  We show that the decay of the density of active particles in the reaction
$A+B \rightarrow 0$ in one dimension, with exclusion interaction, results in
logarithmic corrections to the expected power law decay, when the starting
initial condition (i.c.) is periodic. It is well-known that the late-time
density of surviving particles goes as $t^{-1/4}$ with random initial
conditions, and as $t^{-1/2}$ with alternating initial conditions
($ABABAB$...). We show that the decay for periodic i.c.s made of longer blocks
($A^{n}B^{n}A^{n}B^{n}$...) do not show a pure power-law decay when $n$ is
even. By means of first-passage Monte Carlo simulations, and a mapping to a
q-state coarsening model which can be solved in the Independent Interval
Approximation (IIA), we show that the late-time decay of the density of
surviving particles goes as $t^{-1/2}(\log{(t)})^{-1}$ for $n$ even, but as
$t^{-1/2}$ when $n$ is odd. We relate this kinetic symmetry breaking in the
Glauber Ising model. We also see a very slow crossover from a
$t^{-1/2}(\log{(t)})^{-1}$ regime to eventual $t^{-1/2}$ behaviour for i.c.s
made of mixtures of odd- and even-length blocks.
