  The problem of constructing hazard-free Boolean circuits (those avoiding
electronic glitches) dates back to the 1940s and is an important problem in
circuit design and even in cybersecurity. We show that a DeMorgan circuit is
hazard-free if and only if the circuit produces (purely syntactically) all
prime implicants as well as all prime implicates of the Boolean function it
computes. This extends to arbitrary DeMorgan circuits a classical result of
Eichelberger [IBM J. Res. Develop., 9 (1965)] showing this property for special
depth-two circuits. Via an amazingly simple proof, we also strengthen a recent
result Ikenmeyer et al. [J. ACM, 66:4 (2019)]: not only the complexities of
hazard-free and monotone circuits for monotone Boolean functions do coincide,
but every optimal hazard-free circuit for a monotone Boolean function must be
monotone. Then we show that hazard-free circuit complexity of a very simple
(non-monotone) Boolean function is super-polynomially larger than its
unrestricted circuit complexity. This function accepts a Boolean n x n matrix
iff every row and every column has exactly one 1-entry. Finally, we show that
every Boolean function of n variables can be computed by a hazard-free circuit
of size O(2^n/n).
