DFA and NFA Construction Based on Conditions

Jun 30, 2024

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.