Abstract. Resolution refinements called w-resolution trees with lemmas (WRTL) and with input lemmas (WRTI) are introduced. Dag-like resolution is equivalent to both WRTL and WRTI when there is no regularity condition. For regular proofs, an exponential separation between regular dag-like resolution and both regular WRTL and regular WRTI is given.It is proved that DLL proof search algorithms that use clause learning based on unit propagation can be polynomially simulated by regular WRTI. More generally, non-greedy DLL algorithms with learning by unit propagation are equivalent to regular WRTI. A general form of clause learning, called DLL-Learn, is defined that is equivalent to regular WRTL.A variable extension method is used to give simulations of resolution by regular WRTI, using a simplified form of proof trace extensions. DLL-Learn and non-greedy DLL algorithms with learning by unit propagation can use variable extensions to simulate general resolution without doing restarts.Finally, an exponential lower bound for WRTL where the lemmas are restricted to short clauses is shown.

This paper gives two distinct proofs of an exponential separation between regular resolution and unrestricted resolution. The previous best known separation between these systems was quasi-polynomial.

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The problem of converting deterministic finite automata into (short) regular expressions is considered. It is known that the required expression size is 2 Θ(n) in the worst case for infinite languages, and for finite languages it is n Ω(log log n) and n O(log n) , if the alphabet size grows with the number of states n of the given automaton. A new lower bound method based on communication complexity for regular expression size is developed to show that the required size is indeed n Θ(log n). For constant alphabet size the best lower bound known to date is Ω(n 2), even when allowing infinite languages and nondeterministic finite automata. As the technique developed here works equally well for deterministic finite automata over binary alphabets, the lower bound is improved to n Ω(log n) .

This paper gives two distinct proofs of an exponential separation between regular resolution and unrestricted resolution. The previous best known separation between these systems was quasi-polynomial.