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In this work, we exploit the power of finite ambiguity for the complementation problem of Büchi automata by using reduced run directed acyclic graphs (DAGs) over infinite words, in which each vertex has at most one predecessor; these reduced run DAGs have only a finite number of infinite runs, thus obtaining the finite ambiguity in Büchi complementation. We show how to use this type of reduced run DAGs as a unified tool to optimize both rank-based and slice-based complementation constructions for Büchi automata with a finite degree of ambiguity. As a result, given a Büchi automaton with n states and a finite degree of ambiguity, the number of states in the complementary Büchi automaton constructed by the classical rank-based and slice-based complementation constructions can be improved from 2 O (n log ⁡ n) and O ((3 n) n) to O (6 n) ⊆ 2 O (n) and O (4 n) , respectively. We further show how to construct such reduced run DAGs for limit deterministic Büchi automata and obtain a specialized complementation algorithm, thus demonstrating the generality of the power of finite ambiguity. [ABSTRACT FROM AUTHOR]

The paper demonstrates the non-closure of the family of unambiguous linear languages (that is, those defined by unambiguous linear context-free grammars) under complementation. To be precise, a particular unambiguous linear grammar is presented, and it is proved that the complement of this language is not defined by any context-free grammar. This also constitutes an alternative proof for the result of Hibbard and Ullian ("The independence of inherent ambiguity from complementedness among context-free languages", JACM , 1966) on the non-closure of the unambiguous languages under complementation. [ABSTRACT FROM AUTHOR]

A morphism g is ambiguous with respect to a word u if there exists a second morphism h ≠ g such that g (u) = h (u). Otherwise g is unambiguous with respect to u. Thus unambiguous morphisms are those for which the structure of the morphism is preserved in the image. Ambiguity has so far been studied for morphisms of free monoids, where several characterisations exist for the set of words u permitting an (injective) unambiguous morphism. In the present paper, we consider ambiguity of morphisms of free groups, and consider possible analogies to the existing characterisations in the free monoid. While a direct generalisation results in a trivial situation where all morphisms are ambiguous, we discuss some natural and well-motivated reformulations, and provide a characterisation of words in a free group that permit a morphism which is "as unambiguous as possible". [ABSTRACT FROM AUTHOR]

Probabilistic automata are an extension of nondeterministic finite automata in which transitions are annotated with probabilities. Despite its simplicity, this model is very expressive and many algorithmic questions are undecidable. In this work we focus on the emptiness problem (and its variant the value problem), which asks whether a given probabilistic automaton accepts some word with probability greater than a given threshold. We consider finitely ambiguous probabilistic automata. Our main contributions are to construct efficient algorithms for analysing finitely ambiguous probabilistic automata through a reduction to a multi-objective optimisation problem called the stochastic path problem. We obtain a polynomial time algorithm for approximating the value of probabilistic automata of fixed ambiguity and a quasi-polynomial time algorithm for the emptiness problem for 2-ambiguous probabilistic automata. We complement these positive results by an inapproximability result stating that the value of finitely ambiguous probabilistic automata cannot be approximated unless P = NP. [ABSTRACT FROM AUTHOR]

An automaton is unambiguous if for every input it has at most one accepting computation. An automaton is finitely (respectively, countably) ambiguous if for every input it has at most finitely (respectively, countably) many accepting computations. An automaton is boundedly ambiguous if there is k ∈ N , such that for every input it has at most k accepting computations. We consider nondeterministic Büchi automata (NBA) over infinite trees and prove that it is decidable in polynomial time, whether an automaton is unambiguous, boundedly ambiguous, finitely ambiguous, or countably ambiguous. [ABSTRACT FROM AUTHOR]

Abstract: Nondeterministic finite automata (NFA) with at most one accepting computation on every input string are known as unambiguous finite automata (UFA). This paper considers UFAs over a one-letter alphabet, and determines the exact number of states in DFAs needed to represent unary languages recognized by n-state UFAs in terms of a new number-theoretic function . The growth rate of , and therefore of the UFA–DFA tradeoff, is estimated as . The conversion of an n-state unary NFA to a UFA requires UFAs with states, where is the greatest order of a permutation of n elements, known as Landauʼs function. In addition, it is shown that representing the complement of n-state unary UFAs requires UFAs with at least states in the worst case, while the Kleene star requires up to exactly states. [Copyright &y& Elsevier]

Abstract: Boolean grammars are an extension of context-free grammars, in which conjunction and negation may be explicitly used in the rules. In this paper, the notion of ambiguity in Boolean grammars is defined. It is shown that the known transformation of a Boolean grammar to the binary normal form preserves unambiguity, and that every unambiguous Boolean language can be parsed in time O(n 2). Linear conjunctive languages are shown to be unambiguous, while the existence of languages inherently ambiguous with respect to Boolean grammars is left open. [Copyright &y& Elsevier]

Abstract: Regular expressions with numeric occurrence indicators are an extension of traditional regular expressions, which let the required minimum and the allowed maximum number of iterations of subexpressions be described with numeric parameters. We consider the problem of testing whether a given regular expression E with numeric occurrence indicators is 1-unambiguous or not. This condition means, informally, that any prefix of any word accepted by expression E determines a unique path of matching symbol positions in E. One-unambiguity appears as a validity constraint in popular document schema languages such as SGML and XML DTDs (document type definitions) and XML Schema; the last one both includes numeric occurrence indicators and requires one-unambiguity of expressions. Previously published solutions for testing the one-unambiguity of regular expressions with numeric occurrence indicators are either erroneous or require exponential time. The main contribution of this paper is a polynomial-time method for solving this problem, and a formal proof of its correctness. [Copyright &y& Elsevier]

While deterministic finite automata seem to be well understood, surprisingly many important problems concerning nondeterministic finite automata (nfa''s) remain open. One such problem area is the study of different measures of nondeterminism in finite automata and the estimation of the sizes of minimal nondeterministic finite automata. In this paper the concept of communication complexity is applied in order to achieve progress in this problem area. The main results are as follows:1. Deterministic communication complexity provides lower bounds on the size of nfa''s with bounded unambiguity. Applying this fact, the proofs of several results about nfa''s with limited ambiguity can be simplified and presented in a uniform way.2. There is a family of languages KONk2 with an exponential size gap between nfa''s with polynomial leaf number/ambiguity and nfa''s with ambiguity k. This partially provides an answer to the open problem posed by B. Ravikumar and O. Ibarra (1989, SIAM J. Comput.18, 1263–1282) and H. Leung (1998, SIAM J. Comput.27, 1073–1082). [Copyright &y& Elsevier]