5,730 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
Complexity of Left-Ideal, Suffix-Closed and Suffix-Free Regular Languages
A language over an alphabet is suffix-convex if, for any words
, whenever and are in , then so is .
Suffix-convex languages include three special cases: left-ideal, suffix-closed,
and suffix-free languages. We examine complexity properties of these three
special classes of suffix-convex regular languages. In particular, we study the
quotient/state complexity of boolean operations, product (concatenation), star,
and reversal on these languages, as well as the size of their syntactic
semigroups, and the quotient complexity of their atoms.Comment: 20 pages, 11 figures, 1 table. arXiv admin note: text overlap with
arXiv:1605.0669
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
Logic Meets Algebra: the Case of Regular Languages
The study of finite automata and regular languages is a privileged meeting
point of algebra and logic. Since the work of Buchi, regular languages have
been classified according to their descriptive complexity, i.e. the type of
logical formalism required to define them. The algebraic point of view on
automata is an essential complement of this classification: by providing
alternative, algebraic characterizations for the classes, it often yields the
only opportunity for the design of algorithms that decide expressibility in
some logical fragment.
We survey the existing results relating the expressibility of regular
languages in logical fragments of MSO[S] with algebraic properties of their
minimal automata. In particular, we show that many of the best known results in
this area share the same underlying mechanics and rely on a very strong
relation between logical substitutions and block-products of pseudovarieties of
monoid. We also explain the impact of these connections on circuit complexity
theory.Comment: 37 page
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