  Symbolic knowledge can provide crucial inductive bias for training neural
models, especially in low data regimes. A successful strategy for incorporating
such knowledge involves relaxing logical statements into sub-differentiable
losses for optimization. In this paper, we study the question of how best to
relax logical expressions that represent labeled examples and knowledge about a
problem; we focus on sub-differentiable t-norm relaxations of logic. We present
theoretical and empirical criteria for characterizing which relaxation would
perform best in various scenarios. In our theoretical study driven by the goal
of preserving tautologies, the Lukasiewicz t-norm performs best. However, in
our empirical analysis on the text chunking and digit recognition tasks, the
product t-norm achieves best predictive performance. We analyze this apparent
discrepancy, and conclude with a list of best practices for defining loss
functions via logic.
