  A decision list is an ordered list of rules. Each rule is specified by a
term, which is a conjunction of literals, and a value. Given an input, the
output of a decision list is the value corresponding to the first rule whose
term is satisfied by the input. Decision lists generalize both CNFs and DNFs,
and have been studied both in complexity theory and in learning theory.
  The size of a decision list is the number of rules, and its width is the
maximal number of variables in a term. We prove that decision lists of small
width can always be approximated by decision lists of small size, where we
obtain sharp bounds. This in particular resolves a conjecture of Gopalan, Meka
and Reingold (Computational Complexity, 2013) on DNF sparsification.
  An ingredient in our proof is a new random restriction lemma, which allows to
analyze how DNFs (and more generally, decision lists) simplify if a small
fraction of the variables are fixed. This is in contrast to the more commonly
used switching lemma, which requires most of the variables to be fixed.
