see Dragon Book 3.9, pp.173, pp . 179)

Question

How to minimize a DFA ?

How to convert RE to DFA directly?

Answer 1

Minimizing a DFA involves removing any unnecessary states while preserving the language recognized by the automaton. Here are the general steps for minimizing a DFA:

1. Remove Unreachable States:

   • Identify and remove any states that are not reachable from the start state.

2. Equivalent State Identification:

   • Identify and merge equivalent states. Two states are equivalent if, for any input symbol, they lead to equivalent states. This process is often done iteratively until no more merges are possible.

3. Final State Grouping:

   • Group the states based on whether they are final or non-final. If two states from different groups are equivalent, merge the groups.

4. Iterative Minimization:

   • Iterate the process of merging equivalent states until no more merges can be made.

5. Equivalent State Table:

   • Maintain an equivalent state table to keep track of which states are equivalent at each iteration.

6. Renumbering States:

   • Renumber the states to ensure a consistent and minimal representation.

Converting a Regular Expression (RE) to a DFA directly is typically done through the Thompson Construction or the Subset Construction methods. Here's a brief overview of both:

Thompson Construction:

1. Start with the basic building blocks for regular expressions (symbols, concatenation, union, closure).
2. Construct a set of NFA (Non-deterministic Finite Automaton) fragments for each basic building block.
3. Combine these fragments using the rules of regular expressions.
4. Convert the resulting NFA to a DFA using the subset construction algorithm if needed.

Subset Construction:

1. Start with an initial state representing the empty set of NFA states (corresponding to the epsilon closure of the NFA's start state).
2. For each state in the DFA, determine the set of NFA states it represents by computing the epsilon closure.
3. For each input symbol, compute the set of NFA states that can be reached from the current state using that symbol.
4. Create a new DFA state for each set of NFA states encountered during the process.
5. Repeat steps 2-4 until no new states are added to the DFA.
6. Determine the final and non-final states of the DFA based on the NFA's final states.

Both methods involve a series of transformations and can be complex, but they provide systematic ways to obtain a DFA directly from a regular expression.