Highly Undecidable Problems about Recognizability by Tiling Systems

arXiv:0811.3704v1 [cs.CC] 22 Nov 2008 Fundamenta Informaticae (2008) 1–18 1 IOS Press Highly Undecidable Problems about Recognizability by Tiling Systems Olivier Finkel Equipe Mod`eles de Calcul et Complexit´e Laboratoire de l’Informatique du Parall´elisme CNRS et Ecole Normale Sup´erieure de Lyon 46, All´ee d’Italie 69364 Lyon Cedex 07, France. Olivier.Finkel@ens-lyon.fr Abstract. Altenbernd, Thomas and W¨ohrle have considered acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with usual acceptance conditions, such as the B¨uchi and Muller ones, in [1]. It was proved in [9] that it is undecidable whether a B¨uchi- recognizable language of infinite pictures is E-recognizable (respectively, A-recognizable). We show here that these two decision problems are actually Π1 2-complete, hence located at the second level of the analytical hierarchy, and “highly undecidable”. We give the exact degree of numerous other undecidable problems for B¨uchi-recognizable languages of infinite pictures. In particular, the non- emptiness and the infiniteness problems are Σ1 1-complete, and the universality problem, the inclusion problem, the equivalence problem, the determinizability problem, the complementability problem, are all Π1 2-complete. It is also Π1 2-complete to determine whether a given B¨uchi recognizable lan- guage of infinite pictures can be accepted row by row using an automaton model over ordinal words of length ω2. Keywords: Languages of infinite pictures; recognizability by tiling systems; decision problems; highly undecidable problems; analytical hierarchy. 1. Introduction Languages of infinite words accepted by finite automata were first studied by B¨uchi to prove the de- cidability of the monadic second order theory of one successor over the integers. Since then regular ω-languages have been much studied and many applications have been found for specification and veri- fication of non-terminating systems, see [24, 23, 19] for many results and references. Address for correspondence: E Mail: Olivier.Finkel@ens-lyon.fr 2 Olivier Finkel / Highly Undecidable Problems about Recognizability by Tiling Systems In a recent paper, Altenbernd, Thomas and W¨ohrle have considered acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with the usual acceptance conditions, such as the B¨uchi and Muller ones, firstly used for infinite words. This way they extended both the clas- sical theory of ω-regular languages and the classical theory of recognizable languages of finite pictures, [11], to the case of infinite pictures. Many classical decision problems are studied in formal language theory and in automata theory and arise now naturally about recognizable languages of infinite pictures. We proved in [9] that many decision problems for B¨uchi-recognizable languages of infinite pictures are undecidable. In particular, we showed, using topological arguments, that it is undecidable whether a B¨uchi-recognizable language of infinite pictures is E-recognizable (respectively, A-recognizable), giving the answer to two questions raised in [1]. We proved also several other undecidability results as the following ones: one cannot decide whether a B¨uchi-recognizable language of infinite pictures can be recognized by a deterministic B¨uchi or Muller tiling system, or whether it can be accepted row by row using an automaton model over ordinal words of length ω2. Using the Π1 2-completeness of the universality problem for ω-languages of non deterministic Turing machines which was proved by Castro and Cucker in [3], and some topological arguments, we show in this paper that the above decision problems are actually Π1 2-complete, hence located at the second level of the analytical hierarchy, and “highly undecidable”. Using other results of [3], we give also the exact degree of numerous other undecidable problems for B¨uchi-recognizable languages of infinite pictures. In particular, the non-emptiness and the infiniteness problems are Σ1 1-complete, and the universality problem, the inclusion problem, the equivalence problem, the complementability problem, are all Π1 2- complete. This gives new natural examples of decision problems located at the first or at the second level of the analytical hierarchy. We show also that topological properties of B¨uchi-recognizable languages of infinite pictures are highly undecidable. The paper is organized as follows. In Section 2 we recall definitions for pictures and tiling systems. The definition and properties of the analytical hierarchy are introduced in Section 3. We recall in Sec- tion 4 some notions of topology, including the definitions of Borel and analytic sets. We prove high undecidability results in Section 5. Concluding remarks are given in Section 6. 2. Tiling Systems We assume the reader to be familiar with the theory of formal (ω)-languages [24, 23]. We recall usual notations of formal language theory. When Σ is a

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