Designing NFAs

• embrace the nondeterminism
• good model: guess and check
  • is there some information that you’d really like to have? have the machine nondeterministically guess that information
  • then, have the machine deterministically check that the choice was correct
• the guess phase corresponds to trying lots of different options
• check phase corresponds to filtering out bad guesses or wrong options

Example

$L=\{w \in \{0,1\}^* ~|~ \text{$w$ ends in 010 or 101} \}$

• it would be a lot easier if you knew where the string ends

• let start $\to_\Sigma$ start
• start $\to_0$ node $\to_1$ node $\to_0$ accepting state
• start $\to_1$ node $\to_0$ node $\to_1$ accepting state
  • the genius of this NFA is that it allows the complex part (remembering what the last three characters are) to be part of the nondeterminism

Another Example

$L=\{w \in \{a,b,c\}^*~|~\text{at least one of a, b, and c is not in $w$} \}$

• it would be easier if you knew which character was missing
• lets say you had a node $\to_{a,b}$ same node, node is accepting state
  • this is a NFA that allows for only a and b
• now, we can glue 3 copies of that together
  • start $\to_\varepsilon$ node 1, node 2, node 3 (guess which character is missing)
  • node 1 $\to_{a,b}$ node 1 (accepting)
  • node 2 $\to_{a,c}$ node 2 (accepting)
    • node 3 $\to_{b, c}$ node 3 (accepting)

Implementing DFAs and NFAs

• Tabular DFAs

  • can make a 2D array representing every transition

  

  • can make every row as a state, every column as a particular label for a transition
  • each element in the 2D table is the result of the transition