Given a well-quasi-order $X$ and an ordinal $alpha$, the set $s^F_alpha(X)$ of transfinite strings on $X$ with length less than $alpha$ and with finite image is also a well-quasi-order, as proven by Nash-Williams. Before Nash-Williams proved it for general $alpha$, however, it was proven for $alpha
{"title":"Bounding finite-image strings of length $ω^k$","authors":"Harry Altman","doi":"arxiv-2409.03199","DOIUrl":"https://doi.org/arxiv-2409.03199","url":null,"abstract":"Given a well-quasi-order $X$ and an ordinal $alpha$, the set $s^F_alpha(X)$\u0000of transfinite strings on $X$ with length less than $alpha$ and with finite\u0000image is also a well-quasi-order, as proven by Nash-Williams. Before\u0000Nash-Williams proved it for general $alpha$, however, it was proven for\u0000$alpha<omega^omega$ by ErdH{o}s and Rado. In this paper, we revisit\u0000ErdH{o}s and Rado's proof and improve upon it, using it to obtain upper bounds\u0000on the maximum linearization of $s^F_{omega^k}(X)$ in terms of $k$ and $o(X)$,\u0000where $o(X)$ denotes the maximum linearization of $X$. We show that, for fixed\u0000$k$, $o(s^F_{omega^k}(X))$ is bounded above by a function which can roughly be\u0000described as $(k+1)$-times exponential in $o(X)$. We also show that, for $kle\u00002$, this bound is not far from tight.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142224555","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}
Nicanor Carrasco-Vargas, Valentino Delle Rose, Cristóbal Rojas
We revisit the problem of algorithmically deciding whether a given infinite connected graph has an Eulerian path, namely, a path that uses every edge exactly once. It has been recently observed that this problem is $D_3^0$-complete for graphs that have a computable description, whereas it is $Pi_2^0$-complete for graphs that have a highly computable description, and that this same bound holds for the class of automatic graphs. A closely related problem consists of determining the number of ends of a graph, namely, the maximum number of distinct infinite connected components the graph can be separated into after removing a finite set of edges. The complexity of this problem for highly computable graphs is known to be $Pi_2^0$-complete as well. The connection between these two problems lies in that only graphs with one or two ends can have Eulerian paths. In this paper we are interested in understanding the complexity of the infinite Eulerian path problem in the setting where the input graphs are known to have the right number of ends. We find that in this setting the problem becomes strictly easier, and that its exact difficulty varies according to whether the graphs have one or two ends, and to whether the Eulerian path we are looking for is one-way or bi-infinite. For example, we find that deciding existence of a bi-infinite Eulerian path for one-ended graphs is only $Pi_1^0$-complete if the graphs are highly computable, and that the same problem becomes decidable for automatic graphs. Our results are based on a detailed computability analysis of what we call the Separation Problem, which we believe to be of independent interest. For instance, as a side application, we observe that K"onig's infinity lemma, well known to be non-effective in general, becomes effective if we restrict to graphs with finitely many ends.
{"title":"On the complexity of the Eulerian path problem for infinite graphs","authors":"Nicanor Carrasco-Vargas, Valentino Delle Rose, Cristóbal Rojas","doi":"arxiv-2409.03113","DOIUrl":"https://doi.org/arxiv-2409.03113","url":null,"abstract":"We revisit the problem of algorithmically deciding whether a given infinite\u0000connected graph has an Eulerian path, namely, a path that uses every edge\u0000exactly once. It has been recently observed that this problem is\u0000$D_3^0$-complete for graphs that have a computable description, whereas it is\u0000$Pi_2^0$-complete for graphs that have a highly computable description, and\u0000that this same bound holds for the class of automatic graphs. A closely related\u0000problem consists of determining the number of ends of a graph, namely, the\u0000maximum number of distinct infinite connected components the graph can be\u0000separated into after removing a finite set of edges. The complexity of this\u0000problem for highly computable graphs is known to be $Pi_2^0$-complete as well.\u0000The connection between these two problems lies in that only graphs with one or\u0000two ends can have Eulerian paths. In this paper we are interested in\u0000understanding the complexity of the infinite Eulerian path problem in the\u0000setting where the input graphs are known to have the right number of ends. We\u0000find that in this setting the problem becomes strictly easier, and that its\u0000exact difficulty varies according to whether the graphs have one or two ends,\u0000and to whether the Eulerian path we are looking for is one-way or bi-infinite.\u0000For example, we find that deciding existence of a bi-infinite Eulerian path for\u0000one-ended graphs is only $Pi_1^0$-complete if the graphs are highly\u0000computable, and that the same problem becomes decidable for automatic graphs.\u0000Our results are based on a detailed computability analysis of what we call the\u0000Separation Problem, which we believe to be of independent interest. For\u0000instance, as a side application, we observe that K\"onig's infinity lemma, well\u0000known to be non-effective in general, becomes effective if we restrict to\u0000graphs with finitely many ends.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187739","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 that for a countable discrete group which is locally of finite asymptotic dimension, the generic continuous action on Cantor space has hyperfinite orbit equivalence relation. In particular, this holds for free groups, answering a question of Frisch-Kechris-Shinko-Vidny'anszky.
{"title":"Asymptotic dimension and hyperfiniteness of generic Cantor actions","authors":"Sumun Iyer, Forte Shinko","doi":"arxiv-2409.03078","DOIUrl":"https://doi.org/arxiv-2409.03078","url":null,"abstract":"We show that for a countable discrete group which is locally of finite\u0000asymptotic dimension, the generic continuous action on Cantor space has\u0000hyperfinite orbit equivalence relation. In particular, this holds for free\u0000groups, answering a question of Frisch-Kechris-Shinko-Vidny'anszky.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187740","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 that the orbit equivalence relation of a free action of a locally compact group is hyperfinite (`a la Connes-Feldman-Weiss) precisely when it is 'hypercompact'. This implies an uncountable version of the Ornstein-Weiss Theorem and that every locally compact group admitting a hypercompact probability preserving free action is amenable. We also establish an uncountable version of Danilenko's Random Ratio Ergodic Theorem. From this we deduce the 'Hopf dichotomy' for many nonsingular Bernoulli actions.
我们证明了局部紧密群的自由作用的轨道等价关系是超无限的（`a la Connes-Feldman-Weiss），正是当它是 "超紧密 "时。这意味着奥恩斯坦-魏斯定理的不可数版本，以及每个局部紧密群都接纳超紧密概率保存自由作用是可处理的。我们还建立了丹尼连科随机比率遍历定理的不可数版本。由此，我们推导出许多非星形伯努利作用的 "霍普夫二分法"。
{"title":"Uncountable Hyperfiniteness and The Random Ratio Ergodic Theorem","authors":"Nachi Avraham-Re'em, George Peterzil","doi":"arxiv-2409.02781","DOIUrl":"https://doi.org/arxiv-2409.02781","url":null,"abstract":"We show that the orbit equivalence relation of a free action of a locally\u0000compact group is hyperfinite (`a la Connes-Feldman-Weiss) precisely when it is\u0000'hypercompact'. This implies an uncountable version of the Ornstein-Weiss\u0000Theorem and that every locally compact group admitting a hypercompact\u0000probability preserving free action is amenable. We also establish an\u0000uncountable version of Danilenko's Random Ratio Ergodic Theorem. From this we\u0000deduce the 'Hopf dichotomy' for many nonsingular Bernoulli actions.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187743","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}
Stefania Damato, Thorsten Altenkirch, Axel Ljungström
Containers capture the concept of strictly positive data types in programming. The original development of containers is done in the internal language of Locally Cartesian Closed Categories (LCCCs) with disjoint coproducts and W-types. Although it is claimed that these developments can also be interpreted in extensional Martin-L"of type theory, this interpretation is not made explicit. Moreover, as a result of extensionality, these developments freely assume Uniqueness of Identity Proofs (UIP), so it is not clear whether this is a necessary condition. In this paper, we present a formalisation of the result that `containers preserve least and greatest fixed points' in Cubical Agda, thereby giving a formulation in intensional type theory, and showing that UIP is not necessary. Our main incentive for using Cubical Agda is that its path type restores the equivalence between bisimulation and coinductive equality. Thus, besides developing container theory in a more general setting, we also demonstrate the usefulness of Cubical Agda's path type to coinductive proofs.
{"title":"Formalising inductive and coinductive containers","authors":"Stefania Damato, Thorsten Altenkirch, Axel Ljungström","doi":"arxiv-2409.02603","DOIUrl":"https://doi.org/arxiv-2409.02603","url":null,"abstract":"Containers capture the concept of strictly positive data types in\u0000programming. The original development of containers is done in the internal\u0000language of Locally Cartesian Closed Categories (LCCCs) with disjoint\u0000coproducts and W-types. Although it is claimed that these developments can also\u0000be interpreted in extensional Martin-L\"of type theory, this interpretation is\u0000not made explicit. Moreover, as a result of extensionality, these developments\u0000freely assume Uniqueness of Identity Proofs (UIP), so it is not clear whether\u0000this is a necessary condition. In this paper, we present a formalisation of the\u0000result that `containers preserve least and greatest fixed points' in Cubical\u0000Agda, thereby giving a formulation in intensional type theory, and showing that\u0000UIP is not necessary. Our main incentive for using Cubical Agda is that its\u0000path type restores the equivalence between bisimulation and coinductive\u0000equality. Thus, besides developing container theory in a more general setting,\u0000we also demonstrate the usefulness of Cubical Agda's path type to coinductive\u0000proofs.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187744","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 that the category of countable Borel equivalence relations (CBERs) is dually equivalent to the category of countable $mathcal{L}_{omega_1omega}$ theories which admit a one-sorted interpretation of a particular theory we call $mathcal{T}_mathsf{LN} sqcup mathcal{T}_mathsf{sep}$ that witnesses embeddability into $2^mathbb{N}$ and the Lusin--Novikov uniformization theorem. This allows problems about Borel combinatorial structures on CBERs to be translated into syntactic definability problems in $mathcal{L}_{omega_1omega}$, modulo the extra structure provided by $mathcal{T}_mathsf{LN} sqcup mathcal{T}_mathsf{sep}$, thereby formalizing a folklore intuition in locally countable Borel combinatorics. We illustrate this with a catalogue of the precise interpretability relations between several standard classes of structures commonly used in Borel combinatorics, such as Feldman--Moore $omega$-colorings and the Slaman--Steel marker lemma. We also generalize this correspondence to locally countable Borel groupoids and theories interpreting $mathcal{T}_mathsf{LN}$, which admit a characterization analogous to that of Hjorth--Kechris for essentially countable isomorphism relations.
{"title":"Structurable equivalence relations and $mathcal{L}_{ω_1ω}$ interpretations","authors":"Rishi Banerjee, Ruiyuan Chen","doi":"arxiv-2409.02896","DOIUrl":"https://doi.org/arxiv-2409.02896","url":null,"abstract":"We show that the category of countable Borel equivalence relations (CBERs) is\u0000dually equivalent to the category of countable $mathcal{L}_{omega_1omega}$\u0000theories which admit a one-sorted interpretation of a particular theory we call\u0000$mathcal{T}_mathsf{LN} sqcup mathcal{T}_mathsf{sep}$ that witnesses\u0000embeddability into $2^mathbb{N}$ and the Lusin--Novikov uniformization\u0000theorem. This allows problems about Borel combinatorial structures on CBERs to\u0000be translated into syntactic definability problems in\u0000$mathcal{L}_{omega_1omega}$, modulo the extra structure provided by\u0000$mathcal{T}_mathsf{LN} sqcup mathcal{T}_mathsf{sep}$, thereby formalizing\u0000a folklore intuition in locally countable Borel combinatorics. We illustrate\u0000this with a catalogue of the precise interpretability relations between several\u0000standard classes of structures commonly used in Borel combinatorics, such as\u0000Feldman--Moore $omega$-colorings and the Slaman--Steel marker lemma. We also\u0000generalize this correspondence to locally countable Borel groupoids and\u0000theories interpreting $mathcal{T}_mathsf{LN}$, which admit a characterization\u0000analogous to that of Hjorth--Kechris for essentially countable isomorphism\u0000relations.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"2023 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187741","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}
Gilda Ferreira, Paulo Oliva, Clarence Lewis Protin
Several different proof translations exist between classical and intuitionistic logic (negative translations), and intuitionistic and linear logic (Girard translations). Our aims in this paper are (1) to show that all these systems can be expressed as extensions of a basic logical system (essentially intuitionistic linear logic), and that (2) with this common logical basis, a common approach to devising and simplifying such proof translations can be formalised. Via this process of ``simplification'' we get the most well-known translations in the literature.
{"title":"On the Various Translations between Classical, Intuitionistic and Linear Logic","authors":"Gilda Ferreira, Paulo Oliva, Clarence Lewis Protin","doi":"arxiv-2409.02249","DOIUrl":"https://doi.org/arxiv-2409.02249","url":null,"abstract":"Several different proof translations exist between classical and\u0000intuitionistic logic (negative translations), and intuitionistic and linear\u0000logic (Girard translations). Our aims in this paper are (1) to show that all\u0000these systems can be expressed as extensions of a basic logical system\u0000(essentially intuitionistic linear logic), and that (2) with this common\u0000logical basis, a common approach to devising and simplifying such proof\u0000translations can be formalised. Via this process of ``simplification'' we get\u0000the most well-known translations in the literature.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187742","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}
This article is interested in internality to the constants of systems of autonomous algebraic ordinary differential equations. Roughly, this means determining when can all solutions of such a system be written as a rational function of finitely many fixed solutions (and their derivatives) and finitely many constants. If the system is a single order one equation, the answer was given in an old article of Rosenlicht. In the present work, we completely answer this question for a large class of systems. As a corollary, we obtain a necessary condition for the generic solution to be Liouvillian. We then apply these results to determine exactly when solutions to Poizat equations (a special case of Li'enard equations) are internal, answering a question of Freitag, Jaoui, Marker and Nagloo, and to the classic Lotka-Volterra system, showing that its generic solutions are almost never Liouvillian.
{"title":"Internality of autonomous algebraic differential equations","authors":"Christine Eagles, Léo Jimenez","doi":"arxiv-2409.01863","DOIUrl":"https://doi.org/arxiv-2409.01863","url":null,"abstract":"This article is interested in internality to the constants of systems of\u0000autonomous algebraic ordinary differential equations. Roughly, this means\u0000determining when can all solutions of such a system be written as a rational\u0000function of finitely many fixed solutions (and their derivatives) and finitely\u0000many constants. If the system is a single order one equation, the answer was\u0000given in an old article of Rosenlicht. In the present work, we completely\u0000answer this question for a large class of systems. As a corollary, we obtain a\u0000necessary condition for the generic solution to be Liouvillian. We then apply\u0000these results to determine exactly when solutions to Poizat equations (a\u0000special case of Li'enard equations) are internal, answering a question of\u0000Freitag, Jaoui, Marker and Nagloo, and to the classic Lotka-Volterra system,\u0000showing that its generic solutions are almost never Liouvillian.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"160 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187750","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}
Carlos Martinez-Ranero, Dubraska Salcedo, Javier Utreras
Building on work of J. Robinson and A. Shlapentokh, we develop a general framework to obtain definability and decidability results of large classes of infinite algebraic extensions of $mathbb{F}_p(t)$. As an application, we show that for every odd rational prime $p$ there exist infinitely many primes $r$ such that the fields $mathbb{F}_{p^a}left(t^{r^{-infty}}right)$ have undecidable first-order theory in the language of rings without parameters. Our method uses character theory to construct families of non-isotrivial elliptic curves whose Mordell-Weil group is finitely generated and of positive rank in $mathbb{Z}_r$-towers.
在 J. Robinson 和 A. Shlapentokh 的工作基础上，我们建立了一个一般框架，以获得 $mathbb{F}_p(t)$ 的大类无限代数扩展的可定义性和可判定性结果。作为应用，我们证明了对于每个奇有理素数 $p$，存在无限多的素数 $r$，使得域 $mathbb{F}_{p^a}left(t^{r^{-infty}}right)$ 在无参数环语言中具有可判一阶理论。我们的方法利用特性理论来构造非等离椭圆曲线族，这些族的莫德尔-韦尔群在$mathbb{Z}_r$塔中是有限生成且正秩的。
{"title":"Undecidability of infinite algebraic extensions of $mathbb{F}_p(t)$","authors":"Carlos Martinez-Ranero, Dubraska Salcedo, Javier Utreras","doi":"arxiv-2409.01492","DOIUrl":"https://doi.org/arxiv-2409.01492","url":null,"abstract":"Building on work of J. Robinson and A. Shlapentokh, we develop a general\u0000framework to obtain definability and decidability results of large classes of\u0000infinite algebraic extensions of $mathbb{F}_p(t)$. As an application, we show\u0000that for every odd rational prime $p$ there exist infinitely many primes $r$\u0000such that the fields $mathbb{F}_{p^a}left(t^{r^{-infty}}right)$ have\u0000undecidable first-order theory in the language of rings without parameters. Our\u0000method uses character theory to construct families of non-isotrivial elliptic\u0000curves whose Mordell-Weil group is finitely generated and of positive rank in\u0000$mathbb{Z}_r$-towers.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187799","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 investigate the arithmetical completeness theorems of some extensions of Fitting, Marek, and Truszczy'{n}ski's pure logic of necessitation $mathbf{N}$. For $m,n in omega$, let $mathbf{N}^+ mathbf{A}_{m,n}$, which was introduced by Kurahashi and Sato, be the logic obtained from $mathbf{N}$ by adding the axiom scheme $Box^n A to Box^m A$ and the rule $dfrac{neg Box A}{neg Box Box A}$. In this paper, among other things, we prove that for each $m,n geq 1$, the logic $mathbf{N}^+ mathbf{A}_{m,n}$ becomes a provability logic.
{"title":"Arithmetical completeness for some extensions of the pure logic of necessitation","authors":"Haruka Kogure","doi":"arxiv-2409.00938","DOIUrl":"https://doi.org/arxiv-2409.00938","url":null,"abstract":"We investigate the arithmetical completeness theorems of some extensions of\u0000Fitting, Marek, and Truszczy'{n}ski's pure logic of necessitation\u0000$mathbf{N}$. For $m,n in omega$, let $mathbf{N}^+ mathbf{A}_{m,n}$, which\u0000was introduced by Kurahashi and Sato, be the logic obtained from $mathbf{N}$\u0000by adding the axiom scheme $Box^n A to Box^m A$ and the rule $dfrac{neg\u0000Box A}{neg Box Box A}$. In this paper, among other things, we prove that\u0000for each $m,n geq 1$, the logic $mathbf{N}^+ mathbf{A}_{m,n}$ becomes a\u0000provability logic.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187745","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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