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Generators and bases for algebras over a monad
[article]

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). ...

##
###
Automata on ordinals and automaticity of linear orders

2013
*
Annals of Pure and Applied Logic
*

We investigate structures recognizable by finite state

doi:10.1016/j.apal.2012.11.007
fatcat:dddk6zpt7raubl7pg6cuec7a6y
*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*= Σ*i*ω*α**i*and β = Σjω β j in Cantor normal form, the*commutative*sum ...##
###
Wreath operations in the group of automorphisms of the binary tree

2002
*
Journal of Algebra
*

Indeed, G contains a normal subgroup N which is an infinite direct sum

doi:10.1016/s0021-8693(02)00050-9
fatcat:eghbz6qhybenhov6ezltras5ea
*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*...##
###
Extending the scalars of minimizations
[article]

2006
*
arXiv
*
pre-print

In order

arXiv:math/0607411v1
fatcat:izg3unz6ejdpfhc3h2psjigycu
*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*α*u =*i**α**i*v*i*(*with**α*∈ K − {0}) then the rank*of*Span(V ...##
###
A Bialgebraic Approach to Automata and Formal Language Theory
[article]

2009
*
arXiv
*
pre-print

For example, formal languages correspond

arXiv:0807.4553v4
fatcat:tmkt6hbznzeblf2im25gvuf3za
*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. ...##
###
A Bialgebraic Approach to Automata and Formal Language Theory
[chapter]

2008
*
Lecture Notes in Computer Science
*

We then treat

doi:10.1007/978-3-540-92687-0_31
fatcat:lapdyiao2bdkdiy4ebmj6p2y2i
*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. ...##
###
A bialgebraic approach to automata and formal language theory

2012
*
Annals of Pure and Applied Logic
*

We then treat

doi:10.1016/j.apal.2011.09.019
fatcat:tc36r5rksffqhkiifosaeaa2o4
*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. ...##
###
Synchronous composition of interval weighted automata

2010
*
IFAC Proceedings Volumes
*

Their synchronous composition, which coincides

doi:10.3182/20100830-3-de-4013.00053
fatcat:f2p5tgbprfdrvomhpljfihcbga
*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 ...##
###
Remarks on the structure of Clifford quantum cellular automata
[article]

2008
*
arXiv
*
pre-print

This means that tensor

arXiv:0812.0714v1
fatcat:choo33d77vcj5px5khikpiua5i
*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 ...##
###
Cellular automata over algebraic structures
[article]

2021
*
arXiv
*
pre-print

When A is entropic (i.e. all finitary operations are homomorphisms), we establish that the set EndCA(G;A), consisting

arXiv:1908.09675v3
fatcat:wz2sj7aikfgxhb5vpwhz4pmuru
*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). ...##
###
Heap Models, Composition and Control
[chapter]

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 ...

##
###
Index Theory of One Dimensional Quantum Walks and Cellular Automata

2012
*
Communications in Mathematical Physics
*

If a

doi:10.1007/s00220-012-1423-1
fatcat:crdsr5posfamrbkqpmjnidpemi
*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. ...##
###
Rational and Recognisable Power Series
[chapter]

2009
*
Monographs in Theoretical Computer Science
*

The canonical

doi:10.1007/978-3-642-01492-5_4
fatcat:dki7cvharbcv5hk2xgpr3hhqmq
*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] ...##
###
Generalized Linear Cellular Automata in Groups and Difference Galois Theory
[article]

2013
*
arXiv
*
pre-print

Generalized non-autonomous linear celullar

arXiv:1308.1186v1
fatcat:jyxjvwyz45cqda2qilhi65ia2a
*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. ...
In this paper we show how

doi:10.1145/2603088.2603095
dblp:conf/csl/GrathwohlKM14
fatcat:dft4oqf4gzc5pehfiq4dtaugmq
*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*...
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