Program to Implement NFA with epsilon move to DFA Conversion

Non-deterministic Finite Automata (NFA) : NFA is a finite automaton where for some cases when a single input is given to a single state, the machine goes to more than 1 states, i.e. some of the moves cannot be uniquely determined by the present state and the present input symbol.

An NFA can be represented as M = { Q, ∑, ∂, q0, F}

Q → Finite non-empty set of states. ∑ → Finite non-empty set of input symbols. ∂ → Transitional Function. q0 → Beginning state. F → Final State

NFA with (null) or ∈ move : If any finite automata contains ε (null) move or transaction, then that finite automata is called NFA with ∈ moves

Example : Consider the following figure of NFA with ∈ move : Transition state table for the above NFA 

STATES	0	1	epsilon
A	B, C	A	B
B	–	B	C
C	C	C	–

Epsilon (∈) – closure : Epsilon closure for a given state X is a set of states which can be reached from the states X with only (null) or ε moves including the state X itself. In other words, ε-closure for a state can be obtained by union operation of the ε-closure of the states which can be reached from X with a single ε move in recursive manner. For the above example ∈ closure are as follows :

∈ closure(A) : {A, B, C}
∈ closure(B) : {B, C}
∈ closure(C) : {C}

Deterministic Finite Automata (DFA) : DFA is a finite automata where, for all cases, when a single input is given to a single state, the machine goes to a single state, i.e., all the moves of the machine can be uniquely determined by the present state and the present input symbol.

Steps to Convert NFA with ε-move to DFA :

Step 1 : Take ∈ closure for the beginning state of NFA as beginning state of DFA. Step 2 : Find the states that can be traversed from the present for each input symbol (union of transition value and their closures for each states of NFA present in current state of DFA). Step 3 : If any new state is found take it as current state and repeat step 2. Step 4 : Do repeat Step 2 and Step 3 until no new state present in DFA transition table. Step 5 : Mark the states of DFA which contains final state of NFA as final states of DFA.

Transition State Table for DFA corresponding to above NFA

STATES	0	1
A, B, C	B, C	A, B, C
B, C	C	B, C
C	C	C

DFA STATE DIAGRAM Examples :

Input : 6
        2
        FC - BF
        - C -
        - - D
        E A -
        A - BF
        - - -


Output :
 STATES OF NFA :        A, B, C, D, E, F,

 GIVEN SYMBOLS FOR NFA:     0, 1, eps


 NFA STATE TRANSITION TABLE 


STATES    |0    |1    eps
--------+------------------------------------
A    |FC     |-     |BF     
B    |-     |C     |-     
C    |-     |-     |D     
D    |E     |A     |-     
E    |A     |-     |BF     
F    |-     |-     |-     

 e-Closure (A) :    ABF

 e-Closure (B) :    B

 e-Closure (C) :    CD

 e-Closure (D) :    D

 e-Closure (E) :    BEF

 e-Closure (F) :    F


********************************************************

         DFA TRANSITION STATE TABLE          


 STATES OF DFA :        ABF, CDF, CD, BEF,

 GIVEN SYMBOLS FOR DFA:     0, 1,

STATES    |0    |1    
--------+-----------------------
ABF    |CDF     |CD     
CDF    |BEF     |ABF     
CD    |BEF     |ABF     
BEF    |ABF     |CD     



Input :
9
2
- - BH
- - CE
D - -
- - G
- F -
- - G
- - BH
I - -
- -  -


Output :

STATES OF NFA :        A, B, C, D, E, F, G, H, I,

 GIVEN SYMBOLS FOR NFA:     0, 1, eps


 NFA STATE TRANSITION TABLE 


STATES    |0    |1    eps
--------+------------------------------------
A    |-     |-     |BH     
B    |-     |-     |CE     
C    |D     |-     |-     
D    |-     |-     |G     
E    |-     |F     |-     
F    |-     |-     |G     
G    |-     |-     |BH     
H    |I     |-     |-     
I    |-     |-     |-     

 e-Closure (A) :    ABCEH

 e-Closure (B) :    BCE

 e-Closure (C) :    C

 e-Closure (D) :    BCDEGH

 e-Closure (E) :    E

 e-Closure (F) :    BCEFGH

 e-Closure (G) :    BCEGH

 e-Closure (H) :    H

 e-Closure (I) :    I


********************************************************

         DFA TRANSITION STATE TABLE          


 STATES OF DFA :        ABCEH, BCDEGHI, BCEFGH,

 GIVEN SYMBOLS FOR DFA:     0, 1,

STATES    |0    |1    
--------+-----------------------
ABCEH    |BCDEGHI     |BCEFGH     
BCDEGHI    |BCDEGHI     |BCEFGH     
BCEFGH    |BCDEGHI     |BCEFGH     

Explanation : First line of the input contains the number of states (N) of NFA. Second line of the input says the number of input symbols (S). In example1 number of states of NFA is 6 i.e.( A, B, C, D, E, F) and 2 input symbols i.e. ( 0, 1). Since we are working on NFA with ∈ move, ∈ will be added as an extra input symbol. The next N lines contains the transition values for every state of NFA. The value of ith row, jth column indicates transition value for ith state on jth input symbol. Here in example1 transition(A, 0) : FC. Output contains the NFA, ∈ closure for every states of the corresponding NFA and DFA obtained by converting the input NFA. States and input symbols of the DFA are also specified. Below is the implementation of above approach : 

C

// C Program to illustrate how to convert e-nfa to DFA

#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#define MAX_LEN 100

char NFA_FILE[MAX_LEN];
char buffer[MAX_LEN];
int zz = 0;

// Structure to store DFA states and their
// status ( i.e new entry or already present)
struct DFA {
  char *states;
  int count;
} dfa;

int last_index = 0;
FILE *fp;
int symbols;

/* reset the hash map*/
void reset(int ar[], int size) {
  int i;

  // reset all the values of
  // the mapping array to zero
  for (i = 0; i < size; i++) {
    ar[i] = 0;
  }
}

// Check which States are present in the e-closure

/* map the states of NFA to a hash set*/
void check(int ar[], char S[]) {
  int i, j;

  // To parse the individual states of NFA
  int len = strlen(S);
  for (i = 0; i < len; i++) {

    // Set hash map for the position
    // of the states which is found
    j = ((int)(S[i]) - 65);
    ar[j]++;
  }
}

// To find new Closure States
void state(int ar[], int size, char S[]) {
  int j, k = 0;

  // Combine multiple states of NFA
  // to create new states of DFA
  for (j = 0; j < size; j++) {
    if (ar[j] != 0)
      S[k++] = (char)(65 + j);
  }

  // mark the end of the state
  S[k] = '\0';
}

// To pick the next closure from closure set
int closure(int ar[], int size) {
  int i;

  // check new closure is present or not
  for (i = 0; i < size; i++) {
    if (ar[i] == 1)
      return i;
  }
  return (100);
}

// Check new DFA states can be
// entered in DFA table or not
int indexing(struct DFA *dfa) {
  int i;

  for (i = 0; i < last_index; i++) {
    if (dfa[i].count == 0)
      return 1;
  }
  return -1;
}

/* To Display epsilon closure*/
void Display_closure(int states, int closure_ar[],
                     char *closure_table[],
                     char *NFA_TABLE[][symbols + 1],
                     char *DFA_TABLE[][symbols]) {
  int i;
  for (i = 0; i < states; i++) {
    reset(closure_ar, states);
    closure_ar[i] = 2;

    // to neglect blank entry
    if (strcmp(&NFA_TABLE[i][symbols], "-") != 0) {

      // copy the NFA transition state to buffer
      strcpy(buffer, &NFA_TABLE[i][symbols]);
      check(closure_ar, buffer);
      int z = closure(closure_ar, states);

      // till closure get completely saturated
      while (z != 100)
      {
        if (strcmp(&NFA_TABLE[z][symbols], "-") != 0) {
          strcpy(buffer, &NFA_TABLE[z][symbols]);

          // call the check function
          check(closure_ar, buffer);
        }
        closure_ar[z]++;
        z = closure(closure_ar, states);
      }
    }

    // print the e closure for every states of NFA
    printf("\n e-Closure (%c) :\t", (char)(65 + i));

    bzero((void *)buffer, MAX_LEN);
    state(closure_ar, states, buffer);
    strcpy(&closure_table[i], buffer);
    printf("%s\n", &closure_table[i]);
  }
}

/* To check New States in DFA */
int new_states(struct DFA *dfa, char S[]) {

  int i;

  // To check the current state is already
  // being used as a DFA state or not in
  // DFA transition table
  for (i = 0; i < last_index; i++) {
    if (strcmp(&dfa[i].states, S) == 0)
      return 0;
  }

  // push the new
  strcpy(&dfa[last_index++].states, S);

  // set the count for new states entered
  // to zero
  dfa[last_index - 1].count = 0;
  return 1;
}

// Transition function from NFA to DFA
// (generally union of closure operation )
void trans(char S[], int M, char *clsr_t[], int st,
               char *NFT[][symbols + 1], char TB[]) {
  int len = strlen(S);
  int i, j, k, g;
  int arr[st];
  int sz;
  reset(arr, st);
  char temp[MAX_LEN], temp2[MAX_LEN];
  char *buff;

  // Transition function from NFA to DFA
  for (i = 0; i < len; i++) {

    j = ((int)(S[i] - 65));
    strcpy(temp, &NFT[j][M]);

    if (strcmp(temp, "-") != 0) {
      sz = strlen(temp);
      g = 0;

      while (g < sz) {
        k = ((int)(temp[g] - 65));
        strcpy(temp2, &clsr_t[k]);
        check(arr, temp2);
        g++;
      }
    }
  }

  bzero((void *)temp, MAX_LEN);
  state(arr, st, temp);
  if (temp[0] != '\0') {
    strcpy(TB, temp);
  } else
    strcpy(TB, "-");
}

/* Display DFA transition state table*/
void Display_DFA(int last_index, struct DFA *dfa_states,
                 char *DFA_TABLE[][symbols]) {
  int i, j;
  printf("\n\n********************************************************\n\n");
  printf("\t\t DFA TRANSITION STATE TABLE \t\t \n\n");
  printf("\n STATES OF DFA :\t\t");

  for (i = 1; i < last_index; i++)
    printf("%s, ", &dfa_states[i].states);
  printf("\n");
  printf("\n GIVEN SYMBOLS FOR DFA: \t");

  for (i = 0; i < symbols; i++)
    printf("%d, ", i);
  printf("\n\n");
  printf("STATES\t");

  for (i = 0; i < symbols; i++)
    printf("|%d\t", i);
  printf("\n");

  // display the DFA transition state table
  printf("--------+-----------------------\n");
  for (i = 0; i < zz; i++) {
    printf("%s\t", &dfa_states[i + 1].states);
    for (j = 0; j < symbols; j++) {
      printf("|%s \t", &DFA_TABLE[i][j]);
    }
    printf("\n");
  }
}

// Driver Code
int main() {
  int i, j, states;
  char T_buf[MAX_LEN];

  // creating an array dfa structures
  struct DFA *dfa_states = malloc(MAX_LEN * (sizeof(dfa)));
  states = 6, symbols = 2;

  printf("\n STATES OF NFA :\t\t");
  for (i = 0; i < states; i++)

    printf("%c, ", (char)(65 + i));
  printf("\n");
  printf("\n GIVEN SYMBOLS FOR NFA: \t");

  for (i = 0; i < symbols; i++)

    printf("%d, ", i);
  printf("eps");
  printf("\n\n");
  char *NFA_TABLE[states][symbols + 1];

  // Hard coded input for NFA table
  char *DFA_TABLE[MAX_LEN][symbols];
  strcpy(&NFA_TABLE[0][0], "FC");
  strcpy(&NFA_TABLE[0][1], "-");
  strcpy(&NFA_TABLE[0][2], "BF");
  strcpy(&NFA_TABLE[1][0], "-");
  strcpy(&NFA_TABLE[1][1], "C");
  strcpy(&NFA_TABLE[1][2], "-");
  strcpy(&NFA_TABLE[2][0], "-");
  strcpy(&NFA_TABLE[2][1], "-");
  strcpy(&NFA_TABLE[2][2], "D");
  strcpy(&NFA_TABLE[3][0], "E");
  strcpy(&NFA_TABLE[3][1], "A");
  strcpy(&NFA_TABLE[3][2], "-");
  strcpy(&NFA_TABLE[4][0], "A");
  strcpy(&NFA_TABLE[4][1], "-");
  strcpy(&NFA_TABLE[4][2], "BF");
  strcpy(&NFA_TABLE[5][0], "-");
  strcpy(&NFA_TABLE[5][1], "-");
  strcpy(&NFA_TABLE[5][2], "-");
  printf("\n NFA STATE TRANSITION TABLE \n\n\n");
  printf("STATES\t");

  for (i = 0; i < symbols; i++)
    printf("|%d\t", i);
  printf("eps\n");

  // Displaying the matrix of NFA transition table
  printf("--------+------------------------------------\n");
  for (i = 0; i < states; i++) {
    printf("%c\t", (char)(65 + i));

    for (j = 0; j <= symbols; j++) {
      printf("|%s \t", &NFA_TABLE[i][j]);
    }
    printf("\n");
  }
  int closure_ar[states];
  char *closure_table[states];

  Display_closure(states, closure_ar, closure_table, NFA_TABLE, DFA_TABLE);
  strcpy(&dfa_states[last_index++].states, "-");

  dfa_states[last_index - 1].count = 1;
  bzero((void *)buffer, MAX_LEN);

  strcpy(buffer, &closure_table[0]);
  strcpy(&dfa_states[last_index++].states, buffer);

  int Sm = 1, ind = 1;
  int start_index = 1;

  // Filling up the DFA table with transition values
  // Till new states can be entered in DFA table
  while (ind != -1) {
    dfa_states[start_index].count = 1;
    Sm = 0;
    for (i = 0; i < symbols; i++) {

      trans(buffer, i, closure_table, states, NFA_TABLE, T_buf);

      // storing the new DFA state in buffer
      strcpy(&DFA_TABLE[zz][i], T_buf);

      // parameter to control new states
      Sm = Sm + new_states(dfa_states, T_buf);
    }
    ind = indexing(dfa_states);
    if (ind != -1)
      strcpy(buffer, &dfa_states[++start_index].states);
    zz++;
  }
  // display the DFA TABLE
  Display_DFA(last_index, dfa_states, DFA_TABLE);

  return 0;
}

Use of NFA with ∈ move : If we want to construct an FA which accepts a language, sometimes it becomes very difficult or seems to be impossible to construct a direct NFA or DFA. But if NFA with ∈ moves is used, then the transitional diagram can be constructed and described easily.