Nondeterministic Finite Automata

Nondeterministic Finite Automata Hopcroft, Motawi, Ullman, Chap 2

NFAs:Nondeterministic Finite Automata • Same as a DFA, except: • On input a, state q may have more than one transition out, implying the possibility of multiple choices when processing an input symbol • On input a, state q may have no transition out, implying the possibility of “being stuck” • A string is w acceptable as long as there exists an admissible state sequence for w

NFAs • A nondeterministic finite automaton M is a five-tuple M = (Q, , , q0, F), where: • Q is a finite set of states of M •  is the finite input alphabet of M • : Q    power set of Q, is the state transition function mapping a state-symbol pair to a subset of Q • q0 is the start state of M • F  Q is the set of accepting states or final states of M

Example NFA • NFA that recognizes the language of strings that end in 01 0,1 Exercise:draw the complete transition table for this NFA 1 q2 0 q1 q0 note: (q0,0) = {q0,q1} (q1,0) = {}

Converting NFAs to DFAs • Given a NFA, M = (Q, , , q0, F), build a DFA, M’ = (Q’, , ’, {q0}, F’) as follows. • Q’ contains all subsets S of states in Q. • The initial state of M’ is the set containing q0 • F’ is the set of all subsets of Q that contain at least one element in F (equivalently, the subset contains at least one final state)

Converting NFAs to DFAs • ’ is determined by putting together, for each state in the subset and each symbol, all states that may result from a transition:’(S, a) =  (q, a)qS • May remove “unreachable” states in Q’

Example conversion • NFA • DFA 0,1 1 q2 0 q1 q0 1 0 1 0 {q0,q1} {q0,q2} {q0 } 0 1

NFA with -transitions • NFA that allows the transition of an empty string from a state • Jumping to a state is possible even without input • Revision on NFA definition simply allows the  “symbol” for 

NFA with -transitions • A nondeterministic finite automaton with -transitions (or -NFA) is a five-tupleM = (Q, , , q0, F), where: • Q is a finite set of states of M •  is the finite input alphabet of M • : Q  ( + )  power set of Q, is the state transition function mapping a state-symbol pair to a subset of Q • q0 is the start state of M • F  Q is the set of accepting states or final states of M

Converting -NFAs to NFAs • Task: Given an-NFA M = (Q, , , q0, F), build aNFA M’ = (Q’, , ’, q’0, F’) • Need to eliminate -transitions • Need epsilon closure concept • Add transitions to enable transitions previously allowed by the -transitions • Note: the conversion process in the textbook instead builds a DFA from an -NFA

Equivalence of Finite Automata • Conversion processes betweenDFAs, NFAs, and -NFAs show that no additional expressive capacity (except convenience) is introduced by non-determinism or -transitions • All models represent regular languages • Note: possible exponential explosion of states when converting from DFA to NFA

Closure of Regular Languages under certain operations • Union L1  L2 • Complementation L1 • Intersection L1  L2 • Concatenation L1L2 • Goal: ensure a FA can be produced from the FAs of the “operand” languages

Finite Automata with Output • Moore Machines • Output symbol for each state encountered • Mealy Machines • Output symbol for each transition encountered • Exercise: formally define Moore and Mealy machines

Next: Regular Expressions • Defines languages in terms of symbols and operations • Example • (01)* +(10)* defines all even-length strings of alternating 0s and 1s • Regular expressions also model regular languages and we will demonstrate equivalence with finite automata

Nondeterministic Finite Automata