Christian Ikenmeyer, Balagopal Komarath, C. Lenzen, Vladimir Lysikov, A. Mokhov, Karteek Sreenivasaiah
The problem of constructing hazard-free Boolean circuits dates back to the 1940s and is an important problem in circuit design. Our main lower-bound result unconditionally shows the existence of functions whose circuit complexity is polynomially bounded while every hazard-free implementation is provably of exponential size. Previous lower bounds on the hazard-free complexity were only valid for depth 2 circuits. The same proof method yields that every subcubic implementation of Boolean matrix multiplication must have hazards. These results follow from a crucial structural insight: Hazard-free complexity is a natural generalization of monotone complexity to all (not necessarily monotone) Boolean functions. Thus, we can apply known monotone complexity lower bounds to find lower bounds on the hazard-free complexity. We also lift these methods from the monotone setting to prove exponential hazard-free complexity lower bounds for non-monotone functions. As our main upper-bound result, we show how to efficiently convert a Boolean circuit into a bounded-bit hazard-free circuit with only a polynomially large blow-up in the number of gates. Previously, the best known method yielded exponentially large circuits in the worst case, so our algorithm gives an exponential improvement. As a side result, we establish the NP-completeness of several hazard detection problems.
{"title":"On the Complexity of Hazard-free Circuits","authors":"Christian Ikenmeyer, Balagopal Komarath, C. Lenzen, Vladimir Lysikov, A. Mokhov, Karteek Sreenivasaiah","doi":"10.1145/3320123","DOIUrl":"https://doi.org/10.1145/3320123","url":null,"abstract":"The problem of constructing hazard-free Boolean circuits dates back to the 1940s and is an important problem in circuit design. Our main lower-bound result unconditionally shows the existence of functions whose circuit complexity is polynomially bounded while every hazard-free implementation is provably of exponential size. Previous lower bounds on the hazard-free complexity were only valid for depth 2 circuits. The same proof method yields that every subcubic implementation of Boolean matrix multiplication must have hazards. These results follow from a crucial structural insight: Hazard-free complexity is a natural generalization of monotone complexity to all (not necessarily monotone) Boolean functions. Thus, we can apply known monotone complexity lower bounds to find lower bounds on the hazard-free complexity. We also lift these methods from the monotone setting to prove exponential hazard-free complexity lower bounds for non-monotone functions. As our main upper-bound result, we show how to efficiently convert a Boolean circuit into a bounded-bit hazard-free circuit with only a polynomially large blow-up in the number of gates. Previously, the best known method yielded exponentially large circuits in the worst case, so our algorithm gives an exponential improvement. As a side result, we establish the NP-completeness of several hazard detection problems.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79230538","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}
S. KarthikC., Bundit Laekhanukit, Pasin Manurangsi
We study the parameterized complexity of approximating the k-Dominating Set (DomSet) problem where an integer k and a graph G on n vertices are given as input, and the goal is to find a dominating set of size at most F(k) ⋅ k whenever the graph G has a dominating set of size k. When such an algorithm runs in time T(k) ⋅ poly (n) (i.e., FPT-time) for some computable function T, it is said to be an F(k)-FPT-approximation algorithm for k-DomSet. Whether such an algorithm exists is listed in the seminal book of Downey and Fellows (2013) as one of the “most infamous” open problems in parameterized complexity. This work gives an almost complete answer to this question by showing the non-existence of such an algorithm under W[1] ≠ FPT and further providing tighter running time lower bounds under stronger hypotheses. Specifically, we prove the following for every computable functions T, F and every constant ε > 0: • Assuming W[1] ≠ FPT, there is no F(k)-FPT-approximation algorithm for k-DomSet. • Assuming the Exponential Time Hypothesis (ETH), there is no F(k)-approximation algorithm for k-DomSet that runs in T(k) ⋅ no(k) time. • Assuming the Strong Exponential Time Hypothesis (SETH), for every integer k ≥ 2, there is no F(k)-approximation algorithm for k-DomSet that runs in T(k) ⋅ nk − ε time. • Assuming the k-SUM Hypothesis, for every integer k ≥ 3, there is no F(k)-approximation algorithm for k-DomSet that runs in T(k) ⋅ n⌈ k/2 ⌉ − ε time. Previously, only constant ratio FPT-approximation algorithms were ruled out under sf W[1] ≠ FPT and (log1/4 &minus ε k)-FPT-approximation algorithms were ruled out under ETH [Chen and Lin, FOCS 2016]. Recently, the non-existence of an F(k)-FPT-approximation algorithm for any function F was shown under Gap-ETH [Chalermsook et al., FOCS 2017]. Note that, to the best of our knowledge, no running time lower bound of the form n&delta k for any absolute constant δ > 0 was known before even for any constant factor inapproximation ratio. Our results are obtained by establishing a connection between communication complexity and hardness of approximation, generalizing the ideas from a recent breakthrough work of Abboud et al. [FOCS 2017]. Specifically, we show that to prove hardness of approximation of a certain parameterized variant of the label cover problem, it suffices to devise a specific protocol for a communication problem that depends on which hypothesis we rely on. Each of these communication problems turns out to be either a well-studied problem or a variant of one; this allows us to easily apply known techniques to solve them.
{"title":"On the Parameterized Complexity of Approximating Dominating Set","authors":"S. KarthikC., Bundit Laekhanukit, Pasin Manurangsi","doi":"10.1145/3325116","DOIUrl":"https://doi.org/10.1145/3325116","url":null,"abstract":"We study the parameterized complexity of approximating the k-Dominating Set (DomSet) problem where an integer k and a graph G on n vertices are given as input, and the goal is to find a dominating set of size at most F(k) ⋅ k whenever the graph G has a dominating set of size k. When such an algorithm runs in time T(k) ⋅ poly (n) (i.e., FPT-time) for some computable function T, it is said to be an F(k)-FPT-approximation algorithm for k-DomSet. Whether such an algorithm exists is listed in the seminal book of Downey and Fellows (2013) as one of the “most infamous” open problems in parameterized complexity. This work gives an almost complete answer to this question by showing the non-existence of such an algorithm under W[1] ≠ FPT and further providing tighter running time lower bounds under stronger hypotheses. Specifically, we prove the following for every computable functions T, F and every constant ε > 0: • Assuming W[1] ≠ FPT, there is no F(k)-FPT-approximation algorithm for k-DomSet. • Assuming the Exponential Time Hypothesis (ETH), there is no F(k)-approximation algorithm for k-DomSet that runs in T(k) ⋅ no(k) time. • Assuming the Strong Exponential Time Hypothesis (SETH), for every integer k ≥ 2, there is no F(k)-approximation algorithm for k-DomSet that runs in T(k) ⋅ nk − ε time. • Assuming the k-SUM Hypothesis, for every integer k ≥ 3, there is no F(k)-approximation algorithm for k-DomSet that runs in T(k) ⋅ n⌈ k/2 ⌉ − ε time. Previously, only constant ratio FPT-approximation algorithms were ruled out under sf W[1] ≠ FPT and (log1/4 &minus ε k)-FPT-approximation algorithms were ruled out under ETH [Chen and Lin, FOCS 2016]. Recently, the non-existence of an F(k)-FPT-approximation algorithm for any function F was shown under Gap-ETH [Chalermsook et al., FOCS 2017]. Note that, to the best of our knowledge, no running time lower bound of the form n&delta k for any absolute constant δ > 0 was known before even for any constant factor inapproximation ratio. Our results are obtained by establishing a connection between communication complexity and hardness of approximation, generalizing the ideas from a recent breakthrough work of Abboud et al. [FOCS 2017]. Specifically, we show that to prove hardness of approximation of a certain parameterized variant of the label cover problem, it suffices to devise a specific protocol for a communication problem that depends on which hypothesis we rely on. Each of these communication problems turns out to be either a well-studied problem or a variant of one; this allows us to easily apply known techniques to solve them.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79591434","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 eliminate a key roadblock to efficient verification of nonlinear integer arithmetic using CDCL SAT solvers, by showing how to construct short resolution proofs for many properties of the most widely used multiplier circuits. Such short proofs were conjectured not to exist. More precisely, we give nO(1) size regular resolution proofs for arbitrary degree 2 identities on array, diagonal, and Booth multipliers and nO(log n) size proofs for these identities on Wallace tree multipliers.
{"title":"Toward Verifying Nonlinear Integer Arithmetic","authors":"P. Beame, Vincent Liew","doi":"10.1145/3319396","DOIUrl":"https://doi.org/10.1145/3319396","url":null,"abstract":"We eliminate a key roadblock to efficient verification of nonlinear integer arithmetic using CDCL SAT solvers, by showing how to construct short resolution proofs for many properties of the most widely used multiplier circuits. Such short proofs were conjectured not to exist. More precisely, we give nO(1) size regular resolution proofs for arbitrary degree 2 identities on array, diagonal, and Booth multipliers and nO(log n) size proofs for these identities on Wallace tree multipliers.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76599490","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}
F. Wirth, S. Stüdli, Jia Yuan Yu, M. Corless, R. Shorten
A stochastic algorithm is presented for a class of optimisation problems that arise when a group of agents compete to share a single constrained resource in an optimal manner. The approach uses intermittent single-bit feedback, which indicates a constraint violation and does not require inter-agent communication. The algorithm is based on a positive matrix model of AIMD, which is extended to the nonhomogeneous Markovian case. The key feature is the assignment of back-off probabilities to the individual agents as a function of the past average access to the resource. This leads to a nonhomogeneous Markov chain in an extended state space, and we show almost sure convergence of the average access to the social optimum.
{"title":"Nonhomogeneous Place-dependent Markov Chains, Unsynchronised AIMD, and Optimisation","authors":"F. Wirth, S. Stüdli, Jia Yuan Yu, M. Corless, R. Shorten","doi":"10.1145/3312741","DOIUrl":"https://doi.org/10.1145/3312741","url":null,"abstract":"A stochastic algorithm is presented for a class of optimisation problems that arise when a group of agents compete to share a single constrained resource in an optimal manner. The approach uses intermittent single-bit feedback, which indicates a constraint violation and does not require inter-agent communication. The algorithm is based on a positive matrix model of AIMD, which is extended to the nonhomogeneous Markovian case. The key feature is the assignment of back-off probabilities to the individual agents as a function of the past average access to the resource. This leads to a nonhomogeneous Markov chain in an extended state space, and we show almost sure convergence of the average access to the social optimum.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89418016","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 give a new, strongly polynomial-time algorithm and improved analysis for the metric s-t path Traveling Salesman Problem (TSP). It finds a tour of cost less than 1.53 times the optimum of the subtour elimination linear program (LP), while known examples show that 1.5 is a lower bound for the integrality gap. A key new idea is the deletion of some edges of the spanning trees used in the best-of-many Christofides-Serdyukov-algorithm, which is then accompanied by novel arguments of the analysis: edge-deletion disconnects the trees, and the arising forests are then partly reconnected by “parity correction.” We show that the arising “connectivity correction” can be achieved for a minor extra cost. On the one hand, this algorithm and analysis extend previous tools such as the best-of-many Christofides-Serdyukov-algorithm. On the other hand, powerful new tools are solicited, such as a flow problem for analyzing the reconnection cost, and the construction of a set of more and more restrictive spanning trees, each of which can still be found by the greedy algorithm. We show that these trees, which are easy to compute, can replace the spanning trees of the best-of-many Christofides-Serdyukov-algorithm. These new methods lead to improving the integrality ratio and approximation guarantee below 1.53, as was shown in the preliminary, shortened version of this article that appeared in FOCS 2016. The algorithm and analysis have been significantly simplified in the current article, while details and explanations have been added.
{"title":"The Salesman’s Improved Paths through Forests","authors":"András Sebö, A. V. Zuylen","doi":"10.1145/3326123","DOIUrl":"https://doi.org/10.1145/3326123","url":null,"abstract":"We give a new, strongly polynomial-time algorithm and improved analysis for the metric s-t path Traveling Salesman Problem (TSP). It finds a tour of cost less than 1.53 times the optimum of the subtour elimination linear program (LP), while known examples show that 1.5 is a lower bound for the integrality gap. A key new idea is the deletion of some edges of the spanning trees used in the best-of-many Christofides-Serdyukov-algorithm, which is then accompanied by novel arguments of the analysis: edge-deletion disconnects the trees, and the arising forests are then partly reconnected by “parity correction.” We show that the arising “connectivity correction” can be achieved for a minor extra cost. On the one hand, this algorithm and analysis extend previous tools such as the best-of-many Christofides-Serdyukov-algorithm. On the other hand, powerful new tools are solicited, such as a flow problem for analyzing the reconnection cost, and the construction of a set of more and more restrictive spanning trees, each of which can still be found by the greedy algorithm. We show that these trees, which are easy to compute, can replace the spanning trees of the best-of-many Christofides-Serdyukov-algorithm. These new methods lead to improving the integrality ratio and approximation guarantee below 1.53, as was shown in the preliminary, shortened version of this article that appeared in FOCS 2016. The algorithm and analysis have been significantly simplified in the current article, while details and explanations have been added.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78384653","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 proves the completeness of an axiomatization for differential equation invariants described by Noetherian functions. First, the differential equation axioms of differential dynamic logic are shown to be complete for reasoning about analytic invariants. Completeness crucially exploits differential ghosts, which introduce additional variables that can be chosen to evolve freely along new differential equations. Cleverly chosen differential ghosts are the proof-theoretical counterpart of dark matter. They create a new hypothetical state, whose relationship to the original state variables satisfies invariants that did not exist before. The reflection of these new invariants in the original system then enables its analysis. An extended axiomatization with existence and uniqueness axioms is complete for all local progress properties, and, with a real induction axiom, is complete for all semianalytic invariants. This parsimonious axiomatization serves as the logical foundation for reasoning about invariants of differential equations. Indeed, it is precisely this logical treatment that enables the generalization of completeness to the Noetherian case.
{"title":"Differential Equation Invariance Axiomatization","authors":"André Platzer, Yong Kiam Tan","doi":"10.1145/3380825","DOIUrl":"https://doi.org/10.1145/3380825","url":null,"abstract":"This article proves the completeness of an axiomatization for differential equation invariants described by Noetherian functions. First, the differential equation axioms of differential dynamic logic are shown to be complete for reasoning about analytic invariants. Completeness crucially exploits differential ghosts, which introduce additional variables that can be chosen to evolve freely along new differential equations. Cleverly chosen differential ghosts are the proof-theoretical counterpart of dark matter. They create a new hypothetical state, whose relationship to the original state variables satisfies invariants that did not exist before. The reflection of these new invariants in the original system then enables its analysis. An extended axiomatization with existence and uniqueness axioms is complete for all local progress properties, and, with a real induction axiom, is complete for all semianalytic invariants. This parsimonious axiomatization serves as the logical foundation for reasoning about invariants of differential equations. Indeed, it is precisely this logical treatment that enables the generalization of completeness to the Noetherian case.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88215009","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 Invited Articles section of this issue consists of three papers. The first is “Shellability is NPcomplete,” by Xavier Goaoc, Pavel Paták, Zuzana Patáková, Martin Tancer, and Uli Wagner, which won the best-paper award at the 34th International Symposium on Computational Geometry (SoCG’17). We want to thank the SoGS Program Committee for their help in selecting this invited paper and editor Jean-Daniel Boissonnat for handling the paper. Next is “Towards Verifying Nonlinear Integer Arithmetic,” by Paul Beame and Vincent Liew, invited from the 39th International Conference on Computer Aided Verification (CAV’17). We want to thank the CAV Program Committee for their help in selecting this invited paper and editor Rajeev Alur for handling the paper. Last, but not least, is the paper “On the Computability of Conditional Probability,” by Nathanael L. Ackerman, Cameron E. Freer, and Daniel M. Roy, invited from the 26th Annual IEEE Symposium on Logic in Computer Science (LICS’11). We want to thank the LICS Program Committee for their help in selecting this invited paper and editor Nachum Dershowitz for handling the paper.
本期特邀文章部分由三篇论文组成。第一个是“Shellability is NPcomplete”，作者是Xavier Goaoc、Pavel Paták、Zuzana Patáková、Martin Tancer和Uli Wagner，该论文在第34届国际计算几何研讨会(SoCG ' 17)上获得了最佳论文奖。我们要感谢SoGS项目委员会在选择这篇受邀论文方面的帮助，以及编辑Jean-Daniel Boissonnat对论文的处理。接下来是Paul Beame和Vincent Liew从第39届计算机辅助验证国际会议(CAV ' 17)邀请的“迈向验证非线性整数算法”。我们要感谢CAV项目委员会帮助我们选择这篇受邀论文，并感谢编辑Rajeev Alur处理这篇论文。最后，但并非最不重要的是Nathanael L. Ackerman, Cameron E. Freer和Daniel M. Roy在第26届IEEE计算机科学逻辑研讨会(LICS ' 11)上发表的论文“On the Computability of Conditional Probability”。我们要感谢LICS项目委员会在选择这篇受邀论文方面的帮助，以及编辑Nachum Dershowitz对论文的处理。
{"title":"Invited Articles Foreword","authors":"É. Tardos","doi":"10.1145/3328536","DOIUrl":"https://doi.org/10.1145/3328536","url":null,"abstract":"The Invited Articles section of this issue consists of three papers. The first is “Shellability is NPcomplete,” by Xavier Goaoc, Pavel Paták, Zuzana Patáková, Martin Tancer, and Uli Wagner, which won the best-paper award at the 34th International Symposium on Computational Geometry (SoCG’17). We want to thank the SoGS Program Committee for their help in selecting this invited paper and editor Jean-Daniel Boissonnat for handling the paper. Next is “Towards Verifying Nonlinear Integer Arithmetic,” by Paul Beame and Vincent Liew, invited from the 39th International Conference on Computer Aided Verification (CAV’17). We want to thank the CAV Program Committee for their help in selecting this invited paper and editor Rajeev Alur for handling the paper. Last, but not least, is the paper “On the Computability of Conditional Probability,” by Nathanael L. Ackerman, Cameron E. Freer, and Daniel M. Roy, invited from the 26th Annual IEEE Symposium on Logic in Computer Science (LICS’11). We want to thank the LICS Program Committee for their help in selecting this invited paper and editor Nachum Dershowitz for handling the paper.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78362378","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 to construct temporal testers for the logic MITL, a prominent linear-time logic for real-time systems. A temporal tester is a transducer that inputs a signal holding the Boolean value of atomic propositions and outputs the truth value of a formula along time. Here we consider testers over continuous-time Boolean signals that use clock variables to enforce duration constraints, as in timed automata. We first rewrite the MITL formula into a “simple” formula using a limited set of temporal modalities. We then build testers for these specific modalities and show how to compose testers for simple formulae into complex ones. Temporal testers can be turned into acceptors, yielding a compositional translation from MITL to timed automata. This construction is much simpler than previously known and remains asymptotically optimal. It supports both past and future operators and can easily be extended.
{"title":"From Real-time Logic to Timed Automata","authors":"Thomas Ferrère, O. Maler, D. Ničković, A. Pnueli","doi":"10.1145/3286976","DOIUrl":"https://doi.org/10.1145/3286976","url":null,"abstract":"We show how to construct temporal testers for the logic MITL, a prominent linear-time logic for real-time systems. A temporal tester is a transducer that inputs a signal holding the Boolean value of atomic propositions and outputs the truth value of a formula along time. Here we consider testers over continuous-time Boolean signals that use clock variables to enforce duration constraints, as in timed automata. We first rewrite the MITL formula into a “simple” formula using a limited set of temporal modalities. We then build testers for these specific modalities and show how to compose testers for simple formulae into complex ones. Temporal testers can be turned into acceptors, yielding a compositional translation from MITL to timed automata. This construction is much simpler than previously known and remains asymptotically optimal. It supports both past and future operators and can easily be extended.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89388031","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}
J. Ouaknine, Amaury Pouly, João Sousa-Pinto, J. Worrell
We consider the decidability of the membership problem for matrix-exponential semigroups: Given k∈ N and square matrices A1, … , Ak, C, all of the same dimension and with real algebraic entries, decide whether C is contained in the semigroup generated by the matrix exponentials exp (Ai t), where i∈ { 1,… ,k} and t ≥ 0. This problem can be seen as a continuous analog of Babai et al.’s and Cai et al.’s problem of solving multiplicative matrix equations and has applications to reachability analysis of linear hybrid automata and switching systems. Our main results are that the semigroup membership problem is undecidable in general, but decidable if we assume that A1, … , Ak commute. The decidability proof is by reduction to a version of integer programming that has transcendental constants. We give a decision procedure for the latter using Baker’s theorem on linear forms in logarithms of algebraic numbers, among other tools. The undecidability result is shown by reduction from Hilbert’s Tenth Problem.
{"title":"On the Decidability of Membership in Matrix-exponential Semigroups","authors":"J. Ouaknine, Amaury Pouly, João Sousa-Pinto, J. Worrell","doi":"10.1145/3286487","DOIUrl":"https://doi.org/10.1145/3286487","url":null,"abstract":"We consider the decidability of the membership problem for matrix-exponential semigroups: Given k∈ N and square matrices A1, … , Ak, C, all of the same dimension and with real algebraic entries, decide whether C is contained in the semigroup generated by the matrix exponentials exp (Ai t), where i∈ { 1,… ,k} and t ≥ 0. This problem can be seen as a continuous analog of Babai et al.’s and Cai et al.’s problem of solving multiplicative matrix equations and has applications to reachability analysis of linear hybrid automata and switching systems. Our main results are that the semigroup membership problem is undecidable in general, but decidable if we assume that A1, … , Ak commute. The decidability proof is by reduction to a version of integer programming that has transcendental constants. We give a decision procedure for the latter using Baker’s theorem on linear forms in logarithms of algebraic numbers, among other tools. The undecidability result is shown by reduction from Hilbert’s Tenth Problem.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90225774","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}
Vincent Rahli, M. Bickford, L. Cohen, R. Constable
Powerful yet effective induction principles play an important role in computing, being a paramount component of programming languages, automated reasoning, and program verification systems. The Bar Induction (BI) principle is a fundamental concept of intuitionism, which is equivalent to the standard principle of transfinite induction. In this work, we investigate the compatibility of several variants of BI with Constructive Type Theory (CTT), a dependent type theory in the spirit of Martin-Löf’s extensional theory. We first show that CTT is compatible with a BI principle for sequences of numbers. Then, we establish the compatibility of CTT with a more general BI principle for sequences of name-free closed terms. The formalization of the latter principle within the theory involved enriching CTT’s term syntax with a limit constructor and showing that consistency is preserved. Furthermore, we provide novel insights regarding BI, such as the non-truncated version of BI on monotone bars being intuitionistically false. These enhancements are carried out formally using the Nuprl proof assistant that implements CTT and the formalization of CTT within the Coq proof assistant presented in previous works.
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