86 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
Quotient Complexity of Bifix-, Factor-, and Subword-Free Regular Language
A language is prefix-free if whenever words and are in and is a prefix of , then . Suffix-, factor-, and subword-free languages are defined similarly, where by ``subword" we mean ``subsequence", and a language is bifix-free if it is both prefix- and suffix-free. These languages have important applications in coding theory. The quotient complexity of an operation on regular languages is defined as the number of left quotients of the result of the operation as a function of the numbers of left quotients of the operands. The quotient complexity of a regular language is the same as its state complexity, which is the number of states in the complete minimal deterministic finite automaton accepting the language. The state/quotient complexity of operations in the classes of prefix- and suffix-free languages has been studied before. Here, we study the complexity of operations in the classes of bifix-, factor-, and subword-free languages. We find tight upper bounds on the quotient complexity of intersection, union, difference, symmetric difference, concatenation, star, and reversal in these three classes of languages.Natural Sciences and Engineering Research Council of Canada [OGP0000871]Slovak Research and Development Agency [APVV-0035-10]Algorithms, Automata, and Discrete Data Structures VEGA, [2/0183/11
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
Specular sets
We introduce the notion of specular sets which are subsets of groups called
here specular and which form a natural generalization of free groups. These
sets are an abstract generalization of the natural codings of linear
involutions. We prove several results concerning the subgroups generated by
return words and by maximal bifix codes in these sets.Comment: arXiv admin note: substantial text overlap with arXiv:1405.352
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
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