Marija Boričić Joksimović, Nebojša Ikodinović, Nenad Stojanović
We develop a system of basic probability reasoning founded on two great logical concepts, Gentzen’s natural deduction systems and Carnap–Popper probability of sentences. Our system makes it possible to manipulate with probabilized sentences and justify their causal relationships: if probabilities of sentences $A$ and $B$ are in $[r,1]$ and $[s,1]$, respectively, then the probability of sentence $C$ belongs to $[t,1]$, i.e. $A^{r},B^{s}vdash C^{t}$, for $r,s,tin [0,1]$. We prove that our system is sound and complete with respect to the traditional Carnap–Popper type probability semantics. This approach opens up a new perspective of proof-theoretic treatment of sentence probability, potentially allowing immediate algorithmic use of the pure syntactic convenience of natural deductions in programming.
{"title":"Probability and natural deduction","authors":"Marija Boričić Joksimović, Nebojša Ikodinović, Nenad Stojanović","doi":"10.1093/logcom/exae007","DOIUrl":"https://doi.org/10.1093/logcom/exae007","url":null,"abstract":"We develop a system of basic probability reasoning founded on two great logical concepts, Gentzen’s natural deduction systems and Carnap–Popper probability of sentences. Our system makes it possible to manipulate with probabilized sentences and justify their causal relationships: if probabilities of sentences $A$ and $B$ are in $[r,1]$ and $[s,1]$, respectively, then the probability of sentence $C$ belongs to $[t,1]$, i.e. $A^{r},B^{s}vdash C^{t}$, for $r,s,tin [0,1]$. We prove that our system is sound and complete with respect to the traditional Carnap–Popper type probability semantics. This approach opens up a new perspective of proof-theoretic treatment of sentence probability, potentially allowing immediate algorithmic use of the pure syntactic convenience of natural deductions in programming.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"76 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140054487","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}
Peter Arndt, Hugo Luiz Mariano, Darllan Conceição Pinto
Filter pairs are a tool for creating and analyzing logics. A filter pair can be seen as a presentation of a logic, given by presenting its lattice of theories as the image of a lattice homomorphism, with certain properties ensuring that the resulting logic is substitution invariant. Every substitution invariant logic arises from a filter pair. Particular classes of logics can be characterized as arising from special classes of filter pairs. We consider so-called congruence filter pairs, i.e. filter pairs for which the domain of the lattice homomorphism is a lattice of congruences for some quasivariety. We show that the class of logics admitting a presentation by such a filter pair is exactly the class of logics having an algebraic semantics. We study the properties of a certain Galois connection coming with such filter pairs. We give criteria for a congruence filter pair to present a logic in some classes of the Leibniz hierarchy by means of this Galois connection, and its interplay with the Leibniz operator.
{"title":"Congruence filter pairs, equational filter pairs and adjoints","authors":"Peter Arndt, Hugo Luiz Mariano, Darllan Conceição Pinto","doi":"10.1093/logcom/exae002","DOIUrl":"https://doi.org/10.1093/logcom/exae002","url":null,"abstract":"Filter pairs are a tool for creating and analyzing logics. A filter pair can be seen as a presentation of a logic, given by presenting its lattice of theories as the image of a lattice homomorphism, with certain properties ensuring that the resulting logic is substitution invariant. Every substitution invariant logic arises from a filter pair. Particular classes of logics can be characterized as arising from special classes of filter pairs. We consider so-called congruence filter pairs, i.e. filter pairs for which the domain of the lattice homomorphism is a lattice of congruences for some quasivariety. We show that the class of logics admitting a presentation by such a filter pair is exactly the class of logics having an algebraic semantics. We study the properties of a certain Galois connection coming with such filter pairs. We give criteria for a congruence filter pair to present a logic in some classes of the Leibniz hierarchy by means of this Galois connection, and its interplay with the Leibniz operator.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"134 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139980261","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}
In the context of positive logic expanded with the dual of intuitionistic negation, obtaining intuitionistic negation itself as a consequence, those two connectives allow to introduce modal concepts such as necessity and possibility. We study the resulting modal logic, paying attention to different extensions of it. We provide a syntactic approach and both algebraic and Kripke semantics.
{"title":"Modalities combining two negations","authors":"José Luis Castiglioni, Rodolfo C Ertola-Biraben","doi":"10.1093/logcom/exae004","DOIUrl":"https://doi.org/10.1093/logcom/exae004","url":null,"abstract":"In the context of positive logic expanded with the dual of intuitionistic negation, obtaining intuitionistic negation itself as a consequence, those two connectives allow to introduce modal concepts such as necessity and possibility. We study the resulting modal logic, paying attention to different extensions of it. We provide a syntactic approach and both algebraic and Kripke semantics.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"71 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139757678","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}
Birzhan S Kalmurzayev, Nikolay A Bazhenov, Alibek M Iskakov
The complexity classification of computably enumerable equivalence relations (or ceers, for short) has received much attention in the recent literature. A measure of complexity is typically provided by an appropriate notion of a reduction. Given binary relations $R$ and $S$ on natural numbers, a total function $f$ is a reduction from $R$ to $S$ if for arbitrary $x$ and $y$, the conditions $x~R~y$ and $f(x)~S~f(y)$ are always equivalent. If the function $f$ can be chosen primitive recursive, then we say that $R$ is primitively recursively reducible to $S$, denoted by $R leq _{pr} S$. We investigate the degree structure $(textbf {Ceers},leq _{pr})$ of $leq _{pr}$-degrees of ceers. We examine when pairs of incomparable degrees have an infimum and a supremum. In particular, we show that $(textbf {Ceers},leq _{pr})$ is neither an upper semilattice nor a lower semilattice. We also study first-order definable subclasses of $(textbf {Ceers},leq _{pr})$. In particular, we prove that the set of equivalences that have only finitely many classes is definable in $(textbf {Ceers},leq _{pr})$. Finally, we show that the structure of $leq _{pr}$-degrees of computably enumerable preorders has a hereditarily undecidable theory.
{"title":"Computably enumerable equivalence relations via primitive recursive reductions","authors":"Birzhan S Kalmurzayev, Nikolay A Bazhenov, Alibek M Iskakov","doi":"10.1093/logcom/exad082","DOIUrl":"https://doi.org/10.1093/logcom/exad082","url":null,"abstract":"The complexity classification of computably enumerable equivalence relations (or ceers, for short) has received much attention in the recent literature. A measure of complexity is typically provided by an appropriate notion of a reduction. Given binary relations $R$ and $S$ on natural numbers, a total function $f$ is a reduction from $R$ to $S$ if for arbitrary $x$ and $y$, the conditions $x~R~y$ and $f(x)~S~f(y)$ are always equivalent. If the function $f$ can be chosen primitive recursive, then we say that $R$ is primitively recursively reducible to $S$, denoted by $R leq _{pr} S$. We investigate the degree structure $(textbf {Ceers},leq _{pr})$ of $leq _{pr}$-degrees of ceers. We examine when pairs of incomparable degrees have an infimum and a supremum. In particular, we show that $(textbf {Ceers},leq _{pr})$ is neither an upper semilattice nor a lower semilattice. We also study first-order definable subclasses of $(textbf {Ceers},leq _{pr})$. In particular, we prove that the set of equivalences that have only finitely many classes is definable in $(textbf {Ceers},leq _{pr})$. Finally, we show that the structure of $leq _{pr}$-degrees of computably enumerable preorders has a hereditarily undecidable theory.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"177 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139658733","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}
Ana Luiza Tenorio, Caio de Andrade Mendes, Hugo Luiz Mariano
In this paper, we introduce a new definition of sheaves on semicartesian quantales, providing first examples and categorical properties. We note that our sheaves are similar to the standard definition of a sheaf on a locale; however, we prove that in general it is not an elementary topos—since the lattice of external truth values of $Sh(Q)$, $Sub(1)$, is canonically isomorphic to the quantale $Q$—placing this paper as part of a greater project towards a monoidal (not necessarily cartesian) closed version of elementary topos. To start the study the logical aspects of the category of sheaves we are introducing, we explore the nature of the ‘internal truth value objects’ in such sheaves categories. More precisely, we analyse two candidates for subobject classifier for different subclasses of commutative and semicartesian quantales.
{"title":"On sheaves on semicartesian quantales and their truth values","authors":"Ana Luiza Tenorio, Caio de Andrade Mendes, Hugo Luiz Mariano","doi":"10.1093/logcom/exad081","DOIUrl":"https://doi.org/10.1093/logcom/exad081","url":null,"abstract":"In this paper, we introduce a new definition of sheaves on semicartesian quantales, providing first examples and categorical properties. We note that our sheaves are similar to the standard definition of a sheaf on a locale; however, we prove that in general it is not an elementary topos—since the lattice of external truth values of $Sh(Q)$, $Sub(1)$, is canonically isomorphic to the quantale $Q$—placing this paper as part of a greater project towards a monoidal (not necessarily cartesian) closed version of elementary topos. To start the study the logical aspects of the category of sheaves we are introducing, we explore the nature of the ‘internal truth value objects’ in such sheaves categories. More precisely, we analyse two candidates for subobject classifier for different subclasses of commutative and semicartesian quantales.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"4 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139553095","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 show that the monadic fragment of the modal predicate logic of a single Kripke frame with finitely many possible worlds, but possibly infinite domains, is decidable. This holds true even for multimodal logics with equality, regardless of whether equality is interpreted as identity or as congruence. By the Gödel–Tarski translation, similar results follow for superintuitionistic predicate logics, with or without equality. Using these observations, we establish upper algorithmic bounds, which match the known lower bounds, for monadic fragments of some modal predicate logics. In particular, we prove that, if $L$ is a propositional modal logic contained in $textbf{S5}$, $textbf{GL.3}$ or $textbf{Grz.3}$ and the class of finite Kripke frames validating $L$ is recursively enumerable, then the monadic fragment with equality of the predicate logic of finite Kripke frames validating $L$ is $varPi ^{0}_{1}$-complete; this, in particular, holds if $L$ is one of the following propositional logics: $textbf{K}$, $textbf{T}$, $textbf{D}$, $textbf{KB}$, $textbf{KTB}$, $textbf{K4}$, $textbf{K4.3}$, $textbf{S4}$, $textbf{S4.3}$, $textbf{GL}$, $textbf{Grz}$, $textbf{K5}$, $textbf{K45}$ and $textbf{S5}$. We also prove that monadic fragments with equality of logics $textbf{QAlt}^=_{n}$ and $textbf{QTAlt}^=_{n}$ are decidable. The obtained results are easily extendable to the multimodal versions of the predicate logics we consider and to logics with the Barcan formula.
{"title":"Algorithmic properties of modal and superintuitionistic logics of monadic predicates over finite Kripke frames","authors":"Mikhail Rybakov, Dmitry Shkatov","doi":"10.1093/logcom/exad078","DOIUrl":"https://doi.org/10.1093/logcom/exad078","url":null,"abstract":"We show that the monadic fragment of the modal predicate logic of a single Kripke frame with finitely many possible worlds, but possibly infinite domains, is decidable. This holds true even for multimodal logics with equality, regardless of whether equality is interpreted as identity or as congruence. By the Gödel–Tarski translation, similar results follow for superintuitionistic predicate logics, with or without equality. Using these observations, we establish upper algorithmic bounds, which match the known lower bounds, for monadic fragments of some modal predicate logics. In particular, we prove that, if $L$ is a propositional modal logic contained in $textbf{S5}$, $textbf{GL.3}$ or $textbf{Grz.3}$ and the class of finite Kripke frames validating $L$ is recursively enumerable, then the monadic fragment with equality of the predicate logic of finite Kripke frames validating $L$ is $varPi ^{0}_{1}$-complete; this, in particular, holds if $L$ is one of the following propositional logics: $textbf{K}$, $textbf{T}$, $textbf{D}$, $textbf{KB}$, $textbf{KTB}$, $textbf{K4}$, $textbf{K4.3}$, $textbf{S4}$, $textbf{S4.3}$, $textbf{GL}$, $textbf{Grz}$, $textbf{K5}$, $textbf{K45}$ and $textbf{S5}$. We also prove that monadic fragments with equality of logics $textbf{QAlt}^=_{n}$ and $textbf{QTAlt}^=_{n}$ are decidable. The obtained results are easily extendable to the multimodal versions of the predicate logics we consider and to logics with the Barcan formula.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"12 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139475100","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}
An approach for encoding abstract dialectical frameworks and their semantics into classical higher-order logic is presented. Important properties and semantic relationships are formally encoded and proven using the proof assistant Isabelle/HOL. This approach allows for the computer-assisted analysis of abstract dialectical frameworks using automated and interactive reasoning tools within a uniform logic environment. Exemplary applications include the formal analysis and verification of meta-theoretical properties, and the generation of interpretations and extensions under specific semantic constraints.
{"title":"An encoding of abstract dialectical frameworks into higher-order logic","authors":"Antoine Martina, Alexander Steen","doi":"10.1093/logcom/exad079","DOIUrl":"https://doi.org/10.1093/logcom/exad079","url":null,"abstract":"An approach for encoding abstract dialectical frameworks and their semantics into classical higher-order logic is presented. Important properties and semantic relationships are formally encoded and proven using the proof assistant Isabelle/HOL. This approach allows for the computer-assisted analysis of abstract dialectical frameworks using automated and interactive reasoning tools within a uniform logic environment. Exemplary applications include the formal analysis and verification of meta-theoretical properties, and the generation of interpretations and extensions under specific semantic constraints.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139462348","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}
For a class $varGamma $ of formulas, $varGamma $ local reflection principle $textrm{Rfn}_{varGamma }(T)$ for a theory $T$ of arithmetic is a scheme formalizing the $varGamma $-soundness of $T$. Beklemishev (1997, Theoria, 63, 139–146) proved that for every $varGamma in {varSigma _{n}, varPi _{n+1} mid n geq 1}$, the full local reflection principle $textrm{Rfn}(T)$ is $varGamma $-conservative over $T + textrm{Rfn}_{varGamma }(T)$. We firstly generalize the conservation theorem to nonstandard provability predicates: we prove that the second condition $textbf{D2}$ of the derivability conditions is a sufficient condition for the conservation theorem to hold. We secondly investigate the conservation theorem in terms of Rosser provability predicates. We construct Rosser predicates for which the conservation theorem holds and Rosser predicates for which the theorem does not hold.
{"title":"On the conservation results for local reflection principles","authors":"Haruka Kogure, Taishi Kurahashi","doi":"10.1093/logcom/exad076","DOIUrl":"https://doi.org/10.1093/logcom/exad076","url":null,"abstract":"For a class $varGamma $ of formulas, $varGamma $ local reflection principle $textrm{Rfn}_{varGamma }(T)$ for a theory $T$ of arithmetic is a scheme formalizing the $varGamma $-soundness of $T$. Beklemishev (1997, Theoria, 63, 139–146) proved that for every $varGamma in {varSigma _{n}, varPi _{n+1} mid n geq 1}$, the full local reflection principle $textrm{Rfn}(T)$ is $varGamma $-conservative over $T + textrm{Rfn}_{varGamma }(T)$. We firstly generalize the conservation theorem to nonstandard provability predicates: we prove that the second condition $textbf{D2}$ of the derivability conditions is a sufficient condition for the conservation theorem to hold. We secondly investigate the conservation theorem in terms of Rosser provability predicates. We construct Rosser predicates for which the conservation theorem holds and Rosser predicates for which the theorem does not hold.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"85 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139103871","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 explores how, given a proof, we can systematically transform it into a proof that contains no irrelevancies and which is as strong as possible. I define a weaker and stronger notion of what counts as a proof with no irrelevancies, calling them perfect proofs and gaunt proofs, respectively. Using classical core logic to study classical validities and core logic to study intuitionistic validities, I show that every core proof or classical core proof can be transformed into a perfect proof. In a sequel paper, I show how proofs in core logic can also be transformed into gaunt proofs and I observe that this property fails for classical core logic.
{"title":"Coarsening Natural Deduction Proofs I: Finding Perfect Proofs","authors":"Ethan Brauer","doi":"10.1093/logcom/exad077","DOIUrl":"https://doi.org/10.1093/logcom/exad077","url":null,"abstract":"This paper explores how, given a proof, we can systematically transform it into a proof that contains no irrelevancies and which is as strong as possible. I define a weaker and stronger notion of what counts as a proof with no irrelevancies, calling them perfect proofs and gaunt proofs, respectively. Using classical core logic to study classical validities and core logic to study intuitionistic validities, I show that every core proof or classical core proof can be transformed into a perfect proof. In a sequel paper, I show how proofs in core logic can also be transformed into gaunt proofs and I observe that this property fails for classical core logic.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"100 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139103991","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}
Reinhard Kahle, Isabel Oitavem, Paulo Guilherme Santos
We study numeral forms of completeness and consistency for $mathsf {S}^1_2$ and other weak theories, like $mathsf {EA}$. This gives rise to an exploration of the derivability conditions needed to establish the mentioned results; a presentation of a weak form of Gödel’s Second Incompleteness Theorem without using ‘provability implies provable provability’; a provability predicate that satisfies the mentioned derivability condition for weak theories; and a completeness result via consistency statements. Moreover, the paper includes characterizations of the provability predicates for which the numeral results hold, having $mathsf {EA}$ as the surrounding theory, and results on functions that compute finitist consistency statements.
{"title":"Numeral completeness of weak theories of arithmetic","authors":"Reinhard Kahle, Isabel Oitavem, Paulo Guilherme Santos","doi":"10.1093/logcom/exad075","DOIUrl":"https://doi.org/10.1093/logcom/exad075","url":null,"abstract":"We study numeral forms of completeness and consistency for $mathsf {S}^1_2$ and other weak theories, like $mathsf {EA}$. This gives rise to an exploration of the derivability conditions needed to establish the mentioned results; a presentation of a weak form of Gödel’s Second Incompleteness Theorem without using ‘provability implies provable provability’; a provability predicate that satisfies the mentioned derivability condition for weak theories; and a completeness result via consistency statements. Moreover, the paper includes characterizations of the provability predicates for which the numeral results hold, having $mathsf {EA}$ as the surrounding theory, and results on functions that compute finitist consistency statements.","PeriodicalId":50162,"journal":{"name":"Journal of Logic and Computation","volume":"171 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138690333","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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