Groote and Zantema proved that a particular OBDD computation of the pigeonhole formula has an exponential
size and that limited OBDD derivations cannot simulate resolution polynomially. Here we show that any arbitrary OBDD Apply refutation of the pigeonhole formula has an exponential
size: we prove that the size of one ... more >>>

We consider the problem of testing if a given function
$f : \F_2^n \rightarrow \F_2$ is close to any degree $d$ polynomial
in $n$ variables, also known as the Reed-Muller testing problem.
Alon et al.~\cite{AKKLR} proposed and analyzed a natural
$2^{d+1}$-query test for this property and showed that it accepts
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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 >>>