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非确定有限状态自动机（Nondeterministic Finite Automata，NFA）由以下元素组成：
- 一个有限的状态集合S
- 一个输入符号集合Sigma，并且架设空字符epsilon不属于Sigma
- 一个状态迁移函数，对于所给的每一个状态和每一个属于Sigma或{epsilon}的符号，输出迁移状态的集合。
- 一个S中的状态s0作为开始状态（初始状态）
- S的一个子集F，作为接受状态（结束状态）
例如，我们给定：
- S＝{s0, s1, s2, s3, s4}
- Sigma={a, b}
- 状态迁移函数T，且T(s0, a} = {s1}, T(s1, a) = {s2}, T(s2, b) = {s3}, T(s3, b) = {s4}
- s0为开始状态
- {s4}为接受状态
这样我们就得到一个很简单的NFA，它可以用图来表示，如下图图1：
NFA是一个识别器，例如图1所示的NFA，我们从状态s0开始，按顺序输入aabb，在输入第一个符号a之后，状态将从s0迁移到s1，输入第二个符号a之后，状态迁移到s2，输入第三个符号b之后，状态迁移到s3，输入第四个符号b之后，状态迁移到s4，而s4是接收状态，也就是说对我们刚才输入的aabb字符串说yes，表明本NFA识别了所输入的字符串。
所谓非确定，是指在某个状态输入同一个符号，状态可以迁移到不同的下一个状态，例如图2，在s0处输入字符a，状态既可以迁移为s1，也可以迁移为s3，准确的说是状态迁移到了{s1,s3}，因此图2所示的NFA能够接受的字符串包括aa和ab。
另外，NFA的特点还在于空符号也能进行状态迁移，例如图3的s0，不需要任何输入字符就可以迁移到s1，因此图3的NFA可以识别的语言为*a*b，即0到任意多个a，接着0到任意多个b。
NFA可以识别的语言与正则表达式所表达的语言是等价的，参考
http://en.wikipedia.org/wiki/Nondeterministic_finite_automaton
那么，NFA如何实现呢？我们先来看看NFA状态节点的一种实现：
/*
This file is one of the component a Context-free Grammar Parser Generator,
which accept a piece of text as the input, and generates a parser
for the inputted context-free grammar.
Copyright (C) 2013, Junbiao Pan (Email: panjunbiao@gmail.com)
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
package automata;
import java.util.*;
public class NFAState implements Comparable<NFAState> {
private static int COUNT = 0;
//状态标识，每个NFA状态节点都有唯一的数值标识
private int id;
public int getId() { return this.id; }
//在创建NFA状态对象的时候，通过静态变量生成唯一标识
public NFAState() {
this.id = COUNT ++;
}
//迁移函数，由于迁移函数需要两个输入：当前状态和输入符号，因此在一个状态对象内部，
//迁移函数都是针对本对象的，只需要输入符号就可以了，这里通过Map接口实现迁移函数
protected Map<Integer, Set<NFAState>> transition = new HashMap<Integer, Set<NFAState>>();
public Map<Integer, Set<NFAState>> getTransition() { return this.transition; }
//空字符迁移函数，即从当前节点经过空字符输入所能够到达的下一个状态节点
protected Set<NFAState> epsilonTransition = new HashSet<NFAState>();
public Set<NFAState> getEpsilonTransition() { return this.epsilonTransition; }
//向迁移函数添加一个映射，不给定下一个状态节点
public NFAState addTransit(int input) {
return addTransit(input, new NFAState());
}
//向迁移函数添加一个映射，给定下一个状态节点
public NFAState addTransit(int input, NFAState next) {
Set<NFAState> states = this.transition.get(input);
if (states == null) {
states = new HashSet<NFAState>();
this.transition.put(input, states);
}
states.add(next);
return next;
}
//向迁移函数添加一个映射，不给定下一个状态节点
public NFAState addTransit(char input) {
return addTransit(input, new NFAState());
}
//向迁移函数添加一个映射，给定下一个状态节点
//假定我们的上下文无关文法是大小写不敏感的，当输入字符是char类型并且是字母时，
//生成大写字母和小写字母两个映射
public NFAState addTransit(char input, NFAState next) {
if (Character.isLetter(input)) {
this.addTransit((int) (Character.toUpperCase(input)), next);
this.addTransit((int)(Character.toLowerCase(input)), next);
return next;
}
this.addTransit((int)input, next);
return next;
}
//添加一个空字符的映射
public NFAState addTransit(NFAState next) {
this.epsilonTransition.add(next);
return next;
}
//返回迁移函数
public Set<NFAState> getTransition(int input) {
return this.transition.get(input);
}
}
再来看看NFA的实现：/*
This file is one of the component a Context-free Grammar Parser Generator,
which accept a piece of text as the input, and generates a parser
for the inputted context-free grammar.
Copyright (C) 2013, Junbiao Pan (Email: panjunbiao@gmail.com)
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
package automata;
import java.util.*;
import abnf.CharVal;
import abnf.NumVal;
import abnf.AbnfParser;
import abnf.RangedNumVal;
import abnf.Repeat;
import abnf.Repetition;
import abnf.Rule;
import abnf.RuleName;
public class NFA {
//开始状态startState
private NFAState startState = null;
public NFAState getStartState() { return startState; }
//接收状态acceptingStates
private Set<NFAState> acceptingStates = new HashSet<NFAState>();
public Set<NFAState> getAcceptingStates() { return acceptingStates; }
public boolean accept(NFAState state) {
return this.acceptingStates.contains(state);
}
public void addAcceptingState(NFAState state) {
this.acceptingStates.add(state);
}
public NFA() {
this(new NFAState(), new NFAState());
}
public NFA(NFAState startState) {
this(startState, new NFAState());
}
public NFA(NFAState startState, NFAState acceptingState) {
this.startState = startState;
this.addAcceptingState(acceptingState);
}
//在上面的NFAState类实现中，新的状态节点是在添加迁移映射的过程中生成的，
//这个过程中NFA并没有介入，因此NFA类不能直接得到状态集S的成员
//而是需要从状态startState开始，不断迭代找出所有的状态节点
protected void getStateSet(NFAState current, Set<NFAState> states) {
if (states.contains(current)) return;
states.add(current);
Iterator<NFAState> it;
it = current.getNextStates().iterator();
while (it.hasNext()) {
this.getStateSet(it.next(), states);
}
it = current.getEpsilonTransition().iterator();
while (it.hasNext()) {
this.getStateSet(it.next(), states);
}
}
public Set<NFAState> getStateSet() {
Set<NFAState> states = new HashSet<NFAState>();
this.getStateSet(this.getStartState(), states);
return states;
}
}
这样，我们可以从NFA类中获得一个NFA的开始状态startState和接受状态集合acceptingStates，在每一个状态节点NFAState中可以获得状态迁移函数，因此NFA所定义的各个元素都实现了。