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 $n$ of these languages. We study the syntactic complexity of prefix-, suffix-, bifix-, and factor-free regular languages. We prove that $n^{n-2}$ 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 $(n-1)^{n-2}+(n-2)$, $(n-1)^{n-3} + (n-2)^{n-3} + (n-3)2^{n-3}$, and $(n-1)^{n-3} + (n-3)2^{n-3} + 1$, 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 $Z$ to be isomorphic with the $F$-group of $Z\cap F$, when $F$ is recurrent and $Z\cap F$ is rational. The case where $F$ is uniformly recurrent, which is known to imply the finiteness of $Z\cap F$, receives special attention. The proofs are done by exploring the connections with the structure of the free profinite monoid over the alphabet of $F$

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

Syntactic Complexity of Circular Semi-Flower Automata

We investigate the syntactic complexity of certain types of finitely generated submonoids of a free monoid. In fact, we consider those submonoids which are accepted by circular semi-flower automata (CSFA). Here, we show that the syntactic complexity of CSFA with at most one `branch point going in' (bpi) is linear. Further, we prove that the syntactic complexity of $n$-state CSFA with two bpis over a binary alphabet is $2n(n+1)$

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 $F$ by an $F$-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