31 research outputs found
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 from 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 ) 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
must have at least states. Prior to this work, lower
bounds were known for NTBPs only for fixed heights (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 -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
Pebbling Arguments for Tree Evaluation
The Tree Evaluation Problem was introduced by Cook et al. in 2010 as a
candidate for separating P from L and NL. The most general space lower bounds
known for the Tree Evaluation Problem require a semantic restriction on the
branching programs and use a connection to well-known pebble games to generate
a bottleneck argument. These bounds are met by corresponding upper bounds
generated by natural implementations of optimal pebbling algorithms. In this
paper we extend these ideas to a variety of restricted families of both
deterministic and non-deterministic branching programs, proving tight lower
bounds under these restricted models. We also survey and unify known lower
bounds in our "pebbling argument" framework
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
On the performance and programming of reversible molecular computers
If the 20th century was known for the computational revolution, what will the 21st be known for? Perhaps the recent strides in the nascent fields of molecular programming and biological computation will help bring about the âComing Era of Nanotechnologyâ promised in Drexlerâs âEngines of Creationâ. Though there is still far to go, there is much reason for optimism. This thesis examines the underlying principles needed to realise the computational aspects of such âenginesâ in a performant way. Its main body focusses on the ways in which thermodynamics constrains the operation and design of such systems, and it ends with the proposal of a model of computation appropriate for exploiting these constraints.
These thermodynamic constraints are approached from three different directions. The first considers the maximum possible aggregate performance of a system of computers of given volume, V, with a given supply of free energy. From this perspective, reversible computing is imperative in order to circumvent the Landauer limit. A result of Frank is refined and strengthened, showing that the adiabatic regime reversible computer performance is the best possible for any computerâquantum or classical. This therefore shows a universal scaling law governing the performance of compact computers of ~V^(5/6), compared to ~V^(2/3) for conventional computers. For the case of molecular computers, it is shown how to attain this bound. The second direction extends this performance analysis to the case where individual computational particles or sub-units can interact with one another. The third extends it to interactions with shared, non-computational parts of the system. It is found that accommodating these interactions in molecular computers imposes a performance penalty that undermines the earlier scaling result. Nonetheless, scaling superior to that of irreversible computers can be preserved, and appropriate mitigations and considerations are discussed. These analyses are framed in a context of molecular computation, but where possible more general computational systems are considered.
The proposed model, the Ś-calculus, is appropriate for programming reversible molecular computers taking into account these constraints. A variety of examples and mathematical analyses accompany it. Moreover, abstract sketches of potential molecular implementations are provided. Developing these into viable schemes suitable for experimental validation will be a focus of future work