  We define decision trees for monotone functions on a simplicial complex. We
define homology decidability of monotone functions, and show that various
monotone functions related to semimatroids are homology decidable. Homology
decidability is a generalization of semi-nonevasiveness, a notion due to
Jonsson. The motivating example is the complex of bipartite graphs, whose Betti
numbers are unknown in general.
  We show that these monotone functions have optimum decision trees, from which
we can compute relative Betti numbers of related pairs of simplicial complexes.
Moreover, these relative Betti numbers are coefficients of evaluations of the
Tutte polynomial, and every semimatroid collapses onto its broken circuit
complex.
