We consider first order logic over words and show FO+MOD[<] is contained in MAJ[<] with three variables.
It is known that for the classes FO[<], FO+MOD[<], FO+GROUP[<] three variables suffice. In the case of MOD[<] even two variables are sufficient.

As a consequence we know that if TC^ 0 neq ... more >>>

In the setting known as DLOGTIME-uniformity,
the fundamental complexity classes
$AC^0\subset ACC^0\subseteq TC^0\subseteq NC^1$ have several
robust characterizations.
In this paper we refine uniformity further and examine the impact
of these refinements on $NC^1$ and its subclasses.
When applied to the logarithmic circuit depth characterization of $NC^1$,
some refinements leave ... more >>>

Based on different concepts to obtain a finer notion of language recognition via finite monoids we develop an algebraic structure called typed monoid.
This leads to an algebraic description of regular and non regular languages.

We obtain for each language a unique minimal recognizing typed monoid, the typed syntactic monoid.
more >>>

We consider the regular languages recognized by weighted threshold circuits with a linear number of wires.
We present a simple proof to show that parity cannot be computed by such circuits.
Our proofs are based on an explicit construction to restrict the input of the circuit such that the value ... more >>>