On learning Read-k-Satisfy-j DNF

We study the learnability of Read-<italic>k</italic>-Satisfy-<italic>j</italic> (RkSj) DNF formulae. These are DNF formulae in which the maximal number of occurrences of a variable is bounded by <italic>k</italic>, and the number of terms satisfied by any assignment is at most <italic>j</italic>. We show that this class of functions is learnable in polynomial time, using Equivalence and Membership Queries, as long as <italic>k•j</italic>=<italic>O</italic>(log<italic>n</italic>/loglog<italic>n</italic>). Learnability was previously known only in case that both <italic>k</italic> and <italic>j</italic> are constants. We also present a family of boolean functions that have short (<italic>poly(n)</italic>) Read-2-Satisfy-1 DNF formulae but require CNF formulae of size ><inline-equation> <f> 2<sup><g>W</g><fen lp="par"><rad><rcd><it>n</it></rcd></rad><rp post="par"></fen> </sup></f> </inline-equation>. Therefore, our result does not seem to follow from the recent learnability result of [Bsh93].

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