  In this paper, we show that a generalization of the discrete Burgers equation
can be obtained by a kind of discrete Cole--Hopf transformation to the discrete
diffusion equation corresponding to the correlated random walk, which is also
known as a generalization of the well known random walks. By applying the
technique called ultradiscretization, we obtain the generalization of the
ultradiscrete diffusion equation, the ultradiscrete Cole--Hopf transformation
and the ultradiscrete Burgers equation. Moreover, we show that the resulting
ultradiscrete Burgers equation yields cellular automata. The relationship of
the ultradiscrete Burgers equation to the traffic flow is also discussed.
