20 research outputs found
Constructing Deterministic Parity Automata from Positive and Negative Examples
We present a polynomial time algorithm that constructs a deterministic parity
automaton (DPA) from a given set of positive and negative ultimately periodic
example words. We show that this algorithm is complete for the class of
-regular languages, that is, it can learn a DPA for each regular
-language. For use in the algorithm, we give a definition of a DPA,
that we call the precise DPA of a language, and show that it can be constructed
from the syntactic family of right congruences for that language (introduced by
Maler and Staiger in 1997). Depending on the structure of the language, the
precise DPA can be of exponential size compared to a minimal DPA, but it can
also be a minimal DPA. The upper bound that we obtain on the number of examples
required for our algorithm to find a DPA for is therefore exponential in
the size of a minimal DPA, in general. However we identify two parameters of
regular -languages such that fixing these parameters makes the bound
polynomial.Comment: Changes from v1: - integrate appendix into paper - extend
introduction to cover related work in more detail - add a second (more
involved) example - minor change
A novel family of finite automata for recognizing and learning -regular languages
Families of DFAs (FDFAs) have recently been introduced as a new
representation of -regular languages. They target ultimately periodic
words, with acceptors revolving around accepting some representation . Three canonical FDFAs have been suggested, called periodic,
syntactic, and recurrent. We propose a fourth one, limit FDFAs, which can be
exponentially coarser than periodic FDFAs and are more succinct than syntactic
FDFAs, while they are incomparable (and dual to) recurrent FDFAs. We show that
limit FDFAs can be easily used to check not only whether {\omega}-languages are
regular, but also whether they are accepted by deterministic B\"uchi automata.
We also show that canonical forms can be left behind in applications: the limit
and recurrent FDFAs can complement each other nicely, and it may be a good way
forward to use a combination of both. Using this observation as a starting
point, we explore making more efficient use of Myhill-Nerode's right
congruences in aggressively increasing the number of don't-care cases in order
to obtain smaller progress automata. In pursuit of this goal, we gain
succinctness, but pay a high price by losing constructiveness