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Read-Once Branching Programs, Rectangular Proofs of the Pigeonhole Principle and the Transversal Calculus

We investigate read-once branching programs for the following search problem: given a Boolean m × n matrix with m > n, find either an all-zero row, or two 1’s in some column. Our primary motivation is that this models regular resolution proofs of the pigeonhole principle \( PHP^{m}_{n} \), and that for m > n 2 no lower bounds are known for the length of such proofs. We prove exponential lower bounds (for arbitrarily large m!) if we further restrict this model by requiring the branching program either to finish one row of queries before asking queries about another row (the row model) or put the dual column restriction (the column model).

Then we investigate a special class of resolution proofs for \( PHP^{m}_{n} \) that operate with positive clauses of rectangular shape; we call this fragment the rectangular calculus. We show that all known upper bounds on the size of resolution proofs of \( PHP^{m}_{n} \) actually give rise to proofs in this calculus and, inspired by this fact, also give a remarkably simple “rectangular” reformulation of the Haken–Buss–Turán lower bound for the case m ≪ n 2. Finally we show that the rectangular calculus is equivalent to the column model on the one hand, and to transversal calculus on the other hand, where the latter is a natural proof system for estimating from below the transversal size of set families. In particular, our exponential lower bound for the column model translates both to the rectangular and transversal calculi.

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Correspondence to Alexander Razborov*.

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* Part of the work was done while this author was visiting Special Year on Logic and Algorithms at DIMACS, Princeton. Also supported by Russian Basic Research Foundation grant 96-01-01222.

† Part of this work was done while on sabbatical leave at the Institute for Advanced Study and Princeton University, Princeton. This work was supported by USA-Israel BSF grant 92-00106 and by a Wolfson research award administered by the Israeli Academy of Sciences, as well as a Sloan Foundation grant.

‡ This work was supported in part by National Science Foundation and DARPA under grant CCR-9627819, and by USA-Israel BSF grant 92-00106.

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Razborov*, A., Wigderson†, A. & Yao‡, A. Read-Once Branching Programs, Rectangular Proofs of the Pigeonhole Principle and the Transversal Calculus. Combinatorica 22, 555–574 (2002).

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AMS Subject Classification (2000):

  • 03F20
  • 68Q17