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Generators and bases for algebras over a monad [article]

Stefan Zetzsche, Alexandra Silva, Matteo Sammartino
2021 arXiv   pre-print
On the one hand, we establish purely algebraic results, for instance about the existence and uniqueness of generators and bases, the representation of algebra homomorphisms, and the product of generators  ...  As a result, we are able to recover known constructions from automata theory such as the canonical residual finite state automaton and the minimal xor automaton.  ...  This research has been supported by GCHQ via the VeTSS grant "Automated black-box verification of networking systems" (4207703/RFA 15845).  ... 
arXiv:2010.10223v4 fatcat:vmfekuxqjzckblpgghle34xlyq

Automata on ordinals and automaticity of linear orders

Philipp Schlicht, Frank Stephan
2013 Annals of Pure and Applied Logic  
We investigate structures recognizable by finite state automata with an input tape of length a limit ordinal.  ...  At limits, the set of states which appear unboundedly often before the limit are mapped to a limit state.  ...  For each word |v| of length α, we record the states s(v) of all automata on inputs tuples in {v} ∪ {(b i ) j : 5 Recall that for α = Σiω α i and β = Σjω β j in Cantor normal form, the commutative sum  ... 
doi:10.1016/j.apal.2012.11.007 fatcat:dddk6zpt7raubl7pg6cuec7a6y

Wreath operations in the group of automorphisms of the binary tree

A.M. Brunner, Said Sidki
2002 Journal of Algebra  
Indeed, G contains a normal subgroup N which is an infinite direct sum of copies of the derived group H and the quotient group G/N is isomorphic to H K.  ...  A new operation called tree-wreathing is defined on groups of automorphisms of the binary tree.  ...  The group K is shown to be generated by a certain set {α(i) | 0 i r − 1} such that the closure K of K in A with respect to its pro-2 topology as K = { α(i) | 0 i r − 1} where each α(i) is isomorphic to  ... 
doi:10.1016/s0021-8693(02)00050-9 fatcat:eghbz6qhybenhov6ezltras5ea

Extending the scalars of minimizations [article]

Gérard Duchamp, Jean-Gabriel Luque
2006 arXiv   pre-print
In order to get an efficient encoding, it may be interesting to point out one of them with the smallest number of states.  ...  Minimal automata have been proved to be isomorphic in cases (a) and (b). But the proof given for (b) is not constructive. In fact, it lays on the existence of a basis for a submodule of ^n.  ...  Or again, this is equivalent to respect the following condition: for each n, m ∈ N + , if V = {v i } i∈[1,m] ⊆ K n is linearly independent and αu = i α i v i (with α ∈ K − {0}) then the rank of Span(V  ... 
arXiv:math/0607411v1 fatcat:izg3unz6ejdpfhc3h2psjigycu

A Bialgebraic Approach to Automata and Formal Language Theory [article]

James Worthington
2009 arXiv   pre-print
For example, formal languages correspond to elements of dual algebras of coalgebras, automata are "pointed representation objects" of algebras, automaton morphisms are instances of linear intertwiners,  ...  and a construction from the theory of bialgebras shows how to run two automata in parallel.  ...  By a standard argument, the tensor product is unique up to isomorphism. We now construct the tensor product of semimodules over a commutative semiring.  ... 
arXiv:0807.4553v4 fatcat:tmkt6hbznzeblf2im25gvuf3za

A Bialgebraic Approach to Automata and Formal Language Theory [chapter]

James Worthington
2008 Lecture Notes in Computer Science  
We then treat automata as certain representation objects of algebras and formal languages as elements of dual algebras of coalgebras.  ...  In this paper, we apply the defining diagrams of algebras, coalgebras, and bialgebras to categories of semimodules and semimodule homomorphisms over a commutative semiring.  ...  By a standard argument, the tensor product is unique up to isomorphism. We now construct the tensor product of semimodules over a commutative semiring.  ... 
doi:10.1007/978-3-540-92687-0_31 fatcat:lapdyiao2bdkdiy4ebmj6p2y2i

A bialgebraic approach to automata and formal language theory

James Worthington
2012 Annals of Pure and Applied Logic  
We then treat automata as certain representation objects of algebras and formal languages as elements of dual algebras of coalgebras.  ...  In this paper, we apply the defining diagrams of algebras, coalgebras, and bialgebras to categories of semimodules and semimodule homomorphisms over a commutative semiring.  ...  Acknowledgements The author would like to thank Anil Nerode for many inspiring discussions and the anonymous reviewers of LFCS'09 for their helpful comments and suggestions.  ... 
doi:10.1016/j.apal.2011.09.019 fatcat:tc36r5rksffqhkiifosaeaa2o4

Synchronous composition of interval weighted automata

Jan Komenda, Sébastien Lahaye, Jean-Louis Boimond
2010 IFAC Proceedings Volumes  
Their synchronous composition, which coincides with the synchronous product of underlying (one clock) timed automata, results in multi-event interval weighted automata, whose behaviors are studied.  ...  Interval weighted automata are introduced as automata with weights in a product dioid (idempotent semiring).  ...  CONCLUSION We have shown how techniques based on semirings of interval and tensor linear algebra enable modeling of an  ... 
doi:10.3182/20100830-3-de-4013.00053 fatcat:f2p5tgbprfdrvomhpljfihcbga

Remarks on the structure of Clifford quantum cellular automata [article]

Dirk-Michael Schlingemann
2008 arXiv   pre-print
This means that tensor products of Weyl operators (projective representation of a finite abelian symplectic group) are mapped to multiples of tensor products of Weyl operators.  ...  We characterize these symplectic cellular automata and find that all possible local rules must be, up to some global shift, reflection invariant with respect to the origin.  ...  The group of Clifford quantum cellular automata acting on a ddimensional lattice with single cell algebras M p (C) is isomorphic to the direct product Z d × SL(2, P p,d ) of the lattice translation group  ... 
arXiv:0812.0714v1 fatcat:choo33d77vcj5px5khikpiua5i

Cellular automata over algebraic structures [article]

Alonso Castillo-Ramirez, O. Mata-Gutiérrez, Angel Zaldivar-Corichi
2021 arXiv   pre-print
When A is entropic (i.e. all finitary operations are homomorphisms), we establish that the set EndCA(G;A), consisting of all such cellular automata, is isomorphic to the direct limit of Hom(A^S, A), where  ...  Let G be a group and A a set equipped with a collection of finitary operations.  ...  We endow A G with the prodiscrete topology, which is the product topology of the discrete topology on A (see [13] for a comprehensive introduction to product topologies).  ... 
arXiv:1908.09675v3 fatcat:wz2sj7aikfgxhb5vpwhz4pmuru

Heap Models, Composition and Control [chapter]

Jan Komenda, Sebastien Lahaye, Jean-Louis Boimo
2008 New Developments in Robotics Automation and Control  
One has to increase the number of clocks in order to define a synchronous product of (max,+) automata viewed as 1-clock timed automata.  ...  (Synchronous product of heap models) Let ( ) 1,2 = i , u , l , r , P , A i i i i i i = ℜ be two heap models with ∅ = ∩ 2 1 P P .  ...  In the above computation we use the simple fact that for two series (here polynomial) s and t with t s ≤ , e s ≥ , and e t ≥ we have * * This book represents the contributions of the top researchers in  ... 
doi:10.5772/6271 fatcat:p3ofkdnv3nhchexxqurd26qho4

Index Theory of One Dimensional Quantum Walks and Cellular Automata

D. Gross, V. Nesme, H. Vogts, R. F. Werner
2012 Communications in Mathematical Physics  
If a one-dimensional quantum lattice system is subject to one step of a reversible discrete-time dynamics, it is intuitive that as much "quantum information" as moves into any given block of cells from  ...  the left, has to exit that block to the right.  ...  We gratefully acknowledge the support of the DFG (Forschergruppe 635) and the EU (projects CORNER and QUICS), as well as the Erwin Schrödinger Institute.  ... 
doi:10.1007/s00220-012-1423-1 fatcat:crdsr5posfamrbkqpmjnidpemi

Rational and Recognisable Power Series [chapter]

Jacques Sakarovitch
2009 Monographs in Theoretical Computer Science  
The canonical isomorphisms Polynomials or series in several (commutative) variables can be ordered with respect to one or another variable.  ...  If P is not a word base for A , it exists m coefficients α i such that i=m i=1 α i (λ · p i µ ) = 0 but multiplying this equality on the left by X yields i=m i=1 α i (λ · p i µ) = 0 a contradiction (with  ...  In [19] , it was also observed that Schützenberger's reduction algorithm applies to the case of series on a skew field, but with a reference to a previous theory of non-commutative determinants [37]  ... 
doi:10.1007/978-3-642-01492-5_4 fatcat:dki7cvharbcv5hk2xgpr3hhqmq

Generalized Linear Cellular Automata in Groups and Difference Galois Theory [article]

David Blazquez-Sanz, Weimar Muñoz
2013 arXiv   pre-print
Generalized non-autonomous linear celullar automata are systems of linear difference equations with many variables that can be seen as convolution equations in a discrete group.  ...  We study those systems from the stand point of the Galois theory of difference equations and discrete Fourier transform.  ...  Acknoledgements We would like to thank to G. Casale, M. F. Singer and M. Wibmer for their comments and discussions on the subject of the article.  ... 
arXiv:1308.1186v1 fatcat:jyxjvwyz45cqda2qilhi65ia2a

KAT + B!

Niels Bjørn Bugge Grathwohl, Dexter Kozen, Konstantinos Mamouras
2014 Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) - CSL-LICS '14  
In this paper we show how to axiomatically extend KAT with this extra feature in the form of mutable tests.  ...  We give several results on deductive completeness and complexity of the system, as well as some examples of its use.  ...  If K is a KAT and F is any KAT of binary relations on a finite set S, then the commutative coproduct K) , and k and f compose with this isomorphism to give the canonical injections from K and F , respectively  ... 
doi:10.1145/2603088.2603095 dblp:conf/csl/GrathwohlKM14 fatcat:dft4oqf4gzc5pehfiq4dtaugmq
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