自己动手写编译器:从NFA到DFA

news2024/10/6 8:21:57

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：
​​​​在这里插入图片描述
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：
请添加图片描述
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:
请添加图片描述
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：
请添加图片描述

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

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