Cellular automata are a discrete dynamical system which models massively parallel computation. Much attention is devoted to computations with small time complexity for which the parallelism may provide further possibilities. In this paper, we investigate the ability of cellular automata related to functional computation. We introduce several functional classes of low time complexity which contain "natural" problems. We examine their inclusion relationships and emphasize that several questions arising from this functional framework are related to current ones coming from the recognition context. We also provide a negative result which explicits limits on the information transmission whose consequences go beyond the functional point of view.
{"title":"Linear functional classes over cellular automata","authors":"Anaël Grandjean, G. Richard, V. Terrier","doi":"10.4204/EPTCS.90.15","DOIUrl":"https://doi.org/10.4204/EPTCS.90.15","url":null,"abstract":"Cellular automata are a discrete dynamical system which models massively parallel computation. Much attention is devoted to computations with small time complexity for which the parallelism may provide further possibilities. In this paper, we investigate the ability of cellular automata related to functional computation. We introduce several functional classes of low time complexity which contain \"natural\" problems. We examine their inclusion relationships and emphasize that several questions arising from this functional framework are related to current ones coming from the recognition context. We also provide a negative result which explicits limits on the information transmission whose consequences go beyond the functional point of view.","PeriodicalId":415843,"journal":{"name":"AUTOMATA & JAC","volume":"388 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116657583","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}
While for synchronous deterministic cellular automata there is an accepted definition of reversibility, the situation is less clear for asynchronous cellular automata. We first discuss a few possibilities and then investigate what we call phase space invertible asynchronous cellular automata in more detail. We will show that for each Turing machine there is such a cellular automaton simulating it, and that it is decidable whether an asynchronous cellular automaton has this property or not, even in higher dimensions.
{"title":"Phase Space Invertible Asynchronous Cellular Automata","authors":"S. Wacker, T. Worsch","doi":"10.4204/EPTCS.90.19","DOIUrl":"https://doi.org/10.4204/EPTCS.90.19","url":null,"abstract":"While for synchronous deterministic cellular automata there is an accepted definition of reversibility, the situation is less clear for asynchronous cellular automata. We first discuss a few possibilities and then investigate what we call phase space invertible asynchronous cellular automata in more detail. We will show that for each Turing machine there is such a cellular automaton simulating it, and that it is decidable whether an asynchronous cellular automaton has this property or not, even in higher dimensions.","PeriodicalId":415843,"journal":{"name":"AUTOMATA & JAC","volume":"186 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123275181","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}
Cellular automata are investigated towards their ability to compute transductions, that is, to transform inputs into outputs. The families of transductions computed are classified with regard to the time allowed to process the input and to compute the output. Since there is a particular interest in fast transductions, we mainly focus on the time complexities real time and linear time. We first investigate the computational capabilities of cellular automaton transducers by comparing them to iterative array transducers, that is, we compare parallel input/output mode to sequential input/output mode of massively parallel machines. By direct simulations, it turns out that the parallel mode is not weaker than the sequential one. Moreover, with regard to certain time complexities cellular automaton transducers are even more powerful than iterative arrays. In the second part of the paper, the model in question is compared with the sequential devices single-valued finite state transducers and deterministic pushdown transducers. It turns out that both models can be simulated by cellular automaton transducers faster than by iterative array transducers.
{"title":"Transductions Computed by One-Dimensional Cellular Automata","authors":"Martin Kutrib, Andreas Malcher","doi":"10.4204/EPTCS.90.16","DOIUrl":"https://doi.org/10.4204/EPTCS.90.16","url":null,"abstract":"Cellular automata are investigated towards their ability to compute transductions, that is, to transform inputs into outputs. The families of transductions computed are classified with regard to the time allowed to process the input and to compute the output. Since there is a particular interest in fast transductions, we mainly focus on the time complexities real time and linear time. We first investigate the computational capabilities of cellular automaton transducers by comparing them to iterative array transducers, that is, we compare parallel input/output mode to sequential input/output mode of massively parallel machines. By direct simulations, it turns out that the parallel mode is not weaker than the sequential one. Moreover, with regard to certain time complexities cellular automaton transducers are even more powerful than iterative arrays. In the second part of the paper, the model in question is compared with the sequential devices single-valued finite state transducers and deterministic pushdown transducers. It turns out that both models can be simulated by cellular automaton transducers faster than by iterative array transducers.","PeriodicalId":415843,"journal":{"name":"AUTOMATA & JAC","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121893512","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}
Katsunobu Imai, Takahiro Hatsuda, V. Poupet, Kota Sato
In this paper we investigate certain properties of semi-totalistic cellular automata (CA) on the well known quasi-periodic kite and dart two dimensional tiling of the plane presented by Roger Penrose. We show that, despite the irregularity of the underlying grid, it is possible to devise a semi-totalistic CA capable of simulating any boolean circuit on this aperiodic tiling.
{"title":"A Universal Semi-totalistic Cellular Automaton on Kite and Dart Penrose Tilings","authors":"Katsunobu Imai, Takahiro Hatsuda, V. Poupet, Kota Sato","doi":"10.4204/EPTCS.90.21","DOIUrl":"https://doi.org/10.4204/EPTCS.90.21","url":null,"abstract":"In this paper we investigate certain properties of semi-totalistic cellular automata (CA) on the well known quasi-periodic kite and dart two dimensional tiling of the plane presented by Roger Penrose. We show that, despite the irregularity of the underlying grid, it is possible to devise a semi-totalistic CA capable of simulating any boolean circuit on this aperiodic tiling.","PeriodicalId":415843,"journal":{"name":"AUTOMATA & JAC","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126831775","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 that there do not exist positively expansive cellular automata defined on the full k-ary tree shift (for k ≥ 2). Moreover, we investigate some topological properties of these automata and their relationships, namely permutivity, surjectivity, preinj ectivity, right-closingness and openness.
{"title":"Topological properties of cellular automata on trees","authors":"G. Fici, Francesca Fiorenzi","doi":"10.4204/EPTCS.90.20","DOIUrl":"https://doi.org/10.4204/EPTCS.90.20","url":null,"abstract":"We prove that there do not exist positively expansive cellular automata defined on the full k-ary tree shift (for k ≥ 2). Moreover, we investigate some topological properties of these automata and their relationships, namely permutivity, surjectivity, preinj ectivity, right-closingness and openness.","PeriodicalId":415843,"journal":{"name":"AUTOMATA & JAC","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130255162","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 the computational and structural aspects of countable two-dimensional SFTs and other subshifts. Our main focus is on the topological derivatives and subpattern posets of these objects, and our main results are constructions of two-dimensional countable subshifts with interesting properties. We present an SFT whose iterated derivatives are maximally complex from the computational point of view, a sofic shift whose subpattern poset contains an infinite descending chain, a family of SFTs whose finite subpattern posets contain arbitrary finite posets, and a natural example of an SFT with infinite Cantor-Bendixon rank.
{"title":"On Derivatives and Subpattern Orders of Countable Subshifts","authors":"Ville Salo, Ilkka Törmä","doi":"10.4204/EPTCS.90.3","DOIUrl":"https://doi.org/10.4204/EPTCS.90.3","url":null,"abstract":"We study the computational and structural aspects of countable two-dimensional SFTs and other subshifts. Our main focus is on the topological derivatives and subpattern posets of these objects, and our main results are constructions of two-dimensional countable subshifts with interesting properties. We present an SFT whose iterated derivatives are maximally complex from the computational point of view, a sofic shift whose subpattern poset contains an infinite descending chain, a family of SFTs whose finite subpattern posets contain arbitrary finite posets, and a natural example of an SFT with infinite Cantor-Bendixon rank.","PeriodicalId":415843,"journal":{"name":"AUTOMATA & JAC","volume":"86 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126174042","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}
One of the most interesting aspects in the theory of cellular automata is the study of different typesof nilpotency, that is, different ways in which a cellular automaton can force a particular symbol (usu-ally called 0) to appear frequently in all its spacetime diagrams. The simplest such notion, called sim-ply ‘nilpotency’, is that the cellular automaton c maps every configuration to a uniform configuration:::000:::, on which it behaves as the identity, in a uniformly bounded number of steps, that is, c
{"title":"On Nilpotency and Asymptotic Nilpotency of Cellular Automata","authors":"Ville Salo","doi":"10.4204/EPTCS.90.7","DOIUrl":"https://doi.org/10.4204/EPTCS.90.7","url":null,"abstract":"One of the most interesting aspects in the theory of cellular automata is the study of different typesof nilpotency, that is, different ways in which a cellular automaton can force a particular symbol (usu-ally called 0) to appear frequently in all its spacetime diagrams. The simplest such notion, called sim-ply ‘nilpotency’, is that the cellular automaton c maps every configuration to a uniform configuration:::000:::, on which it behaves as the identity, in a uniformly bounded number of steps, that is, c","PeriodicalId":415843,"journal":{"name":"AUTOMATA & JAC","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125569872","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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