On regular tree languages and deterministic pushdown automata

The theory of formal string languages and of formal tree languages are both important parts of the theory of formal languages. Regular tree languages are recognized by finite tree automata. Trees in their postfix notation can be seen as strings. This paper presents a simple transformation from any given (bottom-up) finite tree automaton recognizing a regular tree language to a deterministic pushdown automaton accepting the same tree language in postfix notation. The resulting deterministic pushdown automaton can be implemented easily by an existing parser generator because it is constructed for an LR(0) grammar, and its size directly corresponds to the size of the deterministic finite tree automaton. The class of regular tree languages in postfix notation is a proper subclass of deterministic context-free string languages. Moreover, the class of tree languages which are in their postfix notation deterministic context-free string languages is a proper superclass of the class of regular tree languages.     

Complementing deterministic tree-walking automata

We consider various kinds of deterministic tree-walking automata, with and without pebbles, over ranked and unranked trees. For each such kind of automata we show that there is an equivalent one which never loops. The main consequence of this result is the closure under complementation of the various types of automata we consider with a focus on the number of pebbles used in order to complement the automata.     

From LTL to deterministic automata

We present a new algorithm to construct a (generalized) deterministic Rabin automaton for an LTL formula \(\varphi \). The automaton is the product of a co-Büchi automaton for \(\varphi \) and an array of Rabin automata, one for each \({\mathbf {G}}\)-subformula of \(\varphi \). The Rabin automaton for \({\mathbf {G}}\psi \) is in charge of recognizing whether \({\mathbf {F}}{\mathbf {G}}\psi \) holds. This information is passed to the co-Büchi automaton that decides on acceptance. As opposed to standard procedures based on Safra’s determinization, the states of all our automata have a clear logical structure, which allows for various optimizations. Experimental results show improvement in the sizes of the resulting automata compared to existing methods.     

On stateless deterministic restarting automata

The transitions of a stateless automaton do not depend on internal states but solely on the symbols currently scanned by its head accessing the input and memory. We investigate stateless deterministic restarting automata that, after executing a rewrite step, continue to read their tape before performing a restart. Even the weakest class thus obtained contains the regular languages properly. The relations between different classes of stateless automata as well as between stateless automata and the corresponding types of automata with states are investigated, and it is shown that the language classes defined by the various types of deterministic stateless restarting automata without auxiliary symbols are anti-AFLs that are not even closed under reversal.     

确定型格值有限自动机的最小化

给出了确定型格值有限自动机的定义,并同时给出了有效终止状态和可达到状态的定义。指出了求取DLFA M=Q,Σ,δ,q₀,σ的实质是求取Q/R_k。由此以可到达状态为基础引入了等价关系R_k、S_k与商集Q/S_k,证明了R_k=R_k-1∩S_k,由此得到Q/R_k的等价类为Q/R_k-1中等价类与Q/S_k中等价类的非空交集全体。引入了H_k,并证明了可由H_k求取Q/S_k,从而得到仅利用集合运算便可求取Q/R_k的算法,最终给出了DLFA最小化算法的一个容易实现的构造型描述和相应示例。     

Equivalence of deterministic pushdown automata revisited

This paper briefly analyzes main ideas underlying the comparison algorithm that made it possible to prove the equivalence of deterministic pushdown automata. An example of using this algorithm is presented. The relationship of this algorithm with other results in this area is shown. Moreover, the decidability of problems associated with some classes of formal grammars is established. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 24–39, March–April 2007.     

From Emerson-Lei automata to deterministic,limit-deterministic or good-for-MDP automata

Innovations in Systems and Software Engineering - The topic of this paper is the determinization problem of $$omega $$ -automata under the transition-based Emerson-Lei acceptance (called TELA),...     

Pushdown tree automata

We define topdown pushdown tree automata (PDTA's) which extend the usual string pushdown automata by allowing trees instead of strings in both the input and the stack. We prove that PDTA's recognize the class of context-free tree languages. (Quasi)realtime and deterministic PDTA's accept the classes of Greibach and deterministic tree languages, respectively. Finally, PDTA's are shown to be equivalent to restricted PDTA's, whose stack is linear: this both yields a more operational way of recognizing context-free tree languages and connects them with the class of indexed languages.     

Amount of nonconstructivity in deterministic finite automata

When D. Hilbert used nonconstructive methods in his famous paper on invariants (1888), P. Gordan tried to prevent the publication of this paper considering these methods as non-mathematical. L.E.J. Brouwer in the early twentieth century initiated intuitionist movement in mathematics. His slogan was “nonconstructive arguments have no value for mathematics”. However, P. Erdös got many exciting results in discrete mathematics by nonconstructive methods. It is widely believed that these results either cannot be proved by constructive methods or the proofs would have been prohibitively complicated. The author (Freivalds, 2008) [10] showed that nonconstructive methods in coding theory are related to the notion of Kolmogorov complexity.We study the problem of the quantitative characterization of the amount of nonconstructiveness in nonconstructive arguments. We limit ourselves to computation by deterministic finite automata. The notion of nonconstructive computation by finite automata is introduced. Upper and lower bounds of nonconstructivity are proved.     

确定型模糊多重集有限自动机的极小化

主要研究确定型模糊多重集有限自动机的状态极小化问题。给出了模糊多重集有限自动机的同余和同态概念,并利用同余和同态关系研究了确定型模糊多重集有限自动机的极小化问题。进一步从确定型模糊多重集有限自动机自身出发,构造出极小模糊多重集有限自动机,并给出了极小化的算法。     

Language recognition by two-way deterministic pushdown automata

It is proved that any bounded context-free language can be recognized by a two-way deterministic automaton with a finite-rotary counter.Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 177–181, November–December 2004.This revised version was published online in April 2005 with a corrected cover date.     

A note on non-singular deterministic pushdown automata

The non-singular deterministic pushdown automata were first defined by Valiant as an example of a class of machines with a decidable equivalence problem [3]. No algorithm currently exist for deciding whether or not a deterministic pushdown automation is non-singular, so the applicability of Valiant's equivalence decision procedure cannot be readily (if ever) determined. In this paper, it is shown that the equivalence problem for non-singular automata is reducible to the problem of deciding whether or not a deterministic pushdown automaton is non-singular.     

On input-revolving deterministic and nondeterministic finite automata

We introduce and investigate input-revolving finite automata, which are (nondeterministic) finite state automata with the additional ability to shift the remaining part of the input. Three different modes of shifting are considered, namely revolving to the left, revolving to the right, and circular interchanging. We investigate the computational capacities of these three types of automata and their deterministic variants, comparing any of the six classes of automata with each other and with further classes of well-known automata. In particular, it is shown that nondeterminism is better than determinism, that is, for all three modes of shifting there is a language accepted by the nondeterministic model but not accepted by any deterministic automaton of the same type. Concerning the closure properties most of the deterministic language families studied are not closed under standard operations. For example, we show that the family of languages accepted by deterministic right-revolving finite automata is an anti-AFL which is not closed under reversal and intersection.