Unambiguous finite automaton - Alchetron, the free social encyclopedia

In automata theory, an unambiguous finite automaton (UFA) is a special kind of a nondeterministic finite automaton (NFA). Each deterministic finite automaton (DFA) is an UFA, but not vice versa. DFA, UFA, and NFA recognize exactly the same class of formal languages. On the one hand, an NFA can be exponentially smaller than an equivalent DFA. On the other hand, some problems are easily solved on DFAs and not on UFAs. For example, given an automaton A, an automaton A' which accepts the complement of A can be computed in linear time when A is a DFA, it is not known whether it can be done in polynomial time for UFA. Hence UFAs are a mix of the worlds of DFA and of NFA; in some cases, they lead to smaller automata than DFA and quicker algorithms than NFA.

Formal definition

An NFA is represented formally by a 5-tuple, A=(Q, Σ, Δ, q₀, F). An UFA is an NFA such that, for each word w = a₁a₂ … a_n, there exists at most one sequence of states r₀,r₁, …, r_n, in Q with the following conditions:

r₀ = q₀
r_i+1 ∈ Δ(r_i, a_i+1), for i = 0, …, n−1
r_n ∈ F.

In words, those conditions state that, if w is accepted by A, there is exactly one accepting path, that is, one path from an initial state to a final state, labelled by w.

Example

Let L be the set of words over the alphabet {a,b} whose nth last letter is an a. The figures show a DFA and a UFA accepting this language for n=2.

The minimal DFA accepting L has 2ⁿ states, one for each subset of {1...n}. There is an UFA of n+1 states which accepts L: it guesses the nth last letter, and then verifies that only n-1 letters remain. It is indeed unambiguous as there exists only one nth last letter.

Three PSPACE-hard problems for general NFA belong to PTIME for DFA and are now considered.

Inclusion

It is decidable in polynomial-time whether an automaton's language is a subset of an automaton of another language.

The problem of universality and of equivalence, also belong to PTIME, by reduction to the inclusion problem.

Checking whether an automaton in unambiguous

For a nondeterministic finite automaton A with n states and an m letter alphabet, it is decidable in time O(n²m) whether A is unambiguous.

Some properties

The cartesian product of two UFAs is a UFA.

The notion of unambiguity extends to finite state transducers and weighted automata. If a finite state transducer T is unambiguous, then each input word is associated by T to at most one output word. If a weighted automaton A is unambiguous, then the set of weight does not need to be a semiring, instead it suffices to consider a monoid. Indeed, there is at most one accepting path.

State complexity

Mathematical proofs that every UFA for a language needs a certain number of states were pioneered by Schmidt. Leung proved that a DFA equivalent to an n -state UFA requires 2 n states in the worst case. and a UFA equivalent to an n -state NFA requires 2 n − 1 states.

Jirásek, Jirásková and Šebej researched the number of states necessary to represent basic operations on languages. They proved in particular that for every n -state UFA the complement of the language it accepts is accepted by a UFA with O ( 2 0.79 n ) states.

For a one-letter alphabet Okhotin proved that a DFA equivalent to an n -state UFA requires e Θ ( n ( ln ⁡ n ) 2 3 ) states in the worst case.

References

Unambiguous finite automaton Wikipedia

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