上一节我们完成了使用NFA来识别字符串的功能。NFA有个问题就是其状态节点太多，使用起来效率不够好。本节我们介绍一种叫“子集构造”的算法，将拥有多个节点的NFA转化为DFA。在上一节我们描述的epsilon闭包操作可以看到，实际上所有由epsilon边连接在一起的节点其实都能看作是一个状态节点，由此我们就能通过epsilon操作将多个节点转化为一个DFA节点，同时epsilon闭包操作所得的节点集合中，每一个节点发出的边都可以看作是新DFA节点发出的边。
我们用上一节完成的NFA状态机来看看具体过程：
从节点0开始做epsilon操作所得结果为：
epsilon-closure(0) = {0, 27, 11, 19, 9, 12, 13}, 由此我们把这些节点合成一个新节点，我们标记为DFA state 0。
接着我们对集合{0, 27, 11, 19, 9, 12}做move操作有：
move({0, 27, 11, 19, 9, 12, 14}, D} = {10, 20}, 于是可以把节点10，20合成新节点，记做"DFA state 1", 因为有:
move({0, 27, 11, 19, 9, 12}, .} = {14}, 于是我们把节点14看做新节点，记做"DFA state 2"，这么一来我们就得到如下DFA状态机：
接下来我们继续对{10, 20}进行epsilon闭包操作，epsilon-closure({10, 20})={10, 20, 9,12,13,21}，然后再对这个结果做move操作有:
move({10, 20, 9,12,13,21}, D) = {10} , 于是我们再产生一个新DFA节点记作DFA state 3, move({10, 20, 9,12,13,21}, . } = {14, 22} 于是我们再产生新的DFA节点记作DFA state 4,于是就有:
这个过程以此类推，这里需要注意的是如果epsilon闭包操作后所得的节点集合中有NFA状态机的终结节点，那么其对应的DFA节点就是一个终结节点。接下来看看代码如何实现，我们添加一个名为nfa_to_dfa.go的文件，然后添加代码如下：
import "fmt"
const (
DFA_MAX = 254 //DFA 最多节点数
F = -1 //用于初始化跳转表
MAX_CHARS = 128 //128个ascii字符
)
type ACCEPT struct {
acceptString string //接收节点对应的执行代码字符串
anchor Anchor
}
type DFA struct {
group int //后面执行最小化算法时有用
mark bool //当前节点是否已经设置好接收字符对应的边
anchor Anchor
set []*NFA //dfa节点对应的nfa节点集合
state int //dfa 节点号码
acceptString string
}
这里我们先定义基本的数据结构，在转换的DFA状态机中，它最多包含254个节点，同时状态机只接收来自ascii表中数值从0到128的字符，这次我们构造的DFA状态机将不像上次构造的NFA状态机那样使用链表结构，这次我们使用跳转表结构，我们将构造一个二维数组dtrans，假设状态节点1接收字符“.“后，跳转到状态节点2，由于字符”."对应的ascii数值为46，那么就有dtrans[1][46] = 2.
在上面代码中我们定义了DFA节点，由于一个DFA节点由一组NFA节点转换而来，因此在它的定义中有一个NFA节点的指针数组。接下来我们设计用于将NFA转换成DFA的类，其代码为：
type NfaDfaConverter struct {
nstates int //当前dfa 节点计数
lastMarked int //下一个需要处理的dfa节点
dtrans [][]int //dfa状态机的跳转表
accepts []*ACCEPT
dstates []DFA //所有dfa节点的集合
}
func NewNfaDfaConverter() *NfaDfaConverter {
n := &NfaDfaConverter{
nstates: 0,
lastMarked: 0,
dtrans: make([][]int, DFA_MAX),
dstates: make([]DFA, DFA_MAX),
}
for i := range n.dtrans {
n.dtrans[i] = make([]int, MAX_CHARS)
}
return n
}
在定义中有几个变量需要注意，其中dtrans是用于构造DFA跳转表的二维数组， nstates用于记录当前已经生成的DFA节点数量，lastMarked用于指向下一个要创建其跳转逻辑的DFA节点编号，dstates用于存储当前已经创建了的DFA节点。下面我们看看转换逻辑的实现:
func (n *NfaDfaConverter) getUnMarked() *DFA {
for ; n.lastMarked < n.nstates; n.lastMarked++ {
debug := 0
if n.dstates[n.lastMarked].state == 5 {
debug = 1
fmt.Printf("debug: %d", debug)
}
if n.dstates[n.lastMarked].mark == false {
return &n.dstates[n.lastMarked]
}
}
return nil
}
func (n *NfaDfaConverter) compareNfaSlice(setOne []*NFA, setTwo []*NFA) bool {
//比较两个集合的元素是否相同
if len(setOne) != len(setTwo) {
return false
}
equal := false
for _, nfaOne := range setOne {
for _, nfaTwo := range setTwo {
if nfaTwo == nfaOne {
equal = true
break
}
}
if equal != true {
return false
}
}
return true
}
func (n *NfaDfaConverter) hasDfaContainsNfa(nfaSet []*NFA) (bool, int) {
//查看是否存在dfa节点它对应的nfa节点集合与输入的集合相同
for _, dfa := range n.dstates {
if n.compareNfaSlice(dfa.set, nfaSet) == true {
return true, dfa.state
}
}
return false, -1
}
func (n *NfaDfaConverter) addDfaState(epsilonResult *EpsilonResult) int {
//根据当前nfa节点集合构造一个新的dfa节点
nextState := F
if n.nstates >= DFA_MAX {
panic("Too many DFA states")
}
nextState = n.nstates
n.nstates += 1
n.dstates[nextState].set = epsilonResult.results
n.dstates[nextState].mark = false
n.dstates[nextState].acceptString = epsilonResult.acceptStr
n.dstates[nextState].anchor = epsilonResult.anchor
n.dstates[nextState].state = nextState //记录当前dfa节点的编号s
n.printDFAState(&n.dstates[nextState])
fmt.Print("\n")
return nextState
}
func (n *NfaDfaConverter) printDFAState(dfa *DFA) {
fmt.Printf("DFA state : %d, it is nfa are: {", dfa.state)
for _, nfa := range dfa.set {
fmt.Printf("%d,", nfa.state)
}
fmt.Printf("}")
}
func (n *NfaDfaConverter) MakeDTran(start *NFA) {
//根据输入的nfa状态机起始节点构造dfa状态机的跳转表
startStates := make([]*NFA, 0)
startStates = append(startStates, start)
statesCopied := make([]*NFA, len(startStates))
copy(statesCopied, startStates)
//先根据起始状态的求Epsilon闭包操作的结果，由此获得第一个dfa节点
epsilonResult := EpsilonClosure(statesCopied)
n.dstates[0].set = epsilonResult.results
n.dstates[0].anchor = epsilonResult.anchor
n.dstates[0].acceptString = epsilonResult.acceptStr
n.dstates[0].mark = false
//debug purpose
n.printDFAState(&n.dstates[0])
fmt.Print("\n")
nextState := 0
n.nstates = 1 //当前已经有一个dfa节点
//先获得第一个没有设置其跳转边的dfa节点
current := n.getUnMarked()
for current != nil {
current.mark = true
for c := 0; c < MAX_CHARS; c++ {
nfaSet := move(current.set, c)
if len(nfaSet) > 0 {
statesCopied = make([]*NFA, len(nfaSet))
copy(statesCopied, nfaSet)
epsilonResult = EpsilonClosure(statesCopied)
nfaSet = epsilonResult.results
}
if len(nfaSet) == 0 {
nextState = F
} else {
//如果当前没有那个dfa节点对应的nfa节点集合和当前nfaSet相同，那么就增加一个新的dfa节点
isExist, state := n.hasDfaContainsNfa(nfaSet)
if isExist == false {
nextState = n.addDfaState(epsilonResult)
} else {
nextState = state
}
}
//设置dfa跳转表
n.dtrans[current.state][c] = nextState
}
current = n.getUnMarked()
}
}
func (n *NfaDfaConverter) PrintDfaTransition() {
for i := 0; i < DFA_MAX; i++ {
if n.dstates[i].mark == false {
break
}
for j := 0; j < MAX_CHARS; j++ {
if n.dtrans[i][j] != F {
n.printDFAState(&n.dstates[i])
fmt.Print(" jump to : ")
n.printDFAState(&n.dstates[n.dtrans[i][j]])
fmt.Printf("by character %s\n", string(j))
}
}
}
}
前面我们看到，一个DFA节点本质上对应一组NFA节点，因此当我们使用move 和epsilon闭包操作得到一组NFA节点后，我们需要看看是不是已经有DFA节点对应到了生成的NFA节点集合，如果有了，说明对应的DFA节点已经生成，这个操作由函数compareNfaSlice和hasDfaContainsNfa完成，如果当前得到的NFA节点集合没有对应的DFA节点，那么就使用addDfaState函数去创建一个新的DFA节点，然后将其加入到dstates数组中。
每新建一个DFA节点时，它的mark标志位会设置成false,这表明我们还没有为它设置跳转边，函数getUnMarked用于将当前所有mark设置为false的DFA节点中找出创建时间最早的那个。上面代码的算法核心在函数MakeDTran,它执行了我们上面提到的算法，首先获得NFA状态机的起始节点，然后通过epsilon闭包操作获得一组NFA节点，用这组节点创建一个对应的DFA节点。接着使用move操作得到第二组NFA节点，然后再次使用epsilon闭包操作获得新一组NFA节点，然后创建第二个DFA节点，最后根据这两个节点对应的编号在二维表dtrans中设置跳转逻辑。
接下来我们在主函数中调用上面实现代码看看结果，在mai.go中输入代码如下：
package main
import (
"nfa"
)
func main() {
lexReader, _ := nfa.NewLexReader("input.lex", "output.py")
lexReader.Head()
parser, _ := nfa.NewRegParser(lexReader)
start := parser.Parse()
parser.PrintNFA(start)
//str := "3.14"
//if nfa.NfaMatchString(start, str) {
// fmt.Printf("string %s is accepted by given regular expression\n", str)
//}
nfaConverter := nfa.NewNfaDfaConverter()
nfaConverter.MakeDTran(start)
nfaConverter.PrintDfaTransition()
}
上面代码运行后输出结果如下:
DFA state : 0, it is nfa are: {
0,27,19,11,12,13,9,}
DFA state : 1, it is nfa are: {
14,15,}
DFA state : 2, it is nfa are: {
10,9,12,13,20,21,}
DFA state : 3, it is nfa are: {
16,28,}
DFA state : 4, it is nfa are: {
22,25,26,28,23,14,15,}
DFA state : 5, it is nfa are: {
10,9,12,13,}
DFA state : 6, it is nfa are: {
16,28,24,23,26,28,}
DFA state : 7, it is nfa are: {
24,23,26,28,}
DFA state : 0, it is nfa are: {
0,27,19,11,12,13,9,} jump to : DFA state : 1, it is nfa are: {
14,15,}by character .
DFA state : 0, it is nfa are: {
0,27,19,11,12,13,9,} jump to : DFA state : 2, it is nfa are: {
10,9,12,13,20,21,}by character 0
DFA state : 0, it is nfa are: {
0,27,19,11,12,13,9,} jump to : DFA state : 2, it is nfa are: {
10,9,12,13,20,21,}by character 1
DFA state : 0, it is nfa are: {
0,27,19,11,12,13,9,} jump to : DFA state : 2, it is nfa are: {
10,9,12,13,20,21,}by character 2
DFA state : 0, it is nfa are: {
0,27,19,11,12,13,9,} jump to : DFA state : 2, it is nfa are: {
10,9,12,13,20,21,}by character 3
DFA state : 0, it is nfa are: {
0,27,19,11,12,13,9,} jump to : DFA state : 2, it is nfa are: {
10,9,12,13,20,21,}by character 4
DFA state : 0, it is nfa are: {
0,27,19,11,12,13,9,} jump to : DFA state : 2, it is nfa are: {
10,9,12,13,20,21,}by character 5
DFA state : 0, it is nfa are: {
0,27,19,11,12,13,9,} jump to : DFA state : 2, it is nfa are: {
10,9,12,13,20,21,}by character 6
DFA state : 0, it is nfa are: {
0,27,19,11,12,13,9,} jump to : DFA state : 2, it is nfa are: {
10,9,12,13,20,21,}by character 7
DFA state : 0, it is nfa are: {
0,27,19,11,12,13,9,} jump to : DFA state : 2, it is nfa are: {
10,9,12,13,20,21,}by character 8
DFA state : 0, it is nfa are: {
0,27,19,11,12,13,9,} jump to : DFA state : 2, it is nfa are: {
10,9,12,13,20,21,}by character 9
DFA state : 1, it is nfa are: {
14,15,} jump to : DFA state : 3, it is nfa are: {
16,28,}by character 0
DFA state : 1, it is nfa are: {
14,15,} jump to : DFA state : 3, it is nfa are: {
16,28,}by character 1
DFA state : 1, it is nfa are: {
14,15,} jump to : DFA state : 3, it is nfa are: {
16,28,}by character 2
DFA state : 1, it is nfa are: {
14,15,} jump to : DFA state : 3, it is nfa are: {
16,28,}by character 3
DFA state : 1, it is nfa are: {
14,15,} jump to : DFA state : 3, it is nfa are: {
16,28,}by character 4
DFA state : 1, it is nfa are: {
14,15,} jump to : DFA state : 3, it is nfa are: {
16,28,}by character 5
DFA state : 1, it is nfa are: {
14,15,} jump to : DFA state : 3, it is nfa are: {
16,28,}by character 6
DFA state : 1, it is nfa are: {
14,15,} jump to : DFA state : 3, it is nfa are: {
16,28,}by character 7
DFA state : 1, it is nfa are: {
14,15,} jump to : DFA state : 3, it is nfa are: {
16,28,}by character 8
DFA state : 1, it is nfa are: {
14,15,} jump to : DFA state : 3, it is nfa are: {
16,28,}by character 9
DFA state : 2, it is nfa are: {
10,9,12,13,20,21,} jump to : DFA state : 4, it is nfa are: {
22,25,26,28,23,14,15,}by character .
DFA state : 2, it is nfa are: {
10,9,12,13,20,21,} jump to : DFA state : 5, it is nfa are: {
10,9,12,13,}by character 0
DFA state : 2, it is nfa are: {
10,9,12,13,20,21,} jump to : DFA state : 5, it is nfa are: {
10,9,12,13,}by character 1
DFA state : 2, it is nfa are: {
10,9,12,13,20,21,} jump to : DFA state : 5, it is nfa are: {
10,9,12,13,}by character 2
DFA state : 2, it is nfa are: {
10,9,12,13,20,21,} jump to : DFA state : 5, it is nfa are: {
10,9,12,13,}by character 3
DFA state : 2, it is nfa are: {
10,9,12,13,20,21,} jump to : DFA state : 5, it is nfa are: {
10,9,12,13,}by character 4
DFA state : 2, it is nfa are: {
10,9,12,13,20,21,} jump to : DFA state : 5, it is nfa are: {
10,9,12,13,}by character 5
DFA state : 2, it is nfa are: {
10,9,12,13,20,21,} jump to : DFA state : 5, it is nfa are: {
10,9,12,13,}by character 6
DFA state : 2, it is nfa are: {
10,9,12,13,20,21,} jump to : DFA state : 5, it is nfa are: {
10,9,12,13,}by character 7
DFA state : 2, it is nfa are: {
10,9,12,13,20,21,} jump to : DFA state : 5, it is nfa are: {
10,9,12,13,}by character 8
DFA state : 2, it is nfa are: {
10,9,12,13,20,21,} jump to : DFA state : 5, it is nfa are: {
10,9,12,13,}by character 9
DFA state : 4, it is nfa are: {
22,25,26,28,23,14,15,} jump to : DFA state : 6, it is nfa are: {
16,28,24,23,26,28,}by character 0
DFA state : 4, it is nfa are: {
22,25,26,28,23,14,15,} jump to : DFA state : 6, it is nfa are: {
16,28,24,23,26,28,}by character 1
DFA state : 4, it is nfa are: {
22,25,26,28,23,14,15,} jump to : DFA state : 6, it is nfa are: {
16,28,24,23,26,28,}by character 2
DFA state : 4, it is nfa are: {
22,25,26,28,23,14,15,} jump to : DFA state : 6, it is nfa are: {
16,28,24,23,26,28,}by character 3
DFA state : 4, it is nfa are: {
22,25,26,28,23,14,15,} jump to : DFA state : 6, it is nfa are: {
16,28,24,23,26,28,}by character 4
DFA state : 4, it is nfa are: {
22,25,26,28,23,14,15,} jump to : DFA state : 6, it is nfa are: {
16,28,24,23,26,28,}by character 5
DFA state : 4, it is nfa are: {
22,25,26,28,23,14,15,} jump to : DFA state : 6, it is nfa are: {
16,28,24,23,26,28,}by character 6
DFA state : 4, it is nfa are: {
22,25,26,28,23,14,15,} jump to : DFA state : 6, it is nfa are: {
16,28,24,23,26,28,}by character 7
DFA state : 4, it is nfa are: {
22,25,26,28,23,14,15,} jump to : DFA state : 6, it is nfa are: {
16,28,24,23,26,28,}by character 8
DFA state : 4, it is nfa are: {
22,25,26,28,23,14,15,} jump to : DFA state : 6, it is nfa are: {
16,28,24,23,26,28,}by character 9
DFA state : 6, it is nfa are: {
16,28,24,23,26,28,} jump to : DFA state : 7, it is nfa are: {
24,23,26,28,}by character 0
DFA state : 6, it is nfa are: {
16,28,24,23,26,28,} jump to : DFA state : 7, it is nfa are: {
24,23,26,28,}by character 1
DFA state : 6, it is nfa are: {
16,28,24,23,26,28,} jump to : DFA state : 7, it is nfa are: {
24,23,26,28,}by character 2
DFA state : 6, it is nfa are: {
16,28,24,23,26,28,} jump to : DFA state : 7, it is nfa are: {
24,23,26,28,}by character 3
DFA state : 6, it is nfa are: {
16,28,24,23,26,28,} jump to : DFA state : 7, it is nfa are: {
24,23,26,28,}by character 4
DFA state : 6, it is nfa are: {
16,28,24,23,26,28,} jump to : DFA state : 7, it is nfa are: {
24,23,26,28,}by character 5
DFA state : 6, it is nfa are: {
16,28,24,23,26,28,} jump to : DFA state : 7, it is nfa are: {
24,23,26,28,}by character 6
DFA state : 6, it is nfa are: {
16,28,24,23,26,28,} jump to : DFA state : 7, it is nfa are: {
24,23,26,28,}by character 7
DFA state : 6, it is nfa are: {
16,28,24,23,26,28,} jump to : DFA state : 7, it is nfa are: {
24,23,26,28,}by character 8
DFA state : 6, it is nfa are: {
16,28,24,23,26,28,} jump to : DFA state : 7, it is nfa are: {
24,23,26,28,}by character 9
我们将上面输出绘制成图形如下：
对比上面的NFA状态图，DFA状态图就要简单很多，此外生成的DFA状态机还可以继续精简，下一节我们再看看相关算法。代码下载地址为：链接: https://pan.baidu.com/s/1kStrJMznrexQkGGBs8vN3w 提取码: dqss