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Author: Admin | 2025-04-28
I to j in G } (1) The computation of transitive closure is essential in the construction of Multi-Character NFAs, where the goal is to enhance the processing throughput by efficiently identifying all reachable states from any given state, considering multiple characters as input [38,39]. This capability is fundamental to optimizing state transitions in NFAs designed to process high volumes of data, thereby significantly reducing the computational overhead associated with single-character NFAs [38,40].In the context of Multi-Character NFA construction, the transitive closure assists in merging multiple cycles of matching into a singular computational step. By doing so, it not only improves the matching speed, but also ensures a compact representation of state transitions, essential for high-throughput computing environments. This concept is illustrated further in the algorithm for constructing Multi-Character NFAs, where the transitive closure facilitates the identification of composite states resulting from the aggregation of individual transitions, thereby enabling the NFA to process multiple input characters simultaneously. 3. Principles of Constructing Multi-Character NFAs 3.1. PreliminariesA NFA is defined as a five-tuple, M = ( Q , Σ , q 0 , δ , F ) , where:Q is a finite set of states; Σ is a finite set of input tokens; q 0 is the initial state; δ : Q × Σ → P ( Q ) is the transition function; F ⊆ Q is the set of accept states.The transition function δ is defined such that δ ( q , a ) = { p 1 , p 2 , … , p m } , indicating possible transitions from state q upon reading character a. 3.2. Multi-Character NFA ConstructionThe core principle behind MC-NFA construction is the exploitation of the transitive closure property in automata, enabling the processing of symbol sequences in bulk. The transitive closure, a well-known concept in mathematics and computer science, is used here to establish a connectivity matrix that represents all possible state transitions over sequences of input symbols. Consider an NFA M = ( Q , Σ , q 0 , δ , F ) , the enhanced transition function δ ∗ for a MC-NFA can be mathematically represented as: δ ∗ ( q , w ) = ⋃ p ∈ δ ( q , a ) δ ∗ ( p , w [ 1 : ] ) (2) where w is a string of characters from Σ ∗ , and w [ 1
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