Space-time causality is one of the fundamental notions of modern physics; however, it is difficult to define in observational physical terms. Intuitively, the fact that a space-time event e=(t,x) can causally influence an event e'=(t',x') means that what we do in the vicinity of e changes what we observe at e'. If we had two copies of the Universe, we could perform some action at e in one copy but not in another copy; if we then observe the difference at e', this would be an indication of causality. However, we only observe one Universe, in which we either perform the action or we do not. At first glance, it may seem that in this case, there is no meaningful way to provide an operational definition of causality. In this paper, we show that such a definition is possible if we use the notions of algorithmic randomness and Kolmogorov complexity. The resulting definition leads to a somewhat unexpected consequence: that space-time causality is a matter of degree.