  We settle two long-standing open problems about Conway's Life, a
two-dimensional cellular automaton. We solve the Generalized grandfather
problem: for all $n \geq 0$, there exists a configuration that has an $n$th
predecessor but not an $(n+1)$st one. We also solve (one interpretation of) the
Unique father problem: there exists a finite stable configuration that contains
a finite subpattern that has no predecessor patterns except itself. In
particular this gives the first example of an unsynthetizable still life. The
new key concept is that of a spatiotemporally periodic configuration (agar)
which has a unique chain of preimages; we show that this property is
semidecidable, and find examples of such agars using a SAT solver.
  Our results about the topological dynamics of Game of Life are as follows: it
never reaches its limit set; its dynamics on its limit set is chain-wandering,
in particular it is not topologically transitive and does not have dense
periodic points; and the spatial dynamics of its limit set is non-sofic, and
does not admit a sublinear gluing radius in the cardinal directions (in
particular it is not block-gluing). Our computability results are that Game of
Life's reachability problem, as well as the language of its limit set, are
PSPACE-hard.
