Nondeterminism
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Nondeterminism Two - Automata and Complexity Theory - Lecture Slides, Slides of Theory of Automata
Some concept of Automata and Complexity Theory are Administrivia, Closure Properties, Context-Free Grammars, Decision Properties, Deterministic Finite Automata, Intractable Problems, More Undecidable Problems. Main points of this lecture are: Nondeterminism Two, Alphabet, Strings, Sketch of Answer, Last Time, Guess, Symbols, Sequence, Accept, Nondeterministic Algorithm
Typology: Slides
2012/2013
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Nondeterminism
Example from last time
- Construct a DFA over alphabet {0, 1} that accepts those strings that end in 101
- Sketch of answer:
0
1
…
…
qε
q 0
q 1
q 00
q 10
q 01
q 11
q 000 q 001
q 101
q 111
0 1 0 1
0 1
1
1
1
1
0
Nondeterminism
- Nondeterminism is the ability to make guesses, which we can later verify
- Informal nondeterministic algorithm for language of strings that end in 101 : 1. Guess if you are approaching end of input 2. If guess is yes, look for 101 and accept if you see it 3. If guess is no, read one more symbol and go to step 1
Nondeterministic finite automaton
- This is a kind of automaton that allows you to make guesses
- Each state can have zero, one, or more transitions out labeled by the same symbol
(^1 0 )
0, 1 q 0 q 1 q 2 q 3
Semantics of guessing
- State q 1 has no transition labeled 1
- Upon reading 1 in q 1 , we die; upon reading 0 , we continue to q (^2)
(^1 0 )
0, 1 q 0 q 1 q 2 q 3
Semantics of guessing
- State q 3 has no transition going out
- Upon reading anything in q 3 , we die
(^1 0 )
0, 1 q 0 q 1 q 2 q 3
Formal definition
- A nondeterministic finite automaton (NFA) is a 5-tuple ( Q , Σ, δ, q 0 , F ) where - Q is a finite set of states - Σ is an alphabet - δ: Q × Σ → subsets of Q is a transition function - q 0 ∈ Q is the initial state - F ⊆ Q is a set of accepting states (or final states).
- Only difference from DFA is that output of δ is a set of states
Example
(^1 0 )
0, 1 q 0 q 1 q 2 q 3
alphabet Σ = {0, 1} start state Q = {q 0 , q 1 , q 2 , q3} initial state q accepting states F = {q 3 }
states
inputs (^0 ) q 0 q (^1) q (^2)
{q 0 , q 1 }
transition function δ:
q 3
{q0} {q 2 } ∅ ∅
∅ {q 3 } ∅
NFAs are as powerful as DFAs
- Obviously, an NFA can do everything a DFA can do
- But can it do more?
NFAs are as powerful as DFAs
- Obviously, an NFA can do everything a DFA can do
- But can it do more?
- Theorem A language L is accepted by some DFA if and only if it is accepted by some NFA.
NO!
Simulation example
(^1 )
0, 1
NFA:^ q 0 q 1 q 2
DFA: q 0 1 q0 or q (^1)
1
q0 or q (^2) 1
0 0 0
General method
NFA DFA
states q 0 , q^1 , …, q n ∅, {q^0 }, {q^1 }, {q^0 ,q^1 }, …, {q^0 ,…,q n } one for each subset of states in the NFA initial state q^0 {q^0 }
transitions δ δ’({q i 1 ,…,q ik },^ a ) = δ(q i 1 , a ) ∪…∪ δ(q ik , a )
accepting states
F ⊆ Q F ’ = { S :^ S^ contains some state in^ F }
Exercises
- Construct NFAs for the following languages over the alphabet {a, b, …, z}: - All strings that contain eat or sea or easy - All strings that contain both sea and tea - All strings that do not contain fool