50 research outputs found
Syntactic Complexity of Prefix-, Suffix-, Bifix-, and Factor-Free Regular Languages
The syntactic complexity of a regular language is the cardinality of its
syntactic semigroup. The syntactic complexity of a subclass of the class of
regular languages is the maximal syntactic complexity of languages in that
class, taken as a function of the state complexity of these languages. We
study the syntactic complexity of prefix-, suffix-, bifix-, and factor-free
regular languages. We prove that is a tight upper bound for
prefix-free regular languages. We present properties of the syntactic
semigroups of suffix-, bifix-, and factor-free regular languages, conjecture
tight upper bounds on their size to be , , and ,
respectively, and exhibit languages with these syntactic complexities.Comment: 28 pages, 6 figures, 3 tables. An earlier version of this paper was
presented in: M. Holzer, M. Kutrib, G. Pighizzini, eds., 13th Int. Workshop
on Descriptional Complexity of Formal Systems, DCFS 2011, Vol. 6808 of LNCS,
Springer, 2011, pp. 93-106. The current version contains improved bounds for
suffix-free languages, new results about factor-free languages, and new
results about reversa
Syntactic Complexities of Nine Subclasses of Regular Languages
The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of the class of regular languages is the maximal syntactic complexity of languages in that class, taken as a function of the state complexity n of these languages.
We study the syntactic complexity of suffix-, bifix-, and factor-free regular languages, star-free languages including three subclasses, and R- and J-trivial regular languages.
We found upper bounds on the syntactic complexities of these classes of languages. For R- and J-trivial regular languages, the upper bounds are n! and ⌊e(n-1)!⌋, respectively, and they are tight for n >= 1. Let C^n_k be the binomial coefficient ``n choose k''. For monotonic languages, the tight upper bound is C^{2n-1}_n. We also found tight upper bounds for partially monotonic and nearly monotonic languages. For the other classes of languages, we found tight upper bounds for languages with small state complexities, and we exhibited languages with maximal known syntactic complexities. We conjecture these lower bounds to be tight upper bounds for these languages.
We also observed that, for some subclasses C of regular languages, the upper bound on state complexity of the reversal operation on languages in C can be met by languages in C with maximal syntactic complexity. For R- and J-trivial regular languages, we also determined tight upper bounds on the state complexity of the reversal operation
On the group of a rational maximal bifix code
We give necessary and sufficient conditions for the group of a rational
maximal bifix code to be isomorphic with the -group of , when
is recurrent and is rational. The case where is uniformly
recurrent, which is known to imply the finiteness of , receives
special attention.
The proofs are done by exploring the connections with the structure of the
free profinite monoid over the alphabet of
Syntactic Complexity of Finite/Cofinite, Definite, and Reverse Definite Languages
We study the syntactic complexity of finite/cofinite, definite and reverse
definite languages. The syntactic complexity of a class of languages is defined
as the maximal size of syntactic semigroups of languages from the class, taken
as a function of the state complexity n of the languages. We prove that (n-1)!
is a tight upper bound for finite/cofinite languages and that it can be reached
only if the alphabet size is greater than or equal to (n-1)!-(n-2)!. We prove
that the bound is also (n-1)! for reverse definite languages, but the minimal
alphabet size is (n-1)!-2(n-2)!. We show that \lfloor e\cdot (n-1)!\rfloor is a
lower bound on the syntactic complexity of definite languages, and conjecture
that this is also an upper bound, and that the alphabet size required to meet
this bound is \floor{e \cdot (n-1)!} - \floor{e \cdot (n-2)!}. We prove the
conjecture for n\le 4.Comment: 10 pages. An error concerning the size of the alphabet has been
corrected in Theorem
Syntactic Complexity of R- and J-Trivial Regular Languages
The syntactic complexity of a regular language is the cardinality of its
syntactic semigroup. The syntactic complexity of a subclass of the class of
regular languages is the maximal syntactic complexity of languages in that
class, taken as a function of the state complexity n of these languages. We
study the syntactic complexity of R- and J-trivial regular languages, and prove
that n! and floor of [e(n-1)!] are tight upper bounds for these languages,
respectively. We also prove that 2^{n-1} is the tight upper bound on the state
complexity of reversal of J-trivial regular languages.Comment: 17 pages, 5 figures, 1 tabl
A profinite approach to complete bifix decodings of recurrent languages
We approach the study of complete bifix decodings of (uniformly) recurrent
languages with the help of the free profinite monoid. We show that the complete
bifix decoding of a uniformly recurrent language by an -charged rational
complete bifix code is uniformly recurrent. An analogous result is obtained for
recurrent languages.Comment: Original Manuscript of article to be published by De Gruyter in Forum
Mathematicum. The last section of the version in Forum Mathematicum is very
different, as there it is not proved that the Sch\"utzenberger group is an
invariant of eventual conjugacy (the argument in the Original Manuscript had
a flaw), but only that its maximal pronilpotent quotient is invariant by
eventual conjugac
Languages convex with respect to binary relations, and their closure properties
A language is prefix-convex if it satisfies the condition that, if a word w and its prefix u are in the language, then so is every prefix of w that has u as a prefix. Prefix-convex languages include prefix-closed languages at one end of the spectrum, and prefix-free languages, which include prefix codes, at the other. In a similar way, we define suffix-, bifix-, factor-, and subword-convex languages and their closed and free counterparts. This provides a common framework for diverse languages such as codes, factorial languages and ideals. We examine the relationships among these languages. We generalize these notions to arbitrary binary relations on the set of all words over a given alphabet, and study the closure properties of such languages