Abstract This paper is intended first for the formal argumentation community (see https://comma.csc.liv.ac.uk/). This community develops logics and systems modelling argumentation and dialogues. The community is in search of major applications areas for their models. One such application area e.g. is Law. The message of this paper is that there is another major application area for formal argumentation. There is an international community of sex offender therapist that is well established and well funded, and their therapy methods use (methods that can be modelled by) formal argumentation and logic. This community presents a natural application area for formal argumentation. We thus describe in this paper how the sex offender therapists work, to give the formal argumentation researcher a view of this application area. What is especially important about this application area is that in order to model it and learn from it, the formal argumentation community have to evolve their formal methods and adapt to this new application. Part of this enhancement is to modify and import certain methods from other areas of Logic e.g. from Non-Monotonic logic. The members of the formal argumentation community are not familiar, on average, with other areas of logic, and so we also describe in this paper, what we need from neighbouring logics. This makes this paper of interest also to sex offender therapist as well. They may be already familiar with their own practices, but the additional logics described will be of interest to them.
{"title":"The use of logic and argumentation in therapy of sex offenders","authors":"Dov Gabbay, Gadi Rozenberg, Lydia Rivlin","doi":"10.1093/jigpal/jzad022","DOIUrl":"https://doi.org/10.1093/jigpal/jzad022","url":null,"abstract":"Abstract This paper is intended first for the formal argumentation community (see https://comma.csc.liv.ac.uk/). This community develops logics and systems modelling argumentation and dialogues. The community is in search of major applications areas for their models. One such application area e.g. is Law. The message of this paper is that there is another major application area for formal argumentation. There is an international community of sex offender therapist that is well established and well funded, and their therapy methods use (methods that can be modelled by) formal argumentation and logic. This community presents a natural application area for formal argumentation. We thus describe in this paper how the sex offender therapists work, to give the formal argumentation researcher a view of this application area. What is especially important about this application area is that in order to model it and learn from it, the formal argumentation community have to evolve their formal methods and adapt to this new application. Part of this enhancement is to modify and import certain methods from other areas of Logic e.g. from Non-Monotonic logic. The members of the formal argumentation community are not familiar, on average, with other areas of logic, and so we also describe in this paper, what we need from neighbouring logics. This makes this paper of interest also to sex offender therapist as well. They may be already familiar with their own practices, but the additional logics described will be of interest to them.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136034586","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 work, we illustrate applications of a semantic framework for non-congruential modal logic based on hyperintensional models. We start by discussing some philosophical ideas behind the approach; in particular, the difference between the set of possible worlds in which a formula is true (its intension) and the semantic content of a formula (its hyperintension), which is captured in a rigorous way in hyperintensional models. Next, we rigorously specify the approach and provide a fundamental completeness theorem. Moreover, we analyse examples of non-congruential systems that can be semantically characterized within this framework in an elegant and modular way. Finally, we compare the proposed framework with some alternatives available in the literature. In the light of the results obtained, we argue that hyperintensional models constitute a basic, general and unifying semantic framework for (non-congruential) modal logic.
{"title":"Hyperintensional models for non-congruential modal logics","authors":"Matteo Pascucci, Igor Sedlár","doi":"10.1093/jigpal/jzad018","DOIUrl":"https://doi.org/10.1093/jigpal/jzad018","url":null,"abstract":"Abstract In this work, we illustrate applications of a semantic framework for non-congruential modal logic based on hyperintensional models. We start by discussing some philosophical ideas behind the approach; in particular, the difference between the set of possible worlds in which a formula is true (its intension) and the semantic content of a formula (its hyperintension), which is captured in a rigorous way in hyperintensional models. Next, we rigorously specify the approach and provide a fundamental completeness theorem. Moreover, we analyse examples of non-congruential systems that can be semantically characterized within this framework in an elegant and modular way. Finally, we compare the proposed framework with some alternatives available in the literature. In the light of the results obtained, we argue that hyperintensional models constitute a basic, general and unifying semantic framework for (non-congruential) modal logic.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136237794","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}
Journal Article Correction to: Decidability of interpretability logics ILM0 and ILW* Get access Luka Mikec, Luka Mikec University of Zagreb Search for other works by this author on: Oxford Academic Google Scholar Tin Perkov, Tin Perkov University of Zagreb E-mail: tin.perkov@ufzg.hr Search for other works by this author on: Oxford Academic Google Scholar Mladen Vukoviĉ Mladen Vukoviĉ University of Zagreb Search for other works by this author on: Oxford Academic Google Scholar Logic Journal of the IGPL, jzad020, https://doi.org/10.1093/jigpal/jzad020 Published: 14 September 2023 Article history Received: 07 September 2023 Published: 14 September 2023
期刊文章更正:可解释性逻辑ILM0和ILW的可决定性*访问Luka Mikec, Luka Mikec萨格勒布大学搜索本作者的其他作品:牛津学术谷歌学者Tin Perkov, Tin Perkov萨格勒布大学E-mail: tin.perkov@ufzg.hr搜索本作者的其他作品:牛津学术谷歌学者Mladen vukovii Mladen vukovii萨格勒布大学搜索本作者的其他作品:牛津学术谷歌学者逻辑IGPL期刊，jzad020, https://doi.org/10.1093/jigpal/jzad020发布日期:2023年9月14日文章历史收稿日期:2023年9月07日发布日期:2023年9月14日
{"title":"Correction to: Decidability of interpretability logics <b>IL</b> <tt>M</tt>0 and <b>IL</b> <tt>W</tt>*","authors":"Luka Mikec, Tin Perkov, Mladen Vukoviĉ","doi":"10.1093/jigpal/jzad020","DOIUrl":"https://doi.org/10.1093/jigpal/jzad020","url":null,"abstract":"Journal Article Correction to: Decidability of interpretability logics ILM0 and ILW* Get access Luka Mikec, Luka Mikec University of Zagreb Search for other works by this author on: Oxford Academic Google Scholar Tin Perkov, Tin Perkov University of Zagreb E-mail: tin.perkov@ufzg.hr Search for other works by this author on: Oxford Academic Google Scholar Mladen Vukoviĉ Mladen Vukoviĉ University of Zagreb Search for other works by this author on: Oxford Academic Google Scholar Logic Journal of the IGPL, jzad020, https://doi.org/10.1093/jigpal/jzad020 Published: 14 September 2023 Article history Received: 07 September 2023 Published: 14 September 2023","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134913976","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}
Wojciech Wieczorek, Łukasz Strąk, Arkadiusz Nowakowski
Abstract This paper presents four state-of-art methods for the finite-state automaton inference based on a sample of labeled strings. The first algorithm is Exbar, and the next three are mathematical models based on ASP, SAT and SMT theories. The potentiality of using multiprocessor computers in the context of automata inference was our research’s primary goal. In a series of experiments, we showed that our parallelization of the exbar algorithm is the best choice when a multiprocessor system is available. Furthermore, we obtained a superlinear speedup for some of the prepared datasets, achieving almost a 5-fold speedup on the median, using 12 and 24 processes.
{"title":"Report on the exact methods for finding minimum-sized DFA","authors":"Wojciech Wieczorek, Łukasz Strąk, Arkadiusz Nowakowski","doi":"10.1093/jigpal/jzad014","DOIUrl":"https://doi.org/10.1093/jigpal/jzad014","url":null,"abstract":"Abstract This paper presents four state-of-art methods for the finite-state automaton inference based on a sample of labeled strings. The first algorithm is Exbar, and the next three are mathematical models based on ASP, SAT and SMT theories. The potentiality of using multiprocessor computers in the context of automata inference was our research’s primary goal. In a series of experiments, we showed that our parallelization of the exbar algorithm is the best choice when a multiprocessor system is available. Furthermore, we obtained a superlinear speedup for some of the prepared datasets, achieving almost a 5-fold speedup on the median, using 12 and 24 processes.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135047734","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":"Editorial: Special issue in honour of John Newsome Crossley","authors":"Guillermo Badia","doi":"10.1093/jigpal/jzad011","DOIUrl":"https://doi.org/10.1093/jigpal/jzad011","url":null,"abstract":"","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44017793","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}
Self-referential sentences have troubled our understanding of language for centuries. The most famous self-referential sentence is probably the Liar, a sentence that says of itself that it is false. The Liar Paradox has encouraged many philosophers to establish theories of truth that manage to give a proper account of the truth predicate in a formal language. Kripke’s Fixed Point Theory from 1975 is one famous example of such a formal theory of truth that aims at giving a plausible notion of truth by allowing truth value gaps. However, not only the concept of truth gives rise to paradoxes. A syntactical treatment of epistemic notions like belief and knowledge leads to contradictions that very much resemble the Liar Paradox. Therefore, it seems to be fruitful to apply the established theories of truth to epistemic concepts. In this paper, I will present one such attempt of solving the epistemic paradoxes: I adapt Kripke’s Fixed Point Theory and interpret truth, knowledge and belief within the framework of a partial logic. Thereby I find not only the fixed point of truth but also the fixed points of knowledge and belief. In this fixed point, the predicates of truth, belief and knowledge find their definite interpretation and the paradoxes are avoided.
{"title":"The fixed points of belief and knowledge","authors":"Daniela Schuster","doi":"10.1093/jigpal/jzad016","DOIUrl":"https://doi.org/10.1093/jigpal/jzad016","url":null,"abstract":"\u0000 Self-referential sentences have troubled our understanding of language for centuries. The most famous self-referential sentence is probably the Liar, a sentence that says of itself that it is false. The Liar Paradox has encouraged many philosophers to establish theories of truth that manage to give a proper account of the truth predicate in a formal language. Kripke’s Fixed Point Theory from 1975 is one famous example of such a formal theory of truth that aims at giving a plausible notion of truth by allowing truth value gaps. However, not only the concept of truth gives rise to paradoxes. A syntactical treatment of epistemic notions like belief and knowledge leads to contradictions that very much resemble the Liar Paradox. Therefore, it seems to be fruitful to apply the established theories of truth to epistemic concepts. In this paper, I will present one such attempt of solving the epistemic paradoxes: I adapt Kripke’s Fixed Point Theory and interpret truth, knowledge and belief within the framework of a partial logic. Thereby I find not only the fixed point of truth but also the fixed points of knowledge and belief. In this fixed point, the predicates of truth, belief and knowledge find their definite interpretation and the paradoxes are avoided.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45430042","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}
Zoran Ognjanović, Angelina Ilić Stepić, Aleksandar Perović
Abstract We study a propositional probabilistic temporal epistemic logic $textbf {PTEL}$ with both future and past temporal operators, with non-rigid set of agents and the operators for agents’ knowledge and for common knowledge and with probabilities defined on the sets of runs and on the sets of possible worlds. A semantics is given by a class ${scriptsize{rm Mod}}$ of Kripke-like models with possible worlds. We prove decidability of $textbf {PTEL}$ by showing that checking satisfiability of a formula in ${scriptsize{rm Mod}}$ is equivalent to checking its satisfiability in a finite set of finitely representable structures. The same procedure can be applied to the class of all synchronous ${scriptsize{rm Mod}}$-models. We give an upper complexity bound for the satisfiability problem for ${scriptsize{rm Mod}}$.
{"title":"A probabilistic temporal epistemic logic: Decidability","authors":"Zoran Ognjanović, Angelina Ilić Stepić, Aleksandar Perović","doi":"10.1093/jigpal/jzac080","DOIUrl":"https://doi.org/10.1093/jigpal/jzac080","url":null,"abstract":"Abstract We study a propositional probabilistic temporal epistemic logic $textbf {PTEL}$ with both future and past temporal operators, with non-rigid set of agents and the operators for agents’ knowledge and for common knowledge and with probabilities defined on the sets of runs and on the sets of possible worlds. A semantics is given by a class ${scriptsize{rm Mod}}$ of Kripke-like models with possible worlds. We prove decidability of $textbf {PTEL}$ by showing that checking satisfiability of a formula in ${scriptsize{rm Mod}}$ is equivalent to checking its satisfiability in a finite set of finitely representable structures. The same procedure can be applied to the class of all synchronous ${scriptsize{rm Mod}}$-models. We give an upper complexity bound for the satisfiability problem for ${scriptsize{rm Mod}}$.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135451718","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 investigate mathematical structures that provide natural semantics for families of (quantified) non-classical logics featuring special unary connectives, known as recovery operators, that allow us to ‘recover’ the properties of classical logic in a controlled manner. These structures are known as topological Boolean algebras, which are Boolean algebras extended with additional operations subject to specific conditions of a topological nature. In this study, we focus on the paradigmatic case of negation. We demonstrate how these algebras are well-suited to provide a semantics for some families of paraconsistent Logics of Formal Inconsistency and paracomplete Logics of Formal Undeterminedness. These logics feature recovery operators used to earmark propositions that behave ‘classically’ when interacting with non-classical negations. Unlike traditional semantical investigations, which are carried out in natural language (extended with mathematical shorthand), our formal meta-language is a system of higher-order logic (HOL) for which automated reasoning tools exist. In our approach, topological Boolean algebras are encoded as algebras of sets via their Stone-type representation. We use our higher-order meta-logic to define and interrelate several transformations on unary set operations, which naturally give rise to a topological cube of opposition. Additionally, our approach enables a uniform characterization of propositional, first-order and higher-order quantification, including restrictions to constant and varying domains. With this work, we aim to make a case for the utilization of automated theorem proving technology for conducting computer-supported research in non-classical logics. All the results presented in this paper have been formally verified, and in many cases obtained, using the Isabelle/HOL proof assistant.
{"title":"Semantical investigations on non-classical logics with recovery operators: negation","authors":"David Fuenmayor","doi":"10.1093/jigpal/jzad013","DOIUrl":"https://doi.org/10.1093/jigpal/jzad013","url":null,"abstract":"Abstract We investigate mathematical structures that provide natural semantics for families of (quantified) non-classical logics featuring special unary connectives, known as recovery operators, that allow us to ‘recover’ the properties of classical logic in a controlled manner. These structures are known as topological Boolean algebras, which are Boolean algebras extended with additional operations subject to specific conditions of a topological nature. In this study, we focus on the paradigmatic case of negation. We demonstrate how these algebras are well-suited to provide a semantics for some families of paraconsistent Logics of Formal Inconsistency and paracomplete Logics of Formal Undeterminedness. These logics feature recovery operators used to earmark propositions that behave ‘classically’ when interacting with non-classical negations. Unlike traditional semantical investigations, which are carried out in natural language (extended with mathematical shorthand), our formal meta-language is a system of higher-order logic (HOL) for which automated reasoning tools exist. In our approach, topological Boolean algebras are encoded as algebras of sets via their Stone-type representation. We use our higher-order meta-logic to define and interrelate several transformations on unary set operations, which naturally give rise to a topological cube of opposition. Additionally, our approach enables a uniform characterization of propositional, first-order and higher-order quantification, including restrictions to constant and varying domains. With this work, we aim to make a case for the utilization of automated theorem proving technology for conducting computer-supported research in non-classical logics. All the results presented in this paper have been formally verified, and in many cases obtained, using the Isabelle/HOL proof assistant.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136283511","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}
A logic $mathcal{L}$ is said to satisfy the descending chain condition, DCC, if any descending chain of formulas in $mathcal{L}$ with ordering induced by $vdash _{mathcal{L}};$ eventually stops. In this short note, we first establish a general theorem, which states that if a propositional logic $mathcal{L}$ satisfies both DCC and has the Craig Interpolation Property, CIP, then it satisfies the Uniform Interpolation Property, UIP, as well. As a result, by using the Nishimura lattice, we give a new simply proof of uniform interpolation for $textbf{IPL}_2$, the two-variable fragment of Intuitionistic Propositional Logic; and one-variable uniform interpolation for $textbf{IPL}$. Also, we will see that the modal logics $textbf{S}_4$ and $textbf{K}_4$ do not satisfy atomic DCC.
{"title":"Remarks on uniform interpolation property","authors":"Majid Alizadeh","doi":"10.1093/jigpal/jzad009","DOIUrl":"https://doi.org/10.1093/jigpal/jzad009","url":null,"abstract":"\u0000 A logic $mathcal{L}$ is said to satisfy the descending chain condition, DCC, if any descending chain of formulas in $mathcal{L}$ with ordering induced by $vdash _{mathcal{L}};$ eventually stops. In this short note, we first establish a general theorem, which states that if a propositional logic $mathcal{L}$ satisfies both DCC and has the Craig Interpolation Property, CIP, then it satisfies the Uniform Interpolation Property, UIP, as well. As a result, by using the Nishimura lattice, we give a new simply proof of uniform interpolation for $textbf{IPL}_2$, the two-variable fragment of Intuitionistic Propositional Logic; and one-variable uniform interpolation for $textbf{IPL}$. Also, we will see that the modal logics $textbf{S}_4$ and $textbf{K}_4$ do not satisfy atomic DCC.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49524979","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}
We study logical reduction (factorization) of relations into relations of lower arity by Boolean or relative products that come from applying conjunctions and existential quantifiers to predicates, i.e. by primitive positive formulas of predicate calculus. Our algebraic framework unifies natural joins and data dependencies of database theory and relational algebra of clone theory with the bond algebra of C.S. Peirce. We also offer new constructions of reductions, systematically study irreducible relations and reductions to them and introduce a new characteristic of relations, ternarity, that measures their ‘complexity of relating’ and allows to refine reduction results. In particular, we refine Peirce’s controversial reduction thesis, and show that reducibility behaviour is dramatically different on finite and infinite domains.
{"title":"Logical reduction of relations: From relational databases to Peirce’s reduction thesis","authors":"Sergiy Koshkin","doi":"10.1093/jigpal/jzad010","DOIUrl":"https://doi.org/10.1093/jigpal/jzad010","url":null,"abstract":"\u0000 We study logical reduction (factorization) of relations into relations of lower arity by Boolean or relative products that come from applying conjunctions and existential quantifiers to predicates, i.e. by primitive positive formulas of predicate calculus. Our algebraic framework unifies natural joins and data dependencies of database theory and relational algebra of clone theory with the bond algebra of C.S. Peirce. We also offer new constructions of reductions, systematically study irreducible relations and reductions to them and introduce a new characteristic of relations, ternarity, that measures their ‘complexity of relating’ and allows to refine reduction results. In particular, we refine Peirce’s controversial reduction thesis, and show that reducibility behaviour is dramatically different on finite and infinite domains.","PeriodicalId":51114,"journal":{"name":"Logic Journal of the IGPL","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41971974","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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