Unambiguous Automata Research Articles

Equivalence, Unambiguity, and Sequentiality of Finitely Ambiguous Max-Plus Tree Automata

We show that the equivalence, unambiguity, and sequentiality problems are decidable for finitely ambiguous max-plus tree automata. A max-plus tree automaton is a weighted tree automaton over the max-plus semiring. A max-plus tree automaton is called finitely ambiguous if the number of accepting runs on every tree is bounded by a global constant and it is called unambiguous if there exists at most one accepting run on every tree. For the equivalence problem, we show that for two finitely ambiguous max-plus tree automata, it is decidable whether both assign the same weight to every tree. For the unambiguity and sequentiality problems, we show that for every finitely ambiguous max-plus tree automaton, both the existence of an equivalent unambiguous automaton and the existence of an equivalent deterministic automaton are decidable.

Markov chains and unambiguous automata

Unambiguous automata are nondeterministic automata in which every word has at most one accepting run. In this paper we give a polynomial-time algorithm for model checking discrete-time Markov chains against ω-regular specifications represented as unambiguous automata. We furthermore show that the complexity of this model checking problem lies in NC: the subclass of P comprising those problems solvable in poly-logarithmic parallel time. These complexity bounds match the known bounds for model checking Markov chains against specifications given as deterministic automata, notwithstanding the fact that unambiguous automata can be exponentially more succinct than deterministic automata. We report on an implementation of our procedure, including an experiment in which the implementation is used to model check LTL formulas on Markov chains.

On the transformation of two-way finite automata to unambiguous finite automata

This paper estimates the number of states in an unambiguous finite automaton (UFA) that is sufficient and in the worst case necessary to simulate an n-state two-way deterministic finite automaton (2DFA) and an n-state two-way unambiguous finite automaton (2UFA). It is proved that a 2UFA with n states can be transformed to a UFA with fewer than 2n⋅n! states. On the other hand, for every n, there is a language recognized by an n-state 2DFA that requires a UFA with at least Ω((2+1)2n⋅n−1)=Ω(5.828n) states. The latter result is obtained as a lower bound on the rank of a certain matrix.

Ambiguity Hierarchy of Regular Infinite Tree Languages

An automaton is unambiguous if for every input it has at most one accepting computation. An automaton is k-ambiguous (for k > 0) if for every input it has at most k accepting computations. An automaton is boundedly ambiguous if it is k-ambiguous for some $k \in \mathbb{N}$. An automaton is finitely (respectively, countably) ambiguous if for every input it has at most finitely (respectively, countably) many accepting computations. The degree of ambiguity of a regular language is defined in a natural way. A language is k-ambiguous (respectively, boundedly, finitely, countably ambiguous) if it is accepted by a k-ambiguous (respectively, boundedly, finitely, countably ambiguous) automaton. Over finite words every regular language is accepted by a deterministic automaton. Over finite trees every regular language is accepted by an unambiguous automaton. Over $\omega$-words every regular language is accepted by an unambiguous B\"uchi automaton and by a deterministic parity automaton. Over infinite trees Carayol et al. showed that there are ambiguous languages. We show that over infinite trees there is a hierarchy of degrees of ambiguity: For every k > 1 there are k-ambiguous languages that are not k - 1 ambiguous; and there are finitely (respectively countably, uncountably) ambiguous languages that are not boundedly (respectively finitely, countably) ambiguous.

The Containment Problem for Unambiguous Register Automata and Unambiguous Timed Automata

We investigate the complexity of the containment problem “Does $L(\mathcal {A})\subseteq L({\mathscr{B}})$ hold?” for register automata and timed automata, where ${\mathscr{B}}$ is assumed to be unambiguous and $\mathcal {A}$ is arbitrary. We prove that the problem is decidable in the case of register automata over $(\mathbb N,=)$ , in the case of register automata over $(\mathbb Q,<)$ when ${\mathscr{B}}$ has a single register, and in the case of timed automata when ${\mathscr{B}}$ has a single clock. We give a 2-EXPSPACE algorithm in the first case, whose complexity is a single exponential in the case that ${\mathscr{B}}$ has a bounded number of registers. In the other cases, we give an EXPSPACE algorithm.

On the length of uncompletable words in unambiguous automata

This paper deals with uncomplete unambiguous automata. In this setting, we investigate the minimal length of uncompletable words. This problem is connected with a well-known conjecture formulated by A. Restivo. We introduce the notion of relatively maximal row for a suitable monoid of matrices. We show that, if M is a monoid of {0, 1}-matrices of dimension n generated by a set S, then there is a matrix of M containing a relatively maximal row which can be expressed as a product of O(n3) matrices of S. As an application, we derive some upper bound to the minimal length of an uncompletable word of an uncomplete unambiguous automaton, in the case that its transformation monoid contains a relatively maximal row which is not maximal. Finally we introduce the maximal row automaton associated with an unambiguous automaton A. It is a deterministic automaton, which is complete if and only if A is. We prove that the minimal length of the uncompletable words of A is polynomially bounded by the number of states of A and the minimal length of the uncompletable words of the associated maximal row automaton.

Operations on Unambiguous Finite Automata

A nondeterministic finite automaton is unambiguous if it has at most one accepting computation on every input string. We investigate the state complexity of basic regular operations on languages represented by unambiguous finite automata. We get tight upper bounds for reversal ([Formula: see text]), intersection ([Formula: see text]), left and right quotients ([Formula: see text]), positive closure ([Formula: see text]), star ([Formula: see text]), shuffle ([Formula: see text]), and concatenation ([Formula: see text]). To prove tightness, we use a binary alphabet for intersection and left and right quotients, a ternary alphabet for star and positive closure, a five-letter alphabet for shuffle, and a seven-letter alphabet for concatenation. For complementation, we reduce the trivial upper bound [Formula: see text] to [Formula: see text]. We also get some partial results for union and square.

On the Strength of Unambiguous Tree Automata

This work is a study of the class of non-deterministic automata on infinite trees that are unambiguous i.e. have at most one accepting run on every tree. The motivating question asks if the fact that an automaton is unambiguous implies some drop in the descriptive complexity of the language recognised by the automaton. As it turns out, such a drop occurs for the parity index and does not occur for the weak parity index.More precisely, given an unambiguous parity automaton [Formula: see text] of index [Formula: see text], we show how to construct an alternating automaton [Formula: see text] which accepts the same language, but is simpler in terms of the acceptance condition. In particular, if [Formula: see text] is a Büchi automaton ([Formula: see text]) then [Formula: see text] is a weak alternating automaton. In general, [Formula: see text] belongs to the class [Formula: see text], what implies that it is simultaneously of alternating index [Formula: see text] and of the dual index [Formula: see text]. The transformation algorithm is based on a separation procedure of Arnold and Santocanale (2005).In the case of non-deterministic automata with the weak parity condition, we provide a separation procedure analogous to the one used above. However, as illustrated by examples, this separation procedure cannot be used to prove a complexity drop in the weak case, as there is no such drop.

(k,l)-Unambiguity and Quasi-Deterministic Structures

We focus on the family of $(k,l)$-unambiguous automata that encompasses the one of deterministic $k$-lookahead automata introduced by Han and Wood. We show that this family presents nice theoretical properties that allow us to compute quasi-deterministic structures. These structures are smaller than DFAs and can be used to solve the membership problem faster than NFAs.

ON THE DISAMBIGUATION OF FINITE AUTOMATA AND FUNCTIONAL TRANSDUCERS

This paper introduces a new disambiguation algorithm for finite automata and functional finite-state transducers. It gives a full description of this algorithm, including a detailed pseudocode and analysis, and several illustrating examples. The algorithm is often more efficient and the result dramatically smaller than the one obtained using determinization for finite automata or the construction of Schützenberger. The unambiguous automaton or transducer created by our algorithm are never larger than those generated by the construction of Schützenberger. In fact, in a variety of cases, the size of the unambiguous transducer returned by our algorithm is only linear in that of the input transducer while the transducer created by the construction of Schützenberger is exponentially larger. Our algorithm can be used effectively in many applications to make automata and transducers more efficient to use.

Lower bounds for the size of deterministic unranked tree automata

Tree automata operating on unranked trees use regular languages, called horizontal languages, to define the transitions of the vertical states that define the bottom-up computation of the automaton. It is well known that the deterministic tree automaton with smallest total number of states, that is, number of vertical states and number of states used to define the horizontal languages, is not unique and it is hard to establish lower bounds for the total number of states. By relying on existing bounds for the size of unambiguous finite automata, we give a lower bound for the size blow-up of determinizing a nondeterministic unranked tree automaton. The lower bound improves the earlier known lower bound that was based on an ad hoc construction.

Unambiguous finite automata over a unary alphabet

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 g˜. The growth rate of g˜(n), and therefore of the UFA–DFA tradeoff, is estimated as eΘ(nln2n3). The conversion of an n-state unary NFA to a UFA requires UFAs with g(n)+O(n2)=e(1+o(1))nlnn states, where g(n) 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 n2−o(1) states in the worst case, while the Kleene star requires up to exactly (n−1)2+1 states.

Some results on the structure of unary unambiguous automata

The paper focuses on deterministic and unambiguous finite automata (DFAʼs and UNFAʼs respectively for short) in the case of a one-letter alphabet. We present a structural characterization of unary UNFAʼs and some considerations relating minimal UNFAʼs with minimum DFAʼs recognizing a given unary language. We also present an algorithm for the construction of a minimal UNFA for a unary regular language. Then we establish a correspondence between pairs of UNFAʼs recognizing a unary language and its complement respectively, and the disjoint covering systems of number theory. It allows us to provide some conditions relating the number of successful simple paths and the lengths of cycles in an UNFA recognizing a unary language with the same parameters in an UNFA recognizing its complement.

Series which are both max-plus and min-plus rational are unambiguous

Consider partial maps from the free monoid into the field of real numbers with a rational domain. We show that two families of such series are actually the same: the unambiguous rational series on the one hand, and the max-plus and min-plus rational series on the other hand. The decidability of equality was known to hold in both families with different proofs, so the above unifies the picture. We give an effective procedure to build an unambiguous automaton from a max-plus automaton and a min-plus one that recognize the same series.

DESCRIPTIONAL COMPLEXITY OF NFA OF DIFFERENT AMBIGUITY

In this paper, we study the tradeoffs in descriptional complexity of NFA (nondeterministic finite automata) of various amounts of ambiguity. We say that two classes of NFA are separated if one class can be exponentially more succinct in descriptional sizes than the other. New results are given for separating DFA (deterministic finite automata) from UFA (unambiguous finite automata), UFA from MDFA (DFA with multiple initial states) and UFA from FNA (finitely ambiguous NFA). We present a family of regular languages that we conjecture to be a good candidate for separating FNA from LNA (linearly ambiguous NFA).

Minimizing finite automata is computationally hard

It is known that deterministic finite automata (DFAs) can be algorithmically minimized, i.e., a DFA M can be converted to an equivalent DFA M ′ which has a minimal number of states. The minimization can be done efficiently (in: Z. Kohavi (Ed.), Theory of Machines and Computations, Academic Press, New York, 1971, pp. 189–196). On the other hand, it is known that unambiguous finite automata and nondeterministic finite automata can be algorithmically minimized too, but their minimization problems turn out to be NP -complete and PSPACE -complete, respectively (SIAM J. Comput. 22(6) (1993) 1117–1141). In this paper, the time complexity of the minimization problem for two restricted types of finite automata is investigated. These automata are nearly deterministic, since they only allow a small amount of nondeterminism to be used. The main result is that the minimization problems for these models are computationally hard, namely NP -complete. Hence, even the slightest extension of the deterministic model towards a nondeterministic one, e.g., allowing at most one nondeterministic move in every accepting computation or allowing two initial states instead of one, results in computationally intractable minimization problems.

Learning Behaviors of Automata from Multiplicity and Equivalence Queries

We consider the problem of identifying the behavior of an unknown automaton with multiplicity in the field Q of rational numbers (Q-automaton) from multiplicity and equivalence queries. We provide an algorithm which is polynomial in the size of the Q-automaton and in the maximum length of the given counterexamples. As a consequence, we have that Q-automata are probably approximately correctly learnable (PAC-learnable) in polynomial time when multiplicity queries are allowed. A corollary of this result is that regular languages are polynomially predictable using membership queries with respect to the representation of unambiguous nondeterministic automata. This is important since there are unambiguous automata such that the equivalent deterministic automaton has an exponentially larger number of states.

On the Equivalence and Containment Problems for Unambiguous Regular Expressions, Regular Grammars and Finite Automata

The known proofs that the equivalence and containment problems for regular expressions, regular grammars and nondeterministic finite automata are PSPACE-complete [SM] depend upon consideration of highly unambiguous expressions, grammars and automata. Here, we prove that such dependence is inherent. Deterministic polynomial-time algorithms are presented for the equivalence and containment problems for unambiguous regular expressions, unambiguous regular grammars and unambiguous finite automata. The algorithms are then extended to ambiguity bounded by a fixed k. Our algorithms depend upon several elementary observations on the solutions of systems of homogeneous linear difference equations with constant coefficients and their relationship with the number of derivations of strings of a given length n by a regular grammar.

Automata that recognize intersections of free submonoids

Unambiguous automata are used to prove B. Tilson's result that the intersection of free submonoids of a free monoid is free. A description of the basis of the resulting intersection is an integral part of our proof.