Prerequisite – Designing finite automata

**To perform the union operation on two deterministic finite automata (DFAs), the following steps can be taken:**

- Create a new DFA with a new set of states, consisting of all the states from both original DFAs.
- Define the initial state of the new DFA to be the tuple (q1, q2), where q1 and q2 are the initial states of the original DFAs.
- For each state in the new DFA, define the transition function by taking the union of the transition functions of the original DFAs. For example, if the transition function for the first DFA is δ1(q1, a) = q2, and the transition function for the second DFA is δ2(q3, a) = q4, then the transition function for the new DFA is δ( (q1,q3), a ) = (q2,q4).
- Define the set of final states of the new DFA to be the union of the sets of final states of the original DFAs.
- The resulting DFA will recognize the language that is the union of the languages recognized by the original DFAs.

Let’s understand the Union process in Deterministic Finite Automata (DFA) with the help of the below example. Designing a DFA for the set of string over {a, b} such that the string of the language start and end with different symbols. These two desired languages will be formed:

L_{1}= {ab, aab, aabab, .......} L_{2}= {ba, bba, bbaba, .......}

L_{1}= {starts with a and ends with b } and L_{2}= {starts with b and ends with a}. Then L= L_{1} ∪ L_{2} or L=L_{1} + L_{2}

**State Transition Diagram for the language L _{1}:**

This DFA accepts all the string starting with a and ending with b. Here, State A is initial state and state C is final state.

**State Transition Diagram for the language L _{2}:**

This DFA accepts all the strings starting with b and ending with a. Here, State A is the initial state and state C is the final state. Now, Taking the union of L_{1} and L_{2} language gives the final result of the language which starts and ends with different elements.

**State Transition Diagram of L _{1} ∪ L_{2}:**

Thus as we see that L_{1} and L_{2} have been combined through the union process and this final DFA accept all the language containing strings starting and ending with different symbols.

**Note:** From the above example, we can also infer that regular languages are closed under union(i.e Union of two regular languages is also regular).

### Advantages:

**Language Combination:** The union operation allows you to combine the languages recognized by two separate DFAs into a single DFA. This is useful when you want to recognize strings that belong to either of the languages. It enables you to handle more complex language requirements by merging existing DFAs.

**Simple Construction:** The union process for DFAs is relatively straightforward. It involves creating a new DFA that combines the states and transitions of the original DFAs. The resulting DFA recognizes the union of the languages without modifying the original DFAs, making it a simple and efficient way to combine languages.

**Deterministic Nature:** DFAs, including the resulting union DFA, are deterministic in nature. This means that for every state and input symbol, there is a unique next state. The determinism allows for easy analysis and behavior prediction of the automaton.

**Efficient Execution: **DFAs, including the union DFA, can be implemented and executed efficiently. The transition function of a DFA can be represented using a table or a set of state-transition diagrams, allowing for fast and constant-time transitions. The efficiency of execution is advantageous when processing input strings to determine membership in the combined language.

### Disadvantages:

**Increased Complexity: **The union process can introduce increased complexity, particularly when combining DFAs with a large number of states and transitions. The resulting union DFA may have a larger number of states and transitions, which can make it more challenging to manage, analyze, and comprehend.

**State Explosion: **Combining two DFAs with a large number of states and transitions may result in a “state explosion” problem. The number of states in the resulting union DFA can grow exponentially, making it more difficult to construct and execute. Managing and representing large union DFAs can become computationally expensive and impractical.

**Lack of Optimization: **The union process does not inherently optimize the resulting DFA. The resulting union DFA may have redundant or unnecessary states and transitions, which can impact the efficiency of execution. Additional optimization techniques may be required to minimize the size and complexity of the union DFA.

**Limited to Regular Languages:** The union process in DFAs is limited to regular languages. While the resulting union DFA can handle the union of regular languages, it cannot handle more complex languages that require context or memory beyond regular languages. For such languages, more powerful models like pushdown automata or Turing machines are needed.