We introduce parametric semilinear data logic (pSDL) for reasoning about data words with numeric data. The logic allows parameters, and Presburger guards on the data and on the Parikh image of equivalence classes (i.e. data counting), allowing us to capture data languages like: (1) each data value occurs at most once in the word and is an even number, (2) the subset of the positions containing data values divisible by 4 has the same number of a’s and b’s, (3) the data value with the highest frequency in the word is divisible by 3, and (4) each data value occurs at most once, and the set of data values forms an interval. We provide decidability and complexity results for the problem of membership and satisfiability checking over these models. In contrast to two-variable logic of data words and data automata (which also permit a form of data counting but no arithmetics over numeric domains and have incomparable inexpressivity), pSDL has elementary complexity of satisfiability checking. We show interesting potential applications of our models in databases and verification.
{"title":"Reasoning on Data Words over Numeric Domains","authors":"Diego Figueira, A. Lin","doi":"10.1145/3531130.3533354","DOIUrl":"https://doi.org/10.1145/3531130.3533354","url":null,"abstract":"We introduce parametric semilinear data logic (pSDL) for reasoning about data words with numeric data. The logic allows parameters, and Presburger guards on the data and on the Parikh image of equivalence classes (i.e. data counting), allowing us to capture data languages like: (1) each data value occurs at most once in the word and is an even number, (2) the subset of the positions containing data values divisible by 4 has the same number of a’s and b’s, (3) the data value with the highest frequency in the word is divisible by 3, and (4) each data value occurs at most once, and the set of data values forms an interval. We provide decidability and complexity results for the problem of membership and satisfiability checking over these models. In contrast to two-variable logic of data words and data automata (which also permit a form of data counting but no arithmetics over numeric domains and have incomparable inexpressivity), pSDL has elementary complexity of satisfiability checking. We show interesting potential applications of our models in databases and verification.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131532959","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 language of Algebraic Geometry combines two complementary and dependent levels of discourse: on the geometric side, schemes define spaces of the same cohesive nature as manifolds ; on the vectorial side, every scheme X comes equipped with a symmetric monoidal category of quasicoherent modules, which may be seen as generalised vector bundles on the scheme X. In this paper, we use the functor of points approach to Algebraic Geometry developed by Grothendieck in the 1970s to establish that every covariant presheaf X on the category of commutative rings — and in particular every scheme X — comes equipped “above it” with a symmetric monoidal closed category PshModX of presheaves of modules. This category PshModX defines moreover a model of intuitionistic linear logic, whose exponential modality is obtained by glueing together in an appropriate way the Sweedler dual construction on ring algebras.
代数几何的语言结合了两个互补和依赖的话语层次:在几何方面，方案定义了具有相同内聚性质的流形空间;在向量侧，每个方案X都配备了一个拟相干模的对称一元范畴，这些准相干模可以看作是方案X上的广义向量束。在本文中，我们利用1970年代由Grothendieck发展的代数几何的点函子方法，建立了交换环范畴上的每个协变preshef X -特别是每个方案X -“上面”配备了一个对称的单轴封闭类别PshModX的预捆模块。该范畴PshModX还定义了一个直观线性逻辑模型，该模型的指数模态是通过将环代数上的Sweedler对偶构造适当地粘合在一起而得到的。
{"title":"A Functorial Excursion Between Algebraic Geometry and Linear Logic","authors":"Paul-André Melliès","doi":"10.1145/3531130.3532488","DOIUrl":"https://doi.org/10.1145/3531130.3532488","url":null,"abstract":"The language of Algebraic Geometry combines two complementary and dependent levels of discourse: on the geometric side, schemes define spaces of the same cohesive nature as manifolds ; on the vectorial side, every scheme X comes equipped with a symmetric monoidal category of quasicoherent modules, which may be seen as generalised vector bundles on the scheme X. In this paper, we use the functor of points approach to Algebraic Geometry developed by Grothendieck in the 1970s to establish that every covariant presheaf X on the category of commutative rings — and in particular every scheme X — comes equipped “above it” with a symmetric monoidal closed category PshModX of presheaves of modules. This category PshModX defines moreover a model of intuitionistic linear logic, whose exponential modality is obtained by glueing together in an appropriate way the Sweedler dual construction on ring algebras.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124041717","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}
R. Lipton, F. Luca, Joris Nieuwveld, J. Ouaknine, David Purser, J. Worrell
It is a longstanding open problem whether there is an algorithm to decide the Skolem Problem for linear recurrence sequences (LRS) over the integers, namely whether a given such sequence has a zero term (i.e., whether un = 0 for some n). A major breakthrough in the early 1980s established decidability for LRS of order 4 or less, i.e., for LRS in which every new term depends linearly on the previous four (or fewer) terms. The Skolem Problem for LRS of order 5 or more, in particular, remains a major open challenge to this day. Our main contributions in this paper are as follows: First, we show that the Skolem Problem is decidable for reversible LRS of order 7 or less. (An integer LRS is reversible if its unique extension to a bi-infinite LRS also takes exclusively integer values; a typical example is the classical Fibonacci sequence, whose bi-infinite extension is ⟨…, 5, −3, 2, −1, 1, 0, 1, 1, 2, 3, 5, …⟩.) Second, assuming the Skolem Conjecture (a central hypothesis in Diophantine analysis, also known as the Exponential Local-Global Principle), we show that the Skolem Problem for LRS of order 5 is decidable, and exhibit a concrete procedure for solving it.
{"title":"On the Skolem Problem and the Skolem Conjecture","authors":"R. Lipton, F. Luca, Joris Nieuwveld, J. Ouaknine, David Purser, J. Worrell","doi":"10.1145/3531130.3533328","DOIUrl":"https://doi.org/10.1145/3531130.3533328","url":null,"abstract":"It is a longstanding open problem whether there is an algorithm to decide the Skolem Problem for linear recurrence sequences (LRS) over the integers, namely whether a given such sequence has a zero term (i.e., whether un = 0 for some n). A major breakthrough in the early 1980s established decidability for LRS of order 4 or less, i.e., for LRS in which every new term depends linearly on the previous four (or fewer) terms. The Skolem Problem for LRS of order 5 or more, in particular, remains a major open challenge to this day. Our main contributions in this paper are as follows: First, we show that the Skolem Problem is decidable for reversible LRS of order 7 or less. (An integer LRS is reversible if its unique extension to a bi-infinite LRS also takes exclusively integer values; a typical example is the classical Fibonacci sequence, whose bi-infinite extension is ⟨…, 5, −3, 2, −1, 1, 0, 1, 1, 2, 3, 5, …⟩.) Second, assuming the Skolem Conjecture (a central hypothesis in Diophantine analysis, also known as the Exponential Local-Global Principle), we show that the Skolem Problem for LRS of order 5 is decidable, and exhibit a concrete procedure for solving it.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124945018","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 quantitative reasoning one compares objects by distances, instead of equivalence relations, so that one can measure how much they are similar, rather than just saying whether they are equivalent or not. In this paper we aim at providing a logical ground to quantitative reasoning with distances in Linear Logic, using the categorical language of Lawvere’s doctrines. The key idea is to see distances as equality predicates in Linear Logic. We use graded modalities to write a resource sensitive substitution rule for equality, which allows us to give it a quantitative meaning by distances. We introduce a deductive calculus for (Graded) Linear Logic with quantitative equality and the notion of Lipschitz doctrine to give it a sound and complete categorical semantics. We also describe a universal construction of Lipschitz doctrines, which generates examples based for instance on metric spaces and quantitative realisability.
{"title":"Logical Foundations of Quantitative Equality","authors":"Francesco Dagnino, Fabio Pasquali","doi":"10.1145/3531130.3533337","DOIUrl":"https://doi.org/10.1145/3531130.3533337","url":null,"abstract":"In quantitative reasoning one compares objects by distances, instead of equivalence relations, so that one can measure how much they are similar, rather than just saying whether they are equivalent or not. In this paper we aim at providing a logical ground to quantitative reasoning with distances in Linear Logic, using the categorical language of Lawvere’s doctrines. The key idea is to see distances as equality predicates in Linear Logic. We use graded modalities to write a resource sensitive substitution rule for equality, which allows us to give it a quantitative meaning by distances. We introduce a deductive calculus for (Graded) Linear Logic with quantitative equality and the notion of Lipschitz doctrine to give it a sound and complete categorical semantics. We also describe a universal construction of Lipschitz doctrines, which generates examples based for instance on metric spaces and quantitative realisability.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129022791","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 polyregular functions are a class of string-to-string functions that have polynomial size outputs, and which can be defined using finite state models. There are many equivalent definitions of this class, with roots in automata theory, programming languages and logic. This paper surveys recent results on polyregular functions. It presents five of the equivalent definitions, and gives self-contained proofs for most of the equivalences. Decision problems as well as restricted subclasses of the polyregular functions are also discussed.
{"title":"Transducers of polynomial growth","authors":"M. Bojanczyk","doi":"10.1145/3531130.3533326","DOIUrl":"https://doi.org/10.1145/3531130.3533326","url":null,"abstract":"The polyregular functions are a class of string-to-string functions that have polynomial size outputs, and which can be defined using finite state models. There are many equivalent definitions of this class, with roots in automata theory, programming languages and logic. This paper surveys recent results on polyregular functions. It presents five of the equivalent definitions, and gives self-contained proofs for most of the equivalences. Decision problems as well as restricted subclasses of the polyregular functions are also discussed.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"115 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114831699","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}
Given a Boolean formula φ and a distribution parameter ε, the problem of almost-uniform generation seeks to design a randomized generator such that every solution of φ is output with probability within (1 + ε)-factor of where sol(φ) is the set of all the solutions of φ. The prior state of the art scheme due to Jerrum, Valiant, and Vazirani, makes calls to a SAT oracle and employs 2 − wise independent hash functions. In this work, we design a new randomized algorithm that makes calls to a SAT oracle and employs 3 − wise independent hash functions. The widely used 2 − wise independent hashing is tabulation hashing proposed by Carter and Wegman. Since this classical scheme is also 3 − wise independent, we observe that practical implementation of our technique does not incur additional overhead. We demonstrate that theoretical improvements translate to practice; in particular, we conduct a comprehensive study over 562 benchmarks and demonstrate that while JVV would time out for 544 out of 562 instances, our proposed scheme can handle all the 562 instances. To the best of our knowledge, this is the first almost-uniform generation scheme that can handle practical instances from real-world applications. We also present a nuanced analysis focusing on the both the size of SAT queries as well as the number of queries.
{"title":"On Almost-Uniform Generation of SAT Solutions: The power of 3-wise independent hashing","authors":"Remi Delannoy, Kuldeep S. Meel","doi":"10.1145/3531130.3533338","DOIUrl":"https://doi.org/10.1145/3531130.3533338","url":null,"abstract":"Given a Boolean formula φ and a distribution parameter ε, the problem of almost-uniform generation seeks to design a randomized generator such that every solution of φ is output with probability within (1 + ε)-factor of where sol(φ) is the set of all the solutions of φ. The prior state of the art scheme due to Jerrum, Valiant, and Vazirani, makes calls to a SAT oracle and employs 2 − wise independent hash functions. In this work, we design a new randomized algorithm that makes calls to a SAT oracle and employs 3 − wise independent hash functions. The widely used 2 − wise independent hashing is tabulation hashing proposed by Carter and Wegman. Since this classical scheme is also 3 − wise independent, we observe that practical implementation of our technique does not incur additional overhead. We demonstrate that theoretical improvements translate to practice; in particular, we conduct a comprehensive study over 562 benchmarks and demonstrate that while JVV would time out for 544 out of 562 instances, our proposed scheme can handle all the 562 instances. To the best of our knowledge, this is the first almost-uniform generation scheme that can handle practical instances from real-world applications. We also present a nuanced analysis focusing on the both the size of SAT queries as well as the number of queries.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116849341","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 λμ-calculus plays a central role in the theory of programming languages as it extends the Curry-Howard correspondence to classical logic. A major drawback is that it does not satisfy Böhm’s Theorem and it lacks the corresponding notion of approximation. On the contrary, we show that Ehrhard and Regnier’s Taylor expansion can be easily adapted, thus providing a resource conscious approximation theory. This produces a sensible λμ-theory with which we prove some advanced properties of the λμ-calculus, such as Stability and Perpendicular Lines Property, from which the impossibility of parallel computations follows.
{"title":"Resource approximation for the λμ-calculus","authors":"Davide Barbarossa","doi":"10.1145/3531130.3532469","DOIUrl":"https://doi.org/10.1145/3531130.3532469","url":null,"abstract":"The λμ-calculus plays a central role in the theory of programming languages as it extends the Curry-Howard correspondence to classical logic. A major drawback is that it does not satisfy Böhm’s Theorem and it lacks the corresponding notion of approximation. On the contrary, we show that Ehrhard and Regnier’s Taylor expansion can be easily adapted, thus providing a resource conscious approximation theory. This produces a sensible λμ-theory with which we prove some advanced properties of the λμ-calculus, such as Stability and Perpendicular Lines Property, from which the impossibility of parallel computations follows.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127242705","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}
Second-order arithmetic has two kinds of computational interpretations: via Spector’s bar recursion of via Girard’s polymorphic lambda-calculus. Bar recursion interprets the negative translation of the axiom of choice which, combined with an interpretation of the negative translation of the excluded middle, gives a computational interpretation of the negative translation of the axiom scheme of comprehension. It is then possible to instantiate universally quantified sets with arbitrary formulas (second-order elimination). On the other hand, polymorphic lambda-calculus interprets directly second-order elimination by means of polymorphic types. The present work aims at bridging the gap between these two interpretations by interpreting directly second-order elimination through update recursion, which is a variant of bar recursion.
{"title":"A direct computational interpretation of second-order arithmetic via update recursion","authors":"Valentin Blot","doi":"10.1145/3531130.3532458","DOIUrl":"https://doi.org/10.1145/3531130.3532458","url":null,"abstract":"Second-order arithmetic has two kinds of computational interpretations: via Spector’s bar recursion of via Girard’s polymorphic lambda-calculus. Bar recursion interprets the negative translation of the axiom of choice which, combined with an interpretation of the negative translation of the excluded middle, gives a computational interpretation of the negative translation of the axiom scheme of comprehension. It is then possible to instantiate universally quantified sets with arbitrary formulas (second-order elimination). On the other hand, polymorphic lambda-calculus interprets directly second-order elimination by means of polymorphic types. The present work aims at bridging the gap between these two interpretations by interpreting directly second-order elimination through update recursion, which is a variant of bar recursion.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"115 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132509202","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 paper resolves two open problems on linear integer arithmetic (LIA), also known as Presburger arithmetic. First, we give a triply exponential geometric decision procedure for LIA, i.e., a procedure based on manipulating semilinear sets. This matches the running time of the best quantifier elimination and automata-based procedures. Second, building upon our first result, we give a doubly exponential upper bound on the Vapnik–Chervonenkis (VC) dimension of sets definable in LIA, proving a conjecture of D. Nguyen and I. Pak [Combinatorica 39, pp. 923–932, 2019]. These results partially rely on an analysis of sets definable in linear real arithmetic (LRA), and analogous results for LRA are also obtained. At the core of these developments are new decomposition results for semilinear and -semilinear sets, the latter being the sets definable in LRA. These results yield new algorithms to compute the complement of (-)semilinear sets that do not cause a non-elementary blowup when repeatedly combined with procedures for other Boolean operations and projection. The existence of such an algorithm for semilinear sets has been a long-standing open problem.
本文解决了线性整数算法(也称为Presburger算法)中的两个开放问题。首先，我们给出了LIA的一个三指数几何决策过程，即一个基于半线性集的操作过程。这与最佳量词消除和基于自动机的程序的运行时间相匹配。其次，在我们的第一个结果的基础上，我们给出了在LIA中可定义集合的Vapnik-Chervonenkis (VC)维的双指数上界，证明了D. Nguyen和I. Pak [Combinatorica 39, pp. 923-932, 2019]的一个猜想。这些结果部分依赖于对线性实算法中可定义集合的分析，并得到了线性实算法的类似结果。这些发展的核心是半线性和-半线性集合的新分解结果，后者是在LRA中可定义的集合。这些结果产生了新的算法来计算(-)半线性集的补，这些补在与其他布尔运算和投影的过程重复组合时不会导致非初等爆炸。这种半线性集算法的存在性是一个长期存在的开放性问题。
{"title":"Geometric decision procedures and the VC dimension of linear arithmetic theories","authors":"D. Chistikov, C. Haase, Alessio Mansutti","doi":"10.1145/3531130.3533372","DOIUrl":"https://doi.org/10.1145/3531130.3533372","url":null,"abstract":"This paper resolves two open problems on linear integer arithmetic (LIA), also known as Presburger arithmetic. First, we give a triply exponential geometric decision procedure for LIA, i.e., a procedure based on manipulating semilinear sets. This matches the running time of the best quantifier elimination and automata-based procedures. Second, building upon our first result, we give a doubly exponential upper bound on the Vapnik–Chervonenkis (VC) dimension of sets definable in LIA, proving a conjecture of D. Nguyen and I. Pak [Combinatorica 39, pp. 923–932, 2019]. These results partially rely on an analysis of sets definable in linear real arithmetic (LRA), and analogous results for LRA are also obtained. At the core of these developments are new decomposition results for semilinear and -semilinear sets, the latter being the sets definable in LRA. These results yield new algorithms to compute the complement of (-)semilinear sets that do not cause a non-elementary blowup when repeatedly combined with procedures for other Boolean operations and projection. The existence of such an algorithm for semilinear sets has been a long-standing open problem.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"244 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128999465","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}
Milner (1984) defined a process semantics for regular expressions. He formulated a sound proof system for bisimilarity of process interpretations of regular expressions, and asked whether this system is complete. We report conceptually on a proof that shows that Milner’s system is complete, by motivating and describing all of its main steps. We substantially refine the completeness proof by Grabmayer and Fokkink (2020) for the restriction of Milner’s system to ‘1-free’ regular expressions. As a crucial complication we recognize that process graphs with empty-step transitions that satisfy the layered loop-existence/elimination property LLEE are not closed under bisimulation collapse (unlike process graphs with LLEE that only have proper-step transitions). We circumnavigate this obstacle by defining a LLEE-preserving ‘crystallization procedure’ for such process graphs. By that we obtain ‘near-collapsed’ process graphs with LLEE whose strongly connected components are either collapsed or of ‘twin-crystal’ shape. Such near-collapsed process graphs guarantee provable solutions for bisimulation collapses of process interpretations of regular expressions.
{"title":"Milner’s Proof System for Regular Expressions Modulo Bisimilarity is Complete: Crystallization: Near-Collapsing Process Graph Interpretations of Regular Expressions","authors":"C. Grabmayer","doi":"10.1145/3531130.3532430","DOIUrl":"https://doi.org/10.1145/3531130.3532430","url":null,"abstract":"Milner (1984) defined a process semantics for regular expressions. He formulated a sound proof system for bisimilarity of process interpretations of regular expressions, and asked whether this system is complete. We report conceptually on a proof that shows that Milner’s system is complete, by motivating and describing all of its main steps. We substantially refine the completeness proof by Grabmayer and Fokkink (2020) for the restriction of Milner’s system to ‘1-free’ regular expressions. As a crucial complication we recognize that process graphs with empty-step transitions that satisfy the layered loop-existence/elimination property LLEE are not closed under bisimulation collapse (unlike process graphs with LLEE that only have proper-step transitions). We circumnavigate this obstacle by defining a LLEE-preserving ‘crystallization procedure’ for such process graphs. By that we obtain ‘near-collapsed’ process graphs with LLEE whose strongly connected components are either collapsed or of ‘twin-crystal’ shape. Such near-collapsed process graphs guarantee provable solutions for bisimulation collapses of process interpretations of regular expressions.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132358130","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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