Pub Date : 2025-11-01DOI: 10.1016/j.ic.2025.105378
Mingyang Gong , Yong Chen , Zhi-Zhong Chen , Guohui Lin , Bing Su , Lusheng Wang
The problem studied in this paper is to find a collection of vertex-disjoint paths in a given graph such that each path has length at least k, called a long path, and the total number of edges on these paths is maximized. The problem is NP-hard for any fixed k or when k is part of the input, by a reduction from the Hamiltonian path problem. Berman and Karpinski presented a 7/6-approximation algorithm for , but for a general , there is no approximation algorithm directly for the problem. We present the first local search -approximation algorithm for any fixed , and a 1.4254-approximation algorithm for built on top of a maximum triangle-free path-cycle cover.
{"title":"Approximation algorithms for the maximum path cover problem using long paths","authors":"Mingyang Gong , Yong Chen , Zhi-Zhong Chen , Guohui Lin , Bing Su , Lusheng Wang","doi":"10.1016/j.ic.2025.105378","DOIUrl":"10.1016/j.ic.2025.105378","url":null,"abstract":"<div><div>The problem studied in this paper is to find a collection of vertex-disjoint paths in a given graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> such that each path has length at least <em>k</em>, called a long path, and the total number of edges on these paths is maximized. The problem is NP-hard for any fixed <em>k</em> or when <em>k</em> is part of the input, by a reduction from the Hamiltonian path problem. Berman and Karpinski presented a 7/6-approximation algorithm for <span><math><mi>k</mi><mo>=</mo><mn>1</mn></math></span>, but for a general <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, there is no approximation algorithm directly for the problem. We present the first local search <span><math><mo>(</mo><mn>0.4394</mn><mi>k</mi><mo>+</mo><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo></math></span>-approximation algorithm for any fixed <span><math><mi>k</mi><mo>≥</mo><mn>1</mn></math></span>, and a 1.4254-approximation algorithm for <span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span> built on top of a maximum triangle-free path-cycle cover.</div></div>","PeriodicalId":54985,"journal":{"name":"Information and Computation","volume":"307 ","pages":"Article 105378"},"PeriodicalIF":1.0,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528772","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01DOI: 10.1016/j.ic.2025.105375
Alexander Shen
In this expository note we present the proof of the constructive version of Birkhoff's ergodic theorem following Vladimir V'yugin, trying to separate and state explicitly the combinatorial statement on which this proof is based. The exposition is based on his papers and explanations given during his visit to LIF (CNRS, Marseille).
{"title":"Ergodic theorem and algorithmic randomness (following V. V'yugin)","authors":"Alexander Shen","doi":"10.1016/j.ic.2025.105375","DOIUrl":"10.1016/j.ic.2025.105375","url":null,"abstract":"<div><div>In this expository note we present the proof of the constructive version of Birkhoff's ergodic theorem following Vladimir V'yugin, trying to separate and state explicitly the combinatorial statement on which this proof is based. The exposition is based on his papers and explanations given during his visit to LIF (CNRS, Marseille).</div></div>","PeriodicalId":54985,"journal":{"name":"Information and Computation","volume":"307 ","pages":"Article 105375"},"PeriodicalIF":1.0,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145416280","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01DOI: 10.1016/j.ic.2025.105381
Ekaterina B. Fokina , Steffen Lempp
We study the learnability of classes of computable structures under models that allow finitely many mind changes. Extending classical notions of explanatory learning from informant and from text, we introduce new paradigms where the information source and the convergence requirements are modified. In particular, we define -learning, where each atomic fact may be presented with finitely many errors before stabilizing, and c.e.- and d.c.e.-learning, where information is restricted to positive atomic facts that may either never be retracted (c.e.) or be retracted at most once (d.c.e.). We provide syntactic characterizations for these notions: -learning coincides with definability by finite existential sentences, and c.e.-learning coincides with TxtEx-learning. For d.c.e.-learning we give partial characterization in restricted languages. Furthermore, for n-learning from informant, where learners are allowed at most n mind changes, we establish a complete syntactic characterization in terms of specific infinitary formulas of bounded depth.
{"title":"Syntactic characterization of learnability of structures with mind changes","authors":"Ekaterina B. Fokina , Steffen Lempp","doi":"10.1016/j.ic.2025.105381","DOIUrl":"10.1016/j.ic.2025.105381","url":null,"abstract":"<div><div>We study the learnability of classes of computable structures under models that allow finitely many mind changes. Extending classical notions of explanatory learning from informant and from text, we introduce new paradigms where the information source and the convergence requirements are modified. In particular, we define <span><math><msubsup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math></span>-learning, where each atomic fact may be presented with finitely many errors before stabilizing, and <em>c.e.- and d.c.e.-learning</em>, where information is restricted to positive atomic facts that may either never be retracted (c.e.) or be retracted at most once (d.c.e.). We provide syntactic characterizations for these notions: <span><math><msubsup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math></span>-learning coincides with definability by finite existential sentences, and c.e.-learning coincides with TxtEx-learning. For d.c.e.-learning we give partial characterization in restricted languages. Furthermore, for <em>n</em>-learning from informant, where learners are allowed at most <em>n</em> mind changes, we establish a complete syntactic characterization in terms of specific infinitary formulas of bounded depth.</div></div>","PeriodicalId":54985,"journal":{"name":"Information and Computation","volume":"307 ","pages":"Article 105381"},"PeriodicalIF":1.0,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145579161","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01DOI: 10.1016/j.ic.2025.105377
Niuniu Qi , Xia Zeng , Banglong Liu , Zhengfeng Yang , Xiaochao Tang , Li Zhang , Chao Peng , Zhenbing Zeng
For many safety-critical systems, there is a pressing need for learning the controllers to endow systems with properties of safety. Reinforcement learning (RL) has emerged as a promising approach to synthesizing controllers that satisfy safety requirements by leveraging user-defined reward functions to encode desired system behavior. However, it remains a significant challenge in synthesizing provably correct controllers with safety requirements. To address this issue, we try to design a special hybrid polynomial-NN controller that is easy to verify without losing its expressiveness and flexibility. This paper proposes an iterative framework to synthesize a hybrid controller based on RL, low-degree polynomial fitting and knowledge distillation. By formulating and solving a constrained optimization problem in which the verification conditions produce the barrier certificates, a computational method is given to ensure that every trajectory starting from the initial set of the system with the resulting controller satisfies the given safety requirement. In addition, we have implemented a tool named SynHC and evaluated its performance over a set of benchmark examples. The experimental results demonstrate that our approach efficiently synthesizes safe DNN controllers.
{"title":"An iterative scheme of hybrid controller synthesis for nonlinear systems subject to safety constraints","authors":"Niuniu Qi , Xia Zeng , Banglong Liu , Zhengfeng Yang , Xiaochao Tang , Li Zhang , Chao Peng , Zhenbing Zeng","doi":"10.1016/j.ic.2025.105377","DOIUrl":"10.1016/j.ic.2025.105377","url":null,"abstract":"<div><div>For many safety-critical systems, there is a pressing need for learning the controllers to endow systems with properties of safety. Reinforcement learning (RL) has emerged as a promising approach to synthesizing controllers that satisfy safety requirements by leveraging user-defined reward functions to encode desired system behavior. However, it remains a significant challenge in synthesizing provably correct controllers with safety requirements. To address this issue, we try to design a special hybrid polynomial-NN controller that is easy to verify without losing its expressiveness and flexibility. This paper proposes an iterative framework to synthesize a hybrid controller based on RL, low-degree polynomial fitting and knowledge distillation. By formulating and solving a constrained optimization problem in which the verification conditions produce the barrier certificates, a computational method is given to ensure that every trajectory starting from the initial set of the system with the resulting controller satisfies the given safety requirement. In addition, we have implemented a tool named SynHC and evaluated its performance over a set of benchmark examples. The experimental results demonstrate that our approach efficiently synthesizes safe DNN controllers.</div></div>","PeriodicalId":54985,"journal":{"name":"Information and Computation","volume":"307 ","pages":"Article 105377"},"PeriodicalIF":1.0,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145528775","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01DOI: 10.1016/j.ic.2025.105368
Bai Xue
This paper studies finite-time safety and reach-avoid verification for stochastic discrete-time dynamical systems. The aim is to ascertain lower and upper bounds of the probability that, within a predefined finite-time horizon, a system starting from an initial state in a safe set will either exit the safe set (safety verification) or reach a target set while remaining within the safe set until the first encounter with the target (reach-avoid verification). We introduce novel barrier-like sufficient conditions for characterizing these bounds, which either complement existing ones or fill gaps. Finally, we demonstrate the efficacy of these conditions on two simple examples.
{"title":"Finite-time safety and reach-avoid verification of stochastic discrete-time systems","authors":"Bai Xue","doi":"10.1016/j.ic.2025.105368","DOIUrl":"10.1016/j.ic.2025.105368","url":null,"abstract":"<div><div>This paper studies finite-time safety and reach-avoid verification for stochastic discrete-time dynamical systems. The aim is to ascertain lower and upper bounds of the probability that, within a predefined finite-time horizon, a system starting from an initial state in a safe set will either exit the safe set (safety verification) or reach a target set while remaining within the safe set until the first encounter with the target (reach-avoid verification). We introduce novel barrier-like sufficient conditions for characterizing these bounds, which either complement existing ones or fill gaps. Finally, we demonstrate the efficacy of these conditions on two simple examples.</div></div>","PeriodicalId":54985,"journal":{"name":"Information and Computation","volume":"307 ","pages":"Article 105368"},"PeriodicalIF":1.0,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145416142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01DOI: 10.1016/j.ic.2025.105373
Periklis Mantenoglou , Alexander Artikis
Composite event recognition (CER) frameworks reason over streams of low-level, symbolic events in order to detect instances of spatio-temporal patterns defining high-level, composite activities. The Event Calculus is a temporal, logical formalism that has been used to define composite activities in CER, while RTEC∘ is a formal CER framework that detects composite activities based on their Event Calculus definitions. RTEC∘, however, cannot handle arbitrary Event Calculus definitions for composite activities, limiting the range of CER applications supported by RTEC∘. We propose RTEC, an extension of RTEC∘ that supports arbitrary composite activity specifications in the Event Calculus. We present the syntax, semantics, reasoning algorithms and time complexity of RTEC. Moreover, we propose a compiler for RTEC, generating the optimal representation of an input set of Event Calculus definitions. We demonstrate the correctness of our compiler and outline its time complexity. We conducted an empirical evaluation of RTEC on synthetic and real data streams from human activity recognition and maritime situational awareness, including a comparison with two state-of-the-art Event Calculus-based systems, which demonstrates the benefits of RTEC.
{"title":"Composite event recognition with arbitrary specifications","authors":"Periklis Mantenoglou , Alexander Artikis","doi":"10.1016/j.ic.2025.105373","DOIUrl":"10.1016/j.ic.2025.105373","url":null,"abstract":"<div><div>Composite event recognition (CER) frameworks reason over streams of low-level, symbolic events in order to detect instances of spatio-temporal patterns defining high-level, composite activities. The Event Calculus is a temporal, logical formalism that has been used to define composite activities in CER, while RTEC<sub>∘</sub> is a formal CER framework that detects composite activities based on their Event Calculus definitions. RTEC<sub>∘</sub>, however, cannot handle arbitrary Event Calculus definitions for composite activities, limiting the range of CER applications supported by RTEC<sub>∘</sub>. We propose RTEC<span><math><msub><mrow></mrow><mrow><mi>f</mi><mi>l</mi></mrow></msub></math></span>, an extension of RTEC<sub>∘</sub> that supports arbitrary composite activity specifications in the Event Calculus. We present the syntax, semantics, reasoning algorithms and time complexity of RTEC<span><math><msub><mrow></mrow><mrow><mi>f</mi><mi>l</mi></mrow></msub></math></span>. Moreover, we propose a compiler for RTEC<span><math><msub><mrow></mrow><mrow><mi>f</mi><mi>l</mi></mrow></msub></math></span>, generating the optimal representation of an input set of Event Calculus definitions. We demonstrate the correctness of our compiler and outline its time complexity. We conducted an empirical evaluation of RTEC<span><math><msub><mrow></mrow><mrow><mi>f</mi><mi>l</mi></mrow></msub></math></span> on synthetic and real data streams from human activity recognition and maritime situational awareness, including a comparison with two state-of-the-art Event Calculus-based systems, which demonstrates the benefits of RTEC<span><math><msub><mrow></mrow><mrow><mi>f</mi><mi>l</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":54985,"journal":{"name":"Information and Computation","volume":"307 ","pages":"Article 105373"},"PeriodicalIF":1.0,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145416141","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01DOI: 10.1016/j.ic.2025.105383
Oscar H. Ibarra , Ian McQuillan
A new model of multicounter machines is introduced where testing the counter status of a counter is optional, rather than existing models where they are always either required (traditional multicounter machines) or no status can be checked (partially-blind multicounter machines). If, in every accepting computation, each counter has a bounded number of occurrences where its status is tested and verified to be zero, then the machine is called finite-testable. One-way nondeterministic finite-testable multicounter machines are shown to be equivalent to partially-blind multicounter machines. However, one-way deterministic finite-testable multicounter machines are strictly more powerful than deterministic partially-blind machines. Interestingly, one-way deterministic finite-testable multicounter machines are shown to have a decidable containment problem. This makes it the most general known model where this problem is decidable, making the class important in the areas of model checking and formal verification. We also study properties of their reachability sets.
{"title":"On the containment problem for deterministic multicounter machine models","authors":"Oscar H. Ibarra , Ian McQuillan","doi":"10.1016/j.ic.2025.105383","DOIUrl":"10.1016/j.ic.2025.105383","url":null,"abstract":"<div><div>A new model of multicounter machines is introduced where testing the counter status of a counter is optional, rather than existing models where they are always either required (traditional multicounter machines) or no status can be checked (partially-blind multicounter machines). If, in every accepting computation, each counter has a bounded number of occurrences where its status is tested and verified to be zero, then the machine is called <em>finite-testable</em>. One-way nondeterministic finite-testable multicounter machines are shown to be equivalent to partially-blind multicounter machines. However, one-way deterministic finite-testable multicounter machines are strictly more powerful than deterministic partially-blind machines. Interestingly, one-way deterministic finite-testable multicounter machines are shown to have a decidable containment problem. This makes it the most general known model where this problem is decidable, making the class important in the areas of model checking and formal verification. We also study properties of their reachability sets.</div></div>","PeriodicalId":54985,"journal":{"name":"Information and Computation","volume":"307 ","pages":"Article 105383"},"PeriodicalIF":1.0,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145624128","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01DOI: 10.1016/j.ic.2025.105379
Qisheng Wang , Ming Yang , Xinrui Zhu
The palindromic tree (a.k.a. eertree) is a data structure that provides access to all palindromic substrings of a string. In this paper, we propose a dynamic version of eertree, called double-ended eertree, which supports online operations on the stored string, including double-ended queue operations, counting distinct palindromic substrings, and finding the longest palindromic prefix/suffix. At the heart of our construction, we identify a new class of substring occurrences, called surfaces, that are palindromic substring occurrences that are neither prefixes nor suffixes of any other palindromic substring occurrences, which is of independent interest. Surfaces characterize the link structure of all palindromic substrings in the eertree, thereby allowing a linear-time implementation of double-ended eertrees through a linear-time maintenance of surfaces.
{"title":"Double-ended palindromic trees in linear time","authors":"Qisheng Wang , Ming Yang , Xinrui Zhu","doi":"10.1016/j.ic.2025.105379","DOIUrl":"10.1016/j.ic.2025.105379","url":null,"abstract":"<div><div>The palindromic tree (a.k.a. eertree) is a data structure that provides access to all palindromic substrings of a string. In this paper, we propose a dynamic version of eertree, called double-ended eertree, which supports online operations on the stored string, including double-ended queue operations, counting distinct palindromic substrings, and finding the longest palindromic prefix/suffix. At the heart of our construction, we identify a new class of substring occurrences, called surfaces, that are palindromic substring occurrences that are neither prefixes nor suffixes of any other palindromic substring occurrences, which is of independent interest. Surfaces characterize the link structure of all palindromic substrings in the eertree, thereby allowing a linear-time implementation of double-ended eertrees through a linear-time maintenance of surfaces.</div></div>","PeriodicalId":54985,"journal":{"name":"Information and Computation","volume":"307 ","pages":"Article 105379"},"PeriodicalIF":1.0,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145624129","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01DOI: 10.1016/j.ic.2025.105369
Michal Hospodár , Jozef Jirásek , Galina Jirásková , Juraj Šebej
<div><div>We examine the complexity of regular operations assuming that the arguments are represented by nondeterministic finite automata, while the resulting language is required to be represented by a deterministic finite automaton. We obtain tight upper bounds on the NFA-to-DFA trade-off for complementation (<span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span>), union and symmetric difference (<span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup></math></span>), intersection (<span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup><mo>−</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><mn>2</mn></math></span>), difference (<span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup><mo>−</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><mn>1</mn></math></span>), left and right quotient (<span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></math></span>), reversal and star (<span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span>), concatenation (<span><math><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup></math></span>), square (<span><math><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup></math></span>), cut (<span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><mn>1</mn><mo>)</mo></math></span>), and shuffle (<span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msup></math></span>), and an upper bound <span><math><msup><mrow><mn>2</mn></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msup></math></span> and a lower bound <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> for square root. Except for square root, all our witnesses are described over a small fixed alphabet of size at most four. We show that the alphabet used to describe witnesses for seven of the considered operations is optimal in the sense that the corresponding upper bounds cannot be met by any languages over a smaller alphabet. We also provide some partial results
{"title":"Operational complexity: NFA-to-DFA trade-off","authors":"Michal Hospodár , Jozef Jirásek , Galina Jirásková , Juraj Šebej","doi":"10.1016/j.ic.2025.105369","DOIUrl":"10.1016/j.ic.2025.105369","url":null,"abstract":"<div><div>We examine the complexity of regular operations assuming that the arguments are represented by nondeterministic finite automata, while the resulting language is required to be represented by a deterministic finite automaton. We obtain tight upper bounds on the NFA-to-DFA trade-off for complementation (<span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span>), union and symmetric difference (<span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup></math></span>), intersection (<span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup><mo>−</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><mn>2</mn></math></span>), difference (<span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup><mo>−</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><mn>1</mn></math></span>), left and right quotient (<span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></math></span>), reversal and star (<span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span>), concatenation (<span><math><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></msup></math></span>), square (<span><math><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup></math></span>), cut (<span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><mn>1</mn><mo>)</mo></math></span>), and shuffle (<span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msup></math></span>), and an upper bound <span><math><msup><mrow><mn>2</mn></mrow><mrow><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msup></math></span> and a lower bound <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span> for square root. Except for square root, all our witnesses are described over a small fixed alphabet of size at most four. We show that the alphabet used to describe witnesses for seven of the considered operations is optimal in the sense that the corresponding upper bounds cannot be met by any languages over a smaller alphabet. We also provide some partial results ","PeriodicalId":54985,"journal":{"name":"Information and Computation","volume":"307 ","pages":"Article 105369"},"PeriodicalIF":1.0,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145416144","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01DOI: 10.1016/j.ic.2025.105372
Tomoyuki Yamakami
Lately, there have been intensive studies on strengths and limitations of nonuniform families of promise decision problems solvable by various types of polynomial-size finite automata families, where “polynomial-size” refers to the polynomially-bounded state complexity of a finite automata family. In this line of study, we further expand the scope of these studies to families of partial counting and gap functions, defined in terms of nonuniform families of polynomial-size nondeterministic finite automata, and their relevant families of promise decision problems. Counting functions have an ability of counting the number of accepting computation paths produced by nondeterministic finite automata. With no unproven hardness assumption, we show numerous separations and collapses of complexity classes of those partial counting and gap function families and their induced promise decision problem families. We also investigate their relationships to pushdown automata families of polynomial stack-state complexity.
{"title":"Power of counting by nonuniform families of polynomial-size finite automata","authors":"Tomoyuki Yamakami","doi":"10.1016/j.ic.2025.105372","DOIUrl":"10.1016/j.ic.2025.105372","url":null,"abstract":"<div><div>Lately, there have been intensive studies on strengths and limitations of nonuniform families of promise decision problems solvable by various types of polynomial-size finite automata families, where “polynomial-size” refers to the polynomially-bounded state complexity of a finite automata family. In this line of study, we further expand the scope of these studies to families of partial counting and gap functions, defined in terms of nonuniform families of polynomial-size nondeterministic finite automata, and their relevant families of promise decision problems. Counting functions have an ability of counting the number of accepting computation paths produced by nondeterministic finite automata. With no unproven hardness assumption, we show numerous separations and collapses of complexity classes of those partial counting and gap function families and their induced promise decision problem families. We also investigate their relationships to pushdown automata families of polynomial stack-state complexity.</div></div>","PeriodicalId":54985,"journal":{"name":"Information and Computation","volume":"307 ","pages":"Article 105372"},"PeriodicalIF":1.0,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145465933","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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