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Yet another proof of the cascade decomposition theorem for finite automata: Correction

Paul Zeiger
1968 Mathematical Systems Theory  
to a cascade of permutation-reset automata, Method IIA finishes the job.  ...  To salvage the proof we eliminate the resets from this first component by modifying the method as follows: Let st B1 = V instead of T, then whenever u is in T, let p' = p (instead of u), and r' = p-1 (  ...  to a cascade of permutation-reset automata, Method IIA finishes the job.  ... 
doi:10.1007/bf01703267 fatcat:7swaj6y44baljnwjja5spnrclm

Classifying Regular Languages via Cascade Products of Automata [chapter]

Marcus Gelderie
2011 Lecture Notes in Computer Science  
Building on the celebrated Krohn-Rhodes Theorem we characterize classes of regular languages in terms of the cascade decompositions of minimal DFA of languages in those classes.  ...  Next, we introduce the concept of the scope of a cascade product of reset automata in order to capture a notion of locality inside a cascade product and show that there exist constant bounds on the scope  ...  Acknowledgments The author thanks his supervisor Wolfgang Thomas and the reviewers of this paper for their advice and many helpful suggestions.  ... 
doi:10.1007/978-3-642-21254-3_22 fatcat:ut5xnkhm4faidmrawqpp5aerjq

On the Krohn-Rhodes Cascaded Decomposition Theorem [chapter]

Oded Maler
2010 Lecture Notes in Computer Science  
The Krohn-Rhodes theorem states that any deterministic automaton is a homomorphic image of a cascade of very simple automata which realize either resets or permutations.  ...  Moreover, if the automaton is counter-free, only reset automata are needed. In this paper we give a very constructive proof of a variant of this theorem due to Eilenberg.  ...  Among the numerous proofs of the Krohn-Rhodes primary decomposition theorem those of Zeiger [15, 16] were more automata oriented.  ... 
doi:10.1007/978-3-642-13754-9_12 fatcat:ei4c25zbcnhmveg4liv7wdn5ye

Decompositions and complexity of linear automata [article]

Boris Plotkin, Tatjana Plotkin
2015 arXiv   pre-print
The wreath product is the terminal object in the Krohn-Rhodes theory for pure automata, while the triangular product provides the terminal object for the cascade connections of linear automata.  ...  The Krohn-Rhodes complexity theory for pure (without linearity) automata is well-known. This theory uses an operation of wreath product as a decomposition tool.  ...  The triangular product of automata is an associative operation, used for the decomposition of linear automata. Another necessary operation is defined in the next section.  ... 
arXiv:1506.06017v1 fatcat:5cacehzihvbuzdh6bjbhsugnjm

Some results on cascade decomposition of automata

B.G. Reynolds
1969 Information and Control  
This paper deals with the conditions causing such permutations in the cascade, and considers decomposition of definite automata.  ...  Zeiger's (1967) procedure for decomposing a finite-state machine M yields a cascade of permutation-reset machines.  ...  DECOMPOSITION OF DEFINITE AUTOMATA Cutlip (Jan. 1968 ) pointed out that prefix automata, i.e., automata which recognize events of the form IM*Z where Z ~ IM k for some positive integer ]c, decompose into  ... 
doi:10.1016/s0019-9958(69)90171-5 fatcat:unapp3xb3fe3xowby54thrt56m

An analysis of errors in finite automata

Philip S. Dauber
1965 Information and Control  
ts .are then given on the partitioning properties of a particular type of error called a finite error.  ...  This p~/p~r Studies errors in finite automatai An error is defi~ed as a pair Of states and errors are then classified according to their pr~b~ ability of being corrected (i.e., being taken into the same  ...  An immediate consequence of this theorem is a decomposition of the automaton as follows.  ... 
doi:10.1016/s0019-9958(65)90223-8 fatcat:juwdqi6zl5bxhk5fpev4b3bv6m

Asynchronous wreath product and cascade decompositions for concurrent behaviours [article]

Bharat Adsul, Paul Gastin, Saptarshi Sarkar, Pascal Weil
2022 arXiv   pre-print
We then propose a decomposition result for recognizable trace languages, analogous to the Krohn-Rhodes theorem, and we prove this decomposition result in the special case of acyclic architectures.  ...  Over distributed alphabets for which the asynchronous Krohn-Rhodes theorem holds, a local cascade product of such automata is sufficient and this, in turn, leads to the identification of a simple temporal  ...  sequences of localized reset asynchronous automata, yet another characterization of this language class.  ... 
arXiv:2105.10897v4 fatcat:ktoojpvkhnekxoii3do4mvp63m

On the Translation of Automata to Linear Temporal Logic [article]

Udi Boker, Karoliina Lehtinen, Salomon Sickert
2022 arXiv   pre-print
an intermediate Krohn-Rhodes cascade decomposition of the automaton.  ...  While the complexity of translating future linear temporal logic (LTL) into automata on infinite words is well-understood, the size increase involved in turning automata back to LTL is not.  ...  We thank Moshe Vardi and Orna Kupferman for suggesting studying the succinctness gap between semantic and syntactic safe formulas, and Miko laj Bojańczyk for answering our questions on algebraic automata  ... 
arXiv:2201.10267v1 fatcat:bbd7ejst6fe6nnkcczbw77zmme

Symmetry structure in discrete models of biochemical systems: natural subsystems and the weak control hierarchy in a new model of computation driven by interactions

Chrystopher L. Nehaniv, John Rhodes, Attila Egri-Nagy, Paolo Dini, Eric Rothstein Morris, Gábor Horváth, Fariba Karimi, Daniel Schreckling, Maria J. Schilstra
2015 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences  
One contribution of 13 to a Theo Murphy meeting issue 'Heterotic computing: exploiting hybrid computational devices' .  ...  Subject Areas: theory of computing, computational biology, computational chemistry, computer modelling and simulation, systems theory  ...  38 We remark that in this Petri net for the cell cycle there are non-trivial automorphisms (symmetries).  ... 
doi:10.1098/rsta.2014.0223 pmid:26078349 fatcat:x6khqxvf4bgidicrsb7gztb67e

Generating the Functions with Regular Graphs under Composition [article]

Thomas Kern
2016 arXiv   pre-print
This will include two clear proofs of the Krohn-Rhodes Theorem in modern notation.  ...  While automata theory often concerns itself with regular predicates, relations corresponding to acceptance by a finite state automaton, in this article we study the regular functions, such relations which  ...  Acknowledgment The author wishes to acknowledge fruitful discussions with Scott Messick. This work would not have been possible without encouragement and guidance from my Ph.D. advisor, Anil Nerode.  ... 
arXiv:1610.07535v1 fatcat:ujqix57ywfcz5e52by23gkqffq

Hierarchical Coordinate Systems for Understanding Complexity and its Evolution, with Applications to Genetic Regulatory Networks

Attila Egri-Nagy, Chrystopher L. Nehaniv
2008 Artificial Life  
Hierarchical decomposition is a useful way of assessing structural complexity changes of organisms modeled as automata, and we show how recently developed computational tools can be used for this purpose  ...  As a proof of concept, we illustrate how the hierarchical complexity analysis reveals symmetries and irreversible structure in biological networks by applying the methods to the lac operon mechanism in  ...  Hierarchical Decomposition: The Krohn-Rhodes Theory Here we present the very basic underlying ideas of algebraic hierarchical decomposition of finite state automata.  ... 
doi:10.1162/artl.2008.14.3.14305 pmid:18489252 fatcat:675h5wpblrfc5ayk5cqvismxhm

Effective theories for circuits and automata

Simon DeDeo
2011 Chaos  
We use the Krohn-Rhodes theorem to show how the composition of underlying mechanisms can lead to innovations in the emergent effective theory.  ...  Here we demonstrate the construction of effective theories in the presence of both irreversibility and noise, in a dynamical model with underlying feedback.  ...  The surprising result of the Krohn-Rhodes theorem 4,20 is that any finite state automaton can be decomposed into a cascade, whose automata contains only the simple groups and the flip-flop.  ... 
doi:10.1063/1.3640747 pmid:21974669 fatcat:t2r4fmtvovhyzhacz2h3thhunu

Topics in the theory of one-dimensional iterative networks

William L. Kilmer
1964 Information and Control  
Consider a cascade of just a parts of Fig. 3 -type automata as shown in Fig. 7 .  ...  Fro. 8 . 8 Partially filled-in network used in theorem proof FIG. 9 . 9 General :aA network used in nondecomposition *heorem proof LE~hCA 2. NOT YET HALTED.  ... 
doi:10.1016/s0019-9958(64)90098-1 fatcat:37gp3yoelfferlwilcxc7ukika

Algebraic properties of automata associated to Petri nets and applications to computation in biological systems

Attila Egri-Nagy, Chrystopher L. Nehaniv
2008 Biosystems (Amsterdam. Print)  
ways in the realm of automata.  ...  We study the algebraic structure of the computations carried out by Petri nets from the viewpoint of algebraic automata theory.  ...  The "excess material" leaves the system, yet another way to represent the environmental output.  ... 
doi:10.1016/j.biosystems.2008.05.019 pmid:18611428 fatcat:j57g7oaspvdi7bsfdzttc5p3im

Exploring the concept of interaction computing through the discrete algebraic analysis of the Belousov–Zhabotinsky reaction

Paolo Dini, Chrystopher L. Nehaniv, Attila Egri-Nagy, Maria J. Schilstra
2013 Biosystems (Amsterdam. Print)  
of the Krohn-Rhodes decomposition of finite-state automata, including the holonomy decomposition of a simple automaton, and of its interpretation as an abstract positional number system.  ...  The method is then applied to the analysis of the algebraic properties of discrete finite-state automata derived from a simplified Petri Net model of the BZ reaction.  ...  The Krohn-Rhodes and Holonomy Theorems The Krohn-Rhodes prime decomposition theorem for finite automata [30] has been discussed, explained, and applied in a large number of books and articles since the  ... 
doi:10.1016/j.biosystems.2013.03.003 pmid:23499885 fatcat:p6povzuzinde7nsj7z7vjakiva
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