1 Introduction to Computability Theory Discussion1: Non-Deterministic Finite Automatons Prof. Amos Israeli

Reminder An NFA is a finite automaton in which: 1.0, 1, many transitions with the same label emanating out of the same state are allowed. 2. transitions are allowed. 2

Exercise (omitted) Construct the following NFA-s: 1.An NFA accepting all words ending with abbab. 2.An NFA accepting all words whose third or second word from the end is a. 3.An NFA accepting all words of the form satisfying 3

Reminder A computation an NFA a tree like collection of paths induced by following all possible transitions and forking when either the current input symbol allows several transitions more than a single transition or when an transition is enabled. 4

Construct the computation of this NFA on some short words. Exercise 5 a b a,b

Reminder Proposition DFA-s and NFA-s are equivalent. Meaning: DFA-s and NFA-s recognize the same class of languages, called the Regular Languages. 6

Reminder Every DFA is (a special case of) an NFA, thus In order to prove equality we have to prove: 7

Reminder The proof works as follows: Given some arbitrary NFA N, we construct a DFA N, such that Yesterday we started to look at the proof. Today we will demonstrate the construction on one particular NFA: 8

Given NFA N Construct the a DFA M satisfying. Exercise 9 a b a,b

Reminder For each set S, the Power Set of S, P( S ), is the set containing all subsets of S. The set S satisfy: Why: Each subset corresponds to a binary vector of elements. 10

Reminder For each set S, the Power Set of S, P( S ), is the set containing all subsets of S. The set S satisfy: Why: Each subset corresponds to a binary vector of elements. 11

Reminder The state set of M is the power set of the state set of N. 12 a b a,b