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}

Alternating Hierarchies for Time-Space Tradeoffs

Nepomnjascii's Theorem states that for all 0 0 the class of languages recognized in nondeterministic time n^k and space n^\epsilon, NTISP[n^k, n^\epsilon ], is contained in the linear time hierarchy. By considering restrictions on the size of the universal quantifiers in the linear time hierarchy, this paper refines Nepomnjascii's result to give a sub- hierarchy, Eu-LinH, of the linear time hierarchy that is contained in NP and which contains NTISP[n^k, n^\epsilon ]. Hence, Eu-LinH contains NL and SC. This paper investigates basic structural properties of Eu-LinH. Then the relationships between Eu-LinH and the classes NL, SC, and NP are considered to see if they can shed light on the NL = NP or SC = NP questions. Finally, a new hierarchy, zeta -LinH, is defined to reduce the space requirements needed for the upper bound on Eu-LinH.Comment: 14 page

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

Better Time-Space Lower Bounds for SAT and Related Problems

We make several improvements on time lower bounds for concrete problems in NP and PH. 1. We present an elementary technique based on “indirect diagonalization ” that uniformly improves upon the known nonlinear time lower bounds for nondeterminism and alternating computation, on both sublinear (n o(1) ) space RAMs and sequential worktape machines with random access to the input. We obtain better lower bounds for SAT, as well as all NP-complete problems that have efficient reductions from SAT, and Σk-SAT, for constant k ≥ 2. For example, SAT cannot be solved by random access machines using n √ 3 time and n o(1) space. The technique is a natural inductive approach, for which previous work is essentially its base case. 2. We show how indirect diagonalization can also yield time-space lower bounds for computation with bounded nondeterminism. One corollary is that for all k, there exists a constant ck> 1 such that satisfiability of Boolean circuits with n inputs and n k gates cannot be solved in deterministic time n k·ck and n o(1) space.