Nondeterministic Finite Automata Research Articles

Deterministic and Unambiguous Families within Recognizable Two-dimensional Languages

Recognizable two-dimensional languages (REC) are defined by tiling systems that generalize to two dimensions non-deterministic finite automata for strings. We introduce the notion of deterministic tiling system and the corresponding family of languages (DREC) and study its structural and closure properties. Furthermore we show that, in contrast with the one-dimensional case, there exist other classes between deterministic and non-deterministic families that we separate by means of examples and decidability properties.

Extended Nondeterministic Finite Automata

We consider a new expansion of nondeterministic finite automata. The goals of this consideration are: to apply some algorithms of such expansion for various problems of minimization of classical nondeterministic automata; to use such automata for describing practical anytime algorithms for the same problems of minimization; using such automata, we often can simplify some proofs for algorithms of simplification of usual nondeterministic automata.

ON THE REGULARITY OF SETS OF MULTI-ACCEPTED STRINGS OF A NON-DETERMINISTIC FINITE AUTOMATON

In this paper we establish the regularity of various sets of multi-accepted strings of nondeterministic finite automata. Regularity follows from the existence of accepting automata constructed by introducing a vector labeling method which generalizes the subset labeling approach. In each set the acceptance levels of the strings correspond to finite sets of additive equivalence classes of non-negative integers.

AUTOMATA SEBAGAI MODEL PENGENAL BAHASA

A deterministic finite automaton as well a nondeterministic finite automaton can be used to model a language recognizer. In computer software technology, language recognizer usually be an integrated part of a compiler, that is a computer program that take responsibility to translate source code into machine code. Comparing with a deterministic finite automaton, a nondeterministic finite automaton is a better model for language recognizer because it might be simpler and less in size than a deterministic one.

THE LENGTH OF SUBSET REACHABILITY IN NONDETERMINISTIC AUTOMATA

We study subset reachability in nondeterministic finite automata and look for bounds of the length of the shortest reaching words for automata with a fixed number of states. We obtain such bounds for nondeterministic automata over 2-letter, 3-letter and arbitrary alphabets.a

NONDETERMINISTIC FINITE AUTOMATA — RECENT RESULTS ON THE DESCRIPTIONAL AND COMPUTATIONAL COMPLEXITY

Nondeterministic finite automata (NFAs) were introduced in [68], where their equivalence to deterministic finite automata was shown. Over the last 50 years, a vast literature documenting the importance of finite automata as an enormously valuable concept has been developed. In the present paper, we tour a fragment of this literature. Mostly, we discuss recent developments relevant to NFAs related problems like, for example, (i) simulation of and by several types of finite automata, (ii) minimization and approximation, (iii) size estimation of minimal NFAs, and (iv) state complexity of language operations. We thus come across descriptional and computational complexity issues of nondeterministic finite automata. We do not prove these results but we merely draw attention to the big picture and some of the main ideas involved.

Testing the Equivalence of Regular Languages

The minimal deterministic finite automaton is generally used to determine regular languages equality. Antimirov and Mosses proposed a rewrite system for deciding regular expressions equivalence of which Almeida et al. presented an improved variant. Hopcroft and Karp pro- posed an almost linear algorithm for testing the equivalence of two de- terministic finite automata that avoids minimisation. In this paper we improve the best-case running time, present an extension of this algo- rithm to non-deterministic finite automaton, and establish a relationship between this algorithm and the one proposed in Almeida et al. We also present some experimental comparative results. All these algorithms are closely related with the recent coalgebraic approach to automata pro- posed by Rutten.

Nondeterministic State Complexity of Positional Addition

Consider nondeterministic finite automata recognizing base-k positional notation of numbers. Assume that numbers are read starting from their least significant digits. It is proved that if two sets of numbers S and T are represented by nondeterministic automata of m and n states, respectively, then their sum {s+t | s in S, t in T} is represented by a nondeterministic automaton with 2mn+2m+2n+1 states. Moreover, this number of states is necessary in the worst case for all k>=9.

Optimization Techniques for RFID Complex Event Processing

One research crucial to wider adoption of Radio Frequency Identification (RFID) technology is how to efficiently transform sequences of RFID readings into meaningful business events. Contrary to traditional events, RFID readings are usually of high volume and velocity, and have the attributes representing their reading objects, occurrence times and spots. Based on these characteristics and the Non-deterministic Finite Automata (NFA) implementation framework, this paper studies the performance issues of RFID complex event processing and proposes corresponding optimization techniques. Our techniques include: (1) taking advantage of negation events or exclusiveness between events to prune intermediate results, thus reduces memory consumption; (2) with different selectivities of complex events, purposefully reordering the join operations between events to improve overall efficiency, achieve higher stream throughput; (3) utilizing the slot-based or B+-tree-based approach to optimizing the processing performance with the time window constraint. We present the analytical results of these techniques and validate their effectiveness through experiments.

Hardware Architecture for High-Performance Regular Expression Matching

This paper presents a bitmap-based hardware architecture for the Glushkov nondeterministic finite automaton (G-NFA), which recognizes a given regular expression. We show that the inductions of the functions needed to construct the G-NFA can be generalized to include other special symbols commonly used in extended regular expressions such as the POSIX 1003.2 format. Our proposed implementation can detect the ending positions of all substrings of an input string T, which start at arbitrary positions of T and belong to the language defined by the given regular expression. To achieve high performance, the implementation is generalized to the NFA, which processes K symbols in each operation cycle. We provide an efficient solution for the boundary condition when the length of the input string is not an integral multiple of K. Compared with previous designs, our proposed architecture is more flexible and programmable because the pattern matching engine uses memory rather than logic.

Determination of finite automata accepting subregular languages

We investigate the descriptional complexity of the nondeterministic finite automaton (NFA) to the deterministic finite automaton (DFA) conversion problem, for automata accepting subregular languages such as combinational languages, definite languages and variants thereof, (strictly) locally testable languages, star-free languages, ordered languages, prefix-, suffix-, and infix-closed languages, and prefix-, suffix-, and infix-free languages. Most of the bounds for the conversion problem are shown to be tight in the exact number of states, that is, the number is sufficient and necessary in the worst case. Otherwise tight bounds in order of magnitude are shown.

Some algorithms for equivalent transformation of nondeterministic finite automata

In this paper we consider algorithms which allow one to combine several states of a nondeterministic finite automaton into one state. Along with the algorithms for combining states, we adduce one more algorithm for the equivalent transformation of a non-deterministic finite automaton, namely, an algorithm for adding cycles. Problems under consideration imply the development of robust computer programs.

On the limits of the communication complexity technique for proving lower bounds on the size of minimal NFA’s

In contrast to the minimization of deterministic finite automata (DFA’s), the task of constructing a minimal nondeterministic finite automaton (NFA) for a given NFA is PSPACE-complete. Moreover, there are no polynomial approximation algorithms with a constant approximation ratio for estimating the number of states of minimal NFA’s. Since one is unable to efficiently estimate the size of a minimal NFA in an efficient way, one should ask at least for developing mathematical proof methods that help to prove good lower bounds on the size of a minimal NFA for a given regular language. Here we consider the robust and most successful lower bound proof technique that is based on communication complexity. In this paper it is proved that even a strong generalization of this method fails for some concrete regular languages. “To fail” is considered here in a very strong sense. There is an exponential gap between the size of a minimal NFA and the achievable lower bound for a specific sequence of regular languages. The generalization of the concept of communication protocols is also strong here. It is shown that cutting the input word into 2 O ( n 1 / 4 ) pieces for a size n of a minimal nondeterministic finite automaton and investigating the necessary communication transfer between these pieces as parties of a multiparty protocol does not suffice to get good lower bounds on the size of minimal nondeterministic automata. It seems that for some regular languages one cannot really abstract from the automata model that cuts the input words into particular symbols of the alphabet and reads them one by one using its input head.

Detecting palindromes, patterns and borders in regular languages

Given a language L and a non-deterministic finite automaton M, we consider whether we can determine efficiently (in the size of M) if M accepts at least one word in L, or infinitely many words. Given that M accepts at least one word in L, we consider how long a shortest word can be. The languages L that we examine include the palindromes, the non-palindromes, the k-powers, the non- k-powers, the powers, the non-powers (also called primitive words), the words matching a general pattern, the bordered words, and the unbordered words.

Nondeterministic state complexity of nested word automata

We study the nondeterministic state complexity of Boolean operations on regular languages of nested words. For union and intersection we obtain matching upper and lower bounds. For complementation of a nondeterministic nested word automaton with n states we establish a lower bound Ω ( n ! ) that is significantly worse than the exponential lower bound for ordinary nondeterministic finite automata (NFA). We develop techniques to prove lower bounds for the size of nondeterministic nested word automata that extend the known techniques used for NFAs.

Efficient probabilistic event stream processing with lineage and Kleene-plus

This paper proposes a working framework and a query language to support probabilistic queries for composite event detection over probabilistic event streams. The language allows users to express Kleene closure patterns for complex event detection in the physical world. Our processing method first detects sequence patterns over probabilistic data streams using AIG, a new data structure, which handles active states with a nondeterministic finite automaton (NFA). Our method then computes the probability of each detected sequence pattern based on its lineage. Through the benefit of lineage, the probability of an output event can be directly calculated without taking into account the query plan. An optimised plan can be selected. Finally, we conducted a performance evaluation of our method and compared the results with the original and optimised query plan. The experiment clearly showed that our proposal outperforms straight-forward query plans.

Software Toolchain for Large‐Scale RE‐NFA Construction on FPGA

We present a software toolchain for constructing large‐scale regular expression matching (REM) on FPGA. The software automates the conversion of regular expressions into compact and high‐performance nondeterministic finite automata (RE‐NFA). Each RE‐NFA is described as an RTL regular expression matching engine (REME) in VHDL for FPGA implementation. Assuming a fixed number of fan‐out transitions per state, an n‐state m‐bytes‐per‐cycle RE‐NFA can be constructed in O(n × m) time and O(n × m) memory by our software. A large number of RE‐NFAs are placed onto a two‐dimensional staged pipeline, allowing scalability to thousands of RE‐NFAs with linear area increase and little clock rate penalty due to scaling. On a PC with a 2 GHz Athlon64 processor and 2 GB memory, our prototype software constructs hundreds of RE‐NFAs used by Snort in less than 10 seconds. We also designed a benchmark generator which can produce RE‐NFAs with configurable pattern complexity parameters, including state count, state fan‐in, loop‐back and feed‐forward distances. Several regular expressions with various complexities are used to test the performance of our RE‐NFA construction software.

The Length of Subset Reachability in Nondeterministic Automata

We study subset reachability in nondeterministic finite automata and look for bounds of the length of the shortest reaching words for automata with a fixed number of states. We obtain such bounds for nondeterministic automata over 2-letter, 3-letter and arbitrary alphabets.

DESCRIPTIONAL COMPLEXITY OF SPLICING SYSTEMS

In this paper, the descriptional complexity of extended finite splicing systems is studied. These systems are known to generate exactly the class of regular languages. Upper and lower bounds are shown relating the size of these splicing systems, defined as the total length of the rules and the initial language of the system, to the size of their equivalent minimal nondeterministic finite automata (NFA). In addition, an accepting model of extended finite splicing systems is studied. Using this variant one can obtain systems which are more than polynomially more succinct than the equivalent NFA or generating extended finite splicing system.

DETERMINISTIC BLOW-UPS OF MINIMAL NONDETERMINISTIC FINITE AUTOMATA OVER A FIXED ALPHABET

We show that for all integers n and α such that n ⩽ α ⩽ 2n, there exists a minimal nondeterministic finite automaton of n states with a four-letter input alphabet whose equivalent minimal deterministic finite automaton has exactly α states. It follows that in the case of a four-letter alphabet, there are no "magic numbers", i.e., the holes in the hierarchy. This improves a similar result obtained by Geffert for a growing alphabet of size n + 2.