6,631 research outputs found
Caterpillar dualities and regular languages
We characterize obstruction sets in caterpillar dualities in terms of regular
languages, and give a construction of the dual of a regular family of
caterpillars. We show that these duals correspond to the constraint
satisfaction problems definable by a monadic linear Datalog program with at
most one EDB per rule
Quotient Complexity of Regular Languages
The past research on the state complexity of operations on regular languages
is examined, and a new approach based on an old method (derivatives of regular
expressions) is presented. Since state complexity is a property of a language,
it is appropriate to define it in formal-language terms as the number of
distinct quotients of the language, and to call it "quotient complexity". The
problem of finding the quotient complexity of a language f(K,L) is considered,
where K and L are regular languages and f is a regular operation, for example,
union or concatenation. Since quotients can be represented by derivatives, one
can find a formula for the typical quotient of f(K,L) in terms of the quotients
of K and L. To obtain an upper bound on the number of quotients of f(K,L) all
one has to do is count how many such quotients are possible, and this makes
automaton constructions unnecessary. The advantages of this point of view are
illustrated by many examples. Moreover, new general observations are presented
to help in the estimation of the upper bounds on quotient complexity of regular
operations
Complexity in Prefix-Free Regular Languages
We examine deterministic and nondeterministic state complexities of regular
operations on prefix-free languages. We strengthen several results by providing
witness languages over smaller alphabets, usually as small as possible. We next
provide the tight bounds on state complexity of symmetric difference, and
deterministic and nondeterministic state complexity of difference and cyclic
shift of prefix-free languages.Comment: In Proceedings DCFS 2010, arXiv:1008.127
Testing the Equivalence of Regular Languages
The minimal deterministic finite automaton is generally used to determine
regular languages equality. Antimirov and Mosses proposed a rewrite system for
deciding regular expressions equivalence of which Almeida et al. presented an
improved variant. Hopcroft and Karp proposed an almost linear algorithm for
testing the equivalence of two deterministic finite automata that avoids
minimisation. In this paper we improve the best-case running time, present an
extension of this algorithm to non-deterministic finite automata, and establish
a relationship between this algorithm and the one proposed in Almeida et al. We
also present some experimental comparative results. All these algorithms are
closely related with the recent coalgebraic approach to automata proposed by
Rutten
Most Complex Non-Returning Regular Languages
A regular language is non-returning if in the minimal deterministic
finite automaton accepting it there are no transitions into the initial state.
Eom, Han and Jir\'askov\'a derived upper bounds on the state complexity of
boolean operations and Kleene star, and proved that these bounds are tight
using two different binary witnesses. They derived upper bounds for
concatenation and reversal using three different ternary witnesses. These five
witnesses use a total of six different transformations. We show that for each
there exists a ternary witness of state complexity that meets the
bound for reversal and that at least three letters are needed to meet this
bound. Moreover, the restrictions of this witness to binary alphabets meet the
bounds for product, star, and boolean operations. We also derive tight upper
bounds on the state complexity of binary operations that take arguments with
different alphabets. We prove that the maximal syntactic semigroup of a
non-returning language has elements and requires at least
generators. We find the maximal state complexities of atoms of
non-returning languages. Finally, we show that there exists a most complex
non-returning language that meets the bounds for all these complexity measures.Comment: 22 pages, 6 figure
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