Minimal DFA Research Articles

Quotients and atoms of reversible languages

We consider several reversible finite automaton models which have been introduced over decades, and study some properties of their languages. In particular, we look at the question whether the quotients and atoms of a specific class of reversible languages also belong to that class or not. We consider bideterministic automata, reversible deterministic finite automata (REV-DFAs), reversible multiple-entry DFAs (REV-MeDFAs), and several variants of reversible nondeterministic finite automata (REV-NFAs). It is known that the class of REV-DFA languages is strictly included in the class of REV-MeDFA languages. We show that the classes of complete REV-DFA languages and complete REV-MeDFA languages are the same. We also prove that differently from the general case of a REV-DFA language, the minimal DFA for a complete REV-DFA language is a complete REV-DFA. We observe that atoms of any regular language are accepted by REV-NFAs with a single initial and a single final state. We also study atoms of atoms of regular languages and show that any atom of an atom of a regular language is a union of atoms of that language.

Minimal consistent DFA from sample strings

We review the grammatical inference problem for regular languages which aims to generate a deterministic finite automaton from a representative set of training sample strings known to be in or not in the language. Although the general problem of producing a minimal DFA consistent with a given sample is known to be NP-hard, it is possible to generate minimal consistent DFA in polynomial time if certain constraints are satisfied by the given samples. In this work we propose a new algorithm which generates minimal DFA if the given training samples satisfy a certain sufficient condition. On the negative side, we also show that this problem is indeed hard, such that even for a more restricted class of training sets, the problem of generating minimal consistent DFA is already intractable.

Syntactic complexity of bifix-free regular languages

We study the properties of syntactic monoids of bifix-free regular languages. In particular, we solve an open problem concerning syntactic complexity: We prove that the cardinality of the syntactic semigroup of a bifix-free language with state complexity n is at most (n−1)n−3+(n−2)n−3+(n−3)2n−3 for n⩾6. The main proof uses a large construction with the method of injective function. Since this bound is known to be reachable, and the values for n⩽5 are known, this completely settles the problem. We also prove that (n−2)n−3+(n−3)2n−3−1 is the minimal size of the alphabet required to meet the bound for n⩾6. Finally, we show that the largest transition semigroups of minimal DFAs which recognize bifix-free languages are unique up to renaming the states.

On automata recognizing birecurrent sets

In this note we study automata recognizing birecurrent sets. A set of words is called birecurrent if the minimal partial DFA recognizing this set and the minimal partial DFA recognizing the reversal of this set are both strongly connected. This notion was introduced by Perrin, and Dolce et al. provided a characterization of such sets. We prove that deciding whether a partial DFA recognizes a birecurrent set is a PSPACE-complete problem. We show that this problem is PSPACE-complete even in the case of binary partial DFAs with all states accepting and in the case of binary complete DFAs.

Minimal Reversible Deterministic Finite Automata

We study reversible deterministic finite automata (REV-DFAs), that are partial deterministic finite automata whose transition function induces an injective mapping on the state set for every letter of the input alphabet. We give a structural characterization of regular languages that can be accepted by REV-DFAs. This characterization is based on the absence of a forbidden pattern in the (minimal) deterministic state graph. Again with a forbidden pattern approach, we also show that the minimality of REV-DFAs among all equivalent REV-DFAs can be decided. Both forbidden pattern characterizations give rise to [Formula: see text]-complete decision algorithms. In fact, our techniques allow us to construct the minimal REV-DFA for a given minimal DFA. These considerations lead to asymptotic upper and lower bounds on the conversion from DFAs to REV-DFAs. Thus, almost all problems that concern uniqueness and the size of minimal REV-DFAs are solved.

Separability by piecewise testable languages is PTime-complete

Piecewise testable languages form the first level of the Straubing–Thérien hierarchy. The membership problem for this level is decidable and testing if the language of a DFA is piecewise testable is NL-complete. So far, this question has not been addressed for NFAs in the literature. We fill in this gap and show that it is PSpace-complete. The main interest of this paper is, however, the lower-bound complexity of separability of regular languages by piecewise testable languages. Two regular languages are separable by a piecewise testable language if the piecewise testable language includes one of them and is disjoint from the other. For languages represented by NFAs, separability by piecewise testable languages is decidable in PTime. We show that it is PTime-hard and that it remains PTime-hard even if the input automata are minimal DFAs. As a result, it is unlikely that separability of regular languages by piecewise testable languages can be solved in a restricted space or effectively parallelized.

Minimal partial languages and automata

Partial words are sequences of characters from an alphabet in which some positions may be marked with a “hole” symbol, ⋄. We can create a ⋄-substitution mapping this symbol to a subset of the alphabet, so that applying such a substitution to a partial word results in a set of total words (ones without holes). This setup allows us to compress regular languages into smaller partial languages. Deterministic finite automata for such partial languages, referred to as ⋄-DFAs, employ a limited non-determinism that can allow them to have lower state complexity than the minimal DFAs for the corresponding total languages. Our paper focuses on algorithms for the construction of minimal partial languages, associated with some ⋄-substitution, as well as approximation algorithms for the construction of minimal ⋄-DFAs.

Minimal consistent DFA revisited

We address the classic problem of polynomial computation of a minimal finite-state representation compatible with two finite input sets that contain those strings that have to be accepted and rejected. We extend the previously defined uniformly-complete sample and prove that it is a special case of the sufficient condition we propose.

On the hierarchy of generalizations of one-unambiguous regular languages

A regular language is lookahead (resp. block) deterministic if it is specified by a lookahead (resp. block) deterministic regular expression. These two subclasses of regular languages have been respectively introduced by Han and Wood for lookahead determinism and by Giammarresi et al. for block determinism, as a possible extension of one-unambiguous languages defined and characterized by Brüggemann-Klein and Wood.In this paper, we study the hierarchies and inclusion links of these families. We first show that each block deterministic language is the alphabetical image of some one-unambiguous language. Moreover, we show that deciding the block determinism of a regular language from its minimal DFA does not only require state elimination. Han and Wood state that there is a proper hierarchy in block deterministic languages, but prove it using this erroneous requirement. However, we prove their statement by giving our own parametrized family. We also prove that there is a proper hierarchy in lookahead deterministic languages by studying particular properties of unary regular expressions. Finally, using our valid results, we confirm that the family of block deterministic languages is strictly included in the one of lookahead deterministic languages by showing that any block deterministic unary language is one-unambiguous.

State Complexity of Basic Operations on Non-Returning Regular Languages

We consider the state complexity of basic operations on non-returning regular languages. For a non-returning minimal DFA, the start state does not have any in-transitions. We establish the precise state complexity of four Boolean operations (union, intersection, difference, symmetric difference), catenation, reversal, and Kleene-star for non-returning regular languages. Our results are usually smaller than the state complexities for general regular languages and larger than the state complexities for suffix-free regular languages.

Deciding determinism of unary languages

In this paper, we investigate the complexity of deciding determinism of unary languages. First, we give a method to derive a set of arithmetic progressions from a regular expression E over a unary alphabet, and establish relations between numbers represented by these arithmetic progressions and words in L(E). Next, we define a problem relating to arithmetic progressions and investigate the complexity of this problem. Then by a reduction from this problem we show that deciding determinism of unary languages is coNP-complete. Finally, we extend our derivation method to expressions with counting, and prove that deciding whether an expression over a unary alphabet with counting defines a deterministic language is in Π2p. We also establish a tight upper bound for the size of the minimal DFA for expressions with counting.

State Complexity of Boundary of Prefix-Free Regular Languages

Recently, researchers studied the state complexity of boundary — [Formula: see text] — of regular languages L motivated from the famous Kuratowski’s 14-theorem. Prefix codes — a set of languages — play an important role in several applications. We consider prefix-free regular languages and investigate the state complexity of two operations, [Formula: see text] and [Formula: see text] for prefix-free regular languages. Based on the unique structural properties of a prefix-free minimal DFA, we compute the precise state complexity of [Formula: see text] and [Formula: see text]. We then present the tight bound over a quaternary alphabet for [Formula: see text] and [Formula: see text]. Our results are smaller than the composition of the state complexity function for individual operations of prefix-free regular languages.

Coalgebraic constructions of canonical nondeterministic automata

For each regular language L we describe a family of canonical nondeterministic acceptors (nfas). Their construction follows a uniform recipe: build the minimal dfa for L in a locally finite variety V, and apply an equivalence between the category of finite V-algebras and a suitable category of finite structured sets and relations. By instantiating this to different varieties, we recover three well-studied canonical nfas: V = boolean algebras yields the átomaton of Brzozowski and Tamm, V = semilattices yields the jiromaton of Denis, Lemay and Terlutte, and V=Z2-vector spaces yields the minimal xor automaton of Vuillemin and Gama. Moreover, we obtain a new canonical nfa called the distromaton by taking V = distributive lattices. Each of these nfas is shown to be minimal relative to a suitable measure, and we derive sufficient conditions for their state-minimality. Our approach is coalgebraic, exhibiting additional structure and universal properties of the canonical nfas.

Determinization and minimization of finite acyclic automata by incremental techniques

The determinization of a nondeterministic finite automaton FA N is the process of generating a deterministic FA DFA D equivalent to sharing the same regular language of N. The minimization of D is the process of generating the minimal DFA i¾? equivalent to D. Classical algorithms for determinization and minimization are available in the literature for several decades. However, they operate monolithically, assuming that the FA to be either determinized or minimized is given once and for all. By contrast, we consider determinization and minimization in a dynamic context, where N augments over time: after each augmentation, determinization and minimization of N into i¾? is required. Using classical monolithic algorithms to solve this problem is bound to poor performance. An algorithm for incremental determinization and minimization of acyclic finite automata, called IDMA, is proposed. Despite being conceived within the narrow domain of model-based diagnosis and monitoring of active systems, the algorithm is general-purpose in nature. Experimental evidence indicates that IDMA is far more efficient than classical algorithms in solving incremental determinization and minimization problems. Copyright © 2015 John Wiley & Sons, Ltd.

On Continuous Nondeterminism and State Minimality

This paper is devoted to the study of nondeterministic closure automata, that is, nondeterministic finite automata (nfas) equipped with a strict closure operator on the set of states and continuous transition structure. We prove that for each regular language L there is a unique minimal nondeterministic closure automaton whose underlying nfa accepts L. Here minimality means no proper sub or quotient automata exist, just as it does in the case of minimal dfas. Moreover, in the important case where the closure operator of this machine is topological, its underlying nfa is shown to be state-minimal. The basis of these results is an equivalence between the categories of finite semilattices and finite strict closure spaces.

Deciding Determinism of Regular Languages

We investigate the problem to decide for DFAs, NFAs and regular expressions whether they describe deterministic regular languages. For DFAs with limited alphabet size we show the NL-completeness (nondeterministic logarithmic space-completeness) of the problem. Bruggemann-Klein and Wood (Inf. Comput. 142(2), 182---206, (1998)) gave an algorithm deciding for a minimal DFA whether it recognizes a deterministic regular language. Their algorithm requires polynomial time. Based on this algorithm we construct an algorithm for those DFAs which are not necessarily minimal, but limited in the size of the alphabet. We first focus on the case that the DFAs are minimal and extend to general DFAs later on. The new algorithm is substantially modified in contrast to the original one and uses structural properties of minimal DFAs and of special subautomata of the DFAs, called orbit automata. This algorithm runs in nondeterministic logarithmic space. After that we show the NL-hardness of the problem. For NFAs and regular expressions with arbitrary alphabets the problem is shown to be in PSPACE by Czerwinski et al. (2013) (and, in fact, to be PSPACE-complete). Their approach resulted in an algorithm working with space ??(x4)$\mathcal {O}(x^{4})$. However, our approach for the NL-algorithm can be modified to get an algorithm for NFAs and regular expressions using just quadratic space.

Regular inference as vertex coloring

This paper is concerned with the problem of supervised learning of deterministic finite state automata, in the technical sense of identification in the limit from complete data, by finding a minimal DFA consistent with a sample.We solve this problem by translating it in its entirety to a vertex coloring problem. Essentially, such a problem consists of two types of constraints that restrict the hypothesis space: inequivalence and equivalence constraints. The inequivalence constraints translate to the vertex coloring problem in a very natural way. The equivalence constraints however greatly complicate the translation to vertex coloring. In previous coloring-based translations, these were therefore encoded either dynamically by modifying the vertex coloring instance on-the-fly, or by encoding them as satisfiability problems.We provide the first translation that encodes both types of constraints together in a pure vertex coloring instance and prove the correctness of the construction. The coloring approach offers many opportunities for applying insights from combinatorial optimization and graph theory to regular inference. We immediately obtain new complexity bounds, as well as a family of new learning algorithms which can be used to obtain exact hypotheses as well as fast approximations.We also demonstrate that satisfying all inequivalence constraints in regular inference and vertex coloring are in some sense equally hard. This immediately implies that regular inference is not approximable within |S|1−ϵ (|S| being the cardinality of the sample) for any ϵ>12, unless P=NP. It also indicates that solving only for inequivalence constraints is easier than the full regular inference problem.

Size lower bounds for quantum automata

We compare the descriptional power of quantum finite automata with control language (qfcs) and deterministic finite automata (dfas). By suitably adapting Rabin's technique, we show how to convert any given qfc to an equivalent dfa, incurring in an at most exponential size increase. This enables us to state a lower bound on the size of qfcs, which is logarithmic in the size of equivalent minimal dfas. In turn, this result yields analogous size lower bounds for several models of quantum finite automata in the literature.

Maximally Atomic Languages

The atoms of a regular language are non-empty intersections of complemented and uncomplemented quotients of the language. Tight upper bounds on the number of atoms of a language and on the quotient complexities of atoms are known. We introduce a new class of regular languages, called the maximally atomic languages, consisting of all languages meeting these bounds. We prove the following result: If L is a regular language of quotient complexity n and G is the subgroup of permutations in the transition semigroup T of the minimal DFA of L, then L is maximally atomic if and only if G is transitive on k-subsets of 1,...,n for 0 <= k <= n and T contains a transformation of rank n-1.

Descriptional Complexity of the Languages KaL: Automata, Monoids and Varieties

The first step when forming the polynomial hierarchies of languages is to consider languages of the form KaL where K and L are over a finite alphabet A and from a given variety V of languages, a being a letter from A. All such KaL's generate the variety of languages BPol1(V). We estimate the numerical parameters of the language KaL in terms of their values for K and L. These parameters include the state complexity of the minimal complete DFA and the size of the syntactic monoids. We also estimate the cardinality of the image of A* in the Schuetzenberger product of the syntactic monoids of K and L. In these three cases we obtain the optimal bounds. Finally, we also consider estimates for the cardinalities of free monoids in the variety of monoids corresponding to BPol1(V) in terms of sizes of the free monoids in the variety of monoids corresponding to V.