CS 461 – Aug. 31 Section 1.2 – Nondeterministic FAs How to trace input √ NFA design makes “union” operation easier Equivalence of NFAs and DFAs

NFA’s using “or” Can you draw NFA for: { begin with 0 or end with 1 } ? New start Old start 1 Old start 2 ε ε

Amazing fact NFA = DFA In other words, the two kinds of machines have the same power. Proof idea: we can always convert a DFA into an NFA, or vice versa. Which do you think is easier to do?

Formal NFA def’n The essential difference with DFA is in the transition function: DFA δ: Q x Σ  Q NFA δ: Q x Σ ε  P(Q) Thus, converting DFA  NFA is easy. We already satisfy the definition!

NFA  DFA construction 1.When creating DFA, states will be all possible subsets of states from NFA. –This takes care of “all possible destinations.” –In practice we won’t need whole subset: only create states as you need them. –“empty set” can be our dead state. 2.DFA start state = NFA’s start state or anywhere you can begin for free. Happy state will be any subset containing NFA’s happy state. 3.Transitions: Please write as a table. Drawing would be too cluttered. When finished, can eliminate useless states.

Example #1 NFA transition table given to the right. DFA start state is {1, 3}, or more simply 13. DFA accept state would be anything containing 1. Could be 1, 12, 13, 123, but we may not need all these states. inputs stateabε  ,

continued The resulting DFA could require 2 n states, but we should only create states as we need them. inputs stateabε  , Let’s begin: If we’re in state 1 or 3, where do we go if we read an ‘a’ or a ‘b’? δ(13, a) = 1, but we can get to 3 for free. δ(13, b) = 2. We need to create a new state “2”. Continue the construction by considering transitions from state 2.

answer NFA DFA inputs stateabε  , inputs stateab    Notice that the DFA is in fact deterministic: it has exactly one destination per transition. Also there is no column for ε.

Example #2 NFA transition table given to the right. DFA start state is A. DFA accept state would be anything containing D. inputs State01ε AAAA,C- BD-C C-B- D BD-

continued Let’s begin. δ(A, 0) = A δ(A, 1) = AC We need new state AC. δ(AC, 0) = A δ(AC, 1) = ABC Continue from ABC… inputs State01ε AAAA,C- BD-C C-B- D BD-

answer NFA DFA inputs State01ε AAAA,C- BD-C C-B- D BD- inputs State01 AAAAC AABC ADABC AD ABCACD ABCABCD

final thoughts NFAs and DFAs have same computational power. NFAs often have fewer states than corresponding DFA. Typically, we want to design a DFA, but NFAs are good for combining 2+ DFAs. After doing NFA  DFA construction, we may see that some states can be combined. –Later in chapter, we’ll see how to simplify FAs.