Causal graph dynamics are transformations over graphs that capture two important symmetries of physics, namely causality and homogeneity. They can be equivalently defined as continuous and translation invariant transformations or functions induced by a local rule applied simultaneously on every vertex of the graph. Intrinsic universality is the ability of an instance of a model to simulate every other instance of the model while preserving the structure of the computation at every step of the simulation. In this work we present the construction of a family of intrinsically universal instances of causal graphs dynamics, each instance being able to simulate a subset of instances.
{"title":"Intrinsic Universality of Causal Graph Dynamics","authors":"S. Martiel, Bruno Martin","doi":"10.4204/EPTCS.128.19","DOIUrl":"https://doi.org/10.4204/EPTCS.128.19","url":null,"abstract":"Causal graph dynamics are transformations over graphs that capture two important symmetries of physics, namely causality and homogeneity. They can be equivalently defined as continuous and translation invariant transformations or functions induced by a local rule applied simultaneously on every vertex of the graph. Intrinsic universality is the ability of an instance of a model to simulate every other instance of the model while preserving the structure of the computation at every step of the simulation. In this work we present the construction of a family of intrinsically universal instances of causal graphs dynamics, each instance being able to simulate a subset of instances.","PeriodicalId":340847,"journal":{"name":"Machines, Computations, and Universality","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114736896","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}
Nondeterministic polynomial-time Blum-Shub-Smale Machines over the reals give rise to a discrete complexity class between NP and PSPACE. Several problems, mostly from real algebraic geometry / polynomial systems, have been shown complete (under many-one reduction by polynomial-time Turing machines) for this class. We exhibit a new one based on questions about expressions built from cross products only.
{"title":"Satisfiability of cross product terms is complete for real nondeterministic polytime Blum-Shub-Smale machines","authors":"C. Herrmann, Johanna Sokoli, M. Ziegler","doi":"10.4204/EPTCS.128.16","DOIUrl":"https://doi.org/10.4204/EPTCS.128.16","url":null,"abstract":"Nondeterministic polynomial-time Blum-Shub-Smale Machines over the reals give rise to a discrete complexity class between NP and PSPACE. Several problems, mostly from real algebraic geometry / polynomial systems, have been shown complete (under many-one reduction by polynomial-time Turing machines) for this class. We exhibit a new one based on questions about expressions built from cross products only.","PeriodicalId":340847,"journal":{"name":"Machines, Computations, and Universality","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128897030","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}
To study groups with small Dehn's function, Olshanskii and Sapir developed a new invariant of bipartite chords diagrams and applied it to hub-free realization of S-machines. In this paper we consider this new invariant together with groups constructed from S-machines containing the hub relation. The idea is to study the links between the topology of the asymptotic cones and polynomial time computations. Indeed it is known that the topology of such metric space depends on diagrams without hubs that do not correspond to the computations of the considered S-machine. This work gives sufficient conditions that avoid this misbehaviour, but as we shall see the method has a significant drawback.
{"title":"Topology and Non-Deterministic Polynomial Time Computation : Avoidance of The Misbehaviour of Hub-Free Diagrams and Consequences","authors":"A. Gasperin","doi":"10.4204/EPTCS.128.15","DOIUrl":"https://doi.org/10.4204/EPTCS.128.15","url":null,"abstract":"To study groups with small Dehn's function, Olshanskii and Sapir developed a new invariant of bipartite chords diagrams and applied it to hub-free realization of S-machines. In this paper we consider this new invariant together with groups constructed from S-machines containing the hub relation. The idea is to study the links between the topology of the asymptotic cones and polynomial time computations. Indeed it is known that the topology of such metric space depends on diagrams without hubs that do not correspond to the computations of the considered S-machine. This work gives sufficient conditions that avoid this misbehaviour, but as we shall see the method has a significant drawback.","PeriodicalId":340847,"journal":{"name":"Machines, Computations, and Universality","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126400457","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}
In [11], it was shown that a few languages constructed from some figures of hyperbolic tilings cannotbe recognized by pushdown automata but they can be recognized by a 2-iterated pushdown automaton.Before, it was known that several tessellations of the hyperbolic plane are generated by substitutions,see [3]. This property is also clear from [7].These substitutions can be also described by the use of grammars. This is rather straightforward. In[6], these substitutions appear as rules of a grammar, although the grammar is not formally described.Iterated pushdown automata were introduced in [4, 12] and we refer the reader to [1] for referencesand for the connection of this topic with sequences of rational numbers. By their definition, iteratedpushdown automata are more powerful than standard pushdown automata but they are far less powerfulthan Turing machines. As Turing machines can be simulated by a finite automaton with two independentstacks, iterated pushdown automata can be viewed as an intermediate device, see also [5] for otherconnections of automata with graph algebras.In this paper, we show an application of this device to the characterization of contour words ofa family of bounded domains in many tilings of the hyperbolic plane. We can do the same kind ofapplication for a tiling of the hyperbolic 3D space and for another one in the hyperbolic 4D space. Thesetwo latter applications cannot be generalized to any dimension as, starting from dimension 5, there is notiling of the hyperbolic space which would be a tessellation generated by a regular polytope.In Section 2, we remember the definition of iterated pushdown automata with an application to thecomputation of the recognition of words of the form a
{"title":"Hyperbolic tilings and formal language theory","authors":"M. Margenstern, K. Subramanian","doi":"10.4204/EPTCS.128.18","DOIUrl":"https://doi.org/10.4204/EPTCS.128.18","url":null,"abstract":"In [11], it was shown that a few languages constructed from some figures of hyperbolic tilings cannotbe recognized by pushdown automata but they can be recognized by a 2-iterated pushdown automaton.Before, it was known that several tessellations of the hyperbolic plane are generated by substitutions,see [3]. This property is also clear from [7].These substitutions can be also described by the use of grammars. This is rather straightforward. In[6], these substitutions appear as rules of a grammar, although the grammar is not formally described.Iterated pushdown automata were introduced in [4, 12] and we refer the reader to [1] for referencesand for the connection of this topic with sequences of rational numbers. By their definition, iteratedpushdown automata are more powerful than standard pushdown automata but they are far less powerfulthan Turing machines. As Turing machines can be simulated by a finite automaton with two independentstacks, iterated pushdown automata can be viewed as an intermediate device, see also [5] for otherconnections of automata with graph algebras.In this paper, we show an application of this device to the characterization of contour words ofa family of bounded domains in many tilings of the hyperbolic plane. We can do the same kind ofapplication for a tiling of the hyperbolic 3D space and for another one in the hyperbolic 4D space. Thesetwo latter applications cannot be generalized to any dimension as, starting from dimension 5, there is notiling of the hyperbolic space which would be a tessellation generated by a regular polytope.In Section 2, we remember the definition of iterated pushdown automata with an application to thecomputation of the recognition of words of the form a","PeriodicalId":340847,"journal":{"name":"Machines, Computations, and Universality","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130337766","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}
In this paper, we explore relationships between two models of systems which are governed by only the local interactions of large collections of simple components: cellular automata (CA) and the abstract Tile Assembly Model (aTAM). While sharing several similarities, the models have fundamental differences, most notably the dynamic nature of CA (in which every cell location is allowed to change state an infinite number of times) versus the static nature of the aTAM (in which tiles are static components that can never change or be removed once they attach to a growing assembly). We work with 2-dimensional systems in both models, and for our results we first define what it means for CA systems to simulate aTAM systems, and then for aTAM systems to simulate CA systems. We use notions of simulate which are similar to those used in the study of intrinsic universality since they are in some sense strict, but also intuitively natural notions of simulation. We then demonstrate a particular nondeterministic CA which can be configured so that it can simulate any arbitrary aTAM system, and finally an aTAM tile set which can be configured so that it can be used to simulate any arbitrary nondeterministic CA system which begins with a finite initial configuration.
{"title":"On the Equivalence of Cellular Automata and the Tile Assembly Model","authors":"Jacob Hendricks, Matthew J. Patitz","doi":"10.4204/EPTCS.128.21","DOIUrl":"https://doi.org/10.4204/EPTCS.128.21","url":null,"abstract":"In this paper, we explore relationships between two models of systems which are governed by only the local interactions of large collections of simple components: cellular automata (CA) and the abstract Tile Assembly Model (aTAM). While sharing several similarities, the models have fundamental differences, most notably the dynamic nature of CA (in which every cell location is allowed to change state an infinite number of times) versus the static nature of the aTAM (in which tiles are static components that can never change or be removed once they attach to a growing assembly). We work with 2-dimensional systems in both models, and for our results we first define what it means for CA systems to simulate aTAM systems, and then for aTAM systems to simulate CA systems. We use notions of simulate which are similar to those used in the study of intrinsic universality since they are in some sense strict, but also intuitively natural notions of simulation. We then demonstrate a particular nondeterministic CA which can be configured so that it can simulate any arbitrary aTAM system, and finally an aTAM tile set which can be configured so that it can be used to simulate any arbitrary nondeterministic CA system which begins with a finite initial configuration.","PeriodicalId":340847,"journal":{"name":"Machines, Computations, and Universality","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130828722","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}
R. Freund, Sergiu Ivanov, Marion Oswald, K. Subramanian
We consider the (one-dimensional) array counterpart of contextual as well as insertion and deletion string grammars and consider the operations of array insertion and deletion in array grammars. First we show that the emptiness problem for P systems with (one-dimensional) insertion rules is undecidable. Then we show computational completeness of P systems using (one-dimensional) array insertion and deletion rules even of norm one only. The main result of the paper exhibits computational completeness of one-dimensional array grammars using array insertion and deletion rules of norm at most two.
{"title":"One-dimensional Array Grammars and P Systems with Array Insertion and Deletion Rules","authors":"R. Freund, Sergiu Ivanov, Marion Oswald, K. Subramanian","doi":"10.4204/EPTCS.128.14","DOIUrl":"https://doi.org/10.4204/EPTCS.128.14","url":null,"abstract":"We consider the (one-dimensional) array counterpart of contextual as well as insertion and deletion string grammars and consider the operations of array insertion and deletion in array grammars. First we show that the emptiness problem for P systems with (one-dimensional) insertion rules is undecidable. Then we show computational completeness of P systems using (one-dimensional) array insertion and deletion rules even of norm one only. The main result of the paper exhibits computational completeness of one-dimensional array grammars using array insertion and deletion rules of norm at most two.","PeriodicalId":340847,"journal":{"name":"Machines, Computations, and Universality","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121840081","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}
Reversible computing is a paradigm of computation that reflects physical reversibility, one of the fundamental microscopic laws of Nature. In this survey, we discuss topics on reversible logic elements with memory (RLEM), which can be used to build reversible computing systems, and their universality. An RLEM is called universal, if any reversible sequential machine (RSM) can be realized as a circuit composed only of it. Since a finite-state control and a tape cell of a reversible Turing machine (RTM) are formalized as RSMs, any RTM can be constructed from a universal RLEM. Here, we investigate 2-state RLEMs, and show that infinitely many kinds of non-degenerate RLEMs are all universal besides only four exceptions. Non-universality of these exceptional RLEMs is also argued.
{"title":"Reversible Logic Elements with Memory and Their Universality","authors":"K. Morita","doi":"10.4204/EPTCS.128.3","DOIUrl":"https://doi.org/10.4204/EPTCS.128.3","url":null,"abstract":"Reversible computing is a paradigm of computation that reflects physical reversibility, one of the fundamental microscopic laws of Nature. In this survey, we discuss topics on reversible logic elements with memory (RLEM), which can be used to build reversible computing systems, and their universality. An RLEM is called universal, if any reversible sequential machine (RSM) can be realized as a circuit composed only of it. Since a finite-state control and a tape cell of a reversible Turing machine (RTM) are formalized as RSMs, any RTM can be constructed from a universal RLEM. Here, we investigate 2-state RLEMs, and show that infinitely many kinds of non-degenerate RLEMs are all universal besides only four exceptions. Non-universality of these exceptional RLEMs is also argued.","PeriodicalId":340847,"journal":{"name":"Machines, Computations, and Universality","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116362043","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}
A universal deterministic inhibitor Petri net with 14 places, 29 transitions and 138 arcs was constructed via simulation of Neary and Woods' weakly universal Turing machine with 2 states and 4 symbols; the total time complexity is exponential in the running time of their weak machine. To simulate the blank words of the weakly universal Turing machine, a couple of dedicated transitions insert their codes when reaching edges of the working zone. To complete a chain of a given Petri net encoding to be executed by the universal Petri net, a translation of a bi-tag system into a Turing machine was constructed. The constructed Petri net is universal in the standard sense; a weaker form of universality for Petri nets was not introduced in this work.
{"title":"A Small Universal Petri Net","authors":"D. Zaitsev","doi":"10.4204/EPTCS.128.22","DOIUrl":"https://doi.org/10.4204/EPTCS.128.22","url":null,"abstract":"A universal deterministic inhibitor Petri net with 14 places, 29 transitions and 138 arcs was constructed via simulation of Neary and Woods' weakly universal Turing machine with 2 states and 4 symbols; the total time complexity is exponential in the running time of their weak machine. To simulate the blank words of the weakly universal Turing machine, a couple of dedicated transitions insert their codes when reaching edges of the working zone. To complete a chain of a given Petri net encoding to be executed by the universal Petri net, a translation of a bi-tag system into a Turing machine was constructed. The constructed Petri net is universal in the standard sense; a weaker form of universality for Petri nets was not introduced in this work.","PeriodicalId":340847,"journal":{"name":"Machines, Computations, and Universality","volume":"8 6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116799804","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}
Whether P systems with only one catalyst can already be computationally complete, is still an open problem. Here we establish computational completeness by using specific variants of additional control mechanisms. At each step using only multiset rewriting rules from one set of a finite number of sets of multiset rewriting rules allows for obtaining computational completeness with one catalyst and only one membrane. If the targets are used for choosing the multiset of rules to be applied, for getting computational completeness with only one catalyst more than one membrane is needed. If the available sets of rules change periodically with time, computational completeness can be obtained with one catalyst in one membrane. Moreover, we also improve existing computational completeness results for P systems with mobile catalysts and for P systems with membrane creation.
{"title":"How to Obtain Computational Completeness in P Systems with One Catalyst","authors":"R. Freund, G. Paun","doi":"10.4204/EPTCS.128.13","DOIUrl":"https://doi.org/10.4204/EPTCS.128.13","url":null,"abstract":"Whether P systems with only one catalyst can already be computationally complete, is still an open problem. Here we establish computational completeness by using specific variants of additional control mechanisms. At each step using only multiset rewriting rules from one set of a finite number of sets of multiset rewriting rules allows for obtaining computational completeness with one catalyst and only one membrane. If the targets are used for choosing the multiset of rules to be applied, for getting computational completeness with only one catalyst more than one membrane is needed. If the available sets of rules change periodically with time, computational completeness can be obtained with one catalyst in one membrane. Moreover, we also improve existing computational completeness results for P systems with mobile catalysts and for P systems with membrane creation.","PeriodicalId":340847,"journal":{"name":"Machines, Computations, and Universality","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129795424","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 exhibit a weakly aperiodic tile set for Baumslag-Solitar groups, and prove that the domino problem is undecidable on these groups. A consequence of our construction is the existence of an arecursive tile set on Baumslag-Solitar groups.
{"title":"Tiling Problems on Baumslag-Solitar groups","authors":"N. Aubrun, J. Kari","doi":"10.4204/EPTCS.128.12","DOIUrl":"https://doi.org/10.4204/EPTCS.128.12","url":null,"abstract":"We exhibit a weakly aperiodic tile set for Baumslag-Solitar groups, and prove that the domino problem is undecidable on these groups. A consequence of our construction is the existence of an arecursive tile set on Baumslag-Solitar groups.","PeriodicalId":340847,"journal":{"name":"Machines, Computations, and Universality","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122777533","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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