Polishness of Some Topologies Related to Automata

We prove that the B\"uchi topology, the automatic topology, the alphabetic topology and the strong alphabetic topology are Polish, and provideconsequences of this. We also show that this cannot be fully extended to the case of a space of infinite labelled binary trees; in particular the B\"uchi and the Muller topologies in that case are not Polish.

[1]  Olivier Finkel,et al.  The Complexity of Infinite Computations In Models of Set Theory , 2009, Log. Methods Comput. Sci..

[2]  Filip Murlak,et al.  On the topological complexity of tree languages , 2008, Logic and Automata.

[3]  Ludwig Staiger.  On the Power of Reading the Whole Infinite Input Tape , 1999, Grammars.

[4]  Victor L. Selivanov,et al.  Wadge degrees of ω-languages of deterministic Turing machines , 2003 .

[5]  Jean-Eric Pin,et al.  Infinite words - automata, semigroups, logic and games , 2004, Pure and applied mathematics series.

[6]  Ines Klimann,et al.  A Connected 3-State Reversible Mealy Automaton Cannot Generate an Infinite Burnside Group , 2014, DLT.

[7]  Pierre Simonnet.  Automates et theorie descriptive , 1992 .

[8]  Thomas Wilke,et al.  Automata logics, and infinite games: a guide to current research , 2002 .

[9]  Calvin C. Elgot,et al.  Review: George H. Mealy, A Method for Synthesizing Sequential Circuits , 1957 .

[10]  Ludwig Staiger,et al.  Topologies refining the Cantor topology on X ω , 2010 .

[11]  Lawrence H. Landweber,et al.  Decision problems forω-automata , 1969, Mathematical systems theory.

[12]  Stevo Todorcevic,et al.  BOREL CHROMATIC NUMBERS , 1999 .

[13]  M. Lothaire.  Algebraic Combinatorics on Words , 2002 .

[14]  Thibault Godin,et al.  On Torsion-Free Semigroups Generated by Invertible Reversible Mealy Automata , 2014, LATA.

[15]  Ludwig Staiger,et al.  Shift-invariant topologies for the Cantor space Xω , 2017, Theor. Comput. Sci..

[16]  A. Kechris.  Classical descriptive set theory , 1987 .

[17]  I. Bondarenko,et al.  On the conjugacy problem for finite-state automorphisms of regular rooted trees , 2010, 1011.2227.

[18]  Ines Klimann,et al.  Automaton Semigroups: The Two-state Case , 2016, Theory of Computing Systems.

[19]  Thibault Godin,et al.  Connected Reversible Mealy Automata of Prime Size Cannot Generate Infinite Burnside Groups , 2016, MFCS.

[20]  Ievgen Bondarenko,et al.  Finite-state self-similar actions of nilpotent groups , 2011, 1105.4969.

[21]  Olivier Carton,et al.  Polishness of Some Topologies Related to Automata , 2017, CSL.

[22]  J. R. Büchi.  On a Decision Method in Restricted Second Order Arithmetic , 1990 .

[23]  Volker Diekert,et al.  Fragments of First-Order Logic over Infinite Words , 2009, Theory of Computing Systems.

[24]  Su Gao.  Invariant Descriptive Set Theory , 2008 .

[25]  Olivier Finkel,et al.  The wadge hierarchy of Petri Nets ω-languages , 2014, Logic, Computation, Hierarchies.

[26]  Olivier Finkel,et al.  Borel ranks and Wadge degrees of context free $\omega$-languages , 2006, Mathematical Structures in Computer Science.

[27]  Laurent Bartholdi,et al.  Algorithmic Decidability of Engel's Property for Automaton Groups , 2015, CSR.

[28]  Olivier Finkel,et al.  Topology and Ambiguity in Omega Context Free Languages , 2008, ArXiv.

[29]  Dominique Lecomte,et al.  Potential Wadge Classes , 2010, 1002.1455.

[30]  Rina S. Cohen,et al.  omega-Computations on Turing Machines , 1978, Theor. Comput. Sci..

[31]  Andrey Nikolaev,et al.  Knapsack problems in products of groups , 2014, J. Symb. Comput..

[32]  R. McNaughton.  Review: J. Richard Buchi, Weak Second-Order Arithmetic and Finite Automata; J. Richard Buchi, On a Decision Method in Restricted second Order Arithmetic , 1963, Journal of Symbolic Logic.

[33]  Wolfgang Thomas,et al.  Automata on Infinite Objects , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[34]  Wolfgang Thomas,et al.  Logical Specifications of Infinite Computations , 1993, REX School/Symposium.

[35]  Markus Lohrey,et al.  Knapsack in graph groups, HNN-extensions and amalgamated products , 2016, STACS.

[36]  Alain Louveau,et al.  Ensembles analytiques et bor?eliens dans les espaces produit , 1980 .

[37]  Équipe de Logique Mathématique.  Classical and Effective Descriptive Complexities of ω-Powers , 2009 .

[38]  Z. Sunic,et al.  The conjugacy problem in automaton groups is not solvable , 2010, 1010.1993.

[39]  Ludwig Staiger,et al.  Subword Metrics for Infinite Words , 2015, CIAA.

[40]  Y. Moschovakis.  Descriptive Set Theory , 1980 .

[41]  Victor L. Selivanov,et al.  Wadge Reducibility and Infinite Computations , 2008, Math. Comput. Sci..

[42]  Olivier Finkel,et al.  Decision problems for Turing machines , 2009, Inf. Process. Lett..

[43]  Olivier Serre.  Games with Winning Conditions of High Borel Complexity , 2004, ICALP.

[44]  Alain Louveau,et al.  A Glimm-Effros dichotomy for Borel equivalence relations , 1990 .

[45]  Dominique Lecomte,et al.  A dichotomy characterizing analytic digraphs of uncountable Borel chromatic number in any dimension , 2007, 0707.1313.

[46]  Christian Choffrut,et al.  Uniformization of Rational Relations , 1999, Jewels are Forever.

[47]  Said Sidki,et al.  Automorphisms of one-rooted trees: Growth, circuit structure, and acyclicity , 2000 .

[48]  Daniele D'Angeli,et al.  Boundary action of automaton groups without singular points and Wang tilings , 2016, ArXiv.

[49]  Markus Lohrey,et al.  Knapsack and subset sum problems in nilpotent, polycyclic, and co-context-free groups , 2015, AMS-EMS-SPM Joint Meeting.

[50]  M. Vorobets,et al.  On a free group of transformations defined by an automaton , 2006, math/0601231.