Hardness of Function Composition for Semantic Read once Branching Programs

In this work, we study time/space trade-offs for function composition. We prove asymptotically optimal lower bounds for function composition in the setting of nondeterministic read once branching programs, for the syntactic model as well as the stronger semantic model of read-once nondeterministic computation. We prove that such branching programs for solving the tree evaluation problem over an alphabet of size k requires size roughly k^{Omega(h)}, i.e space Omega(h log k). Our lower bound nearly matches the natural upper bound which follows the best strategy for black-white pebbling the underlying tree. While previous super-polynomial lower bounds have been proven for read-once nondeterministic branching programs (for both the syntactic as well as the semantic models), we give the first lower bounds for iterated function composition, and in these models our lower bounds are near optimal

Lower Bounds for Nondeterministic Semantic Read-Once Branching Programs

We prove exponential lower bounds on the size of semantic read-once 3-ary nondeterministic branching programs. Prior to our result the best that was known was for D-ary branching programs with |D| >= 2^{13}

Branching Program Lower Bounds

A longstanding open problem in complexity theory is whether the class Polytime (P) is the same as LogSpace (L) or nondeterministic LogSpace (NL). In this thesis, we explore this problem by studying time/space tradeoffs for problems in P . As in, for some natural problem in P , does the addition of a space restriction prevent a polynomial time solution ? To begin, we prove exponential lower bounds on the size of a restricted model of branching programs called semantic read-once 3-ary nondeterministic branching programs solving a polytime computable function, in particular the polynomial evaluation problem. In the second part, we prove lower bounds against branching programs solving Function Composition. In particular, we show that the amount of space required for computing the composed function grows as is seen in the straight forward algorithm (where space grows additively). If shown true for general branching programs, this would separate L and P. We show this is indeed true in the restricted setting of nondeterministic read once branching programs, for the syntactic model as well as the stronger semantic model. We prove that such branching programs for solving the tree evaluation problem over an alphabet of size 'k' requires size roughly k^{Omega(h)}, i.e space Omega(h log k) where 'h' is the number of compositions. Then we focus entirely on general branching programs sticking to the theme of lower bounds against function composition. We give a better lower bound than is possible by using Nechiporuk's method for k-way branching programs solving a specific composition problem. Using essentially the same method we give a matching lower bound to that achievable by using Nechiporuk's method for binary branching programs. Any marginal improvement here would be consequential towards beating Nechiporuk's method for binary branching programs, a longstanding open problem. We then proceed to give some surprising upper bounds based on communication complexity protocols that are different from naive upper bounds. Our aim here is to improve our understanding of a possible approach to prove the suspected lower bound just mentioned, but the connections to communication complexity therein might themselves be of independent interest.Ph.D