Logical depth and physical complexity

Some mathematical and natural objects (a random sequence, a sequence of zeros, a perfect crystal, a gas) are intuitively trivial, while others (e.g. the human body, the digits of π) contain internal evidence of a nontrivial causal history. We formalize this distinction by defining an object’s “logical depth” as the time required by a standard universal Turing machine to generate it from an input that is algorithmically random (i.e. Martin-Lof random). This definition of depth is shown to be reasonably machineindependent, as well as obeying a slow-growth law: deep objects cannot be quickly produced from shallow ones by any deterministic process, nor with much probability by a probabilistic process, but can be produced slowly. Next we apply depth to the physical problem of “self-organization,” inquiring in particular under what conditions (e.g. noise, irreversibility, spatial and other symmetries of the initial conditions and equations of motion) statistical-mechanical model systems can imitate computers well enough to undergo unbounded increase of depth in the limit of infinite space and time.