On Computational Power of Quantum Read-Once Branching Programs

In this paper we review our current results concerning the computational power of quantum read-once branching programs. First of all, based on the circuit presentation of quantum branching programs and our variant of quantum fingerprinting technique, we show that any Boolean function with linear polynomial presentation can be computed by a quantum read-once branching program using a relatively small (usually logarithmic in the size of input) number of qubits. Then we show that the described class of Boolean functions is closed under the polynomial projections.Comment: In Proceedings HPC 2010, arXiv:1103.226

Categorical invariance and structural complexity in human concept learning

An alternative account of human concept learning based on an invariance measure of the categorical\ud stimulus is proposed. The categorical invariance model (CIM) characterizes the degree of structural\ud complexity of a Boolean category as a function of its inherent degree of invariance and its cardinality or\ud size. To do this we introduce a mathematical framework based on the notion of a Boolean differential\ud operator on Boolean categories that generates the degrees of invariance (i.e., logical manifold) of the\ud category in respect to its dimensions. Using this framework, we propose that the structural complexity\ud of a Boolean category is indirectly proportional to its degree of categorical invariance and directly\ud proportional to its cardinality or size. Consequently, complexity and invariance notions are formally\ud unified to account for concept learning difficulty. Beyond developing the above unifying mathematical\ud framework, the CIM is significant in that: (1) it precisely predicts the key learning difficulty ordering of\ud the SHJ [Shepard, R. N., Hovland, C. L.,&Jenkins, H. M. (1961). Learning and memorization of classifications.\ud Psychological Monographs: General and Applied, 75(13), 1-42] Boolean category types consisting of three\ud binary dimensions and four positive examples; (2) it is, in general, a good quantitative predictor of the\ud degree of learning difficulty of a large class of categories (in particular, the 41 category types studied\ud by Feldman [Feldman, J. (2000). Minimization of Boolean complexity in human concept learning. Nature,\ud 407, 630-633]); (3) it is, in general, a good quantitative predictor of parity effects for this large class of\ud categories; (4) it does all of the above without free parameters; and (5) it is cognitively plausible (e.g.,\ud cognitively tractable)

On the Role of Canonicity in Bottom-up Knowledge Compilation

We consider the problem of bottom-up compilation of knowledge bases, which is usually predicated on the existence of a polytime function for combining compilations using Boolean operators (usually called an Apply function). While such a polytime Apply function is known to exist for certain languages (e.g., OBDDs) and not exist for others (e.g., DNNF), its existence for certain languages remains unknown. Among the latter is the recently introduced language of Sentential Decision Diagrams (SDDs), for which a polytime Apply function exists for unreduced SDDs, but remains unknown for reduced ones (i.e. canonical SDDs). We resolve this open question in this paper and consider some of its theoretical and practical implications. Some of the findings we report question the common wisdom on the relationship between bottom-up compilation, language canonicity and the complexity of the Apply function

New Developments in Quantum Algorithms

In this survey, we describe two recent developments in quantum algorithms. The first new development is a quantum algorithm for evaluating a Boolean formula consisting of AND and OR gates of size N in time O(\sqrt{N}). This provides quantum speedups for any problem that can be expressed via Boolean formulas. This result can be also extended to span problems, a generalization of Boolean formulas. This provides an optimal quantum algorithm for any Boolean function in the black-box query model. The second new development is a quantum algorithm for solving systems of linear equations. In contrast with traditional algorithms that run in time O(N^{2.37...}) where N is the size of the system, the quantum algorithm runs in time O(\log^c N). It outputs a quantum state describing the solution of the system.Comment: 11 pages, 1 figure, to appear as an invited survey talk at MFCS'201

Optimal Recombination in Genetic Algorithms

This paper surveys results on complexity of the optimal recombination problem (ORP), which consists in finding the best possible offspring as a result of a recombination operator in a genetic algorithm, given two parent solutions. We consider efficient reductions of the ORPs, allowing to establish polynomial solvability or NP-hardness of the ORPs, as well as direct proofs of hardness results

Nonlinear Models of Neural and Genetic Network Dynamics:\ud \ud Natural Transformations of Łukasiewicz Logic LM-Algebras in a Łukasiewicz-Topos as Representations of Neural Network Development and Neoplastic Transformations \ud

A categorical and Łukasiewicz-Topos framework for Algebraic Logic models of nonlinear dynamics in complex functional systems such as Neural Networks, Cell Genome and Interactome Networks is introduced. Łukasiewicz Algebraic Logic models of both neural and genetic networks and signaling pathways in cells are formulated in terms of nonlinear dynamic systems with n-state components that allow for the generalization of previous logical models of both genetic activities and neural networks. An algebraic formulation of variable next-state/transfer functions is extended to a Łukasiewicz Topos with an N-valued Łukasiewicz Algebraic Logic subobject classifier description that represents non-random and nonlinear network activities as well as their transformations in developmental processes and carcinogenesis.\u