Pebbling, Entropy and Branching Program Size Lower Bounds

We contribute to the program of proving lower bounds on the size of branching programs solving the Tree Evaluation Problem introduced by Cook et. al. (2012). Proving a super-polynomial lower bound for the size of nondeterministic thrifty branching programs (NTBP) would separate $NL$ from $P$ for thrifty models solving the tree evaluation problem. First, we show that {\em Read-Once NTBPs} are equivalent to whole black-white pebbling algorithms thus showing a tight lower bound (ignoring polynomial factors) for this model. We then introduce a weaker restriction of NTBPs called {\em Bitwise Independence}. The best known NTBPs (of size $O(k^{h/2+1})$) for the tree evaluation problem given by Cook et. al. (2012) are Bitwise Independent. As our main result, we show that any Bitwise Independent NTBP solving $TEP_{2}^{h}(k)$ must have at least $\frac{1}{2}k^{h/2}$ states. Prior to this work, lower bounds were known for NTBPs only for fixed heights $h=2,3,4$ (See Cook et. al. (2012)). We prove our results by associating a fractional black-white pebbling strategy with any bitwise independent NTBP solving the Tree Evaluation Problem. Such a connection was not known previously even for fixed heights. Our main technique is the entropy method introduced by Jukna and Z{\'a}k (2001) originally in the context of proving lower bounds for read-once branching programs. We also show that the previous lower bounds given by Cook et. al. (2012) for deterministic branching programs for Tree Evaluation Problem can be obtained using this approach. Using this method, we also show tight lower bounds for any $k$-way deterministic branching program solving Tree Evaluation Problem when the instances are restricted to have the same group operation in all internal nodes.Comment: 25 Pages, Manuscript submitted to Journal in June 2013 This version includes a proof for tight size bounds for (syntactic) read-once NTBPs. The proof is in the same spirit as the proof for size bounds for bitwise independent NTBPs present in the earlier version of the paper and is included in the journal version of the paper submitted in June 201

Karchmer-Wigderson Games for Hazard-free Computation

We present a Karchmer-Wigderson game to study the complexity of hazard-free formulas. This new game is both a generalization of the monotone Karchmer-Wigderson game and an analog of the classical Boolean Karchmer-Wigderson game. Therefore, it acts as a bridge between the existing monotone and general games. Using this game, we prove hazard-free formula size and depth lower bounds that are provably stronger than those possible by the standard technique of transferring results from monotone complexity in a black-box fashion. For the multiplexer function we give (1) a hazard-free formula of optimal size and (2) an improved low-depth hazard-free formula of almost optimal size and (3) a hazard-free formula with alternation depth $2$ that has optimal depth. We then use our optimal constructions to obtain an improved universal worst-case hazard-free formula size upper bound. We see our results as a significant step towards establishing hazard-free computation as an independent missing link between Boolean complexity and monotone complexity.Comment: 34 pages, To appear in ITCS 202

Karchmer-Wigderson Games for Hazard-Free Computation

We present a Karchmer-Wigderson game to study the complexity of hazard-free formulas. This new game is both a generalization of the monotone Karchmer-Wigderson game and an analog of the classical Boolean Karchmer-Wigderson game. Therefore, it acts as a bridge between the existing monotone and general games. Using this game, we prove hazard-free formula size and depth lower bounds that are provably stronger than those possible by the standard technique of transferring results from monotone complexity in a black-box fashion. For the multiplexer function we give (1) a hazard-free formula of optimal size and (2) an improved low-depth hazard-free formula of almost optimal size and (3) a hazard-free formula with alternation depth 2 that has optimal depth. We then use our optimal constructions to obtain an improved universal worst-case hazard-free formula size upper bound. We see our results as a step towards establishing hazard-free computation as an independent missing link between Boolean complexity and monotone complexity

Karchmer-Wigderson games for hazard-free computation

We present a Karchmer-Wigderson game to study the complexity of hazard-free formulas. This new game is both a generalization of the monotone Karchmer-Wigderson game and an analog of the classical Boolean Karchmer-Wigderson game. Therefore, it acts as a bridge between the existing monotone and general games. Using this game, we prove hazard-free formula size and depth lower bounds that are provably stronger than those possible by the standard technique of transferring results from monotone complexity in a black-box fashion. For the multiplexer function we give (1) a hazard-free formula of optimal size and (2) an improved low-depth hazard-free formula of almost optimal size and (3) a hazard-free formula with alternation depth 2 that has optimal depth. We then use our optimal constructions to obtain an improved universal worst-case hazard-free formula size upper bound. We see our results as a significant step towards establishing hazard-free computation as an independent missing link between Boolean complexity and monotone complexity

Karchmer-Wigderson Games for Hazard-free Computation

We present a Karchmer-Wigderson game to study the complexity of hazard-free formulas. This new game is both a generalization of the monotone Karchmer-Wigderson game and an analog of the classical Boolean Karchmer-Wigderson game. Therefore, it acts as a bridge between the existing monotone and general games. Using this game, we prove hazard-free formula size and depth lower bounds that are provably stronger than those possible by the standard technique of transferring results from monotone complexity in a black-box fashion. For the multiplexer function we give (1) a hazard-free formula of optimal size and (2) an improved low-depth hazard-free formula of almost optimal size and (3) a hazard-free formula with alternation depth $2$ that has optimal depth. We then use our optimal constructions to obtain an improved universal worst-case hazard-free formula size upper bound. We see our results as a significant step towards establishing hazard-free computation as an independent missing link between Boolean complexity and monotone complexity