DFA and NFA for Specific Conditions (2a 2b)

Introduction

• Understanding how to construct DFA (Deterministic Finite Automaton) and NFA (Nondeterministic Finite Automaton).
• Focus on different conditions:
  1. At least 2a and 2b
  2. Exactly 2a and 2b
  3. At most 2a and 2b

Part 1: At least 2a and 2b

Language

• Any combination with at least aa and bb.
• Example: aa, bb, aabb, ...
• Infinite combinations

Number of States

• Total states required: 3 (for different combinations of a and b)

NFA Construction

• States: q0, q1, q2, q3, q4, q5, q6, q7, q8
• q8 is the final state
• Transitions:
  • From q0: Taking aa and then bb leads to q8
  • Can also start with bb and proceed with aa
  • Each state moves forward based on remaining a and b requirements

DFA Construction

• Similar states as NFA
• Each state needs every input considered to transition:
  • From q0 with a or b, etc.
• Transitions are copied from NFA to DFA for each state

Part 2: Exactly 2a and 2b

Language

• Sequences containing exactly two as and two bs

NFA Construction

• States: q0, q1, q2, q3, q4, q5, q6, q7, q8
• q8 is the final state
• Transitions must end at q8 after exactly two as and two bs
• Over limit inputs lead to a dead state
• Inclusion of dead state: q9

DFA Construction

• Add dead state q9 for excess inputs
• Transitions are copied from NFA
• States remain same

Part 3: At most 2a and 2b

Language

• Contains combinations of a and b limited to 2a and 2b
• Example: a, b, aa, bb, aab, bba... up to combinations of two

NFA Construction

• States: q0, q1, q2, q3, q4, q5, q6, q7, q8
• All states can be final states
• Transitions follow limited number of a and b

DFA Construction

• Not detailed, but adds states ensuring transitions within specified limits

Conclusion

• Provided methods to construct NFA and DFA for given conditions.
• Key points include transitions between states, final states, and handling of excess or lacking inputs.
• Questions and further clarifications suggested to be asked in Q&A or comments section.