東京大学 情報理工学系研究科 コンピュータ科学専攻 2015年2月実施 問題2

标签：

• Tokyo-University
• Nondeterministic-Finite-Automaton
• Nondeterministic-Finite-Automaton-to-Deterministic-Finite-Automaton-Conversion
• Automata-State-Complexity

Author

kainoj, 祭音Myyura

Description

Let us consider nondeterministic finite automata (NFA) and deterministic finite automata (DFA) over the alphabet $\Sigma = \{a, b\}$. For example, the NFA $M_1$ shown below

recognizes the language

$$L_1 = \{ w \in \Sigma^* \mid \text{the last letter of } w \text{ is } a \}$$

Here $\Sigma^*$ denotes the set of words over $\Sigma$.

Answer the following questions.

(1) Present an NFA, with at most four states, that recognizes the language

$$L_3 = \{ w \in \Sigma^* \mid |w| \geq 3 \text{ and the third last letter of } w \text{ is } a \}$$

Here $|w|$ denotes the length of the word $w$.

(2) Present a DFA that recognizes the language $L_3$ in Question (1).

(3) A DFA that recognizes the language $L_3$ has no less than $2^3 = 8$ states. Prove this fact.

(4) Argue for the following statement:

• To recognize the same language, a DFA possibly needs the number of states that is exponentially bigger than an NFA does.

Kai

(1)

(2)

(3) + (4)

I'll prove that DFA that recognizes $L_n$:

$$L_n = \{ w\in \Sigma^* | \: |w| \geq 3 \text{ and n-th to the last letter of $w$ is $a$}\}$$

has no less than $2^n$ states. This is almost a copy-paste from Introduction to Automata Theory, Languages and Computation 3rd ed., Chapter 2.3.6.

Intuitively we need $2^n$ states to remember every possible ending of length $n$: we can encode it as a binary string of length $n$. Now, I'll show that there's no such DFA with less than $2^n$ states.

Suppose that there's DFA $D$ recognizing $L_n$ with less tan $2^n$ states. If so, then tere must exist a state $q$ in which $D$ is after reading two different sequences, say $x = x_1x_2\cdots x_n$ and $y = y_1y_2\cdots y_n$. Since they are different, let $i\in N$ be the last position on which they differ. By symmetry, assume that $x_i = a$ and $y_i = b$.

• if $i = 1$, then $x = a\:x_2\cdots x_n$ and $y = b\:y_2\cdots y_n$. Which means that state $q$ is both accepting and non-accepting.
• if $i > 1$, then we can append $(n-i)$ $b$'s to both $x$ and $y$. Then a state $p$ in which $D$ is after reading $ax_{i+1}\cdots x_nb\cdots b$ and $by_{i+1}\cdots y_nb\cdots b$ is both accepting and non-accepting.

In both cases we get lead to contradiction. Thus, DFA recognizing $L_n$ has at least $2^n$ states.

• (3): Let $n = 3$. Then DFA recognizing $L_3$ has no less than $8$ states.
• (4): NFA recognizing $L_n$ has exactly $n+1$ states. Equivalent DFA has $2^n$ states.