Reachability problems in infinite-state systems are often subject to extremely high complexity. This motivates the investigation of efficient overapproximations, where we add transitions to obtain a system in which reachability can be decided more efficiently. We consider bidirected infinite-state systems, where for every transition there is a transition with opposite effect. We study bidirected reachability in the framework of valence systems, an abstract model featuring finitely many control states and an infinite-state storage that is specified by a finite graph. By picking suitable graphs, valence systems can uniformly model counters as in vector addition systems, pushdowns, integer counters, and combinations thereof. We provide a comprehensive complexity landscape for bidirected reachability and show that the complexity drops (often to polynomial time) from that of general reachability, for almost every storage mechanism where reachability is known to be decidable.
{"title":"The Complexity of Bidirected Reachability in Valence Systems","authors":"Moses Ganardi, R. Majumdar, Georg Zetzsche","doi":"10.1145/3531130.3533345","DOIUrl":"https://doi.org/10.1145/3531130.3533345","url":null,"abstract":"Reachability problems in infinite-state systems are often subject to extremely high complexity. This motivates the investigation of efficient overapproximations, where we add transitions to obtain a system in which reachability can be decided more efficiently. We consider bidirected infinite-state systems, where for every transition there is a transition with opposite effect. We study bidirected reachability in the framework of valence systems, an abstract model featuring finitely many control states and an infinite-state storage that is specified by a finite graph. By picking suitable graphs, valence systems can uniformly model counters as in vector addition systems, pushdowns, integer counters, and combinations thereof. We provide a comprehensive complexity landscape for bidirected reachability and show that the complexity drops (often to polynomial time) from that of general reachability, for almost every storage mechanism where reachability is known to be decidable.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125807855","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 present two active learning algorithms for sound deterministic negotiations. Sound deterministic negotiations are models of distributed systems, a kind of Petri nets or Zielonka automata with additional structure. We show that this additional structure allows to minimize such negotiations. The two active learning algorithms differ in the type of membership queries they use. Both have similar complexity to Angluin’s L* algorithm, in particular, the number of queries is polynomial in the size of the negotiation, and not in the number of configurations.
{"title":"Active learning for sound negotiations✱","authors":"A. Muscholl, I. Walukiewicz","doi":"10.1145/3531130.3533342","DOIUrl":"https://doi.org/10.1145/3531130.3533342","url":null,"abstract":"We present two active learning algorithms for sound deterministic negotiations. Sound deterministic negotiations are models of distributed systems, a kind of Petri nets or Zielonka automata with additional structure. We show that this additional structure allows to minimize such negotiations. The two active learning algorithms differ in the type of membership queries they use. Both have similar complexity to Angluin’s L* algorithm, in particular, the number of queries is polynomial in the size of the negotiation, and not in the number of configurations.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134631905","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}
Circular (or cyclic) proofs have received increasing attention in recent years, and have been proposed as an alternative setting for studying (co)inductive reasoning. In particular, now several type systems based on circular reasoning have been proposed. However, little is known about the complexity theoretic aspects of circular proofs, which exhibit sophisticated loop structures atypical of more common ‘recursion schemes’. This paper attempts to bridge the gap between circular proofs and implicit computational complexity (ICC). Namely we introduce a circular proof system based on Bellantoni and Cook’s famous safe-normal function algebra, and we identify proof theoretical constraints, inspired by ICC, to characterise the polynomial-time and elementary computable functions. Along the way we introduce new recursion theoretic implicit characterisations of these classes that may be of interest in their own right.
{"title":"Cyclic Implicit Complexity","authors":"Gianluca Curzi, Anupam Das","doi":"10.1145/3531130.3533340","DOIUrl":"https://doi.org/10.1145/3531130.3533340","url":null,"abstract":"Circular (or cyclic) proofs have received increasing attention in recent years, and have been proposed as an alternative setting for studying (co)inductive reasoning. In particular, now several type systems based on circular reasoning have been proposed. However, little is known about the complexity theoretic aspects of circular proofs, which exhibit sophisticated loop structures atypical of more common ‘recursion schemes’. This paper attempts to bridge the gap between circular proofs and implicit computational complexity (ICC). Namely we introduce a circular proof system based on Bellantoni and Cook’s famous safe-normal function algebra, and we identify proof theoretical constraints, inspired by ICC, to characterise the polynomial-time and elementary computable functions. Along the way we introduce new recursion theoretic implicit characterisations of these classes that may be of interest in their own right.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123298406","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 set of indices that correspond to the positive entries of a sequence of numbers is called its positivity set. In this paper, we study the density of the positivity set of a given linear recurrence sequence, that is the question of how much more frequent are the positive entries compared to the non-positive ones. We show that one can compute this density to arbitrary precision, as well as decide whether it is equal to zero (or one). If the sequence is diagonalisable, we prove that its positivity set is finite if and only if its density is zero. Lastly, arithmetic properties of densities are treated, in particular we prove that it is decidable whether the density is a rational number, given that the recurrence sequence has at most one pair of dominant complex roots.
{"title":"Computing the Density of the Positivity Set for Linear Recurrence Sequences","authors":"Edon Kelmendi","doi":"10.1145/3531130.3532399","DOIUrl":"https://doi.org/10.1145/3531130.3532399","url":null,"abstract":"The set of indices that correspond to the positive entries of a sequence of numbers is called its positivity set. In this paper, we study the density of the positivity set of a given linear recurrence sequence, that is the question of how much more frequent are the positive entries compared to the non-positive ones. We show that one can compute this density to arbitrary precision, as well as decide whether it is equal to zero (or one). If the sequence is diagonalisable, we prove that its positivity set is finite if and only if its density is zero. Lastly, arithmetic properties of densities are treated, in particular we prove that it is decidable whether the density is a rational number, given that the recurrence sequence has at most one pair of dominant complex roots.","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":"2021-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134398715","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 Eckmann-Hilton argument shows that any two monoid structures on the same set satisfying the interchange law are in fact the same operation, which is moreover commutative. When the monoids correspond to the vertical and horizontal composition of a sufficiently higher-dimensional category, the Eckmann-Hilton argument itself appears as a higher cell. This cell is often required to satisfy an additional piece of coherence, which is known as the syllepsis. We show that the syllepsis can be constructed from the elimination rule of intensional identity types in Martin-Löf type theory.
{"title":"Syllepsis in Homotopy Type Theory","authors":"Kristina Sojakova","doi":"10.1145/3531130.3533347","DOIUrl":"https://doi.org/10.1145/3531130.3533347","url":null,"abstract":"The Eckmann-Hilton argument shows that any two monoid structures on the same set satisfying the interchange law are in fact the same operation, which is moreover commutative. When the monoids correspond to the vertical and horizontal composition of a sufficiently higher-dimensional category, the Eckmann-Hilton argument itself appears as a higher cell. This cell is often required to satisfy an additional piece of coherence, which is known as the syllepsis. We show that the syllepsis can be constructed from the elimination rule of intensional identity types in Martin-Löf type theory.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114194877","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}
is an extension of first-order logic with a logarithmic recursion operator. It was introduced by Grohe et al. and shown to capture the complexity class L over trees and interval graphs. It does not capture L in general as it is contained in —fixed-point logic with counting. We show that this containment is strict. In particular, we show that the path systems problem, a classic P-complete problem which is definable in —fixed-point logic—is not definable in . This shows that the logarithmic recursion mechanism is provably weaker than general least fixed points.
{"title":"Separating LREC from LFP","authors":"A. Dawar, Felipe Ferreira Santos","doi":"10.1145/3531130.3533368","DOIUrl":"https://doi.org/10.1145/3531130.3533368","url":null,"abstract":"is an extension of first-order logic with a logarithmic recursion operator. It was introduced by Grohe et al. and shown to capture the complexity class L over trees and interval graphs. It does not capture L in general as it is contained in —fixed-point logic with counting. We show that this containment is strict. In particular, we show that the path systems problem, a classic P-complete problem which is definable in —fixed-point logic—is not definable in . This shows that the logarithmic recursion mechanism is provably weaker than general least fixed points.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129084846","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}
Jakub Gajarsk'y, Michal Pilipczuk, Szymon Toruńczyk
We prove that every class of graphs that is monadically stable and has bounded twin-width can be transduced from some class with bounded sparse twin-width. This generalizes analogous results for classes of bounded linear cliquewidth [Nešetřil et al. 2021b] and of bounded cliquewidth [Nešetřil et al. 2021a]. It also implies that monadically stable classes of bounded twin-width are linearly χ-bounded.
我们证明了每一类单根稳定且双宽有界的图都可以由一类稀疏双宽有界的图转化而来。这推广了有界线性cliquewidth类[Nešetřil et al. 2021b]和有界cliquewidth类[Nešetřil et al. 2021a]的类似结果。它还表明有界双宽度的单根稳定类是线性χ-有界的。
{"title":"Stable graphs of bounded twin-width","authors":"Jakub Gajarsk'y, Michal Pilipczuk, Szymon Toruńczyk","doi":"10.1145/3531130.3533356","DOIUrl":"https://doi.org/10.1145/3531130.3533356","url":null,"abstract":"We prove that every class of graphs that is monadically stable and has bounded twin-width can be transduced from some class with bounded sparse twin-width. This generalizes analogous results for classes of bounded linear cliquewidth [Nešetřil et al. 2021b] and of bounded cliquewidth [Nešetřil et al. 2021a]. It also implies that monadically stable classes of bounded twin-width are linearly χ-bounded.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125462886","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 study how the complexity of modular circuits computing AND depends on the depth of the circuits and the prime factorization of the modulus they use. In particular our construction of subexponential circuits of depth 2 for AND helps us to classify (modulo Exponential Time Hypothesis) modular circuits with respect to the complexity of their satisfiability. We also study a precise correlation between this complexity and the sizes of modular circuits realizing AND. In particular we use the superlinear lower bound from [10] to check satisfiability of CC0 circuits in probabilistic 2O(n/ε(n)) time, where ε is some extremely slowly increasing function. Moreover we show that AND can be computed by a polynomial size modular circuit of depth 2 (with O(log n) random bits) providing a probabilistic computational model that can not be derandomized. We apply our methods to determine (modulo ETH) the complexity of solving equations over groups of symmetries of regular polygons with an odd number of sides. These groups form a paradigm for some of the remaining cases in characterizing finite groups with respect to the complexity of their equation solving.
{"title":"Complexity of Modular Circuits","authors":"P. Idziak, Piotr Kawalek, Jacek Krzaczkowski","doi":"10.1145/3531130.3533350","DOIUrl":"https://doi.org/10.1145/3531130.3533350","url":null,"abstract":"We study how the complexity of modular circuits computing AND depends on the depth of the circuits and the prime factorization of the modulus they use. In particular our construction of subexponential circuits of depth 2 for AND helps us to classify (modulo Exponential Time Hypothesis) modular circuits with respect to the complexity of their satisfiability. We also study a precise correlation between this complexity and the sizes of modular circuits realizing AND. In particular we use the superlinear lower bound from [10] to check satisfiability of CC0 circuits in probabilistic 2O(n/ε(n)) time, where ε is some extremely slowly increasing function. Moreover we show that AND can be computed by a polynomial size modular circuit of depth 2 (with O(log n) random bits) providing a probabilistic computational model that can not be derandomized. We apply our methods to determine (modulo ETH) the complexity of solving equations over groups of symmetries of regular polygons with an odd number of sides. These groups form a paradigm for some of the remaining cases in characterizing finite groups with respect to the complexity of their equation solving.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114710713","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 prove normalization for MTT, a general multimodal dependent type theory capable of expressing modal type theories for guarded recursion, internalized parametricity, and various other prototypical modal situations. We prove that deciding type checking and conversion in MTT can be reduced to deciding the equality of modalities in the underlying modal situation, immediately yielding a type checking algorithm for all instantiations of MTT in the literature. This proof follows from a generalization of synthetic Tait computability—an abstract approach to gluing proofs—to account for modalities. This extension is based on MTT itself, so that this proof also constitutes a significant case study of MTT.
{"title":"Normalization for Multimodal Type Theory","authors":"Daniel Gratzer","doi":"10.1145/3531130.3532398","DOIUrl":"https://doi.org/10.1145/3531130.3532398","url":null,"abstract":"We prove normalization for MTT, a general multimodal dependent type theory capable of expressing modal type theories for guarded recursion, internalized parametricity, and various other prototypical modal situations. We prove that deciding type checking and conversion in MTT can be reduced to deciding the equality of modalities in the underlying modal situation, immediately yielding a type checking algorithm for all instantiations of MTT in the literature. This proof follows from a generalization of synthetic Tait computability—an abstract approach to gluing proofs—to account for modalities. This extension is based on MTT itself, so that this proof also constitutes a significant case study of MTT.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125929345","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}
Magnus Baunsgaard Kristensen, Rasmus Ejlers Møgelberg, Andrea Vezzosi
We present Clocked Cubical Type Theory, the first type theory combining multi-clocked guarded recursion with the features of Cubical Type Theory. Guarded recursion is an abstract form of step-indexing, which can be used for construction of advanced programming language models. In its multi-clocked version, it can also be used for coinductive programming and reasoning, encoding productivity in types. Combining this with Higher Inductive Types (HITs) the encoding extends to coinductive types that are traditionally hard to represent in type theory, such as the type of finitely branching labelled transition systems. Among our technical contributions is a new principle of induction under clocks, providing computational content to one of the main axioms required for encoding coinductive types. This principle is verified using a denotational semantics in a presheaf model.
{"title":"Greatest HITs: Higher inductive types in coinductive definitions via induction under clocks","authors":"Magnus Baunsgaard Kristensen, Rasmus Ejlers Møgelberg, Andrea Vezzosi","doi":"10.1145/3531130.3533359","DOIUrl":"https://doi.org/10.1145/3531130.3533359","url":null,"abstract":"We present Clocked Cubical Type Theory, the first type theory combining multi-clocked guarded recursion with the features of Cubical Type Theory. Guarded recursion is an abstract form of step-indexing, which can be used for construction of advanced programming language models. In its multi-clocked version, it can also be used for coinductive programming and reasoning, encoding productivity in types. Combining this with Higher Inductive Types (HITs) the encoding extends to coinductive types that are traditionally hard to represent in type theory, such as the type of finitely branching labelled transition systems. Among our technical contributions is a new principle of induction under clocks, providing computational content to one of the main axioms required for encoding coinductive types. This principle is verified using a denotational semantics in a presheaf model.","PeriodicalId":373589,"journal":{"name":"Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130366379","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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