Two-Way Deterministic Finite Automata

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There is a variation of finite automata which is known as two-way deterministic finite automata or 2DFA. This is an extension of the deterministic finite automata (DFA). Unlike the DFA, which only allows the reading head to move in right direction only, a 2DFA allows movement in both directions (both left and right).

A 2DFA provides more flexibility in recognizing patterns in input strings. In this chapter, we will explain the concept of 2DFA and its components, functionality, and the acceptance and rejection conditions, with some examples.

Components of a 2DFA

A two-way finite automata can also be defined through a quintuple M = (Q,Σ,δ,Q0,F) where −

• Q is the set of states
• Σ is the input alphabet
• Q0 is the initial state
• F is the set of final states
• δ is the transition function

But in the transition function, it is different. For a DFA, the transition function δ maps Q × Σ to Q.

In the case of a 2DFA, the transition function δ maps Q × Σ to Q{L,R}, where L stands for a left move and R for a right move.

The Operations of 2DFA

Like DFA, the 2DFA also starts at the leftmost symbol of the input string in the initial state 0. Depending on the current state and the symbol read, it transitions to a new state and moves the reading head either left or right. We can understand it further through the configuration.

Configuration for 2DFA

Consider an input string w = a₁ a₂ … a_n If the automaton is in state Q and reads symbol a_i −

• The current configuration can be described as w = a₁ a₂ … a_(i-1) Qa_i a_(i+1) … a_n.
• If δ(Q,a_i ) = (P,R), where the next state will be P, and the head will move to the right.
• If δ(Q,a_i ) = (P,L), where the next state will be P, and the head will move to the left.

Acceptance and Rejection Conditions

After the configuration, we must understand the acceptance and rejection of the machine. Let us understand them one by one.

An input is accepted when, all symbols are processed, and the automaton moves off the right end of the tape while in a final state.

An input can be rejected in three ways −

• Moving off the right end of the tape in a non-final state
• Moving off the left end of the tape
• Entering a loop (Repeatedly revisiting the same configuration without making progress)

Instantaneous Description (ID)

To track the working of 2DFA, we must know the instantaneous description (ID) for 2DFA. These are nothing but the snapshot of its current state, the remaining input, and the position of the reading head.

It can be denoted as a string in Σ^* QΣ^*. Let us understand one example ID

For an input string w = a₁ a₂…a_n and current state Q reading a_i, the ID is,

$$\mathrm{a_{1} \:a_{2}\: \dotso\: a_{(i-1)} \: Qa_{i} \: a_{(i+1)}\:\dotso\: a_{n}}$$

If δ(Q,a_i ) = (P,R), the head moves right, and the new ID is a₁ a₂ … a_(i-1) a_i Pa_(i+1) … a_n.

Language Accepted by a 2DFA

After knowing the basics, let us now understand which language can be accepted by 2DFA.

The language accepted by a 2DFA is M = (Q, Σ, δ, Q₀, F) the set of strings w such that starting from the initial state Q₀, the w can be processed and the automaton reaches a final state in F.

This can be defined formally like −

L(M) = { w ∈ Σ^* | starting in Q₀, w leads to some Q_f ∈ F in finite steps}

Sample 2DFA Example

So far, we have seen some theoretical concepts. Let's understand a sample 2DFA with its definition and transition in action.

Consider the following 2DFA, M −

• States: Q = { Q₀, Q₁, Q₂, Q₃ }
• Input alphabet: Σ = { a,b }
• Initial state: Q₀
• Final state: F = { Q₂ }

Transition Table

Its transition table will be like the one shown here −

Conclusion

In this chapter, we explained the basics of 2DFA. The 2DFA offers enhanced capabilities over a traditional Right-only DFA. It allows bidirectional movement of the reading head, which helps for more complex string processing. We explained their properties, definition, formation, and examples with the help of instantaneous description and an example with full detail.