Nondeterministic Finite Automata Research Articles

SIMULATIONS OF UNARY ONE-WAY MULTI-HEAD FINITE AUTOMATA

We investigate the descriptional complexity of deterministic one-way multi-head finite automata accepting unary languages. It is known that in this case the languages accepted are regular. Thus, we study the increase of the number of states when an n-state k-head finite automaton is simulated by a classical (one-head) deterministic or nondeterministic finite automaton. In the former case upper and lower bounds that are tight in the order of magnitude are shown. For the latter case we obtain an upper bound of O(n2k) and a lower bound of Ω(nk) states. We investigate also the costs for the conversion of one-head nondeterministic finite automata to deterministic k-head finite automata, that is, we trade nondeterminism for heads. In addition, we study how the conversion costs vary in the special case of finite and, in particular, of singleton unary lanuages. Finally, as an application of the simulation results, we show that decidability problems for unary deterministic k-head finite automata such as emptiness or equivalence are LOGSPACE-complete.

An embedded automaton to monitor the glycolysis process in pancreatic β-cells.

An embedded automaton is introduced to monitor the whole glycolysis process in pancreatic β-cell and it is a hybridization of both non-deterministic finite automaton and push-down automaton. The set of irreversible and reversible reactions in the glycolysis process are related to non-deterministic finite automaton and push-down automaton respectively. The embedded automaton is used to observe the glucose metabolism with the states of acceptance and rejection. The acceptance state of the embedded automaton depicts the normal level of glycolysis and insulin secretion. The rejection state of this machine shows inhibition of glycolysis which obstructs the secretion of insulin. The subsequent low level of insulin leads to the high blood glucose level also known as hyperglycemia. In this study, the designed machine can be used to regulate the process of glycolysis through a group of regulatory glycolytic enzymes for the treatment of Diabetes mellitus at molecular level.

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.

A Local Logic for Realizability in Web Service Choreographies

Web service choreographies specify conditions on observable interactions among the services. An important question in this regard is realizability: given a choreography C, does there exist a set of service implementations I that conform to C ? Further, if C is realizable, is there an algorithm to construct implementations in I ? We propose a local temporal logic in which choreographies can be specified, and for specifications in the logic, we solve the realizability problem by constructing service implementations (when they exist) as communicating automata. These are nondeterministic finite state automata with a coupling relation. We also report on an implementation of the realizability algorithm and discuss experimental results.

Semi-tensor product of matrices approach to reachability of finite automata with application to language recognition

This paper investigates the transition function and the reachability conditions of finite automata by using a semi-tensor product of matrices, which is a new powerful matrix analysis tool. The states and input symbols are first expressed in vector forms, then the transition function is described in an algebraic form. Using this algebraic representation, a sufficient and necessary condition of the reachability of any two states is proposed, based on which an algorithm is developed for discovering all the paths from one state to another. Furthermore, a mechanism is established to recognize the language acceptable by a finite automaton. Finally, illustrative examples show that the results/algorithms presented in this paper are suitable for both deterministic finite automata (DFA) and nondeterministic finite automata (NFA).

Computational model for monitoring cholesterol metabolism

A non-deterministic finite automaton is designed to observe the cholesterol metabolism with the states of acceptance and rejection. The acceptance state of the automaton depicts the normal level of metabolism and production of good cholesterol as an end product. The rejection state of this machine shows the inhibition of enzymatic activity in cholesterol synthesis and removal of free fatty acids. The deficiency in human cholesterol metabolism pathway results in abnormal accumulation of cholesterol in plasma, arterial tissues leading to diseases such as hypercholesterolemia, atherosclerosis respectively and formation of gallstones. The designed machine can be used to monitor the cholesterol metabolism at molecular level through regulation of enzymes involved in the biosynthesis and metabolism of cholesterol for the treatment of diseases incident due to the respective metabolic disorder. In addition, an algorithm for this machine has been developed to compare the programmed string with the given string. This study demonstrates the construction of a machine that is used for the development of molecular targeted therapy for the disorders in cholesterol metabolism.

A comment on “Construction of fuzzy automata from fuzzy regular expressions”

The aim of this comment is to point out that the fuzzy language recognized by a nondeterministic finite automaton, which is associated with a fuzzy regular expression, in the context of Stamenković–Ćirić's method (Stamenković and Ćirić (2012) [11]), is recognized by a fuzzy finite automaton with ε-moves. Every fuzzy automaton with ε-moves is also equivalent to a fuzzy automaton and there are effective methods for removing ε-moves in order to obtain the equivalent fuzzy automaton. In this way, our proposal to convert a fuzzy regular expression to a fuzzy finite automaton is based on the construction of a fuzzy automaton with ε-moves for the fuzzy regular expression and the construction of an equivalent fuzzy finite automaton by means of ε-removal operation.

Simplifying Nondeterministic Finite Cover Automata

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 finite cover automaton (l-NFCA), we show that minimization can be as hard as minimizing NFAs for regular languages, even in the case of NFCAs using unary alphabets. Moreover, we show how we can adapt the methods used to reduce, or minimize the size of NFAs/DFCAs/l-DFCAs, for simplifying NFCAs/l-NFCAs.

Theory of átomata

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.

Exact and Approximate Determinization of Discounted-Sum Automata

A discounted-sum automaton (NDA) is a nondeterministic finite automaton with edge weights, valuing a run by the discounted sum of visited edge weights. More precisely, the weight in the i-th position of the run is divided by $\lambda^i$, where the discount factor $\lambda$ is a fixed rational number greater than 1. The value of a word is the minimal value of the automaton runs on it. Discounted summation is a common and useful measuring scheme, especially for infinite sequences, reflecting the assumption that earlier weights are more important than later weights. Unfortunately, determinization of NDAs, which is often essential in formal verification, is, in general, not possible. We provide positive news, showing that every NDA with an integral discount factor is determinizable. We complete the picture by proving that the integers characterize exactly the discount factors that guarantee determinizability: for every nonintegral rational discount factor $\lambda$, there is a nondeterminizable $\lambda$-NDA. We also prove that the class of NDAs with integral discount factors enjoys closure under the algebraic operations min, max, addition, and subtraction, which is not the case for general NDAs nor for deterministic NDAs. For general NDAs, we look into approximate determinization, which is always possible as the influence of a word's suffix decays. We show that the naive approach, of unfolding the automaton computations up to a sufficient level, is doubly exponential in the discount factor. We provide an alternative construction for approximate determinization, which is singly exponential in the discount factor, in the precision, and in the number of states. We also prove matching lower bounds, showing that the exponential dependency on each of these three parameters cannot be avoided. All our results hold equally for automata over finite words and for automata over infinite words.

On a compact encoding of the swap automaton

Given a string P of length m over an alphabet Σ of size σ, a swapped version of P is a string derived from P by a series of local swaps, i.e., swaps of adjacent symbols, such that each symbol can participate in at most one swap. We present a theoretical analysis of the nondeterministic finite automaton for the language ⋃P′∈ΠPΣ⁎P′ (swap automaton, for short), where ΠP is the set of swapped versions of P. Our study is based on the bit-parallel simulation of the same automaton due to Fredriksson, and reveals an interesting combinatorial property that links the automaton to the one for the language Σ⁎P. By exploiting this property and the method presented by Cantone et al. (2012), we obtain a bit-parallel encoding of the swap automaton which takes O(σ2⌈k/w⌉) space and allows one to simulate the automaton on a string of length n in time O(n⌈k/w⌉), where ⌈m/σ⌉⩽k⩽m is the size of a specific factorization of P defined by Cantone et al. (2012) and w is the word size in bits.

A novel algorithm for the conversion of shuffle regular expressions into non-deterministic finite automata

The aim of the paper is to concoct a novel algorithm for the metamorphosis of parallel regular expressions to e-free non- deterministic finite automata. For a given parallel regular expression r, let m be the number of symbols that occur in r and let C denote the number of concatenation operators in r. In the worst case, 2 m+1 states are required for the construction of the non-deterministic finite automaton using the novel algorithm. In the earlier existing approaches, th e number of states of the non-deterministic finite automaton in the worst case is equal to 2 2jrj−3C.

Computational regulatory model for detoxification of ammonia from urea cycle in liver

A nondeterministic finite automaton was designed to monitor enzymatic regulation and detoxification of excess ammonia in the urea cycle and its disorders. The designed machine is used for the diagnosis of deficiency and for regulating the expression of any of the enzymes involved with acceptance and rejection states in the urea cycle. The urea cycle is the metabolism of excess nitrogen produced by the breakdown of protein and other nitrogen-containing molecules in liver. Disorder in the urea cycle may lead to the accumulation of toxic ammonia in the blood, which leads to hyperammonemia. The elevation of plasma ammonia concentration may ultimately lead to cerebral edema in infants and severe brain damage due to the toxicity of ammonia. The diagnosis of urea cycle disorder is based on evaluation of clinical, biochemical, and molecular data. In this study, a new therapeutic approach for urea cycle disorders is developed based on a computational model. It is used to observe the normal process of the cycle through the state of acceptance. The state of rejection denotes a deficiency in the respective enzymatic activity. Subsequently, it assists in the creation of targeted treatment for brain damage and related enzymatic deficiency disorders.

Hardware acceleration of regular expression repetitions in deep packet inspection

Network Intrusion Detection Systems (NIDS) make extensive use of regular expressions (regexes) as attack signatures. Such expressions can be handled in hardware using a bit-parallel (BP) architecture based on the Glushkov non-deterministic finite automata (NFA). However, many expressions contain constrained {min, max} repetitions which first need to be unrolled so that they can be handled by the standard BP system. Such unrolling often leads to an excessive memory requirement which makes handling of such regexes unfeasible. This study presents a solution, based on the standard BP architecture, which incorporates a counting mechanism that renders unrolling unnecessary. As a result, many regexes, which were previously unsuitable for the standard BP system, can now be efficiently handled. Unlike many other approaches, this architecture is dynamically reconfigurable thanks to its memory, rather than logic, based engine. This is important as NIDS rule sets are regularly updated. It can also handle repetition of both single and multi-symbol sub-expressions.

An efficient multicharacter transition string-matching engine based on the aho-corasick algorithm

A string-matching engine capable of inspecting multiple characters in parallel can multiply the throughput. However, the space required for implementing a matching engine that can process multiple characters in parallel generally grows exponentially with respect to the characters to be processed in parallel. Based on the Aho-Corasick algorithm (AC-algorithm), this work presents a novel multicharacter transition Nondeterministic Finite Automaton (NFA) approach, called multicharacter AC-NFA , to allow for the inspection of multiple characters in parallel. This approach first converts an AC-trie to an AC-NFA by allowing for the simultaneous activation of multiple states and then converts the AC-NFA to a k -character AC-NFA by an algorithm with concatenation operations and assistant transitions. Additionally, the alignment problem, which occurs while multiple characters are being inspected in parallel, is solved using assistant transitions. Moreover, a corresponding output is provided for each inspected character by introducing priority multiplexers to determine the final matching outputs during implementation of the multicharacter AC-NFA. Consequently, the number of derived k -character transitions grows linearly with respect to the number k . Furthermore, the derived multicharacter AC-NFA is implemented on FPGAs for evaluation. The resulting throughput grows approximately 14 times and the hardware cost grows about 18 times for 16-character AC-NFA implementation, as compared with that for 1-character AC-NFA implementation. The achievable throughput is 21.4Gbps for the 16-character AC-NFA implementation operating at a 167.36MHz clock.

Computational Complexity of Certain Problems Related to Carefully Synchronizing Words for Partial Automata and Directing Words for Nondeterministic Automata

We show that the problem of checking careful synchronizability of partial finite automata is PSPACE-complete. Also the problems of checking D 1-, D 2-, and D 3-directability of nondeterministic finite automata are PSPACE-complete; moreover, the restrictions of all these problems to automata with two input letters remain PSPACE-complete.

Deciding unique decodability of bigram counts via finite automata

We revisit the problem of deciding by means of a finite automaton whether a string is uniquely decodable from its bigram counts. An efficient algorithm for constructing a polynomial-size Nondeterministic Finite Automaton (NFA) that decides unique decodability is given. This NFA may be simulated efficiently in time and space. Conversely, we show that the minimum deterministic finite automaton for deciding unique decodability has exponentially many states in alphabet size, and compute the correct order of magnitude of the exponent.

Complex event processing over distributed probabilistic event streams

With the rapid development of Internet of Things (IoT), enormous events are produced every day. Complex Event Processing (CEP), which can be used to extract high level patterns from raw data, becomes the key part of the IoT middleware. In large-scale IoT applications, the current CEP technology encounters the challenge of massive distributed data which cannot be handled by most of the current methods efficiently. Another challenge is the uncertainty of the data caused by noise, sensor error or wireless communication techniques. In order to solve these challenges, in this paper a high-performance complex event processing method over distributed probabilistic event streams is proposed. With the ability to report confidence for processed complex events over uncertain data, this method uses probabilistic nondeterministic finite automaton and active instance stacks to process a complex event in both single and distributed probabilistic event streams. A parallel algorithm is designed to improve the performance. A query plan-based method is used to process the hierarchical complex event from distributed event streams. Query plan optimization is proposed based on the query optimization technology of probabilistic databases. The experimental study shows that this method is efficient in processing complex events over distributed probabilistic event streams.

On the open problem of Ginsburg concerning semilinear sets and related problems

In his 1966 book, “The Mathematical Theory of Context-Free Languages”, S. Ginsburg posed the following open problem: Find a decision procedure for determining if an arbitrary semilinear set is a finite union of stratified linear sets. It turns out that this problem (which remains open) is equivalent to the problem of synchronizability of multitape machines. Given an n-tape automaton M of a given type (e.g., nondeterministic pushdown automaton (NPDA), nondeterministic finite automaton (NFA), etc.) with a one-way read-only head per tape and a right end marker on each tape, we say that M is 0-synchronized (or aligned) if for every n-tuple x=(x1,…,xn) that is accepted, there is an accepting computation on x such that at any time during the computation, all heads except those that have reached the end marker are on the same position (i.e., aligned). One of our main contributions is to show that Ginsburgʼs problem is equivalent to deciding for an arbitrary n-tape NFA M accepting L(M)⊆a1⁎×⋯×an⁎ (where n⩾1 and a1,…,an are distinct symbols) whether there exists an equivalent 0-synchronized n-tape NPDA M′. Ginsburgʼs problem is decidable if M′ is required to be an NFA, and we will generalize this decidability over bounded inputs.It is known that if the inputs are unrestricted, the problem is undecidable. We also show several other related decidability and undecidability results.It may appear, as one of the referees suggested in an earlier version of this paper, that our main result can be written in first order logic. We explain why this is not the case in the Introduction.

Verification of initial-state opacity in security applications of discrete event systems

In this paper, we formulate and analyze methodologies for verifying the notion of initial-state opacity in discrete event systems that are modeled as non-deterministic finite automata with partial observation on their transitions. A system is initial-state opaque if the membership of its true initial state to a set of secret states remains opaque (i.e., uncertain) to an intruder who observes system activity through some projection map. Initial-state opacity can be used to characterize security requirements in a variety of applications, including tracking problems in sensor networks. In order to model and analyze the intruder capabilities regarding initial-state opacity, we first address the initial-state estimation problem in a non-deterministic finite automaton via the construction of an initial-state estimator. We analyze the properties and complexity of the initial-state estimator, and show how the complexity of the verification method can be greatly reduced in the special case when the set of secret states is invariant. We also establish that the verification of initial-state opacity is a PSPACE-complete problem.