Lower Bounds on OBDD Proofs with Several Orders

This paper is motivated by seeking lower bounds on OBDD(∧,w, r) refutations, namely OBDD refutations that allow weakening and arbitrary reorderings. We first work with 1-NBP(∧) refutations based on read-once nondeterministic branching programs. These generalize OBDD(∧, r) refutations. There are polynomial size 1-NBP(∧) refutations of the pigeonhole principle, hence 1-NBP(∧) is strictly stronger than OBDD(∧, r). There are also formulas that have polynomial size tree-like resolution refutations but require exponential size 1-NBP(∧) refutations. As a corollary, OBDD(∧, r) does not simulate tree-like resolution, answering a previously open question. The system 1-NBP(∧, ∃) uses projection inferences instead of weakening. 1-NBP(∧, ∃k) is the system restricted to projection on at most k distinct variables. We construct explicit constant degree graphs Gn on n vertices and an > 0, such that 1-NBP(∧, ∃ n) refutations of the Tseitin formula for Gn require exponential size. Second, we study the proof system OBDD(∧,w, r`) which allows ` different variable orders in a refutation. We prove an exponential lower bound on the complexity of tree-like OBDD(∧,w, r`) refutations for ` = logn, where n is the number of variables and > 0 is a constant. The lower bound is based on multiparty communication complexity.

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