![]() Finite automata is widely used in areas such as text processing, compilation, pattern matching, network intrusion detection and protection, image analysis and spatial dynamics. Regular grammar and regular expressions generate regular languages, and finite automata is a computation model of speech recognition for regular languages. A comparison to Hopcroft’s algorithm demonstrates experimentally that the algorithm runs faster than traditional algorithms.įinite automata, regular grammar, and regular expressions are three dissimilar representations for regular languages. Overall, the proposal has three advantages: lower time complexity, greater generality, and scalability. The proposed algorithm can be applied not only to the minimization of acyclic automata or simple cyclic automata, but also to automata with high topological complexity. This method achieves greater generality than previous methods because building the backward depth information is independent of the topological complexity of the DFA. Few states need to be refined by the hash table, because most states have been partitioned by the backward depth information in the coarse partition. The minimization algorithm has a lower time complexity O( n) than a naive comparison of transitions O( n 2). ![]() In the second phase, the state set is refined using a hash table. In the first phase, the backward depth information is built, and the state set of the DFA is partitioned into many blocks. ![]() A minimization algorithm is presented in this paper that consists of two main phases. Thus, it is clear that every formal language that can be recognized by a DFA can be recognized by an NFA.Ĭonversely, for each NFA, there is a DFA such that it recognizes the same formal language.Obtaining a minimal automaton is a fundamental issue in the theory and practical implementation of deterministic finite automatons (DFAs). ( Q, Σ, δ, q 0, F ) can be reached after consuming the initial "1", this does not mean that the input "10" is accepted rather, it means that an input string "1" would be accepted.Ī deterministic finite automaton (DFA) can be seen as a special kind of NFA, in which for each state and symbol, the transition function has exactly one state. : 19–20 : 48 : 56 Formal definition įor a more elementary introduction of the formal definition, see automata theory.Īn NFA is represented formally by a 5- tuple, If, after consuming the complete input, any of the copies is in an accept state, the input is accepted, else, it is rejected. If no transition is applicable, the current copy is in a dead end, and it "dies". In each step, whenever two or more transitions are applicable, it "clones" itself into appropriately many copies, each one following a different transition. In the second way, the NFA consumes a string of input symbols, one by one. if no choice sequence at all can consume all the input and lead to an accepting state, the input is rejected. some sequence of choices leading to an accepting state after completely consuming the input, it is accepted. If there exists at least one "lucky run", i.e. In each step, the automaton nondeterministically "chooses" one of the applicable transitions. For each input symbol, the NFA transitions to a new state until all input symbols have been consumed. The first way makes use of the nondeterminism in the name of an NFA. There are two ways to describe the behavior of an NFA, and both of them are equivalent. NFAs have been generalized in multiple ways, e.g., nondeterministic finite automata with ε-moves, finite-state transducers, pushdown automata, alternating automata, ω-automata, and probabilistic automata.īesides the DFAs, other known special cases of NFAsĪnd self-verifying finite automata (SVFA). Conversely, Kleene's algorithm can be used to convert an NFA into a regular expression (whose size is generally exponential in the input automaton). NFAs are used in the implementation of regular expressions: Thompson's construction is an algorithm for compiling a regular expression to an NFA that can efficiently perform pattern matching on strings. ![]() Rabin and Dana Scott, who also showed their equivalence to DFAs. NFAs were introduced in 1959 by Michael O. Like DFAs, NFAs only recognize regular languages. Using the subset construction algorithm, each NFA can be translated to an equivalent DFA i.e., a DFA recognizing the same formal language. Sometimes the term NFA is used in a narrower sense, referring to an NFA that is not a DFA, but not in this article.
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