  It has been suggested that innovations occur mainly by combination: the more
inventions accumulate, the higher the probability that new inventions are
obtained from previous designs. Additionally, it has been conjectured that the
combinatorial nature of innovations naturally leads to a singularity: at some
finite time, the number of innovations should diverge. Although these ideas are
certainly appealing, no general models have been yet developed to test the
conditions under which combinatorial technology should become explosive. Here
we present a generalised model of technological evolution that takes into
account two major properties: the number of previous technologies needed to
create a novel one and how rapidly technology ages. Two different models of
combinatorial growth are considered, involving different forms of ageing. When
long-range memory is used and thus old inventions are available for novel
innovations, singularities can emerge under some conditions with two phases
separated by a critical boundary. If the ageing has a characteristic time
scale, it is shown that no singularities will be observed. Instead, a "black
hole" of old innovations appears and expands in time, making the rate of
invention creation slow down into a linear regime.
