We show that every regular language defines a unique nondeterministic finite automaton (NFA), which we call "átomaton", whose states are the "atoms" of the language, that is, non-empty intersections of complemented or uncomplemented left quotients of the language. We describe methods of constructing the átomaton, and prove that it is isomorphic to the reverse automaton of the minimal deterministic finite automaton (DFA) of the reverse language. We study "atomic" NFAs in which the right language of every state is a union of atoms. We generalize Brzozowski's double-reversal method for minimizing a deterministic finite automaton (DFA), showing that the result of applying the subset construction to an NFA is a minimal DFA if and only if the reverse of the NFA is atomic. We prove that Sengoku's claim that his method always finds a minimal NFA is false.
In this paper, a compressed membership problem for finite automata, both deterministic and non-deterministic, with compressed transition labels is studied. The compression is represented by straight-line programs (SLPs), i.e. context-free grammars generating exactly one string. A novel technique of dealing with SLPs is introduced: the SLPs are recompressed, so that substrings of the input text are encoded in SLPs labelling the transitions of the NFA (DFA) in the same way, as in the SLP representing the input text. To this end, the SLPs are locally decompressed and then recompressed in a unifor…
Deterministic finite automata (DFAs) are constructed for various purposes in computational biology. Little attention, however, has been given to the efficient construction of minimal DFAs. In this article, we define simple non-deterministic finite automata (NFAs) and prove that the standard subset construction transforms NFAs of this type into minimal DFAs. Furthermore, we show how simple NFAs can be constructed from two types of patterns popular in bioinformatics, namely (sets of) generalized strings and (generalized) strings with a Hamming neighborhood.
Linear-time pattern matching engines have seen promising results using Finite Automata (FA) as their computation model. Among different FA variants, deterministic (DFA) and non-deterministic (NFA) are the most commonly used computation models for FA-based pattern matching engines. Moreover, NFA is the prevalent model in pattern matching engines on spatial architectures. The reasons are: i) DFA size, as in #states, can be exponential compared to equivalent NFA, ii) DFA cannot exploit the massive parallelism available on spatial architectures. This paper performs an empirical study on the #state…
We introduce the category of dependency automata. A dependency automaton consists of two nondeterministic finite automata, with a relation between their states satisfying conditions. This category is equivalent to deterministic finite automata interpreted in join-semilattices i.e. JSL-dfas. The canonical dependency automaton accepting $L$ amounts to the state-minimal dfas for $L$ and $rev(L)$ connected by the `dependency relation'. We describe many canonical JSL-dfas as dependency automata and also explain/extend Brzozowski's algorithm. Call an nfa `subatomic' if its individual states accept a…
A word w is called a synchronizing (recurrent, reset) word of a deterministic finite automaton (DFA) if w brings all states of the automaton to some state; a DFA that has a synchronizing word is said to be synchronizing. Cerny conjectured in 1964 that every n-state synchronizing DFA possesses a synchronizing word of length at most (n -1)2. We consider automaton with aperiodic transition monoid (such automaton is called aperiodic). We show that every synchronizing n-state aperiodic automaton has a synchronizing word of length at most n(n-2)+1. Thus, for aperiodic automaton as well as for automa…
We study the following natural variation on the classical universality problem: given a language $L(M)$ represented by $M$ (e.g., a DFA/RE/NFA/PDA), does there exist an integer $\ell \geq 0$ such that $Σ^\ell \subseteq L(M)$? In the case of an NFA, we show that this problem is NEXPTIME-complete, and the smallest such $\ell$ can be doubly exponential in the number of states. This particular case was formulated as an open problem in 2009, and our solution uses a novel and involved construction. In the case of a PDA, we show that it is recursively unsolvable, while the smallest such $\ell$ is not…
The work presents some new algorithms realized recently in the package TESTAS. They decide whether or not deterministic finite automaton (DFA) is synchronizing, several procedures find relatively short synchronizing words and a synchronizing word of the minimal length. We check the existence of a coloring of directed graph that turns the graph into a synchronizing DFA. The algorithm finds the coloring (better known as the road coloring) if it exists. Otherwise, the k-synchronizing road coloring can be found. We use a linear visualization of the graph of an automaton based on its structural pro…
Here we propose a variant of the nondeterministic finite automaton with translucent letters (NFAwtl) which, after reading and deleting a letter, does not return to the left end of its tape, but rather continues from the position of the letter just deleted. When the end-of-tape marker is reached, our automaton can decide whether to accept, to reject, or to continue, which means that it again reads the remaining tape contents from the beginning. This type of automaton, called a non-returning finite automaton with translucent letters or an nrNFAwtl, is strictly more expressive than the NFAwtl. We…
In this paper we define a new descriptional complexity measure for Deterministic Finite Automata, BC-complexity, as an alternative to the state complexity. We prove that for two DFAs with the same number of states BC-complexity can differ exponentially. In some cases minimization of DFA can lead to an exponential increase in BC-complexity, on the other hand BC-complexity of DFAs with a large state space which are obtained by some standard constructions (determinization of NFA, language operations), is reasonably small. But our main result is the analogue of the "Shannon effect" for finite auto…
A regular language is $k$-piecewise testable if it is a finite boolean combination of languages of the form $Σ^* a_1 Σ^* \cdots Σ^* a_n Σ^*$, where $a_i\inΣ$ and $0\le n \le k$. Given a DFA $A$ and $k\ge 0$, it is an NL-complete problem to decide whether the language $L(A)$ is piecewise testable and, for $k\ge 4$, it is coNP-complete to decide whether the language $L(A)$ is $k$-piecewise testable. It is known that the depth of the minimal DFA serves as an upper bound on $k$. Namely, if $L(A)$ is piecewise testable, then it is $k$-piecewise testable for $k$ equal to the depth of $A$. In this pa…
The concept of Deterministic Finite Cover Automata (DFCA) was introduced at WIA '98, as a more compact representation than Deterministic Finite Automata (DFA) for finite languages. In some cases representing a finite language, Nondeterministic Finite Automata (NFA) may significantly reduce the number of states used. The combined power of the succinctness of the representation of finite languages using both cover languages and non-determinism has been suggested, but never systematically studied. In the present paper, for nondeterministic finite cover automata (NFCA) and l-nondeterministic finit…
A word $w$ of letters on edges of underlying graph $Γ$ of deterministic finite automaton (DFA) is called synchronizing if $w$ sends all states of the automaton to a unique state. J. Černy discovered in 1964 a sequence of $n$-state complete DFA possessing a minimal synchronizing word of length $(n-1)^2$. The hypothesis, well known today as Černy conjecture, claims that $(n-1)^2$ is a precise upper bound on the length of such a word over alphabet $Σ$ of letters on edges of $Γ$ for every complete $n$-state DFA. The hypothesis was formulated distinctly in 1966 by Starke. A special classes of matri…
Affine finite automata (AfA) can be more succinct than probabilistic and quantum finite automata when recognizing some regular languages with bounded-error. In this paper, we improve previously known constructions given for the succinctness of AfAs in three ways. First, we replace some of fixed error bounds with arbitrarily small error bounds. Second, we present new constructions by using less states than the previous constructions. Third, we show that any language recognized by a nondeterministic finite automaton (NFA) is also recognized by bounded-error AfAs having one more state, and so, Af…
In this paper we consider the computational complexity of the following problems: given a DFA or NFA representing a regular language L over a finite alphabet Sigma is the set of all prefixes (resp., suffixes, factors, subwords) of all words of L equal to Sigma*? In the case of testing universality for factors of languages represented by DFA's, we find an interesting connection to Cerny's conjecture on synchronizing words.
We study the computational and descriptional complexity of the following transformation: Given a one-counter automaton (OCA) A, construct a nondeterministic finite automaton (NFA) B that recognizes an abstraction of the language L(A): its (1) downward closure, (2) upward closure, or (3) Parikh image. For the Parikh image over a fixed alphabet and for the upward and downward closures, we find polynomial-time algorithms that compute such an NFA. For the Parikh image with the alphabet as part of the input, we find a quasi-polynomial time algorithm and prove a completeness result: we construct a s…
The automation of decision procedures makes certification essential. We suggest to use determinacy of turn-based two-player games with regular winning conditions in order to generate certificates for the number of states that a deterministic finite automaton (DFA) needs in order to recognize a given language. Given a language $L$ and a bound $k$, recognizability of $L$ by a DFA with $k$ states is reduced to a game between Prover and Refuter. The interaction along the game then serves as a certificate. Certificates generated by Prover are minimal DFAs. Certificates generated by Refuter are faul…
The state complexity of a Deterministic Finite-state automaton (DFA) is the number of states in its minimal equivalent DFA. We study the state complexity of random $n$-state DFAs over a $k$-symbol alphabet, drawn uniformly from the set $[n]^{[n]\times[k]}\times2^{[n]}$ of all such automata. We show that, with high probability, the latter is $α_k n + O(\sqrt n\log n)$ for a certain explicit constant $α_k$.
The ambiguity of a nondeterministic finite automaton (NFA) N for input size n is the maximal number of accepting computations of N for an input of size n. For all k, r 2 N we construct languages Lr,k which can be recognized by NFA's with size k poly(r) and ambiguity O(nk), but Lr,k has only NFA's with exponential size, if ambiguity o(nk) is required. In particular, a hierarchy for polynomial ambiguity is obtained, solving a long standing open problem (Ravikumar and Ibarra, 1989, Leung, 1998).
A word $s$ of letters on edges of underlying graph $Γ$ of deterministic finite automaton (DFA) is called synchronizing if $s$ sends all states of the automaton to a unique state. J. Černy discovered in 1964 a sequence of $n$-state complete DFA possessing a minimal synchronizing word of length $(n-1)^2$. The hypothesis, mostly known today as Černy conjecture, claims that $(n-1)^2$ is a precise upper bound on the length of such a word over alphabet $Σ$ of letters on edges of $Γ$ for every complete $n$-state DFA. The hypothesis was formulated in 1966 by Starke. Algebra with nonstandard operation …
Given an order of the underlying alphabet we can lift it to the states of a finite deterministic automaton: to compare states we use the order of the strings reaching them. When the order on strings is the co-lexicographic one \emph{and} this order turns out to be total, the DFA is called Wheeler. This recently introduced class of automata -- the \emph{Wheeler automata} -- constitute an important data-structure for languages, since it allows the design and implementation of a very efficient tool-set of storage mechanisms for the transition function, supporting a large variety of substring quer…
The class of row monomial matrices (one unit and rest zeros in every row) with some non-standard operations of summation and usual multiplication is our main object. These matrices generate a space with respect to the mentioned operations. A word w of letters on edges of underlying graph of deterministic finite automaton (DFA) is called synchronizing if w sends all states of the automaton to a unique state J. Černy discovered in 1964 a sequence of n-state complete DFA possessing a minimal synchronizing word of length (n-1)(n-1). The hypothesis, well known today as the Černy conjecture, claims …
Minimal deterministic finite automata (DFAs) can be reduced further at the expense of a finite number of errors. Recently, such minimization algorithms have been improved to run in time O(n log n), where n is the number of states of the input DFA, by [Gawrychowski and Jeż: Hyper-minimisation made efficient. Proc. MFCS, LNCS 5734, 2009] and [Holzer and Maletti: An n log n algorithm for hyper-minimizing a (minimized) deterministic automaton. Theor. Comput. Sci. 411, 2010]. Both algorithms return a DFA that is as small as possible, while only committing a finite number of errors. These algorithms…
Université Paris-Saclay（巴黎萨克雷大学）；CNRS（法国国家科学研究中心）；ENS Paris-Saclay（巴黎萨克雷高等师范学院）；INRIA（法国国家信息与自动化研究所）；CentraleSupélec（中央理工-高等电力学院）；Université Paris-Est Créteil（巴黎东克雷泰伊大学）；Universität zu Lübeck（吕贝克大学）；Carl von Ossietzky University of Oldenburg（奥尔登堡卡尔·冯·奥西茨基大学）；Max Planck Institute for Software Systems（马克斯·普朗克软件系统研究所）
Angluin's L* algorithm learns the minimal (complete) deterministic finite automaton (DFA) of a regular language using membership and equivalence queries. Its probabilistic approximatively correct (PAC) version substitutes an equivalence query by a large enough set of random membership queries to get a high level confidence to the answer. Thus it can be applied to any kind of (also non-regular) device and may be viewed as an algorithm for synthesizing an automaton abstracting the behavior of the device based on observations. Here we are interested on how Angluin's PAC learning algorithm behaves…
A deterministic finite automaton (DFA) separates two strings $w$ and $x$ if it accepts $w$ and rejects $x$. The minimum number of states required for a DFA to separate $w$ and $x$ is denoted by $sep(w,x)$. The present paper shows that the difference $|sep(w,x)-sep(w^R,x^R)|$ is unbounded for a binary alphabet; here $w^R$ stands for the mirror image of $w$. This solves an open problem stated in [Demaine, Eisenstat, Shallit, Wilson: Remarks on separating words. DCFS 2011. LNCS vol. 6808, pp. 147-157.]
We present a uniform method for translating an arbitrary nondeterministic finite automaton (NFA) into a deterministic mass action input/output chemical reaction network (I/O CRN) that simulates it. The I/O CRN receives its input as a continuous time signal consisting of concentrations of chemical species that vary to represent the NFA's input string in a natural way. The I/O CRN exploits the inherent parallelism of chemical kinetics to simulate the NFA in real time with a number of chemical species that is linear in the size of the NFA. We prove that the simulation is correct and that it is ro…
Ordering the collection of states of a given automaton starting from an order of the underlying alphabet is a natural move towards a computational treatment of the language accepted by the automaton. Along this path, Wheeler \emph{graphs} have been recently introduced as an extension/adaptation of the Burrows-Wheeler Transform (the now famous BWT, originally defined on strings) to graphs. These graphs constitute an important data-structure for languages, since they allow a very efficient storage mechanism for the transition function of an automaton, while providing a fast support to all sorts …
A word w of letters on edges of underlying graph Gamma of deterministic finite automaton (DFA) is called the synchronizing word if w sends all states of the automaton to a unique state. J. Cerny discovered in 1964 a sequence of n-state complete DFA possessing a minimal synchronizing word of length (n-1)^2. The hypothesis, well known today as the Cerny conjecture, claims that it is also precise upper bound on the length of such a word for a complete DFA. This simple looking conjecture is arguably the most fascinating and longstanding open problem in the theory of finite automaton. The hypothesi…
The state complexity of basic operations on finite languages (considering complete DFAs) has been in studied the literature. In this paper we study the incomplete (deterministic) state and transition complexity on finite languages of boolean operations, concatenation, star, and reversal. For all operations we give tight upper bounds for both description measures. We correct the published state complexity of concatenation for complete DFAs and provide a tight upper bound for the case when the right automaton is larger than the left one. For all binary operations the tightness is proved using fa…
In this paper, we propose an architecture of oritatami systems with which one can simulate an arbitrary nondeterministic finite automaton (NFA) in a unified manner. The oritatami system is known to be Turing-universal but the simulation available so far requires 542 bead types and $O(t^4 \log^2 t)$ steps in order to simulate $t$ steps of a Turing machine. The architecture we propose employs only 329 bead types and requires just $O(t |Q|^4 |Σ|^2)$ steps to simulate an NFA over an input alphabet $Σ$ with a state set $Q$ working on a word of length $t$.
Brüggemann-Klein and Wood define a one-unambiguous regular language as a language that can be recognized by a deterministic Glushkov automaton. They give a procedure performed on the minimal DFA, the BW-test, to decide whether a language is one-unambiguous. Block determinism is an extension of one-unambiguity while considering non-empty words as symbols and prefix-freeness as determinism. A block automaton is compact if it does not have two equivalent states (same right language). We showed that a language is $k$-block deterministic if it is recognized by some deterministic $k$-block automaton…
The problem DFA-Intersection-Nonemptiness asks if a given number of deterministic automata accept a common word. In general, this problem is PSPACE-complete. Here, we investigate this problem for the subclasses of commutative automata and automata recognizing sparse languages. We show that in both cases DFA-Intersection-Nonemptiness is complete for NP and for the parameterized class $W[1]$, where the number of input automata is the parameter, when the alphabet is fixed. Additionally, we establish the same result for Tables Non-Empty Join, a problem that asks if the join of several tables (poss…
Department of Computing Science, Umeå University（乌梅大学计算机科学系）；Department of Computer Science, Stellenbosch University（斯坦福博芝大学计算机科学系）；National Institute for Theoretical and Computational Sciences, South Africa（南非理论与计算科学国家研究所）
A regular language is almost fully characterized by its right congruence relation. Indeed, a regular language can always be recognized by a DFA isomorphic to the automaton corresponding to its right congruence, henceforth the Rightcon automaton. The same does not hold for regular omega-languages. The right congruence of a regular omega-language is not informative enough; many regular omega-languages have a trivial right congruence, and in general it is not always possible to define an omega-automaton recognizing a given language that is isomorphic to the rightcon automaton. The class of weak r…
In this paper we combine the algebraic properties of Mealy machines generating self-similar groups and the combinatorial properties of the corresponding deterministic finite automata (DFA). In particular, we relate bounded automata to finitely generated synchronizing automata and characterize finite automata groups in terms of nilpotency of the corresponding DFA. Moreover, we present a decidable sufficient condition to have free semigroups in an automaton group. A series of examples and applications is widely discussed, in particular we show a way to color the De Bruijn automata into Mealy aut…
While finite automata have minimal DFAs as a simple and natural normal form, deterministic omega-automata do not currently have anything similar. One reason for this is that a normal form for omega-regular languages has to speak about more than acceptance - for example, to have a normal form for a parity language, it should relate every infinite word to some natural color for this language. This raises the question of whether or not a concept such as a natural color of an infinite word (for a given language) exists, and, if it does, how it relates back to automata. We define the natural color …
We investigate the magic number problem, that is, the question whether there exists a minimal n-state nondeterministic finite automaton (NFA) whose equivalent minimal deterministic finite automaton (DFA) has alpha states, for all n and alpha satisfying n less or equal to alpha less or equal to exp(2,n). A number alpha not satisfying this condition is called a magic number (for n). It was shown in [11] that no magic numbers exist for general regular languages, while in [5] trivial and non-trivial magic numbers for unary regular languages were identified. We obtain similar results for automata a…
Minimal forbidden factors are a useful tool for investigating properties of words and languages. Two factorial languages are distinct if and only if they have different (antifactorial) sets of minimal forbidden factors. There exist algorithms for computing the minimal forbidden factors of a word, as well as of a regular factorial language. Conversely, Crochemore et al. [IPL, 1998] gave an algorithm that, given the trie recognizing a finite antifactorial language $M$, computes a DFA recognizing the language whose set of minimal forbidden factors is $M$. In the same paper, they showed that the o…
