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Nondeterministic Finite
Automata
Nondeterminism
Subset Construction
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Nondeterministic Finite Automata: Understanding NFA and ε-NFA, Slides of Theory of Automata
An in-depth exploration of nondeterministic finite automata (nfa), focusing on their properties, examples, and the relationship between nfa and deterministic finite automata (dfa). Concepts such as nondeterminism, subset construction, and ε-transitions, using a chessboard example to illustrate the concepts.
Typology: Slides
2012/2013
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Nondeterministic Finite
AutomataNondeterminismSubset Construction
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Nondeterminism
A
nondeterministic finite automatonhas the ability to be in several states atonce.
Transitions from a state on an inputsymbol can be to any set of states.
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Example: Moves on a
Chessboard
States = squares. Inputs = r (move to an adjacent redsquare) and b (move to an adjacentblack square). Start state, final state are in oppositecorners.
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Example: Chessboard – (2) 1 2 5 7
r^1
b^
b 2 4
r^
b 1 2,
- Accept, since final state reached
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Transition Function of an NFA δ(q, a) is a set of states. Extend to strings as follows: Basis:
δ(q,
ε) = {q}
Induction:
δ(q, wa) = the union over
all states p in
δ(q, w) of
δ(p, a)
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Language of an NFA
A string w is accepted by an NFA if^ δ(q
, w) contains at least one final 0 state.
The language of the NFA is the set ofstrings it accepts.
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Equivalence of DFA’s, NFA’s A DFA can be turned into an NFA thataccepts the same language. If^ δD
(q, a) = p, let the NFA have
δ(q, a) = {p}.N^ Then the NFA is always in a setcontaining exactly one state – the statethe DFA is in after reading the sameinput.
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Equivalence – (2)
Surprisingly, for any NFA there is a DFAthat accepts the same language. Proof is the
subset construction.
The number of states of the DFA canbe exponential in the number of statesof the NFA. Thus, NFA’s accept exactly the regularlanguages.
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Critical Point
The DFA states have
names
that are
sets of NFA states. But as a DFA state, an expression like{p,q} must be read as a single symbol,not as a set. Analogy: a class of objects whosevalues are sets of objects of anotherclass.
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Subset Construction – (2) The transition function
δis defined by:D^
δ({qD
,…,q 1
}, a) is the union over all i =k
1,…,k
of^ δ
(q^ , a).N i
Example: We’ll construct the DFAequivalent of our “chessboard” NFA.
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Example: Subset Constructionr^
b 1 2,
r^
b
{1}{2,4}^ {5}{2,4,6,8}
{2,4}^
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Example: Subset Constructionr^
b 1 2,
r^
b
{1} {2,4,6,8}* {1,3,7,9}
{2,4}^ {5}
{2,4}^
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Example: Subset Constructionr^
b 1 2,
r^
b
{1} ^ {1,3,7,9} {1,3,5,7,9}
{2,4}^ {5}
{2,4}^
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Example: Subset Constructionr^
b 1 2,
r^
b
{1} ^ {1,3,7,9} {1,3,5,7,9}