Concatenation and Kleene Star on Deterministic Finite Automata Guo-Qiang Zhang ∗ , Xiangnan Zhou † , Robert Fraser ‡ , Licong Cui ∗ ∗ Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, Ohio 44106, USA Email: { gq, licong.cui } @case.edu † College of Mathematics and Econometrics, Hunan University, Changsha 410012, China Email: xnzhou81026@163.com ‡ Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106, USA Email: rgf11@case.edu matrix product ∆ a 1 ∆ a 2 · · · ∆ a n . The language accepted by Abstract —This paper presents direct, explicit algebraic con- M , denoted L ( M ) , is the set { w | I q 0 ∆ w I t structions of concatenation and Kleene star on deterministic F = 1 } . We refer finite automata (DFA), using the Boolean-matrix method of more details of the utility of this matrix approach to [5]. Zhang [5] and ideas of Kozen [2]. The consequence is trifold: (1) Example 1.1: The matrix system of the following DFA is �� 0 � 1 it provides an alternative proof of the classical Kleene’s Theorem � �� 1 0 on the equivalence of regular expressions and DFAs without using , . 1 0 0 1 nondeterministic finite automata (NFA); (2) it demonstrates how the language constructions of concatenation and Kleene star can b b be captured elegantly as algebraic laws in the form of “binomial a theorems;” (3) it provides a demonstration of the (tight) upper bounds of the state complexity of concatenation and Kleene star, 1 2 start but offers a way to study the state complexity of NFA also. I. M ATRIX -A PPROACH TO A UTOMATA T HEORY a A Boolean matrix is a matrix (of size m × n ) whose elements With the use of Boolean matrices, it is straightforward to are either 0 or 1 , where the internal operations are carried out describe a wide spectrum of constructions on DFA in a simple, over the Boolean algebra. We write B m × n for the set of all algebraic manner [5], with their correctness established by Boolean matrices of size m × n . A Boolean (row) vector of induction and algebraic manipulation. Here we briefly treat dimension n is an n -tuple ( b 1 , b 2 , . . . , b n ) of 0 s and 1 s. We Brzozowski’s derivation [1], as an example. Given a string write B n for the set of all Boolean vectors of dimension n . u and a language L , the Brzozowski derivative u − 1 L is the A column vector is the transpose ( ) t of a row vector. The language { w | uw ∈ L } . Suppose L is accepted by an n -state DFA M = ( Q, Σ , δ, F ) , with { ∆ a | a ∈ Σ } its characteristic vector of a subset A of { 1 , · · · , n } is the row vector I n A ∈ B n such that the p -th component of I n A is a 1 matrix system. Then a DFA accepting u − 1 L can be given as M ′ = ( Q ′ , Σ , δ ′ , q ′ if and only if p ∈ A . The characteristic vector of a singleton 0 , F ′ ) , where set { p } is written as I n p , or simply I p . O m × n stands for an Q ′ = { A | A ∈ B n × n } , ( m × n ) -matrix, all of its elements are 0 . When dimension is q ′ ∆ u , = fixed by context, we abuse notion and write O n × n as 0 . 0 δ ′ ( A, a ) A ∆ a , = A deterministic finite automaton (DFA) is a 5 -tuple M = F ′ { A | I 1 A I t = F = 1 } . ( Q, Σ , δ, q 0 , F ) , where Q is the finite set of states, Σ is the alphabet, δ : Q × Σ → Q is the transition function, One can see that w is accepted by M ′ if and only if q 0 is the start state, and F is the set of final states. For δ ′ (∆ u , w ) = ∆ uw ∈ F ′ , i.e., uw is accepted by M . notational convenience, we use initial segments of natural In the remainder of this short paper, we present the con- numbers { 1 , 2 , · · · , n } to denote the set of states, and fix 1 to structions of concatenation and Kleene star on DFA, and be the start state, for base/background DFAs. When there is analyze the state complexity of such constructions. It turns out no confusion, we omit the indication of the start state (which that, without additional effort, these algebraic constructions are is assumed to be state 1 by default). already optimal in the number of states used after projecting Each n -state DFA determines a (associated) matrix system to the first row. Due to space limitation, we leave the detailed { ∆ a | a ∈ Σ } , where ∆ a is the ( n × n ) adjacency matrix proofs in the appendix. of the a -labeled subgraph associated with the DFA. In other words, the ( i, j ) entry of ∆ a is 1 if and only if δ ( i, a ) = j . II. C ONCATENATION Since M is a DFA, each ∆ a is row-stochastic (i.e., every row Theorem 2.1: Suppose matrix systems { ∆ a contains precisely a single 1 ). The (Boolean) sum ∆ of all 1 | a ∈ Σ } and members ∆ a in the matrix system is the adjacency matrix . { ∆ a 2 | a ∈ Σ } are associated with m - and n -state DFAs M 1 = For a string w = a 1 a 2 · · · a n over Σ , we write ∆ w for the ( Q 1 , Σ , δ 1 , F 1 ) and M 2 = ( Q 2 , Σ , δ 2 , F 2 ) , respectively. The
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