Constructing a DFA for Strings Beginning with Three 1s and Ending with Three 0s

In the realm of theoretical computer science and computational linguistics, constructing a Deterministic Finite Automaton (DFA) is a foundational task. This article delves into building a DFA to recognize strings that begin with three 1s and end with three 0s. We'll explore the transition logic, state structure, and offer a step-by-step guide to ensure a comprehensive understanding.

Introduction to DFA

A Deterministic Finite Automaton (DFA) is a mathematical model in the theory of computation. It consists of a finite set of states, a set of input symbols, and a transition function that maps each state and input symbol to a new state. The DFA processes an input string and moves through these states based on the transitions until the end of the string is reached. The recognition of a string is successful if the DFA reaches an accepting state.

Designing the DFA

To design a DFA that recognizes strings beginning with three 1s and ending with three 0s, we need to carefully account for the states that track the first three 1s and the last three 0s. Here’s a detailed breakdown of the DFA:

State Designation and Transitions

We will use the following states:

State 0: Initial state, no 1s or 0s read yet. State 1: One 1 has been read. State 2: Two 1s have been read. State 2: Three 1s have been read, starting the count for 0s. State -3: Initial state for tracking trailing 0s. State -2: Two 0s have been read. State -1: Three 0s have been read, transitioning to the accepting state. State A: Accepting state, reached after reading three 0s following three 1s. State F: Failing state, where invalid inputs lead to failure.

The transition table will now be explained:

Current State Input Symbol New State 0 1 1 1 1 2 2 1 -3 0 0 F 1 0 -3 2 0 -2 -3 0 -2 -2 0 -1 -1 0 A -3 1 F -2 1 F -1 1 F A 0 A A 1 A F 0 F F 1 F

Explanation of Transitions

In the DFA, the transition logic can be summarized as:

From any state 0 to 1, input 1 transitions the DFA to state 1. From state 1 to 2, input 1 transitions the DFA to state 2. From state 2 to -3, input 1 transitions the DFA to state -3, signaling the start of the trailing zeros. From state 0 to F, input 0 transitions the DFA to the failing state F. From state 1 to -3, input 0 transitions the DFA to state -3. From state 2 to -2, input 0 transitions the DFA to state -2, continuing the count of trailing zeros. From state -3 to -2, input 0 maintains state -2. From state -2 to -1, input 0 transitions the DFA to the accepting state -1. From state -1 to A, input 0 transitions the DFA to the accepting state A. Any input on states -3, -2, -1, A, or A will transition back to state 2 if a 0 is input, or to F if a 1 is input.

Conclusion

The constructed DFA follows a systematic approach to recognize strings that begin with three 1s and end with three 0s. The DFA comprises a total of eight states, including one accepting state (A) and one failing state (F). This structured approach ensures that the DFA can accurately determine if input strings match the specified pattern.

Keywords

DFA, Deterministic Finite Automaton, Regex