Weak index versus Borel rank

arXiv:0802.2842v1 [cs.IT] 20 Feb 2008 Symposium on Theoretical Aspects of Computer Science 2008 (Bordeaux), pp. 573-584 www.stacs-conf.org WEAK INDEX VERSUS BOREL RANK FILIP MURLAK 1 1 Warsaw University E-mail address: fmurlak@mimuw.edu.pl Abstract. We investigate weak recognizability of deterministic languages of infinite trees. We prove that for deterministic languages the Borel hierarchy and the weak index hierarchy coincide. Furthermore, we propose a procedure computing for a deterministic automaton an equivalent minimal index weak automaton with a quadratic number of states. The algorithm works within the time of solving the emptiness problem. 1. Introduction Finite automata on infinite trees are one of the basic tools in the verification of non- terminating programs. Practical applicability of this approach relies on the simplicity of the automata used to express the specifications. On the other hand it is convenient to write the specifications in an expressive language, e. g. µ-calculus. This motivates the search for auto- matic simplifications of automata. An efficient, yet reasonably expressive, model is offered by weak alternating automata. It was essentially showed by Rabin [18] that a language L can be recognized by a weak automaton if and only if both L and L∁can be recognized by nondeterministic B¨uchi automata. Arnold and Niwi´nski [2] proposed an algorithm that, given two B¨uchi automata recognizing a language and its complement, constructs a dou- bly exponential alternation free µ-calculus formula defining L, which essentially provides an equally effective translation to a weak automaton. Kupferman and Vardi [7] gave an immensely improved construction that involves only quadratic blow-up. A more refined construction could also simplify an automaton in terms of different complexity measures. A measure that is particularly important for theoretical and practi- cal reasons is the Mostowski–Rabin index. This measure reflects the alternation depth of positive and negative events in the behaviour of a verified system. The index orders au- tomata into a hierarchy that was proved strict for deterministic [21], nondeterministic [13], alternating [4, 8], and weak alternating automata [9]. Computing the least possible index for a given automaton is called the index problem. Unlike for ω-words, where the solution was essentially given already by Wagner [21], for trees this problem in its general form 1998 ACM Subject Classification: F.1.1, F.4.1, F.4.3. Key words and phrases: weak index, Borel rank, deterministic tree automata. Supported by the Polish government grant no. N206 008 32/0810. c ⃝ Filip Murlak CC ⃝ Creative Commons Attribution-NoDerivs License 574 FILIP MURLAK remains unsolved. For deterministic languages, Niwi´nski and Walukiewicz gave algorithms to compute the deterministic and nondeterministic indices [14, 16]. The theoretical significance of the weak index is best reflected by its coincidence with the quantifier alternation depth in the weak monadic second order logic [9]. Further interesting facts are revealed by the comparison with the Borel rank. In 1993 Skurczy´nski gave examples of Π0 n and Σ0 n-complete languages recognized by weak alternating automata with index (0, n) and (1, n + 1) accordingly [19]. In [5] it was shown that weak (0, n)-automata can only recognize Π0 n languages (and dually, (1, n + 1)-automata can only recognize Σ0 n languages), and it was conjectured that the weak index and the Borel hierarchies actually coincide. Here we prove that the conjecture holds for deterministic languages. Consequently, the algorithm calculating the Borel rank for deterministic languages [11] can be also used to compute the weak index. Since all deterministic languages are at the first level of the alternating hierarchy, this completes the picture for the deterministic case. We also provide an effective translation to a weak automaton with a quadratic number of states and the minimal index. 2. Automata We will be working with deterministic and weak automata, but to have a uniform framework, we first define automata in their most general alternating form. A parity game is a perfect information game of possibly infinite duration played by two players, Adam and Eve. We present it as a tuple (V∃, V∀, E, v0, rank), where V∃and V∀are (disjoint) sets of positions of Eve and Adam, respectively, E ⊆V × V is the relation of possible moves, with V = V∃∪V∀, p0 ∈V is a designated initial position, and rank : V → {0, 1, . . . , n} is the ranking function. The players start a play in the position v0 and then move a token according to relation E (always to a successor of the current position), thus forming a path in the graph (V, E). The move is selected by Eve or Adam, depending on who is the owner of the current position. If a player cannot move, she/he looses. Otherwise, the result of the play is an infinite path in the graph, v0, v1, v2, . . .. Eve wins the play if the highest rank visited infinitely often is even, oth

…(Full text truncated)…

This content is AI-processed based on ArXiv data.