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 $2n(n+1)$

State Complexity of Reversals of Deterministic Finite Automata with Output

We investigate the worst-case state complexity of reversals of deterministic finite automata with output (DFAOs). In these automata, each state is assigned some output value, rather than simply being labelled final or non-final. This directly generalizes the well-studied problem of determining the worst-case state complexity of reversals of ordinary deterministic finite automata. If a DFAO has $n$ states and $k$ possible output values, there is a known upper bound of $k^n$ for the state complexity of reversal. We show this bound can be reached with a ternary input alphabet. We conjecture it cannot be reached with a binary input alphabet except when $k = 2$, and give a lower bound for the case $3 \le k < n$. We prove that the state complexity of reversal depends solely on the transition monoid of the DFAO and the mapping that assigns output values to states.Comment: 18 pages, 3 tables. Added missing affiliation/funding informatio

Complexity of Proper Prefix-Convex Regular Languages

The final publication is available at Springer via http:/dx.doi.org/10.1007/978-3-319-60134-2_5A language L over an alphabet Σ is prefix-convex if, for any words x,y,z∈Σ∗, whenever x and xyz are in L, then so is xy. Prefix-convex languages include right-ideal, prefix-closed, and prefix-free languages, which were studied elsewhere. Here we concentrate on prefix-convex languages that do not belong to any one of these classes; we call such languages proper. We exhibit most complex proper prefix-convex languages, which meet the bounds for the size of the syntactic semigroup, reversal, complexity of atoms, star, product, and Boolean operations.Natural Sciences and Engineering Research Council of Canada [grant no. OGP0000871

On Nonpermutational Transformation Semigroups with an Application to Syntactic Complexity

We give an upper bound of n((n-1)!-(n-3)!) for the possible largest size of a subsemigroup of the full transformational semigroup over n elements consisting only of nonpermutational transformations. As an application we gain the same upper bound for the syntactic complexity of (generalized) definite languages as well

Syntactic Complexities of Six Classes of Star-Free Languages

© Otto-von-Guericke-Universit¨at Magdeburg. This is an accepted manuscript. Details about the final published version are available here: http://theo.cs.ovgu.de/jalc/1996-2015/The syntactic complexity of a regular language is the cardinality of its syntactic semi-group. The syntactic complexity of a subclass of regular languages is the maximal syntactic complexity of languages in that subclass, taken as a function of the state complexity n of these languages. We study the syntactic complexity of six subclasses of star-free languages. We find a tight upper bound of (n−1)! for finite/cofinite and re-verse definite languages, and a lower bound of ⌊e·(n−1)!⌋ for definite languages, where e is the base of the natural logarithms. We also find tight upper bounds for languages accepted by monotonic, partially monotonic and “nearly monotonic” automata. All these bounds are significantly lower than the bound nn for arbitrary regular languages. Also, witness languages reaching these bounds require alphabets that grow with n. The syntactic complexity of arbitrary star-free languages remains open.Natural Sciences and Engineering Research Council of Canada [OGP0000871

Syntactic Complexities of Six Classes of Star-Free Languages

© Otto-von-Guericke-Universit¨at Magdeburg. This is an accepted manuscript. Details about the final published version are available here: http://theo.cs.ovgu.de/jalc/1996-2015/The syntactic complexity of a regular language is the cardinality of its syntactic semi-group. The syntactic complexity of a subclass of regular languages is the maximal syntactic complexity of languages in that subclass, taken as a function of the state complexity n of these languages. We study the syntactic complexity of six subclasses of star-free languages. We find a tight upper bound of (n−1)! for finite/cofinite and re-verse definite languages, and a lower bound of ⌊e·(n−1)!⌋ for definite languages, where e is the base of the natural logarithms. We also find tight upper bounds for languages accepted by monotonic, partially monotonic and “nearly monotonic” automata. All these bounds are significantly lower than the bound nn for arbitrary regular languages. Also, witness languages reaching these bounds require alphabets that grow with n. The syntactic complexity of arbitrary star-free languages remains open.Natural Sciences and Engineering Research Council of Canada [OGP0000871