  We generalize the simplest kinetically constrained model of a glass-forming
liquid by softening kinetic constraints, allowing them to be violated with a
small finite rate. We demonstrate that this model supports a first-order
dynamical (space-time) phase transition, similar to those observed with hard
constraints. In addition, we find that the first-order phase boundary in this
softened model ends in a finite-temperature dynamical critical point, which we
expect to be present in natural systems. We discuss links between this critical
point and quantum phase transitions, showing that dynamical phase transitions
in $d$ dimensions map to quantum transitions in the same dimension, and hence
to classical thermodynamic phase transitions in $d+1$ dimensions. We make these
links explicit through exact mappings between master operators, transfer
matrices, and Hamiltonians for quantum spin chains.
