CDMTCS Research Report Series A Simple Example of an ω-language Topologically Inequivalent to a Regular One Ludwig Staiger Martin-Luther-Universität Halle-Wittenerg Hideki Yamasaki Hitotsuashi University CDMTCS-9 July 2002 Centre for Discrete Mathematics and Theoretical Computer Science
A Simple Example of an ω-language Topologically Inequivalent to a Regular One Ludwig Staiger Martin-Luther-Universität Halle-Wittenerg Institut für Informatik von-seckendorff-platz, D 06099 Halle (Saale), Germany Electronic Mail: staiger@informatik.uni-halle.de Hideki Yamasaki Department of Mathematics, Hitotsuashi University Kunitachi, Tokyo 86-860, Japan Electronic Mail: yamasaki@math.hit-u.ac.jp Landweers s paper [La69] and the susequent ones [SW74, TY83] proved a strong relationship etween acceptance conditions imposed on finite automata on ω-words and the first classes of the Borel hierarchy in the Cantor space of all ω-words, (X ω,ρ), over a finite alphaet X. In Theorem 5 of [SW74] it is shown that an ω-language accepted y a finite automaton eing simultaneously an F σ - and a G δ -set elongs already to the Boolean closure of the class of all open (or, equivalently, closed) susets of (X ω,ρ), B(G). Thus, an ω-language F X ω which is simultaneously an F σ - and a G δ -set ut not a Boolean comination of open sets cannot e accepted y a finite automaton. For a more detailed discussion see e.g. [EH93, St97] or [Th90], for the notation used here see [St97]. The aim of this note is to provide a simple example that a proposition analogous to Theorem 5 of [SW74] is no longer true if we increase the computational power of the accepting device slightly: We augment the finite control y a so-called lind counter (cf. [EH93, Fi0], these automata are also known as partially lind counter automata [Gr78]), that is, y a counter which has no influence on the computational ehavior of the automaton except This paper was written during my visit to Institut für Informatik, Martin-Luther-Universität Halle- Wittenerg, 999, supported y grant 0-KO-87 from Ministry of Education, Culture, Sports, Science and Technology of Japan. The meaning of the word simple here is twofold: on the one hand, as explained aove, the topological complexity of our counterexample is the simplest possile one, and, on the other hand, the accepting device has a power only slightly increasing the power of a finite automaton.
2 L. Staiger and H. Yamasaki that the automaton gets stuck when the counter is decremented elow zero. Moreover we require the counter to e one-turn, that is, once we decrease the value of the counter we cannot increase it afterwards. We are not going to define these one-turn lind one-counter automata in full detail, instead we proceed with the announced example. On reading the first lock of a s a,+ a,±0 a,±0,±0,±0 s 0 s s 2, Figure : A Büchi automaton accepting the ω-language of Eq. () the automaton stores the lock length in the counter, and after reading the first the automaton switches to the cycle of states s,s 2 where the counter is decremented after every second. Thus the automaton gets stuck when the input a i w contains at least 2i + 2 s and, consequently, if the automaton does not get stuck it will finally stay in one of its states. (Looping etween s and s 2 is ounded y the numer of initial as.) Thus depending on the infinite input word ξ our automaton will stay in s 0, if ξ = a ω s = s, if ξ = a n ξ and ξ contains an even numer of s less than 2n + s 2, if ξ = a n ξ and ξ contains an odd numer of s less than 2n + 2. Our acceptance condition is Büchi s condition and the set of final states is {s 2 }, that is, an ω-word ξ {a,} ω is accepted if and only if the automaton runs infinitely often through state s 2 when reading ξ. Thus the ω-language accepted y our automaton is F = a n (a ) 2i+ a ω. () n N i n This ω-language is a countale suset of {a,} ω, thus an F σ -set. Since it is accepted y a deterministic automaton using Büchi acceptance it is also a G δ -suset of {a,} ω (cf. [EH93, St97]). We are going to show that our ω-language F cannot e represented as a Boolean comination of open (or closed) ω-languages. Thus, according to Theorem 5 of [SW74] (an even stronger version is Corollary 23 of [St83]), it cannot e accepted y a finite automaton. Assume the contrary, that is, let E i,e i e open susets of {a,}ω such that F = k i= E i E i. (2)
A Simple Example 3 Consequently, every suset F w {a,} ω has a similar representation F w {a,} ω = k i= ( Ei w {a,} ω) ( E i w {a,} ω). (3) as a Boolean comination of open sets E i w {a,} ω and E i w {a,}ω. Consider the ω-languages F n := F a n {a,} ω = a n i n (a ) 2i+ a ω. Every single ω-language F n is accepted y a finite automaton. Moreover, F n is essentially (up to the prefix a n and the encoding: a and 0 ) Wagner s ω-language c 2n := i<n (a ) 2i+ a ω taken over the alphaet {a,}. It is shown in Lemma of [Wa79] that c 2n+ Ĉ 2n+ Ĉ 2n and, consequently, F n Ĉ 2n Ĉ 2n 3. For the definition of Wagner classes see [Wa79, p. 39] or Definition 4. of [St97]. At the same time Eq. (3) and the fact that the ω-languages F n are accepted y finite automata imply F n Ĉ 2k+ for all n N, a contradiction. Thus Eqs. (3) and (2) cannot hold true, and F is not a Boolean comination of open susets of Cantor space (X ω,ρ). Finally, we present the Petri net derived from the automaton in Fig. which accepts the same ω-language. For acceptance of ω-languages y Petri nets see [HR86, Va83]. a a a Figure 2: A Petri net accepting F In Fig. 2, the initial marking is represented y a lack dot. We also adopt Büchi s acceptance condition with the set of accepting markings having at least one token in the douly circled place. Likewise we may adopt the co-büchi acceptance condition where ultimately all accepting markings have a token in the douly circled place.
4 L. Staiger and H. Yamasaki References [EH93] [Fi0] J.Engelfriet and H.J. Hoogeoom, X-automata on ω-words, Theoret. Comput. Sci. 0 (993), 5. O. Finkel, Wadge hierarchy of omega context-free languages. Theoret. Comput. Sci. 269 (200), 283 35. [La69] L.H. Landweer, Decision prolems for ω-automata, Math. Syst. Theory 3 (969) 4, 376 384. [Gr78] [HR86] S. Greiach. Remarks on lind and partially lind one-way multicounter machines. Theoret. Comput. Sci. 7 (978), 3 324. H.J. Hoogeoom and G. Rozenerg, Infinitary languages: Basic theory and applications to concurrent systems. In: Current Trends in Concurrency. Overviews and Tutorials (eds. J.W. de Bakker, W.-P. de Roever and G. Rozenerg), Lect. Notes Comput. Sci. 224, Springer-Verlag, Berlin 986, 266 342. [St83] L. Staiger, Finite-state ω-languages. J. Comput. System Sci. 27 (983) 3, 434 448. [St97] L. Staiger, ω-languages, in: Handook of Formal Languages, Vol. 3, G. Rozenerg and A. Salomaa (Eds.), Springer-Verlag, Berlin 997, 339 387. [SW74] [TY83] [Th90] L. Staiger und K. Wagner, Automatentheoretische und automatenfreie Charakterisierungen topologischer Klassen regulärer Folgenmengen. Elektron. Informationsverar. Kyernetik EIK 0 (974) 7, 379 392. M. Takahashi and H. Yamasaki, A note on ω-regular languages. Theoret. Comput. Sci. 23 (983), 27 225. W. Thomas, Automata on infinite ojects, in: Handook of Theoretical Computer Science, Vol. B, J. Van Leeuwen (Ed.), Elsevier, Amsterdam 990, 33 9. [Va83] R. Valk, Infinite ehaviour of Petri nets. Theoret. Comput. Sci. 25 (983), 3 34. [Wa79] K. Wagner, On ω-regular sets. Inform. and Control 43 (979), 23 77.