Polynomial Certificates for Propositional Classes

This paper studies the complexity of learning classes of expressions in propositional logic from equivalence queries and membership queries. In particular, we focus on bounding the number of queries that are required to learn the class ignoring computational complexity. This quantity is known to be captured by a combinatorial measure of concept classes known as the certificate complexity. The paper gives new constructions of polynomial size certificates for monotone expressions in conjunctive normal form (CNF), for unate CNF functions where each variable affects the function either positively or negatively but not both ways, and for Horn CNF functions. Lower bounds on certificate size for these classes are derived showing that for some parameter settings the new certificates constructions are optimal. Finally, the paper gives an exponential lower bound on the certificate size for a natural generalization of these classes known as renamable Horn CNF functions, thus implying that the class is not learnable from a polynomial number of queries. This work has been partly supported by NSF Grant IIS-0099446 (M.A. and R.K.) and by an NSF Mathematical Sciences Postdoctoral Research Fellowship (R.S). Work done while A.F. was at the Department of Electrical and Computer Engineering, Northwestern University, and while R.S. was at the Division of Engineering and Applied Sciences, Harvard University.