The Wadge Hierarchy of Max-Regular Languages

LIPIcs Leibniz International Proceedings in Informatics The Wadge Hierarchy of Max-Regular Languages Jer´ emie´ Cabessa1, Jacques Duparc2, Alessandro Facchini2,3, Filip Murlak4 1 Grenoble Institute of Neuroscience, Joseph Fourier University, France 2 Faculty of Business and Economics, University of Lausanne, Switzerland 3 LaBRI, University of Bordeaux 1, France 4 University of Warsaw, Poland ABSTRACT. Recently, Mikołaj Bojanczyk´ introduced a class of max-regular languages, an extension of regular languages of infinite words preserving many of its usual properties. This new class can be seen as a different way of generalising the notion of regularity from finite to infinite words. This paper compares regular and max-regular languages in terms of topological complexity. It is proved 0 that up to Wadge equivalence the classes coincide. Moreover, when restricted to D2-languages, the classes contain virtually the same languages. On the other hand, separating examples of arbitrary 0 complexity exceeding D2 are constructed. Introduction Until recently, the notion of regularity for languages of infinite words developed by Buchi¨ [2] seemed to be universally accepted. Buchi’s¨ class has various characterisations, most notably in terms of automata and monadic second order logic, and enjoys a multitude of elegant properties, like closure by Boolean operations (including negation). Nowadays however some doubt has been cast by Mikołaj Bojanczyk´ [1], who presented a richer class of max- regular languages, arguably as much regular as Buchi’s¨ languages. This new class has a characterisation via weak monadic second-order logic with the unbounding quantifier, and a suitable automaton model with decidable emptiness. It also exhibits the usual closure properties. In this paper we would like to shed some more light on the relations between the two classes. A typical max-regular language is defined by the property “the distance between consecutive b’s is unbounded”, n1 n2 n3 K = fa ba ba ...: 8m 9i ni > mg . 0 This language is not regular, but it is P2-complete. In fact, as Bojanczyk´ notes, all max- 0 regular languages are Boolean combinations of S2-sets, just like regular languages. Is this a coincidence, or does the similarity go further? How big is the new class? The ultimate tool for this kind of questions is the Wadge hierarchy [13, 14]. Ordering the sets based on the existence of continuous reductions (Wadge reductions) between them, the Wadge c J. Cabessa, J. Duparc, A. Facchini, F. Murlak; licensed under Creative Commons License-NC-ND. Foundations of Software Technology and Theoretical Computer Science (Kanpur) 2009. Editors: Ravi Kannan and K. Narayan Kumar; pp 121–132 Leibniz International Proceedings in Informatics (LIPIcs), Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Germany. Digital Object Identifier: 10.4230/LIPIcs.FSTTCS.2009.2312 122 THE WADGE HIERARCHY OF MAX-REGULAR LANGUAGES hierarchy is the most refined complexity measure in descriptive set theory. For classical regular languages, it coincides exactly with automata-based Wagner hierarchy, and is well- understood [15]. Here we investigate the Wadge hierarchy of max-regular languages. As was shown by Finkel’s work on blind counter automata [10], adding very restricted counters already makes the Wadge hierarchy much richer. Surprisingly, even though max- automata do involve counters, the Wadge hierarchy they induce actually coincides with the Wagner hierarchy. In other words, for each max-regular language, there exists a Wadge- equivalent regular language. Topologically, Bojanczyk’s´ extension is very conservative. On the other hand, there is an abundance of separating languages: we provide one for each level beginning from w. This shows that the difference between the two classes spans orthogonally to the topological complexity. 0 0 Below the level w, which corresponds exactly to the languages complete for P2 or S2, the levels contain the same languages. Hence, the exemplary language K is as simple as possible: every max-regular language strictly lower than K in the Wadge hierarchy is necessarily regular. 1 Preliminaries 1.1 Languages A set of finite words is called a language, and a set of infinite words an w-language. Given a finite set A, called the alphabet, then A∗, A+, Aw, and A¥ denote respectively the sets of finite words, nonempty finite words, infinite words, and finite or infinite words, all of them over the alphabet A. The empty word is denoted by #. Given a finite word u and a finite or infinite word v, we write uv to denote the concatenation of u and v. Given X ⊆ A∗ and Y ⊆ A¥, the concatenation of X and Y is defined by XY = fxy : x 2 X and y 2 Yg, the ∗ finite iteration of X is X = fx1 ··· xn : n ≥ 0 and x1,..., xn 2 Xg, and the infinite iteration w ∗ w −1 of X is X = fx0x1x2 ··· : xi 2 X, for all i 2 Ng. Given u 2 A and X ⊆ A , the set u X −1 w −1 w is defined as u X = fx 2 A : ux 2 Xg, and Xu is u(u X) = uA \ X. The w-regular languages are exactly the ones recognised by finite Buchi,¨ or equivalently, by finite Muller automata. We refer to [11, p.15] for further details. Finally, for any alphabet A, the set Aw can be equipped with the product topology of the discrete topology on A. The open sets of Aw are thus of the form WAw, for some W ⊆ A∗. 1.2 The Wadge hierarchy The Wadge hierarchy is a very refined topological classification of w-languages. This classification is obtained by means of Wadge (or continuous) reduction, which is a partial ordering defined via the Wadge games [13] presented below. Let A and B be two finite alphabets, and let X ⊆ Aw and Y ⊆ Bw. The Wadge game W((A, X), (B, Y)) is a two-player infinite game with perfect information, where player I is in charge of the subset X and player II is in charge of the subset Y. Players I and II alternately play letters from the alphabets A and B, respectively. Player I begins. Player II is allowed to skip her turn, formally denoted by the symbol “−”, provided she plays infinitely many letters, whereas player I is not allowed to do so. After w turns, players I J. CABESSA, J. DUPARC, A. FACCHINI, F. MURLAK FSTTCS 2009 123 and II have produced two infinite words, a 2 Aw and b 2 Bw respectively. Player II wins W ((A, X), (B, Y)) if and only if (a 2 X , b 2 Y). From this point onward, the Wadge game W ((A, X), (B, Y)) will be denoted W(X, Y) and the alphabets involved will always be clear from the context. Along the play, the finite sequence of all previous moves of a given player is called the current position of this player. A strategy for player I is a mapping from (B [ f−g)∗ into A.A strategy for player II is a mapping from A+ into B [ f−g. A strategy is winning if the player following it must necessarily win, no matter what his opponent plays. The Wadge reduction is defined via the Wadge game as follows: a set X is said to be Wadge reducible to Y, denoted by X ≤W Y, if and only if player II has a winning strategy in W(X, Y). This relation ≤W is reflexive and transitive. The corresponding equivalence relation and strict reduction are defined by X ≡W Y if and only if both X ≤W Y and Y ≤W X hold, and X <W Y if and only if X ≤W Y and X 6≡W Y. In addition, the sets X and Y are said to be Wadge incomparable, denoted as X?WY, if and only if both X 6≤W Y and Y 6≤W X. w c c Besides, a set X ⊆ A is called self-dual if X ≡W X , and non-self-dual if X 6≡W X . Let us point out that Wadge games were designed so that the Wadge reduction corre- spond precisely to the continuous reduction. Indeed, it holds that X ≤W Y if and only if there exists a continuous function f : Aw ! Bw such that f −1(Y) = X [13]. The Wadge hierarchy consists of the collection of all w-languages ordered by the Wadge reduction, and the Borel Wadge hierarchy is the restriction of the Wadge hierarchy to Borel w-languages. As a consequence of Martin’s Borel determinacy theorem, for any two Borel w-languages X and Y, there exists a winning strategy for one of the players in W(X, Y). This key property induces the following strong consequences on the Borel Wadge hierarchy. First, the ≤W-antichains have length at most 2, and the only incomparable w-languages are, up to Wadge equivalence, of the form X and Xc, for X non-self-dual. Furthermore, the Wadge reduction is well-founded on Borel sets, meaning that there is no infinite strictly descending sequence of Borel w-languages X0 >W X1 >W X2 >W . These results ensure that, up to complementation and Wadge equivalence, the Borel Wadge hierarchy is actually a well ordering. Therefore, there exist a unique ordinal, called the height of the Borel Wadge hierarchy, and a mapping dW from the Borel Wadge hierarchy onto its height, called the Wadge degree, such that dW (X) < dW (Y) if and only if X <W Y, and dW (X) = dW (Y) if and only if either c X ≡W Y or X ≡W Y , for every Borel w-languages X and Y. Actually, it is usually convenient to consider another definition of the Wadge degree which makes the non-self dual sets and the first self dual ones that strictly reduce these latter always share the same degree, namely: 8 1 if X = Æ or X = Æc, <> dW (X) = sup fdW (Y) + 1 : Y n.s.d.

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