Graph communication protocols are a generalization of classical communi- cation protocols to the case when the underlying graph is a directed acyclic graph. Motivated by potential applications in proof complexity, we study variants of graph communication protocols and relations between them. The main result is a comparison of the strength of two types of protocols, protocols with equality and protocols with a conjunction of a constant num- ber of inequalities. We prove that protocols of the first type are at least as strong as protocols of the second type in the following sense: For a Boolean function f, if there is a protocol with a conjunction of a constant number of inequalities of polynomial size solving f, then there is a protocol with equality of polynomial size solving f. We also introduce two new types of graph communication protocols, protocols with disjointness and protocols with non-disjointness, and prove that the first type is at least as strong as the previously considered protocols and that the second type is too strong to be useful for applications.

The primary result from this dissertation is following inequality: d(B) ≤ min(2^< wd(B),sup{λ^c(B): λ < wd(B)}) in ZFC, where B is a homogeneous complete Boolean algebra, d(B) is the density, wd(B) is the weak density, and c(B) is the cellularity of B. Chapter II of this dissertation is a general overview of homogeneous complete Boolean algebras. Assuming the existence of a weakly inaccessible cardinal, we give an example of a homogeneous complete Boolean algebra which does not attain its cellularity. In chapter III, we prove that for any integer n > 1, wd_2(B) = wd_n(B). Also in this chapter, we show that if X⊂B is κ—weakly dense for 1 < κ < sat(B), then sup{wd_κ(B):κ < sat(B)} = d(B). In chapter IV, we address the following question: If X is weakly dense in a homogeneous complete Boolean algebra B, does there necessarily exist b € B\{0} such that {x∗b: x ∈ X} is dense in B|b = {c € B: c ≤ b}? We show that the answer is no for collapsing algebras. In chapter V, we give new proofs to some well known results concerning supporting antichains. A direct consequence of these results is the relation c(B) < wd(B), i.e., the weak density of a homogeneous complete Boolean algebra B is at least as big as the cellularity. Also in this chapter, we introduce discernible sets. We prove that a discernible set of cardinality no greater than c(B) cannot be weakly dense. In chapter VI, we prove the main result of this dissertation, i.e., d(B) ≤ min(2^< wd(B),sup{λ^c(B): λ < wd(B)}). In chapter VII, we list some unsolved problems concerning this dissertation.

M. H. A. Stone showed in 1937 and subsequently that many interesting and important results of general topology involve latices and Boolean rings. This type of result forms the substance of this thesis. Theorem 4, page 11, states that for any r ≠ 0 in a Boolean ring, there exists a homomorphism h into I2 , (the field of integers modulo 2), such that h(r) = 1. Theorem 3, page 6, states that any subring of a characteristic ring of a Boolean space X is the whole ring if it has the two points property (that is, given x, y in X and a, b in I2, there exists a g such that g(x) = a and g(y) = b). From these two theorems follows the Stone Representation theorem which states that any Boolean ring is isomorphic to the characteristic ring of its Stone space. Theorem 1, page 11, is independent of other theorems. It states that any compact Hausdorff space is the continuous image of some closed subset in a Cantor space. Theorem 5, page 23, states that a topological space can be embedded in a Cantor space as a subspace if and only if it is Boolean. This theorem uses the Dual Representation theorem as its sufficient part. It states that any Boolean space is homomorphic to the Stone space of its characteristic ring.

The approach to a theory of non-finitary matroids, as outlined by the author in [20], is here extended to the case in which the relevant closure operators are defined on arbitrary complete Boolean algebras, rather than on the power sets of sets. As a preliminary to this study, the theory of derivatives of operators on complete Boolean algebras is developed and the notion, having interest in its own right, of an analytic closure operator is introduced . The class of B-matroidal closure operators is singled out for especial attention and it is proved that this class is closed under Whitney duality. Also investigated is the class of those closure operators which are both matroidal and topological. / Thesis / Doctor of Philosophy (PhD)

published_or_final_version / Mathematics / Master / Master of Philosophy

Subdivision surfaces are capable of modeling and representing complex shapes of arbi-trary topology. However, methods on how to build the control mesh of a complex surfaceare not studied much. Currently, most meshes of complicated objects come from trian-gulation and simplification of raster scanned data points, like the Stanford 3D ScanningRepository. This approach is costly and leads to very dense meshes.Subdivision surface based one-piece representation means to represent the final objectin a design process with only one subdivision surface, no matter how complicated theobject's topology or shape. Hence the number of parts in the final representation isalways one.In this dissertation we present necessary mathematical theories and geometric algo-rithms to support subdivision surface based one-piece representation. First, an explicitparametrization method is presented for exact evaluation of Catmull-Clark subdivisionsurfaces. Based on it, two approaches are proposed for constructing the one-piece rep-resentation of a given object with arbitrary topology. One approach is to construct theone-piece representation by using the interpolation technique. Interpolation is a naturalway to build models, but the fairness of the interpolating surface is a big concern inprevious methods. With similarity based interpolation technique, we can obtain bet-ter modeling results with less undesired artifacts and undulations. Another approachis through performing Boolean operations. Up to this point, accurate Boolean oper-ations over subdivision surfaces are not approached yet in the literature. We presenta robust and error controllable Boolean operation method which results in a one-piecerepresentation. Because one-piece representations resulting from the above two methodsare usually dense, error controllable simplification of one-piece representations is needed.Two methods are presented for this purpose: adaptive tessellation and multiresolutionanalysis. Both methods can significantly reduce the complexity of a one-piece represen-tation and while having accurate error estimation.A system that performs subdivision surface based one-piece representation was im-plemented and a lot of examples have been tested. All the examples show that our ap-proaches can obtain very good subdivision based one-piece representation results. Eventhough our methods are based on Catmull-Clark subdivision scheme, we believe they canbe adapted to other subdivision schemes as well with small modifications.

Chu-Keung Chung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 88-92). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgments --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Motivation and Aims --- p.1 / Chapter 1.2 --- Contributions --- p.3 / Chapter 1.3 --- Structure of the Thesis --- p.4 / Chapter 2 --- Literature Review --- p.6 / Chapter 2.1 --- Introduction --- p.6 / Chapter 2.2 --- Complete Algorithms --- p.7 / Chapter 2.2.1 --- Parallel Checking --- p.7 / Chapter 2.2.2 --- Mom's --- p.8 / Chapter 2.2.3 --- Davis-Putnam --- p.9 / Chapter 2.2.4 --- Nonchronological Backtracking --- p.9 / Chapter 2.2.5 --- Iterative Logic Array (ILA) --- p.10 / Chapter 2.3 --- Incomplete Algorithms --- p.11 / Chapter 2.3.1 --- GENET --- p.11 / Chapter 2.3.2 --- GSAT --- p.12 / Chapter 2.4 --- Summary --- p.13 / Chapter 3 --- Algorithms --- p.14 / Chapter 3.1 --- Introduction --- p.14 / Chapter 3.2 --- Tree Search Techniques --- p.14 / Chapter 3.2.1 --- Depth First Search --- p.15 / Chapter 3.2.2 --- Forward Checking --- p.16 / Chapter 3.2.3 --- Davis-Putnam --- p.17 / Chapter 3.2.4 --- GRASP --- p.19 / Chapter 3.3 --- Incomplete Algorithms --- p.20 / Chapter 3.3.1 --- GENET --- p.20 / Chapter 3.3.2 --- GSAT Algorithm --- p.22 / Chapter 3.4 --- Summary --- p.23 / Chapter 4 --- Field Programmable Gate Arrays --- p.24 / Chapter 4.1 --- Introduction --- p.24 / Chapter 4.2 --- FPGA --- p.24 / Chapter 4.2.1 --- Xilinx 4000 series FPGAs --- p.26 / Chapter 4.2.2 --- Bitstream --- p.31 / Chapter 4.3 --- Giga Operations Reconfigurable Computing Platform --- p.32 / Chapter 4.4 --- Annapolis Wildforce PCI board --- p.33 / Chapter 4.5 --- Summary --- p.35 / Chapter 5 --- Implementation --- p.36 / Chapter 5.1 --- Parallel Graph Coloring Machine --- p.36 / Chapter 5.1.1 --- System Architecture --- p.38 / Chapter 5.1.2 --- Evaluator --- p.39 / Chapter 5.1.3 --- Finite State Machine (FSM) --- p.42 / Chapter 5.1.4 --- Memory --- p.43 / Chapter 5.1.5 --- Hardware Resources --- p.43 / Chapter 5.2 --- Serial Graph Coloring Machine --- p.44 / Chapter 5.2.1 --- System Architecture --- p.44 / Chapter 5.2.2 --- Input Memory --- p.46 / Chapter 5.2.3 --- Solution Store --- p.46 / Chapter 5.2.4 --- Constraint Memory --- p.47 / Chapter 5.2.5 --- Evaluator --- p.48 / Chapter 5.2.6 --- Input Mapper --- p.49 / Chapter 5.2.7 --- Output Memory --- p.49 / Chapter 5.2.8 --- Backtrack Checker --- p.50 / Chapter 5.2.9 --- Word Generator --- p.51 / Chapter 5.2.10 --- State Machine --- p.51 / Chapter 5.2.11 --- Hardware Resources --- p.54 / Chapter 5.3 --- Serial Boolean Satisfiability Solver --- p.56 / Chapter 5.3.1 --- System Architecture --- p.58 / Chapter 5.3.2 --- Solutions --- p.59 / Chapter 5.3.3 --- Solution Generator --- p.59 / Chapter 5.3.4 --- Evaluator --- p.60 / Chapter 5.3.5 --- AND/OR --- p.62 / Chapter 5.3.6 --- State Machine --- p.62 / Chapter 5.3.7 --- Hardware Resources --- p.64 / Chapter 5.4 --- GSAT Solver --- p.65 / Chapter 5.4.1 --- System Architecture --- p.65 / Chapter 5.4.2 --- Variable Memory --- p.65 / Chapter 5.4.3 --- Flip-Bit Vector --- p.66 / Chapter 5.4.4 --- Clause Evaluator --- p.67 / Chapter 5.4.5 --- Adder --- p.70 / Chapter 5.4.6 --- Random Bit Generator --- p.71 / Chapter 5.4.7 --- Comparator --- p.71 / Chapter 5.4.8 --- Sum Register --- p.71 / Chapter 5.5 --- Summary --- p.71 / Chapter 6 --- Results --- p.73 / Chapter 6.1 --- Introduction --- p.73 / Chapter 6.2 --- Parallel Graph Coloring Machine --- p.73 / Chapter 6.3 --- Serial Graph Coloring Machine --- p.74 / Chapter 6.4 --- Serial SAT Solver --- p.74 / Chapter 6.5 --- GSAT Solver --- p.75 / Chapter 6.6 --- Summary --- p.76 / Chapter 7 --- Conclusion --- p.77 / Chapter 7.1 --- Future Work --- p.78 / Chapter A --- Software Implementation of Graph Coloring in CHIP --- p.79 / Chapter B --- Density Improvements Using Xilinx RAM --- p.81 / Chapter C --- Bit stream Configuration --- p.83 / Bibliography --- p.88 / Publications --- p.93

by Chun Gu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves 69-71). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.3 / Chapter 2 --- Definitions --- p.7 / Chapter 3 --- Some Previous Work on OBDD --- p.13 / Chapter 3.1 --- The Work of Bryant --- p.13 / Chapter 3.2 --- Some Variations of the OBDD --- p.14 / Chapter 3.3 --- Previous Work on Variable Ordering of OBDD --- p.16 / Chapter 3.3.1 --- The FIH Heuristic --- p.16 / Chapter 3.3.2 --- The Dynamic Variable Ordering --- p.17 / Chapter 3.3.3 --- The Interleaving method --- p.19 / Chapter 4 --- Two Level Logic Function and OBDD --- p.21 / Chapter 5 --- DSCF Algorithm --- p.25 / Chapter 6 --- Thin Boolean Function --- p.33 / Chapter 6.1 --- The Structure and Properties of thin Boolean functions --- p.33 / Chapter 6.1.1 --- The construction of Thin OBDDs --- p.33 / Chapter 6.1.2 --- Properties of Thin Boolean Functions --- p.38 / Chapter 6.1.3 --- Thin Factored Functions --- p.49 / Chapter 6.2 --- The Revised DSCF Algorithm --- p.52 / Chapter 6.3 --- Experimental Results --- p.54 / Chapter 7 --- A Pattern Merging Algorithm --- p.59 / Chapter 7.1 --- Merging of Patterns --- p.60 / Chapter 7.2 --- The Algorithm --- p.62 / Chapter 7.3 --- Experiments and Conclusion --- p.65 / Chapter 8 --- Conclusions --- p.67

We present attribute-based encryption (ABE) schemes for Boolean formulas that are adaptively secure under simple assumptions. Notably, our KP-ABE scheme enjoys a ciphertext size that is linear in the attribute vector length and independent of the formula size (even when attributes can be used multiple times), and we achieve an analogous result for CP-ABE. This resolves the central open problem in attribute-based encryption posed by Lewko and Waters. Along the way, we develop a theory of modular design for unbounded ABE schemes and answer an open question regarding the adaptive security of Yao’s Secret Sharing scheme for NC1 circuits.

This thesis discusses the topic of Boolean algebras. In order to build intuitive understanding of the topic, research began with the investigation of Boolean algebras in the area of Abstract Algebra. The content of this initial research used a particular notation. The ideas of partially ordered sets, lattices, least upper bounds, and greatest lower bounds were used to define the structure of a Boolean algebra. From this fundamental understanding, we were able to study atoms, Boolean algebra isomorphisms, and Stone’s Representation Theorem for finite Boolean algebras. We also verified and proved many properties involving Boolean algebras and related structures. We then expanded our study to more thoroughly developed theory. This comprehensive theory was more abstract and required the use of a different, more universal, notation. We continued examining least upper and greatest lower bounds but extended our knowledge to subalgebras and families of subsets. The notions of cardinality, cellularity, and pairwise disjoint families were investigated, defined, and then used to understand the Erdös-Tarski Theorem. Lastly, this study concluded with the investigation of denseness and incomparability as well as normal forms and the completion of Boolean algebras.