Dfa Based Division In C++

In this tutorial, you'll explore the DFA-driven division concept in C++ along with an illustrative example.

Using a Deterministic Finite Automaton (DFA) to Check Divisibility

Implementing integer division in hardware through deterministic finite automata (DFA) is an effective method. This approach involves creating a DFA designed to identify strings that symbolize the step-by-step division process.

If you intend to partition two n-bit integers A and B, you have the option to create a Deterministic Finite Automaton (DFA) comprising 2n+1 states. This DFA is designed to calculate the division A/B by processing one bit at a time, starting from the most significant bit and progressing towards the least significant bit.

The steps are:

• ✅ Construct a DFA with 2n+1 states labelled 0 to 2n. State 0 is the start state, and State 2n is the accepted State.
• ✅ The transition function of the DFA is defined based on the division algorithm: If the current State + the next bit of A is less than B, the transition to the new State is obtained by shifting left and adding the next bit of A. Otherwise, transition to the new State obtained by shifting left, adding the next bit of A and subtracting B.
• ✅ The DFA accepts when it reaches the accept State 2n, at which point the division is complete.
• ✅ The quotient is obtained by recording the number of subtractions performed in each State. The remainder is the last State reached.
• ✅ If the current State + the next bit of A is less than B, the transition to the new State is obtained by shifting left and adding the next bit of A.
• ✅ Otherwise, transition to the new State obtained by shifting left, adding the next bit of A and subtracting B.

It is possible to incorporate this functionality in C++ by utilizing a DFA class that includes a transition table, current state, lookup functions, and more. The process of dividing can be achieved by progressing through the DFA bit by bit and simultaneously monitoring the quotient and remainder values.

The benefit of employing the DFA method lies in its reliance on basic operations like comparisons, additions, and subtractions, which contribute to its high speed when implemented in hardware. Using a fixed-size DFA, it is capable of performing n-bit division in O(n) time efficiently.

Example:

Here is an alternative illustration of DFA-driven division using the instance of dividing by 3:

The fundamental concept involves building a Deterministic Finite Automaton (DFA) comprising k states, where k denotes the divisor. These states symbolize the potential remainders ranging from 0 to k-1.

The transition function is based on the following:

• ✅ Current State = the remainder so far
• ✅ Next input bit = 0 or 1
• ✅ Next state = (2 * current state + input bit) % k

It calculates the updated remainder by left-shifting and incorporating the new bit. The modulo operation with K then assigns it to one of the K states.

For example, to check if a binary number is divisible by 3 (k=3):

• ✅ States = {0, 1, 2}
• ✅ start state = 0
• ✅ final State = 0 (remainder 0 means divisible by 3)

Transitions:

• ✅ State 0, input 0 -> 0 (2*0 + 0 = 0 % 3 = 0)
• ✅ State 0, input 1 -> 1 (2*0 + 1 = 1 % 3 = 1)
• ✅ State 1, input 0 -> 2 (2*1 + 0 = 2 % 3 = 2)
• ✅ State 1, input 1 -> 0 (2*1 + 1 = 3 % 3 = 0)
• ✅ State 2, input 0 -> 1 (2*2 + 0 = 4 % 3 = 1)
• ✅ State 2, input 1 -> 2 (2*2 + 1 = 5 % 3 = 2)

To divide 6 (110 in binary) by 3:

• ✅ Start at State 0
• ✅ Input 1 -> new state 1
• ✅ Input 1 -> new state 0
• ✅ Input 0 -> new state 0

Reach final state 0, so 6 is divisible by 3.

For 4 (100 in binary):

• ✅ Start at State 0
• ✅ Input 1 -> new state 1
• ✅ Input 0 -> new state 2
• ✅ Input 0 -> new state 1

End in non-zero State, so remainder is 1.

This deterministic finite automaton method enables division to be performed in linear time complexity with uncomplicated state changes. The final state reflects the remainder, determining divisibility of the number.

Implementation of C++ Code:

#include <iostream>
#include <string>
using namespace std;
// Define the states for the DFA
enum class State {
START,
INTEGER,
DECIMAL,
ERROR
};

int main() {
string input;
int divisor;
cout << "Enter the dividend (as an integer): ";
cin >> input;
cout << "Enter the divisor (as an integer): ";
cin >> divisor;

State state = State::START;
int result = 0;
int decimalPosition = 0;

for (char c: input) {
switch (State) {
case State::START:
if (c >= '0' && c <= '9') {
result = c - '0';
state = State::INTEGER;
} else {
state = State::ERROR;
}
break;
case State::INTEGER:
if (c >= '0' && c <= '9') {
result = result * 10 + (c - '0');
} else if (c == '.') {
state = State::DECIMAL;
} else {
state = State::ERROR;
}
break;
case State::DECIMAL:
if (c >= '0' && c <= '9') {
result = result * 10 + (c - '0');
decimalPosition++;
} else {
state = State::ERROR;
}
break;
case State::ERROR:
break;
}
}

if (state == State::ERROR || divisor == 0) {
cout << "Invalid input or division by zero." << endl;
} else {
// Perform integer division
result /= divisor;
cout << "Result: " << result;
if (decimalPosition > 0) {
cout << "." << string(decimalPosition, '0');
}
cout << endl;
}

return 0;
}

Output:

Enter the dividend (as an integer): 1234
Enter the divisor (as an integer): 2
Result: 617