We investigate the cyclic proof theory of extensions of Peano Arithmetic by (finitely iterated) inductive definitions. Such theories are essential to proof theoretic analyses of certain `impredicative' theories; moreover, our cyclic systems naturally subsume Simpson's Cyclic Arithmetic. Our main result is that cyclic and inductive systems for arithmetical inductive definitions are equally powerful. We conduct a metamathematical argument, formalising the soundness of cyclic proofs within second-order arithmetic by a form of induction on closure ordinals, thence appealing to conservativity results. This approach is inspired by those of Simpson and Das for Cyclic Arithmetic, however we must further address a difficulty: the closure ordinals of our inductive definitions (around Church-Kleene) far exceed the proof theoretic ordinal of the appropriate metatheory (around Bachmann-Howard), so explicit induction on their notations is not possible. For this reason, we rather rely on formalisation of the theory of (recursive) ordinals within second-order arithmetic.
{"title":"Cyclic Proofs for Arithmetical Inductive Definitions","authors":"Anupam Das, Lukas Melgaard","doi":"10.4230/LIPIcs.FSCD.2023.27","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2023.27","url":null,"abstract":"We investigate the cyclic proof theory of extensions of Peano Arithmetic by (finitely iterated) inductive definitions. Such theories are essential to proof theoretic analyses of certain `impredicative' theories; moreover, our cyclic systems naturally subsume Simpson's Cyclic Arithmetic. Our main result is that cyclic and inductive systems for arithmetical inductive definitions are equally powerful. We conduct a metamathematical argument, formalising the soundness of cyclic proofs within second-order arithmetic by a form of induction on closure ordinals, thence appealing to conservativity results. This approach is inspired by those of Simpson and Das for Cyclic Arithmetic, however we must further address a difficulty: the closure ordinals of our inductive definitions (around Church-Kleene) far exceed the proof theoretic ordinal of the appropriate metatheory (around Bachmann-Howard), so explicit induction on their notations is not possible. For this reason, we rather rely on formalisation of the theory of (recursive) ordinals within second-order arithmetic.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"85 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124683267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The essence of compiling with continuations is that conversion to continuation-passing style (CPS) is equivalent to a source language transformation converting to administrative normal form (ANF). Taking as source language Moggi's computational lambda-calculus (lbc), we define an alternative to the CPS-translation with target in the sequent calculus LJQ, named value-filling style (VFS) translation, and making use of the ability of the sequent calculus to represent contexts formally. The VFS-translation requires no type translation: indeed, double negations are introduced only when encoding the VFS target language in the CPS target language. This optional encoding, when composed with the VFS-translation reconstructs the original CPS-translation. Going back to direct style, the"essence"of the VFS-translation is that it reveals a new sublanguage of ANF, the value-enclosed style (VES), next to another one, the continuation-enclosing style (CES): such an alternative is due to a dilemma in the syntax of lbc, concerning how to expand the application constructor. In the typed scenario, VES and CES correspond to an alternative between two proof systems for call-by-value, LJQ and natural deduction with generalized applications, confirming proof theory as a foundation for intermediate representations.
{"title":"The Logical Essence of Compiling With Continuations","authors":"J. E. Santo, Filipa Mendes","doi":"10.48550/arXiv.2304.14752","DOIUrl":"https://doi.org/10.48550/arXiv.2304.14752","url":null,"abstract":"The essence of compiling with continuations is that conversion to continuation-passing style (CPS) is equivalent to a source language transformation converting to administrative normal form (ANF). Taking as source language Moggi's computational lambda-calculus (lbc), we define an alternative to the CPS-translation with target in the sequent calculus LJQ, named value-filling style (VFS) translation, and making use of the ability of the sequent calculus to represent contexts formally. The VFS-translation requires no type translation: indeed, double negations are introduced only when encoding the VFS target language in the CPS target language. This optional encoding, when composed with the VFS-translation reconstructs the original CPS-translation. Going back to direct style, the\"essence\"of the VFS-translation is that it reveals a new sublanguage of ANF, the value-enclosed style (VES), next to another one, the continuation-enclosing style (CES): such an alternative is due to a dilemma in the syntax of lbc, concerning how to expand the application constructor. In the typed scenario, VES and CES correspond to an alternative between two proof systems for call-by-value, LJQ and natural deduction with generalized applications, confirming proof theory as a foundation for intermediate representations.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117073881","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show how confluence criteria based on decreasing diagrams are generalized to ones composable with other criteria. For demonstration of the method, the confluence criteria of orthogonality, rule labeling, and critical pair systems for term rewriting are recast into composable forms. We also show how such a criterion can be used for a reduction method that removes rewrite rules unnecessary for confluence analysis. In addition to them, we prove that Toyama's parallel closedness result based on parallel critical pairs subsumes his almost parallel closedness theorem.
{"title":"Compositional Confluence Criteria","authors":"Kiraku Shintani, Nao Hirokawa","doi":"10.4230/LIPIcs.FSCD.2022.28","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2022.28","url":null,"abstract":"We show how confluence criteria based on decreasing diagrams are generalized to ones composable with other criteria. For demonstration of the method, the confluence criteria of orthogonality, rule labeling, and critical pair systems for term rewriting are recast into composable forms. We also show how such a criterion can be used for a reduction method that removes rewrite rules unnecessary for confluence analysis. In addition to them, we prove that Toyama's parallel closedness result based on parallel critical pairs subsumes his almost parallel closedness theorem.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123675126","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tennenbaum's theorem states that the only countable model of Peano arithmetic (PA) with computable arithmetical operations is the standard model of natural numbers. In this paper, we use constructive type theory as a framework to revisit, analyze and generalize this result. The chosen framework allows for a synthetic approach to computability theory, exploiting that, externally, all functions definable in constructive type theory can be shown computable. We then build on this viewpoint and furthermore internalize it by assuming a version of Church's thesis, which expresses that any function on natural numbers is representable by a formula in PA. This assumption provides for a conveniently abstract setup to carry out rigorous computability arguments, even in the theorem's mechanization. Concretely, we constructivize several classical proofs and present one inherently constructive rendering of Tennenbaum's theorem, all following arguments from the literature. Concerning the classical proofs in particular, the constructive setting allows us to highlight differences in their assumptions and conclusions which are not visible classically. All versions are accompanied by a unified mechanization in the Coq proof assistant.
{"title":"An Analysis of Tennenbaum's Theorem in Constructive Type Theory","authors":"Marc Hermes, Dominik Kirst","doi":"10.4230/LIPIcs.FSCD.2022.9","DOIUrl":"https://doi.org/10.4230/LIPIcs.FSCD.2022.9","url":null,"abstract":"Tennenbaum's theorem states that the only countable model of Peano arithmetic (PA) with computable arithmetical operations is the standard model of natural numbers. In this paper, we use constructive type theory as a framework to revisit, analyze and generalize this result. The chosen framework allows for a synthetic approach to computability theory, exploiting that, externally, all functions definable in constructive type theory can be shown computable. We then build on this viewpoint and furthermore internalize it by assuming a version of Church's thesis, which expresses that any function on natural numbers is representable by a formula in PA. This assumption provides for a conveniently abstract setup to carry out rigorous computability arguments, even in the theorem's mechanization. Concretely, we constructivize several classical proofs and present one inherently constructive rendering of Tennenbaum's theorem, all following arguments from the literature. Concerning the classical proofs in particular, the constructive setting allows us to highlight differences in their assumptions and conclusions which are not visible classically. All versions are accompanied by a unified mechanization in the Coq proof assistant.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114739925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we adapt previous work on rewriting string diagrams using hypergraphs to the case where the underlying category has a traced comonoid structure, in which wires can be forked and the outputs of a morphism can be connected to its input. Such a structure is particularly interesting because any traced Cartesian (dataflow) category has an underlying traced comonoid structure. We show that certain subclasses of hypergraphs are fully complete for traced comonoid categories: that is to say, every term in such a category has a unique corresponding hypergraph up to isomorphism, and from every hypergraph with the desired properties, a unique term in the category can be retrieved up to the axioms of traced comonoid categories. We also show how the framework of double pushout rewriting (DPO) can be adapted for traced comonoid categories by characterising the valid pushout complements for rewriting in our setting. We conclude by presenting a case study in the form of recent work on an equational theory for sequential circuits: circuits built from primitive logic gates with delay and feedback. The graph rewriting framework allows for the definition of an operational semantics for sequential circuits.
{"title":"Rewriting modulo traced comonoid structure","authors":"D. Ghica, G. Kaye","doi":"10.48550/arXiv.2302.09631","DOIUrl":"https://doi.org/10.48550/arXiv.2302.09631","url":null,"abstract":"In this paper we adapt previous work on rewriting string diagrams using hypergraphs to the case where the underlying category has a traced comonoid structure, in which wires can be forked and the outputs of a morphism can be connected to its input. Such a structure is particularly interesting because any traced Cartesian (dataflow) category has an underlying traced comonoid structure. We show that certain subclasses of hypergraphs are fully complete for traced comonoid categories: that is to say, every term in such a category has a unique corresponding hypergraph up to isomorphism, and from every hypergraph with the desired properties, a unique term in the category can be retrieved up to the axioms of traced comonoid categories. We also show how the framework of double pushout rewriting (DPO) can be adapted for traced comonoid categories by characterising the valid pushout complements for rewriting in our setting. We conclude by presenting a case study in the form of recent work on an equational theory for sequential circuits: circuits built from primitive logic gates with delay and feedback. The graph rewriting framework allows for the definition of an operational semantics for sequential circuits.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"94 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131395704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Higher-order unification (HOU) concerns unification of (extensions of) $lambda$-calculus and can be seen as an instance of equational unification ($E$-unification) modulo $betaeta$-equivalence of $lambda$-terms. We study equational unification of terms in languages with arbitrary variable binding constructions modulo arbitrary second-order equational theories. Abstract syntax with general variable binding and parametrised metavariables allows us to work with arbitrary binders without committing to $lambda$-calculus or use inconvenient and error-prone term encodings, leading to a more flexible framework. In this paper, we introduce $E$-unification for second-order abstract syntax and describe a unification procedure for such problems, merging ideas from both full HOU and general $E$-unification. We prove that the procedure is sound and complete.
{"title":"E-unification for Second-Order Abstract Syntax","authors":"Nikolai Kudasov","doi":"10.48550/arXiv.2302.05815","DOIUrl":"https://doi.org/10.48550/arXiv.2302.05815","url":null,"abstract":"Higher-order unification (HOU) concerns unification of (extensions of) $lambda$-calculus and can be seen as an instance of equational unification ($E$-unification) modulo $betaeta$-equivalence of $lambda$-terms. We study equational unification of terms in languages with arbitrary variable binding constructions modulo arbitrary second-order equational theories. Abstract syntax with general variable binding and parametrised metavariables allows us to work with arbitrary binders without committing to $lambda$-calculus or use inconvenient and error-prone term encodings, leading to a more flexible framework. In this paper, we introduce $E$-unification for second-order abstract syntax and describe a unification procedure for such problems, merging ideas from both full HOU and general $E$-unification. We prove that the procedure is sound and complete.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127801453","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we are concerned with understanding the nature of program metrics for calculi with higher-order types, seen as natural generalizations of program equivalences. Some of the metrics we are interested in are well-known, such as those based on the interpretation of terms in metric spaces and those obtained by generalizing observational equivalence. We also introduce a new one, called the interactive metric, built by applying the well-known Int-Construction to the category of metric complete partial orders. Our aim is then to understand how these metrics relate to each other, i.e., whether and in which cases one such metric refines another, in analogy with corresponding well-studied problems about program equivalences. The results we obtain are twofold. We first show that the metrics of semantic origin, i.e., the denotational and interactive ones, lie emph{in between} the observational and equational metrics and that in some cases, these inclusions are strict. Then, we give a result about the relationship between the denotational and interactive metrics, revealing that the former is less discriminating than the latter. All our results are given for a linear lambda-calculus, and some of them can be generalized to calculi with graded comonads, in the style of Fuzz.
{"title":"On the Lattice of Program Metrics","authors":"Ugo Dal Lago, Naohiko Hoshino, Paolo Pistone","doi":"10.48550/arXiv.2302.05022","DOIUrl":"https://doi.org/10.48550/arXiv.2302.05022","url":null,"abstract":"In this paper we are concerned with understanding the nature of program metrics for calculi with higher-order types, seen as natural generalizations of program equivalences. Some of the metrics we are interested in are well-known, such as those based on the interpretation of terms in metric spaces and those obtained by generalizing observational equivalence. We also introduce a new one, called the interactive metric, built by applying the well-known Int-Construction to the category of metric complete partial orders. Our aim is then to understand how these metrics relate to each other, i.e., whether and in which cases one such metric refines another, in analogy with corresponding well-studied problems about program equivalences. The results we obtain are twofold. We first show that the metrics of semantic origin, i.e., the denotational and interactive ones, lie emph{in between} the observational and equational metrics and that in some cases, these inclusions are strict. Then, we give a result about the relationship between the denotational and interactive metrics, revealing that the former is less discriminating than the latter. All our results are given for a linear lambda-calculus, and some of them can be generalized to calculi with graded comonads, in the style of Fuzz.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"12 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120911172","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Metatheorems about type theories are often proven by interpreting the syntax into models constructed using categorical gluing. We propose to use only sconing (gluing along a global section functor) instead of general gluing. The sconing is performed internally to a presheaf category, and we recover the original glued model by externalization. Our method relies on constructions involving two notions of models: first-order models (with explicit contexts) and higher-order models (without explicit contexts). Sconing turns a displayed higher-order model into a displayed first-order model. Using these, we derive specialized induction principles for the syntax of type theory. The input of such an induction principle is a boilerplate-free description of its motives and methods, not mentioning contexts. The output is a section with computation rules specified in the same internal language. We illustrate our framework by proofs of canonicity, normalization and syntactic parametricity for type theory.
{"title":"For the Metatheory of Type Theory, Internal Sconing Is Enough","authors":"Rafael Bocquet, A. Kaposi, Christian Sattler","doi":"10.48550/arXiv.2302.05190","DOIUrl":"https://doi.org/10.48550/arXiv.2302.05190","url":null,"abstract":"Metatheorems about type theories are often proven by interpreting the syntax into models constructed using categorical gluing. We propose to use only sconing (gluing along a global section functor) instead of general gluing. The sconing is performed internally to a presheaf category, and we recover the original glued model by externalization. Our method relies on constructions involving two notions of models: first-order models (with explicit contexts) and higher-order models (without explicit contexts). Sconing turns a displayed higher-order model into a displayed first-order model. Using these, we derive specialized induction principles for the syntax of type theory. The input of such an induction principle is a boilerplate-free description of its motives and methods, not mentioning contexts. The output is a section with computation rules specified in the same internal language. We illustrate our framework by proofs of canonicity, normalization and syntactic parametricity for type theory.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132415946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
As shown by Tsukada and Ong, normal (extensional) simply-typed resource terms correspond to plays in Hyland-Ong games, quotiented by Melli`es' homotopy equivalence. Though inspiring, their proof is indirect, relying on the injectivity of the relational model w.r.t. both sides of the correspondence - in particular, the dynamics of the resource calculus is taken into account only via the compatibility of the relational model with the composition of normal terms defined by normalization. In the present paper, we revisit and extend these results. Our first contribution is to restate the correspondence by considering causal structures we call augmentations, which are canonical representatives of Hyland-Ong plays up to homotopy. This allows us to give a direct and explicit account of the connection with normal resource terms. As a second contribution, we extend this account to the reduction of resource terms: building on a notion of strategies as weighted sums of augmentations, we provide a denotational model of the resource calculus, invariant under reduction. A key step - and our third contribution - is a categorical model we call a resource category, which is to the resource calculus what differential categories are to the differential {lambda}-calculus.
{"title":"Strategies as Resource Terms, and their Categorical Semantics","authors":"Lison Blondeau-Patissier, P. Clairambault, Lionel Vaux Auclair","doi":"10.48550/arXiv.2302.04685","DOIUrl":"https://doi.org/10.48550/arXiv.2302.04685","url":null,"abstract":"As shown by Tsukada and Ong, normal (extensional) simply-typed resource terms correspond to plays in Hyland-Ong games, quotiented by Melli`es' homotopy equivalence. Though inspiring, their proof is indirect, relying on the injectivity of the relational model w.r.t. both sides of the correspondence - in particular, the dynamics of the resource calculus is taken into account only via the compatibility of the relational model with the composition of normal terms defined by normalization. In the present paper, we revisit and extend these results. Our first contribution is to restate the correspondence by considering causal structures we call augmentations, which are canonical representatives of Hyland-Ong plays up to homotopy. This allows us to give a direct and explicit account of the connection with normal resource terms. As a second contribution, we extend this account to the reduction of resource terms: building on a notion of strategies as weighted sums of augmentations, we provide a denotational model of the resource calculus, invariant under reduction. A key step - and our third contribution - is a categorical model we call a resource category, which is to the resource calculus what differential categories are to the differential {lambda}-calculus.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"88 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133249348","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop the formal theory of monads, as established by Street, in univalent foundations. This allows us to formally reason about various kinds of monads on the right level of abstraction. In particular, we define the bicategory of monads internal to a bicategory, and prove that it is univalent. We also define Eilenberg-Moore objects, and we show that both Eilenberg-Moore categories and Kleisli categories give rise to Eilenberg-Moore objects. Finally, we relate monads and adjunctions in arbitrary bicategories. Our work is formalized in Coq using the UniMath library.
{"title":"The Formal Theory of Monads, Univalently","authors":"N. V. D. Weide","doi":"10.48550/arXiv.2212.08515","DOIUrl":"https://doi.org/10.48550/arXiv.2212.08515","url":null,"abstract":"We develop the formal theory of monads, as established by Street, in univalent foundations. This allows us to formally reason about various kinds of monads on the right level of abstraction. In particular, we define the bicategory of monads internal to a bicategory, and prove that it is univalent. We also define Eilenberg-Moore objects, and we show that both Eilenberg-Moore categories and Kleisli categories give rise to Eilenberg-Moore objects. Finally, we relate monads and adjunctions in arbitrary bicategories. Our work is formalized in Coq using the UniMath library.","PeriodicalId":284975,"journal":{"name":"International Conference on Formal Structures for Computation and Deduction","volume":"12 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131754047","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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