Theory of Computation Lecture 4

Study Notes

Nondeterminism expands deterministic computation, allowing multiple transitions from a state based on the input.
Deterministic finite automata (DFA) have a single unique transition for every symbol in the alphabet from each state, while nondeterministic finite automata (NFA) may have none, one, or multiple transitions per symbol.
In NFAs, transitions can also include epsilon (ε) transitions, which allow the machine to move to a new state without consuming an input symbol.

Every state in a DFA has exactly one outgoing transition for each symbol in the alphabet.
NFAs can have states with zero, one, or many outgoing transitions for any symbol, including ε transitions.
Transition labels in DFAs strictly come from the alphabet, whereas NFAs may include ε as a transition label.

When an NFA processes an input string, it can split into multiple copies based on available transitions, effectively exploring all possible states in parallel.
If a duplicate of the machine reaches an invalid transition (no matching symbol), that copy is eliminated.
The input string is accepted if at least one copy of the NFA ends in an accept state after processing the entire input.
If ε transitions are available, the NFA can remain in the same state while branching to other states simultaneously.

Consider NFA N1; it splits into branches upon encountering multiple transition options after reading an input symbol.
NFA N2 recognizes strings comprising a '1' in the third position from the end, utilizing states q1, q2, q3, and q4 for computation.
NFA N2 remains in the initial state until it approaches the third-to-last input, then requires a '1' to proceed and validate its guess through branches.

Converting an NFA to an equivalent DFA can result in a significantly larger number of states due to the increase in combinatorial possibilities for transitions.
For example, a simple NFA example yielded a DFA with eight total states.

The NFA accepts specific strings such as ε, a, baba, and baa, while rejecting others like b, bb, and babba.
This process illustrates the practical applications and advantages of NFAs in recognizing complex patterns within input strings.

Nondeterminism expands deterministic computation, allowing multiple transitions from a state based on the input.
Deterministic finite automata (DFA) have a single unique transition for every symbol in the alphabet from each state, while nondeterministic finite automata (NFA) may have none, one, or multiple transitions per symbol.
In NFAs, transitions can also include epsilon (ε) transitions, which allow the machine to move to a new state without consuming an input symbol.

Every state in a DFA has exactly one outgoing transition for each symbol in the alphabet.
NFAs can have states with zero, one, or many outgoing transitions for any symbol, including ε transitions.
Transition labels in DFAs strictly come from the alphabet, whereas NFAs may include ε as a transition label.

When an NFA processes an input string, it can split into multiple copies based on available transitions, effectively exploring all possible states in parallel.
If a duplicate of the machine reaches an invalid transition (no matching symbol), that copy is eliminated.
The input string is accepted if at least one copy of the NFA ends in an accept state after processing the entire input.
If ε transitions are available, the NFA can remain in the same state while branching to other states simultaneously.

Consider NFA N1; it splits into branches upon encountering multiple transition options after reading an input symbol.
NFA N2 recognizes strings comprising a '1' in the third position from the end, utilizing states q1, q2, q3, and q4 for computation.
NFA N2 remains in the initial state until it approaches the third-to-last input, then requires a '1' to proceed and validate its guess through branches.

Converting an NFA to an equivalent DFA can result in a significantly larger number of states due to the increase in combinatorial possibilities for transitions.
For example, a simple NFA example yielded a DFA with eight total states.

The NFA accepts specific strings such as ε, a, baba, and baa, while rejecting others like b, bb, and babba.
This process illustrates the practical applications and advantages of NFAs in recognizing complex patterns within input strings.