State complexity
Open-knowledge reference entry
State complexity is an area of theoretical computer science
dealing with the size of abstract automata,
such as different kinds of finite automata.
The classical result in the area is that
simulating an
n
{\displaystyle n}
-state
nondeterministic finite automaton
by a deterministic finite automaton
requires exactly
2
n
{\displaystyle 2^{n}}
states in the worst case.
Transformation between variants of finite automata
Finite automata can be
deterministic and
nondeterministic,
one-way (DFA, NFA)
and two-way
(2DFA, 2NFA).
Other related classes are
unambiguous (UFA),
self-verifying (SVFA)
and alternating (AFA) finite automata.
These automata can also be two-way (2UFA, 2SVFA, 2AFA).
All these machines can accept exactly the regular languages.
However, the size of different types of automata
necessary to accept the same language
(measured in the number of their states)
may be different.
For any two types of finite automata,
the state complexity tradeoff between them
is an integer function
f
{\displaystyle f}
where
f
(
n
)
{\displaystyle f(n)}
is the least number of states in automata of the second type
sufficient to recognize every language
recognized by an
n
{\displaystyle n}
-state automaton of the first type.
The following results are known.
NFA to DFA:
2
n
{\displaystyle 2^{n}}
states.
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