Minimal Deterministic Finite Automaton Research Articles

Yet another canonical nondeterministic automaton

Several canonical forms of finite automata have been introduced over the decades. In particular, if one considers the minimal deterministic finite automaton (DFA), the canonical residual finite state automaton (RFSA), and the átomaton of a language, then the átomaton can be seen as the dual automaton of the minimal DFA, but no such dual has been presented for the canonical RFSA so far. We fill this gap by introducing a new canonical automaton that we call the maximized prime átomaton, and study its properties. We also describe how these four automata can be extracted from suitable observation tables used in the automata learning context.

Analyzing Robustness of Angluin's L$^*$ Algorithm in Presence of Noise

Angluin's L$^*$ algorithm learns the minimal deterministic finite automaton (DFA) of a regular language using membership and equivalence queries. Its probabilistic approximatively correct (PAC) version substitutes an equivalence query by numerous random membership queries to get a high level confidence to the answer. Thus it can be applied to any kind of 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 for devices which are obtained from a DFA by introducing some noise. More precisely we study whether Angluin's algorithm reduces the noise and produces a DFA closer to the original one than the noisy device. We propose several ways to introduce the noise: (1) the noisy device inverts the classification of words w.r.t. the DFA with a small probability, (2) the noisy device modifies with a small probability the letters of the word before asking its classification w.r.t. the DFA, (3) the noisy device combines the classification of a word w.r.t. the DFA and its classification w.r.t. a counter automaton, and (4) the noisy DFA is obtained by a random process from two DFA such that the language of the first one is included in the second one. Then when a word is accepted (resp. rejected) by the first (resp. second) one, it is also accepted (resp. rejected) and in the remaining cases, it is accepted with probability 0.5. Our main experimental contributions consist in showing that: (1) Angluin's algorithm behaves well whenever the noisy device is produced by a random process, (2) but poorly with a structured noise, and, that (3) is able to eliminate pathological behaviours specified in a regular way. Theoretically, we show that randomness almost surely yields systems with non-recursively enumerable languages.

The Range of State Complexities of Languages Resulting from the Cascade Product — The Unary Case

We investigate the state complexity of languages resulting from the cascade product of two minimal deterministic finite automata with [Formula: see text] and [Formula: see text] states, respectively. More precisely we study the magic number problem of the cascade product operation and show what range of complexities can be produced in case the left automaton is unary, that is, has only a singleton letter alphabet. Here we distinguish the cases when the involved automata are reset automata, permutation automata, permutation-reset automata, or do not have any restriction on their structure. It turns out that the picture on the obtained state complexities of the cascade product is diverse, and for all cases, except where the left automaton is a unary permutation(-reset) or a deterministic finite automaton without structural restrictions, and the right one is a reset automaton or a deterministic finite automaton without structural restrictions, we are able to identify state sizes that cannot be reached — these numbers are called “magic.”

Canonicalizing BigSMILES for Polymers with Defined Backbones.

BigSMILES, a line notation for encapsulating the molecular structure of stochastic molecules such as polymers, was recently proposed as a compact and readable solution for writing macromolecules. While BigSMILES strings serve as useful identifiers for reconstructing the molecular connectivity for polymers, in general, BigSMILES allows the same polymer to be codified into multiple equally valid representations. Having a canonicalization scheme that eliminates the multiplicity would be very useful in reducing time-intensive tasks like structural comparison and molecular search into simple string-matching tasks. Motivated by this, in this work, two strategies for deriving canonical representations for linear polymers are proposed. In the first approach, a canonicalization scheme is proposed to standardize the expression of BigSMILES stochastic objects, thereby standardizing the expression of overall BigSMILES strings. In the second approach, an analogy between formal language theory and the molecular ensemble of polymer molecules is drawn. Linear polymers can be converted into regular languages, and the minimal deterministic finite automaton uniquely associated with each prescribed language is used as the basis for constructing the unique text identifier associated with each distinct polymer. Overall, this work presents algorithms to convert linear polymers into unique structure-based text identifiers. The derived identifiers can be readily applied in chemical information systems for polymers and other polymer informatics applications.

Analyzing Robustness of Angluin's L* Algorithm in Presence of Noise

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 for devices which are obtained from a DFA by introducing some noise. More precisely we study whether Angluin's algorithm reduces the noise and produces a DFA closer to the original one than the noisy device. We propose several ways to introduce the noise: (1) the noisy device inverts the classification of words w.r.t. the DFA with a small probability, (2) the noisy device modifies with a small probability the letters of the word before asking its classification w.r.t. the DFA, and (3) the noisy device combines the classification of a word w.r.t. the DFA and its classification w.r.t. a counter automaton. Our experiments were performed on several hundred DFAs. Our main contributions, bluntly stated, consist in showing that: (1) Angluin's algorithm behaves well whenever the noisy device is produced by a random process, (2) but poorly with a structured noise, and, that (3) almost surely randomness yields systems with non-recursively enumerable languages.

Descriptional Complexity of Non-Unary Self-Verifying Symmetric Difference Automata

Unary self-verifying symmetric difference automata have a known tight bound of [Formula: see text] for their state complexity. We now consider the non-unary case and show that, for every [Formula: see text], there is a regular language [Formula: see text] accepted by a non-unary self-verifying symmetric difference nondeterministic automaton with [Formula: see text] states, such that its equivalent minimal deterministic finite automaton has [Formula: see text] states. Furthermore, given any SV-XNFA with [Formula: see text] states, it is possible, up to isomorphism, to find at most another [Formula: see text] equivalent SV-XNFA. Finally, we show that for a certain set of non-unary SV-XNFA, [Formula: see text] is a tight bound on the state complexity.

Finitely generated subgroups of free groups as formal languages and their cogrowth

For finitely generated subgroups $H$ of a free group $F_m$ of finite rank $m$, we study the language $L_H$ of reduced words that represent $H$ which is a regular language. Using the (extended) core of Schreier graph of $H$, we construct the minimal deterministic finite automaton that recognizes $L_H$. Then we characterize the f.g. subgroups $H$ for which $L_H$ is irreducible and for such groups explicitly construct ergodic automaton that recognizes $L_H$. This construction gives us an efficient way to compute the cogrowth series $L_H(z)$ of $H$ and entropy of $L_H$. Several examples illustrate the method and a comparison is made with the method of calculation of $L_H(z)$ based on the use of Nielsen system of generators of $H$.

Finitely generated subgroups of free groups as formal languages and their cogrowth

For finitely generated subgroups $H$ of a free group $F_m$ of finite rank $m$, we study the language $L_H$ of reduced words that represent $H$ which is a regular language. Using the (extended) core of Schreier graph of $H$, we construct the minimal deterministic finite automaton that recognizes $L_H$. Then we characterize the f.g. subgroups $H$ for which $L_H$ is irreducible and for such groups explicitly construct ergodic automaton that recognizes $L_H$. This construction gives us an efficient way to compute the cogrowth series $L_H(z)$ of $H$ and entropy of $L_H$. Several examples illustrate the method and a comparison is made with the method of calculation of $L_H(z)$ based on the use of Nielsen system of generators of $H$.

L^*-based learning of Markov decision processes (extended version)

Automata learning techniques automatically generate systemmodels fromtest observations. Typically, these techniques fall into two categories: passive and active. On the one hand, passive learning assumes no interaction with the system under learning and uses a predetermined training set, e.g., system logs. On the other hand, active learning techniques collect training data by actively querying the system under learning, allowing one to steer the discovery ofmeaningful information about the systemunder learning leading to effective learning strategies. A notable example of active learning technique for regular languages is Angluin’s L^*-algorithm. The L^*-algorithm describes the strategy of a student who learns the minimal deterministic finite automaton of an unknown regular language L by asking a succinct number of queries to a teacher who knows L.In this work, we study L^* -based learning of deterministic Markov decision processes, a class of Markov decision processes where an observation following an action uniquely determines a successor state. For this purpose, we first assume an ideal setting with a teacher who provides perfect information to the student. Then, we relax this assumption and present a novel learning algorithm that collects information by sampling execution traces of the system via testing.Experiments performed on an implementation of our sampling-based algorithm suggest that our method achieves better accuracy than state-of-the-art passive learning techniques using the same amount of test obser vations. In contrast to existing learning algorithms which assume a predefined number of states, our algorithm learns the complete model structure including the state space.

Accepting networks of evolutionary processors with resources restricted and structure limited filters

In this paper, we continue the research on accepting networks of evolutionary processors where the filters belong to several special families of regular languages. We consider families of codes or ideals and subregular families which are defined by restricting the resources needed for generating or accepting them (the number of states of the minimal deterministic finite automaton accepting a language of the family as well as the number of non-terminal symbols or the number of production rules of a right-linear grammar generating such a language). We insert the newly defined language families into the hierachy of language families obtained by using as filters languages of other subregular families (such as ordered, non-counting, power-separating, circular, suffix-closed regular, union-free, definite, combinational, finite, monoidal, nilpotent, or commutative languages).

State Complexity of Overlap Assembly

The state complexity of a regular language [Formula: see text] is the number [Formula: see text] of states in a minimal deterministic finite automaton (DFA) accepting [Formula: see text]. The state complexity of a regularity-preserving binary operation on regular languages is defined as the maximal state complexity of the result of the operation where the two operands range over all languages of state complexities [Formula: see text] and [Formula: see text], respectively. We determine, for [Formula: see text], [Formula: see text], the exact value of the state complexity of the binary operation overlap assembly on regular languages. This operation was introduced by Csuhaj-Varjú, Petre, and Vaszil to model the process of self-assembly of two linear DNA strands into a longer DNA strand, provided that their ends “overlap”. We prove that the state complexity of the overlap assembly of languages [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text], is at most [Formula: see text]. Moreover, for [Formula: see text] and [Formula: see text] there exist languages [Formula: see text] and [Formula: see text] over an alphabet of size [Formula: see text] whose overlap assembly meets the upper bound and this bound cannot be met with smaller alphabets. Finally, we prove that [Formula: see text] is the state complexity of the overlap assembly in the case of unary languages and that there are binary languages whose overlap assembly has exponential state complexity at least [Formula: see text].

Fundamental results for learning deterministic extended finite state machines from queries

Regular language inference, initiated by Angluin, has many developments, including applications in software engineering and testing. However, the capability of finite automata to model the system data is quite limited and, in many cases, extended finite state machine formalisms, that combine the system control with data structures, are used instead. The application of Angluin-style inference algorithms to extended state machines would involve constructing a minimal deterministic extended finite state machine consistent with a deterministic 3-valued deterministic finite automaton. In addition to the usual, accepting and rejecting, states of finite automaton, a 3-valued deterministic finite automaton may have “don't care” states; the sequences of inputs that reach such states may be considered as accepted or rejected, as is convenient. The aforementioned construction reduces to finding a minimal deterministic finite automaton consistent with a 3-valued deterministic finite automaton, that preserves the deterministic nature of the extended model that also handles the data structure associated with it. This paper investigates fundamental properties of extended finite state machines in relation to Angluin's language inference problem and provides an inference algorithm for such models.

Most Complex Non-Returning Regular Languages

A regular language [Formula: see text] is non-returning if in the minimal deterministic finite automaton accepting it there are no transitions into the initial state. Eom, Han and Jirásková derived upper bounds on the state complexity of boolean operations and Kleene star, and proved that these bounds are tight using two different binary witnesses. They derived tight upper bounds for concatenation and reversal using three different ternary witnesses. These five witnesses use a total of six different transformations. We show that for each [Formula: see text], there exists a ternary witness of state complexity [Formula: see text] that meets the bound for reversal, and restrictions of this witness to binary alphabets meet the bounds for star, product, and boolean operations. Hence all of these operations can be handled simultaneously with a single witness, using only three different transformations. We also derive tight upper bounds on the state complexity of binary operations that take arguments with different alphabets. We prove that the maximal syntactic semigroup of a non-returning language has [Formula: see text] elements and requires at least [Formula: see text] generators. We find the maximal state complexities of atoms of non-returning languages. We show that there exists a most complex sequence of non-returning languages that meet the bounds for all of these complexity measures. Furthermore, we prove there is a most complex sequence that meets all the bounds using alphabets of minimal size.

Stochastic Analysis of Minimal Automata Growth for Generalized Strings

Generalized strings describe various biological motifs that arise in molecular and computational biology. In this manuscript, we introduce an alternative but efficient algorithm to construct the minimal deterministic finite automaton (DFA) associated with any generalized string. We exploit this construction to characterize the typical growth of the minimal DFA (i.e., with the least number of states) associated with a random generalized string of increasing length. Even though the worst-case growth may be exponential, we characterize a point in the construction of the minimal DFA when it starts to grow linearly and conclude it has at most a polynomial number of states with asymptotically certain probability. We conjecture that this number is linear.

State complexity of pattern matching in regular languages

In a simple pattern matching problem one has a pattern w and a text t, which are words over a finite alphabet Σ. One may ask whether w occurs in t, and if so, where? More generally, we may have a set P of patterns and a set T of texts, where P and T are regular languages. We are interested whether any word of T begins with a word of P, ends with a word of P, has a word of P as a factor, or has a word of P as a subsequence. Thus we are interested in the languages (PΣ⁎)∩T, (Σ⁎P)∩T, (Σ⁎PΣ⁎)∩T, and ▪, where ▪ is the shuffle operation. The state complexity κ(L) of a regular language L is the number of states in the minimal deterministic finite automaton recognizing L. We derive the following upper bounds on the state complexities of our pattern-matching languages, where κ(P)⩽m, and κ(T)⩽n: κ((PΣ⁎)∩T)⩽mn; κ((Σ⁎P)∩T)⩽2m−1n; κ((Σ⁎PΣ⁎)∩T)⩽(2m−2+1)n; and ▪. We prove that these bounds are tight, and that to meet them, the alphabet must have at least two letters in the first three cases, and at least m−1 letters in the last case.

Primitivity, Uniform Minimality, and State Complexity of Boolean Operations

A minimal deterministic finite automaton (DFA) is uniformly minimal if it always remains minimal when the final state set is replaced by a non-empty proper subset of the state set. We prove that a permutation DFA is uniformly minimal if and only if its transition monoid is a primitive group. We use this to study boolean operations on group languages, which are recognized by direct products of permutation DFAs. A direct product cannot be uniformly minimal, except in the trivial case where one of the DFAs in the product is a one-state DFA. However, non-trivial direct products can satisfy a weaker condition we call uniform boolean minimality, where only final state sets used to recognize boolean operations are considered. We give sufficient conditions for a direct product of two DFAs to be uniformly boolean minimal, which in turn gives sufficient conditions for pairs of group languages to have maximal state complexity under all binary boolean operations ("maximal boolean complexity"). In the case of permutation DFAs with one final state, we give necessary and sufficient conditions for pairs of group languages to have maximal boolean complexity. Our results demonstrate a connection between primitive groups and automata with strong minimality properties.

Minimal auxiliary Markov chains through sequential elimination of states

ABSTRACTWhen using an auxiliary Markov chain to compute the distribution of a pattern statistic, the computational complexity is directly related to the number of Markov chain states. Theory related to minimal deterministic finite automata have been applied to large state spaces to reduce the number of Markov chain states so that only a minimal set remains. In this paper, a characterization of equivalent states is given so that extraneous states are deleted during the process of forming the state space, improving computational efficiency. The theory extends the applicability of Markov chain based methods for computing the distribution of pattern statistics.

Complexity of Infimal Observable Superlanguages

The infimal prefix-closed, controllable, and observable superlanguage plays an essential role in the relationship between controllability, observability, and co-observability—the central notions of supervisory control theory. Existing algorithms for its computation are exponential and it is not known whether a polynomial algorithm exists. We answer the question by studying the state complexity of this language. State complexity of a language is the number of states of its minimal deterministic finite automaton (DFA). For a language with state complexity $n$ , we show that the upper bound state complexity on the infimal prefix-closed and observable superlanguage is $2^n + 1$ and that this bound is asymptotically tight. Hence, there is no algorithm computing a DFA of the infimal prefix-closed and observable superlanguage in polynomial time. Our construction shows that such a DFA can be computed in time $O(2^n)$ . The construction involves nondeterministic finite automata (NFAs) and a computation of the supremal prefix-closed sublanguage. We study the computation of supremal prefix-closed sublanguages and show that there is no polynomial-time algorithm computing an NFA of the supremal prefix-closed sublanguage of a language given as an NFA even if the language is unary.

Descriptional Complexity of Non-Unary Self-Verifying Symmetric Difference Automata

Previously, self-verifying symmetric difference automata were defined and a tight bound of 2^n-1-1 was shown for state complexity in the unary case. We now consider the non-unary case and show that, for every n at least 2, there is a regular language L_n accepted by a non-unary self-verifying symmetric difference nondeterministic automaton with n states, such that its equivalent minimal deterministic finite automaton has 2^n-1 states. Also, given any SV-XNFA with n states, it is possible, up to isomorphism, to find at most another |GL(n,Z_2)|-1 equivalent SV-XNFA.

Syntactic Complexity of Regular Ideals

The state complexity of a regular language is the number of states in a minimal deterministic finite automaton accepting the language. The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of regular languages is the worst-case syntactic complexity taken as a function of the state complexity n of languages in that class. We prove that nn−1, nn−1 + n − 1, and nn−2 + (n − 2)2n−2 + 1 are tight upper bounds on the syntactic complexities of right ideals and prefix-closed languages, left ideals and suffix-closed languages, and two-sided ideals and factor-closed languages, respectively. Moreover, we show that the transition semigroups meeting the upper bounds for all three types of ideals are unique, and the numbers of generators (4, 5, and 6, respectively) cannot be reduced.