q$-algebras. The first one is based on strong $M_3$-valued $Q$-algebra homomorphisms, and the second way is based on strong $M_6$-valued $Q$-algebra homomorphisms. For the first way, we first construct a spatial and semiunital $Q$-algebra $M_3$ containing three elements and show that prime elements of semiunital $Q$-algebras are identified with strong $Q$-algebra homomorphisms taking their values in $M_3$. Then we prove that a semiunital $Q$-algebra is spatial iff strong $Q$-algebra homomorphisms with values in $M_3$ separate elements. Based on this, we obtain that every spatial and semiunital $Q$-algebra can be identified with an $M_3$-enriched sober space. For the second enriched topological representation, we construct a spatial and semiunital $Q$-algebra $M_6$ containing exactly six elements and prove that every spatial and semiunital $Q$-algebra can be identified with an $M_6$-enriched sober space.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"83 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135325098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Paola Cappanera, Marco Gavanelli, Maddalena Nonato, Marco Roma
Abstract Chronic patients suffering from non-communicable diseases are often enrolled into a diagnostic and therapeutic care program featuring a personalized care plan. Healthcare is mostly provided at the patient’s home, but those examinations and treatments that must be delivered at the hospital have to be explicitly booked. Booking is not trivial due to, on the one hand, the several time constraints that become particularly tight in the case of comorbidity, on the other hand, the limited availability of both staff and equipment at the hospital care units. This suggests that the scheduling of the clinical pathways for enrolled outpatients should be managed in a centralized manner, taking advantage of the fact that demand for services is known well in advance. The aim is to serve as many requests as possible (unattended requests are supplied by contracted private health facilities) in a timely manner, taking patients priority into account. Booking involves setting a date and a time for each selected health service, which is rather complex. In this work, we provide a declarative approach by encoding the problem in Answer Set Programming (ASP). In order to improve the scalability of the ASP approach, we present and compare two heuristic approaches, respectively based on service demand and time decomposition. All approaches are tested on instances of increasing size to assess scalability with respect to time horizon and number of requests.
{"title":"Decomposition approaches for scheduling chronic outpatients’ clinical pathways in Answer Set Programming","authors":"Paola Cappanera, Marco Gavanelli, Maddalena Nonato, Marco Roma","doi":"10.1093/logcom/exad038","DOIUrl":"https://doi.org/10.1093/logcom/exad038","url":null,"abstract":"Abstract Chronic patients suffering from non-communicable diseases are often enrolled into a diagnostic and therapeutic care program featuring a personalized care plan. Healthcare is mostly provided at the patient’s home, but those examinations and treatments that must be delivered at the hospital have to be explicitly booked. Booking is not trivial due to, on the one hand, the several time constraints that become particularly tight in the case of comorbidity, on the other hand, the limited availability of both staff and equipment at the hospital care units. This suggests that the scheduling of the clinical pathways for enrolled outpatients should be managed in a centralized manner, taking advantage of the fact that demand for services is known well in advance. The aim is to serve as many requests as possible (unattended requests are supplied by contracted private health facilities) in a timely manner, taking patients priority into account. Booking involves setting a date and a time for each selected health service, which is rather complex. In this work, we provide a declarative approach by encoding the problem in Answer Set Programming (ASP). In order to improve the scalability of the ASP approach, we present and compare two heuristic approaches, respectively based on service demand and time decomposition. All approaches are tested on instances of increasing size to assess scalability with respect to time horizon and number of requests.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"148 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135214983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Computational Argumentation provides tools for both modelling and reasoning with controversial information. Abstract Argumentation Frameworks represent the building blocks in this field and allow one to model the relationships between arguments with the ultimate goal of establishing their acceptability. Arguments can be evaluated through sets of criteria, called semantics, which distinguish among various justification states. For example, an argument may be accepted, rejected, ignored or even marked as undecided. This paper considers Weighted Argumentation Frameworks and proposes a novel labelling semantics that differentiates four states and generalizes existing approaches.
{"title":"On the study of acceptability in weighted argumentation frameworks through four-state labelling semantics","authors":"Stefano Bistarelli, Carlo Taticchi","doi":"10.1093/logcom/exad039","DOIUrl":"https://doi.org/10.1093/logcom/exad039","url":null,"abstract":"Abstract Computational Argumentation provides tools for both modelling and reasoning with controversial information. Abstract Argumentation Frameworks represent the building blocks in this field and allow one to model the relationships between arguments with the ultimate goal of establishing their acceptability. Arguments can be evaluated through sets of criteria, called semantics, which distinguish among various justification states. For example, an argument may be accepted, rejected, ignored or even marked as undecided. This paper considers Weighted Argumentation Frameworks and proposes a novel labelling semantics that differentiates four states and generalizes existing approaches.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135325056","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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