Sofic Tree-Shifts

We introduce the notion of sofic tree-shifts which corresponds to symbolic dynamical systems of infinite ranked trees accepted by finite tree automata. We show that, contrary to shifts of infinite sequences, there is no unique reduced deterministic irreducible tree automaton accepting an irreducible...

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Veröffentlicht in:	Theory of computing systems Jg. 53; H. 4; S. 621 - 644
Hauptverfasser:	Aubrun, Nathalie, Béal, Marie-Pierre
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Boston Springer US 01.11.2013 Springer Nature B.V Springer Verlag
Schlagworte:	Algorithms Analysis Automation Computer Science Data Structures and Algorithms Delay Design engineering Dynamical systems Invariants Mathematical analysis Studies Symbolism Theory of Computation Trees Symbolic dynamics Tree-shift Tree automata
ISSN:	1432-4350, 1433-0490
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Abstract	We introduce the notion of sofic tree-shifts which corresponds to symbolic dynamical systems of infinite ranked trees accepted by finite tree automata. We show that, contrary to shifts of infinite sequences, there is no unique reduced deterministic irreducible tree automaton accepting an irreducible sofic tree-shift, but that there is a unique synchronized one, called the Fischer automaton of the tree-shift. We define the notion of almost of finite type tree-shift which are sofic tree-shifts accepted by a tree automaton which is both deterministic and co-deterministic with a finite delay. It is a meaningful intermediate dynamical class in between irreducible finite type tree-shifts and irreducible sofic tree-shifts. We characterize the Fischer automaton of an almost of finite type tree-shift and we design an algorithm to check whether a sofic tree-shift is almost of finite type. We prove that the Fischer automaton is a topological conjugacy invariant of the underlying irreducible sofic tree-shift.
AbstractList	We introduce the notion of sofic tree-shifts which corresponds to symbolic dynamical systems of infinite ranked trees accepted by finite tree automata. We show that, contrary to shifts of infinite sequences, there is no unique reduced deterministic irreducible tree automaton accepting an irreducible sofic tree-shift, but that there is a unique synchronized one, called the Fischer automaton of the tree-shift. We define the notion of almost of finite type tree-shift which are sofic tree-shifts accepted by a tree automaton which is both deterministic and co-deterministic with a finite delay. It is a meaningful intermediate dynamical class in between irreducible finite type tree-shifts and irreducible sofic tree-shifts. We characterize the Fischer automaton of an almost of finite type tree-shift and we design an algorithm to check whether a sofic tree-shift is almost of finite type. We prove that the Fischer automaton is a topological conjugacy invariant of the underlying irreducible sofic tree-shift. We introduce the notion of Sofic Tree-Shifts which corresponds to symbolic dynamical systems of infinite ranked trees accepted by finite tree automata. We show that, contrary to shifts of infinite sequences, there is no unique reduced deterministic irreducible tree automaton accepting an irreducible sofic tree-shift, but that there is a unique synchronized one, called the Fischer automaton of the tree-shift. We define the notion of almost of finite type tree-shift which are Sofic Tree-Shifts accepted by a tree automaton which is both deterministic and co-deterministic with a finite delay. It is a meaningful intermediate dynamical class in between irreducible finite type tree-shifts and irreducible Sofic Tree-Shifts. We characterize the Fischer automaton of an almost of finite type tree-shift and we design an algorithm to check whether a sofic tree-shift is almost of finite type. We prove that the Fischer automaton is a topological conjugacy invariant of the underlying irreducible sofic tree-shift.[PUBLICATION ABSTRACT] We introduce the notion of soﬁc tree-shifts which corresponds to symbolic dynamical systems of inﬁnite ranked trees accepted by ﬁnite tree automata. We show that, contrary to shifts of inﬁnite sequences, there is no unique reduced deterministic irreducible tree automaton accepting an irreducible soﬁc tree-shift, but that there is a unique synchronized one, called the Fischer automaton of the tree-shift. We deﬁne the notion of almost of ﬁnite type tree-shift which are soﬁc tree-shifts accepted by a tree automaton which is both deterministic and co-deterministic with a ﬁnite delay. It is a meaningful intermediate dynamical class in between irreducible ﬁnite type tree-shifts and irreducible soﬁc tree-shifts. We characterize the Fischer automaton of an almost of ﬁnite type tree-shift and we design an algorithm to check whether a soﬁc tree-shift is almost of ﬁnite type. We prove that the Fischer automaton is a topological conjugacy invariant of the underlying irreducible soﬁc shift.
Author	Béal, Marie-Pierre Aubrun, Nathalie
Author_xml	– sequence: 1 givenname: Nathalie surname: Aubrun fullname: Aubrun, Nathalie organization: LIP, UMR 5668, ENS de Lyon - CNRS - UCBL - INRIA – sequence: 2 givenname: Marie-Pierre surname: Béal fullname: Béal, Marie-Pierre email: beal@univ-mlv.fr organization: Laboratoire d’informatique Gaspard-Monge, UMR 8049, CNRS, Université Paris-Est
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Cites_doi	10.1090/S0002-9939-1988-0958076-3 10.1109/TIT.1985.1057037 10.1090/S0002-9939-99-04678-X 10.1016/j.tcs.2005.07.007 10.1007/s11856-011-0107-4 10.1007/978-3-662-07675-0_10 10.1007/3-540-51498-8_38 10.1109/TPAMI.2005.144 10.1109/18.340457 10.1016/j.tcs.2012.07.020 10.1007/BF02760631 10.1007/978-3-642-59126-6_7 10.1016/S0020-0190(02)00285-5 10.1017/S014338570200113X 10.1007/BFb0061747 10.1090/S0002-9939-1985-0806078-7 10.1007/978-3-642-02927-1_13 10.1017/CBO9780511626302 10.1007/978-3-642-58822-8 10.1017/CBO9781139195218 10.1007/978-3-642-13182-0_2
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Keywords	Symbolic dynamics Tree-shift Tree automata
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Snippet	We introduce the notion of sofic tree-shifts which corresponds to symbolic dynamical systems of infinite ranked trees accepted by finite tree automata. We show... We introduce the notion of Sofic Tree-Shifts which corresponds to symbolic dynamical systems of infinite ranked trees accepted by finite tree automata. We show... We introduce the notion of soﬁc tree-shifts which corresponds to symbolic dynamical systems of inﬁnite ranked trees accepted by ﬁnite tree automata. We show...
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SubjectTerms	Algorithms Analysis Automation Computer Science Data Structures and Algorithms Delay Design engineering Dynamical systems Invariants Mathematical analysis Studies Symbolism Theory of Computation Trees
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Title	Sofic Tree-Shifts
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