Estimation of the length of reset words for automata with simple idempotents

A quadratic upper bound on the length of a minimal reset word is obtained for finite automata with simple idempotents. Each input symbol of the automata considered induces a transformation that is an idempotent with the unit defect or a bijection on the set of states. This bound is only twice as large as the well-known lower bound of this length. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 32–39, May–June, 2000.     

Enumeration and generation with a string automata representation

The representation of combinatorial objects is decisive for the feasibility of several enumerative tasks. In this work, we present a unique string representation for complete initially-connected deterministic automata (ICDFAs) with n$n$ states over an alphabet of k$k$ symbols. For these strings we give a regular expression and show how they are adequate for exact and random generation, allow an alternative way for enumeration and lead to an upper bound for the number of ICDFAs. The exact generation algorithm can be used to partition the set of ICDFAs in order to parallelize the counting of minimal automata, and thus of regular languages. A uniform random generator for ICDFAs is presented that uses a table of pre-calculated values. Based on the same table, an optimal coding for ICDFAs is obtained.     

Improved upper bounds on synchronizing nondeterministic automata

We show that i-directable nondeterministic automata can be i-directed with a word of length O(2ⁿ) for i=1,2, where n stands for the number of states. Since for i=1,2 there exist i-directable automata having i-directing words of length Ω(2ⁿ), these upper bounds are asymptotically optimal. We also show that a 3-directable nondeterministic automaton with n states can be 3-directed with a word of length , improving the previously known upper bound O(2ⁿ). Here the best known lower bound is .     

关于同步部分规约的有限自动机的优化问题的近似难度

自动机是可同步的是指它具有满足以下性质的同步字:不论自动机当前所处的状态,以同步字为输入执行后它一定会到达某个特定状态。同步自动机问题的核心是计算最短同步字。聚焦于这一核心问题,文中就一类称为部分规约的确定的有限自动机的最短同步字问题,研究了近似计算这类自动机的最短同步字的复杂性,即近似计算它的难度,该工作有助于其近似算法的分析与设计。通过建立由两个优化问题(MAX SAT问题以及MAX FA-INT问题)到最短同步字长度计算这一问题(即Shortest-Syn)的归约,利用与概率可检验证明(Probabilistically Checkable Proofs,PCP)定理和概率可检验辩论(Probabilistically Checkable Debate,PCD)定理有关的若干结果证明了文中的主要结论:对于部分规约的确定的有限自动机,在某个近似因子内Shortest-Syn的近似难度是NP-难的和PSPACE-难的,除非NP和PSPACE分别坍塌到P。     

On the structure of bispecial Sturmian words

A balanced word is one in which any two factors of the same length contain the same number of each letter of the alphabet up to one. Finite binary balanced words are called Sturmian words. A Sturmian word is bispecial if it can be extended to the left and to the right with both letters remaining a Sturmian word. There is a deep relation between bispecial Sturmian words and Christoffel words, that are the digital approximations of Euclidean segments in the plane. In 1997, J. Berstel and A. de Luca proved that palindromic bispecial Sturmian words are precisely the maximal internal factors of primitive Christoffel words. We extend this result by showing that bispecial Sturmian words are precisely the maximal internal factors of all Christoffel words. Our characterization allows us to give an enumerative formula for bispecial Sturmian words. We also investigate the minimal forbidden words for the language of Sturmian words.     

Operational state complexity of nested word automata

We introduce techniques to prove lower bounds for the number of states needed by finite automata operating on nested words. We study the state complexity of Boolean operations and obtain lower bounds that are tight within an additive constant. The results for union and complementation differ from corresponding bounds for ordinary finite automata. For reversal and concatenation, we establish lower bounds that are of a different order than the worst-case bounds for ordinary finite automata.     

Weighted automata with discounting

We investigate weighted automata with discounting and their behaviors over semirings and finitely generated graded monoids. We characterize the discounted behaviors of weighted automata precisely as rational formal power series with a discounted form of the Cauchy product. This extends a classical result of Kleene-Schützenberger. Here we show that the very special case of Schützenberger's result for free monoids over singleton alphabets suffices to deduce our generalization.     

From Nerode’s congruence to suffix automata with mismatches

In this paper we focus on the minimal deterministic finite automaton _Sk$S_k$ that recognizes the set of suffixes of a word w$w$ up to k$k$ errors. As first result we give a characterization of the Nerode’s right-invariant congruence that is associated with _Sk$S_k$. This result generalizes the classical characterization described in [A. Blumer, J. Blumer, D. Haussler, A. Ehrenfeucht, M. Chen, J. Seiferas, The smallest automaton recognizing the subwords of a text, Theoretical Computer Science, 40, 1985, 31–55]. As second result we present an algorithm that makes use of _Sk$S_k$ to accept in an efficient way the language of all suffixes of w$w$ up to k$k$ errors in every window of size r$r$ of a text, where r$r$ is the repetition index of w$w$. Moreover, we give some experimental results on some well-known words, like prefixes of Fibonacci and Thue-Morse words. Finally, we state a conjecture and an open problem on the size and the construction of the suffix automaton with mismatches.     

A general type‐2 fuzzy model for computing with words

As a methodology, computing with words (CW) allows the use of words, instead of numbers or symbols, in the process of computing and reasoning and thus conforms more to humans’ inference when it is used to describe real‐world problems. In the line of developing a computational theory for CW, in this paper we develop a formal general type‐2 fuzzy model of CW by exploiting general type‐2 fuzzy sets (GT2 FSs) since GT2 FSs bear greater potential to model the linguistic uncertainty. On the one hand, we generalize the interval type‐2 fuzzy sets (IT2 FSs)‐based formal model of CW into general type‐2 fuzzy environments. Concretely, we present two kinds of general type‐2 fuzzy automata (i.e., general type‐2 fuzzy finite automata and general type‐2 fuzzy pushdown automata) as computational models of CW. On the other hand, we also give a somewhat universally general type‐2 fuzzy model of computing with (some special) words and establish a retraction principle from computing with words to computing with values for handling crisp inputs in general type‐2 fuzzy setting and a generalized extension principle from computing with words to computing with all words for handling general type‐2 fuzzy inputs.     

Scheduling with timed automata

In this work, we present timed automata as a natural tool for posing and solving scheduling problems. We show how efficient shortest path algorithms for timed automata can find optimal schedules for the classical job-shop problem. We then extend these results to synthesize adaptive scheduling strategies for problems with uncertainty in task durations.     

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.     

Approximating Wardrop equilibria with finitely many agents

We present efficient algorithms for computing approximate Wardrop equilibria in a distributed and concurrent fashion. Our algorithms are exexuted by a finite number of agents each of which controls the flow of one commodity striving to balance the induced latency over all utilised paths. The set of allowed paths is represented by a DAG. Our algorithms are based on previous work on policies for infinite populations of agents. These policies achieve a convergence time which is independent of the underlying network and depends mildly on the latency functions. These policies can neither be applied to a finite set of agents nor can they be simulated directly due to the exponential number of paths. Our algorithms circumvent these problems by computing a randomised path decomposition in every communication round. Based on this decomposition, flow is shifted from overloaded to underloaded paths. This way, our algorithm can handle exponentially large path collections in polynomial time. Our algorithms are stateless, and the number of communication rounds depends polynomially on the approximation quality and is independent of the topology and size of the network. Supported by DFG grant Vo889/1–3 and by DFG through German excellence cluster UMIC at RWTH Aachen. Supported by the DFG GK/1298 “AlgoSyn”. Supported in part by the the EU within the 6th Framework Programme under contract 001907 (DELIS) and by DFG through German excellence cluster UMIC at RWTH Aachen. A preliminary version of this work has appeared in [14].     

Limitations of lower bound methods for deterministic nested word automata

Finite automata operating on nested words were introduced by Alur and Madhusudan in 2006. While nested word automata retain many of the desirable properties of ordinary finite automata, there is no known efficient minimization algorithm for deterministic nested word automata and, interestingly, state complexity bounds for nested word automata turn out to differ significantly from the corresponding bounds for ordinary finite automata. Consequently lower bounds for the state complexity of nested word languages need to rely on fooling set type techniques. We discuss limitations of the techniques and show that, even in the deterministic case, the bounds given by the lower bound methods may be arbitrarily far away from the actual state complexity of the nested word language.     

Applications of finite automata representing large vocabularies

The construction of minimal acyclic deterministic partial finite automata to represent large natural language vocabularies is described. Applications of such automata include spelling checkers and advisers, multilanguage dictionaries, thesauri, minimal perfect hashing and text compression.     

A note on the Markoff condition and central words

We define Markoff words as certain factors appearing in bi-infinite words satisfying the Markoff condition. We prove that these words coincide with central words, yielding a new characterization of Christoffel words.