Pub Date : 2023-08-16DOI: 10.48550/arXiv.2308.08374
Pamela Fleischmann, Sungmin Kim, Tore Koss, F. Manea, Dirk Nowotka, Stefan Siemer, Max Wiedenhöft
We introduce and investigate a series of matching problems for patterns with variables under Simon's congruence. Our results provide a thorough picture of these problems' computational complexity.
{"title":"Matching Patterns with Variables Under Simon's Congruence","authors":"Pamela Fleischmann, Sungmin Kim, Tore Koss, F. Manea, Dirk Nowotka, Stefan Siemer, Max Wiedenhöft","doi":"10.48550/arXiv.2308.08374","DOIUrl":"https://doi.org/10.48550/arXiv.2308.08374","url":null,"abstract":"We introduce and investigate a series of matching problems for patterns with variables under Simon's congruence. Our results provide a thorough picture of these problems' computational complexity.","PeriodicalId":404583,"journal":{"name":"Reachability Problems","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127770389","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}
Pub Date : 2023-07-11DOI: 10.48550/arXiv.2307.05283
Paul C. Bell, R. Niskanen, I. Potapov, P. Semukhin
We study the Identity Problem, the problem of determining if a finitely generated semigroup of matrices contains the identity matrix; see Problem 3 (Chapter 10.3) in ``Unsolved Problems in Mathematical Systems and Control Theory'' by Blondel and Megretski (2004). This fundamental problem is known to be undecidable for $mathbb{Z}^{4 times 4}$ and decidable for $mathbb{Z}^{2 times 2}$. The Identity Problem has been recently shown to be in polynomial time by Dong for the Heisenberg group over complex numbers in any fixed dimension with the use of Lie algebra and the Baker-Campbell-Hausdorff formula. We develop alternative proof techniques for the problem making a step forward towards more general problems such as the Membership Problem. We extend our techniques to show that the fundamental problem of determining if a given set of Heisenberg matrices generates a group, can also be decided in polynomial time.
{"title":"On the Identity and Group Problems for Complex Heisenberg Matrices","authors":"Paul C. Bell, R. Niskanen, I. Potapov, P. Semukhin","doi":"10.48550/arXiv.2307.05283","DOIUrl":"https://doi.org/10.48550/arXiv.2307.05283","url":null,"abstract":"We study the Identity Problem, the problem of determining if a finitely generated semigroup of matrices contains the identity matrix; see Problem 3 (Chapter 10.3) in ``Unsolved Problems in Mathematical Systems and Control Theory'' by Blondel and Megretski (2004). This fundamental problem is known to be undecidable for $mathbb{Z}^{4 times 4}$ and decidable for $mathbb{Z}^{2 times 2}$. The Identity Problem has been recently shown to be in polynomial time by Dong for the Heisenberg group over complex numbers in any fixed dimension with the use of Lie algebra and the Baker-Campbell-Hausdorff formula. We develop alternative proof techniques for the problem making a step forward towards more general problems such as the Membership Problem. We extend our techniques to show that the fundamental problem of determining if a given set of Heisenberg matrices generates a group, can also be decided in polynomial time.","PeriodicalId":404583,"journal":{"name":"Reachability Problems","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126372236","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}
Pub Date : 2023-07-04DOI: 10.48550/arXiv.2307.01897
D. Auger, Pierre Coucheney, Loric Duhaz'e, Kossi Roland Etse
Rotor walks are cellular automata that determine deterministic traversals of particles in a directed multigraph using simple local rules, yet they can generate complex behaviors. Furthermore, these trajectories exhibit statistical properties similar to random walks. In this study, we investigate a generalized version of the reachability problem known as ARRIVAL in Path Multigraphs, which involves predicting the number of particles that will reach designated target vertices. We show that this problem is in NP and co-NP in the general case. However, we exhibit algebraic invariants for Path Multigraphs that allow us to solve the problem efficiently, even for an exponential configuration of particles. These invariants are based on harmonic functions and are connected to the decomposition of integers in rational bases.
{"title":"Generalized ARRIVAL Problem for Rotor Walks in Path Multigraphs","authors":"D. Auger, Pierre Coucheney, Loric Duhaz'e, Kossi Roland Etse","doi":"10.48550/arXiv.2307.01897","DOIUrl":"https://doi.org/10.48550/arXiv.2307.01897","url":null,"abstract":"Rotor walks are cellular automata that determine deterministic traversals of particles in a directed multigraph using simple local rules, yet they can generate complex behaviors. Furthermore, these trajectories exhibit statistical properties similar to random walks. In this study, we investigate a generalized version of the reachability problem known as ARRIVAL in Path Multigraphs, which involves predicting the number of particles that will reach designated target vertices. We show that this problem is in NP and co-NP in the general case. However, we exhibit algebraic invariants for Path Multigraphs that allow us to solve the problem efficiently, even for an exponential configuration of particles. These invariants are based on harmonic functions and are connected to the decomposition of integers in rational bases.","PeriodicalId":404583,"journal":{"name":"Reachability Problems","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127142289","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}
Pub Date : 2022-07-19DOI: 10.48550/arXiv.2207.09201
Maria Kosche, Tore Koss, F. Manea, V. Pak
In this paper, we consider a variant of the classical algorithmic problem of checking whether a given word $v$ is a subsequence of another word $w$. More precisely, we consider the problem of deciding, given a number $p$ (defining a range-bound) and two words $v$ and $w$, whether there exists a factor $w[i:i+p-1]$ (or, in other words, a range of length $p$) of $w$ having $v$ as subsequence (i.,e., $v$ occurs as a subsequence in the bounded range $w[i:i+p-1]$). We give matching upper and lower quadratic bounds for the time complexity of this problem. Further, we consider a series of algorithmic problems in this setting, in which, for given integers $k$, $p$ and a word $w$, we analyse the set $p$-Subseq$_{k}(w)$ of all words of length $k$ which occur as subsequence of some factor of length $p$ of $w$. Among these, we consider the $k$-universality problem, the $k$-equivalence problem, as well as problems related to absent subsequences. Surprisingly, unlike the case of the classical model of subsequences in words where such problems have efficient solutions in general, we show that most of these problems become intractable in the new setting when subsequences in bounded ranges are considered. Finally, we provide an example of how some of our results can be applied to subsequence matching problems for circular words.
{"title":"Subsequences in Bounded Ranges: Matching and Analysis Problems","authors":"Maria Kosche, Tore Koss, F. Manea, V. Pak","doi":"10.48550/arXiv.2207.09201","DOIUrl":"https://doi.org/10.48550/arXiv.2207.09201","url":null,"abstract":"In this paper, we consider a variant of the classical algorithmic problem of checking whether a given word $v$ is a subsequence of another word $w$. More precisely, we consider the problem of deciding, given a number $p$ (defining a range-bound) and two words $v$ and $w$, whether there exists a factor $w[i:i+p-1]$ (or, in other words, a range of length $p$) of $w$ having $v$ as subsequence (i.,e., $v$ occurs as a subsequence in the bounded range $w[i:i+p-1]$). We give matching upper and lower quadratic bounds for the time complexity of this problem. Further, we consider a series of algorithmic problems in this setting, in which, for given integers $k$, $p$ and a word $w$, we analyse the set $p$-Subseq$_{k}(w)$ of all words of length $k$ which occur as subsequence of some factor of length $p$ of $w$. Among these, we consider the $k$-universality problem, the $k$-equivalence problem, as well as problems related to absent subsequences. Surprisingly, unlike the case of the classical model of subsequences in words where such problems have efficient solutions in general, we show that most of these problems become intractable in the new setting when subsequences in bounded ranges are considered. Finally, we provide an example of how some of our results can be applied to subsequence matching problems for circular words.","PeriodicalId":404583,"journal":{"name":"Reachability Problems","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114331925","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}
Pub Date : 2022-03-16DOI: 10.48550/arXiv.2203.08416
Kazuyuki Asada, H. Katsura, N. Kobayashi
We consider the reachability problem for higher-order functional programs and study the relationship between reachability games (i.e., the reachability problem for programs with angelic and demonic nondeterminism) and may-reachability (i.e., the reachability problem for programs with only angelic nondeterminism). We show that reachability games for order-n programs can be reduced to may-reachability problems for order(n + 1) programs, and vice versa. We formalize the reductions by using higher-order fixpoint logic and prove their correctness. We also discuss applications of the reductions to higher-order program verification.
{"title":"On Higher-Order Reachability Games vs May Reachability","authors":"Kazuyuki Asada, H. Katsura, N. Kobayashi","doi":"10.48550/arXiv.2203.08416","DOIUrl":"https://doi.org/10.48550/arXiv.2203.08416","url":null,"abstract":"We consider the reachability problem for higher-order functional programs and study the relationship between reachability games (i.e., the reachability problem for programs with angelic and demonic nondeterminism) and may-reachability (i.e., the reachability problem for programs with only angelic nondeterminism). We show that reachability games for order-n programs can be reduced to may-reachability problems for order(n + 1) programs, and vice versa. We formalize the reductions by using higher-order fixpoint logic and prove their correctness. We also discuss applications of the reductions to higher-order program verification.","PeriodicalId":404583,"journal":{"name":"Reachability Problems","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129649187","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}
Pub Date : 2021-08-31DOI: 10.1007/978-3-030-89716-1_8
Maria Kosche, Tore Koss, F. Manea, Stefan Siemer
{"title":"Absent Subsequences in Words","authors":"Maria Kosche, Tore Koss, F. Manea, Stefan Siemer","doi":"10.1007/978-3-030-89716-1_8","DOIUrl":"https://doi.org/10.1007/978-3-030-89716-1_8","url":null,"abstract":"","PeriodicalId":404583,"journal":{"name":"Reachability Problems","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129314185","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}
Pub Date : 2020-10-19DOI: 10.1007/978-3-030-61739-4_4
Sergiy Bogomolov, M. Forets, Kostiantyn Potomkin
{"title":"Case Study: Reachability and Scalability in a Unified Combat-Command-and-Control Model","authors":"Sergiy Bogomolov, M. Forets, Kostiantyn Potomkin","doi":"10.1007/978-3-030-61739-4_4","DOIUrl":"https://doi.org/10.1007/978-3-030-61739-4_4","url":null,"abstract":"","PeriodicalId":404583,"journal":{"name":"Reachability Problems","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122214304","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}
Pub Date : 2020-10-19DOI: 10.1007/978-3-030-61739-4_10
Ronny Tredup
{"title":"The Complexity of the Label-Splitting-Problem for Flip-Flop-Nets","authors":"Ronny Tredup","doi":"10.1007/978-3-030-61739-4_10","DOIUrl":"https://doi.org/10.1007/978-3-030-61739-4_10","url":null,"abstract":"","PeriodicalId":404583,"journal":{"name":"Reachability Problems","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124965124","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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