DFA Transition Tables
A DFA Transition Table is a tabular representation of the transition function of a Deterministic Finite Automaton (DFA). It explicitly shows how the DFA transitions from one state to another based on the input symbols. Transition tables are a compact and systematic way to define the behavior of a DFA.
Structure of a Transition Table
The transition table is organized as follows:
- Rows: Represent the states of the DFA.
- Columns: Represent the input symbols from the alphabet (Σ).
- Cells: Indicate the next state the DFA transitions to when it is in a given state and reads a specific input symbol.
Example: DFA Transition Table for Strings Containing ‘ab’
Let’s construct a transition table for a DFA that recognizes strings over the alphabet Σ = {a, b} and accepts those that contain the substring ‘ab’.
- States (Q): {q₀, q₁, q₂}
- Alphabet (Σ): {a, b}
- Start State: q₀
- Accept State: q₂
The transition function (δ) is defined as:
δ(q₀, 'a') = q₁
δ(q₀, 'b') = q₀
δ(q₁, 'a') = q₁
δ(q₁, 'b') = q₂
δ(q₂, 'a') = q₂
δ(q₂, 'b') = q₂The corresponding transition table is:
| State | Input: a | Input: b |
|---|---|---|
| q₀ | q₁ | q₀ |
| q₁ | q₁ | q₂ |
| q₂ | q₂ | q₂ |
Explanation: Each cell in the table shows the next state of the DFA for a given state and input symbol. For example, if the DFA is in state q₀ and reads the input ‘a’, it transitions to state q₁.
Advantages of Transition Tables
Transition tables are particularly useful because:
- They provide a clear and systematic way to describe the behavior of a DFA.
- They are easy to implement programmatically in algorithms and simulations.
- They help verify the correctness of the DFA design by explicitly showing all transitions.