6,221 research outputs found
Algorithms and Lower Bounds in Circuit Complexity
Computational complexity theory aims to understand what problems can be efficiently solved by computation. This thesis studies computational complexity in the model of Boolean circuits. Boolean circuits provide a basic mathematical model for computation and play a central role in complexity theory, with important applications in separations of complexity classes, algorithm design, and pseudorandom constructions. In this thesis, we investigate various types of circuit models such as threshold circuits, Boolean formulas, and their extensions, focusing on obtaining complexity-theoretic lower bounds and algorithmic upper bounds for these circuits. (1) Algorithms and lower bounds for generalized threshold circuits: We extend the study of linear threshold circuits, circuits with gates computing linear threshold functions, to the more powerful model of polynomial threshold circuits where the gates can compute polynomial threshold functions. We obtain hardness and meta-algorithmic results for this circuit model, including strong average-case lower bounds, satisfiability algorithms, and derandomization algorithms for constant-depth polynomial threshold circuits with super-linear wire complexity. (2) Algorithms and lower bounds for enhanced formulas: We investigate the model of Boolean formulas whose leaf gates can compute complex functions. In particular, we study De Morgan formulas whose leaf gates are functions with "low communication complexity". Such gates can capture a broad class of functions including symmetric functions and polynomial threshold functions. We obtain new and improved results in terms of lower bounds and meta-algorithms (satisfiability, derandomization, and learning) for such enhanced formulas. (3) Circuit lower bounds for MCSP: We study circuit lower bounds for the Minimum Circuit Size Problem (MCSP), the fundamental problem of deciding whether a given function (in the form of a truth table) can be computed by small circuits. We get new and improved lower bounds for MCSP that nearly match the best-known lower bounds against several well-studied circuit models such as Boolean formulas and constant-depth circuits
Neural computation of arithmetic functions
A neuron is modeled as a linear threshold gate, and the network architecture considered is the layered feedforward network. It is shown how common arithmetic functions such as multiplication and sorting can be efficiently computed in a shallow neural network. Some known results are improved by showing that the product of two n-bit numbers and sorting of n n-bit numbers can be computed by a polynomial-size neural network using only four and five unit delays, respectively. Moreover, the weights of each threshold element in the neural networks require O(log n)-bit (instead of n -bit) accuracy. These results can be extended to more complicated functions such as multiple products, division, rational functions, and approximation of analytic functions
Functional lower bounds for arithmetic circuits and connections to boolean circuit complexity
We say that a circuit over a field functionally computes an
-variate polynomial if for every we have that . This is in contrast to syntactically computing , when as
formal polynomials. In this paper, we study the question of proving lower
bounds for homogeneous depth- and depth- arithmetic circuits for
functional computation. We prove the following results :
1. Exponential lower bounds homogeneous depth- arithmetic circuits for a
polynomial in .
2. Exponential lower bounds for homogeneous depth- arithmetic circuits
with bounded individual degree for a polynomial in .
Our main motivation for this line of research comes from our observation that
strong enough functional lower bounds for even very special depth-
arithmetic circuits for the Permanent imply a separation between and
. Thus, improving the second result to get rid of the bounded individual
degree condition could lead to substantial progress in boolean circuit
complexity. Besides, it is known from a recent result of Kumar and Saptharishi
[KS15] that over constant sized finite fields, strong enough average case
functional lower bounds for homogeneous depth- circuits imply
superpolynomial lower bounds for homogeneous depth- circuits.
Our proofs are based on a family of new complexity measures called shifted
evaluation dimension, and might be of independent interest
Programmable neural logic
Circuits of threshold elements (Boolean input, Boolean output neurons) have been shown to be surprisingly powerful. Useful functions such as XOR, ADD and MULTIPLY can be implemented by such circuits more efficiently than by traditional AND/OR circuits. In view of that, we have designed and built a programmable threshold element. The weights are stored on polysilicon floating gates, providing long-term retention without refresh. The weight value is increased using tunneling and decreased via hot electron injection. A weight is stored on a single transistor allowing the development of dense arrays of threshold elements. A 16-input programmable neuron was fabricated in the standard 2 μm double-poly, analog process available from MOSIS.
We also designed and fabricated the multiple threshold element introduced in [5]. It presents the advantage of reducing the area of the layout from O(n^2) to O(n); (n being the number of variables) for a broad class of Boolean functions, in particular symmetric Boolean functions such as PARITY.
A long term goal of this research is to incorporate programmable single/multiple threshold elements, as building blocks in field programmable gate arrays
Spectral Norm of Symmetric Functions
The spectral norm of a Boolean function is the sum
of the absolute values of its Fourier coefficients. This quantity provides
useful upper and lower bounds on the complexity of a function in areas such as
learning theory, circuit complexity, and communication complexity. In this
paper, we give a combinatorial characterization for the spectral norm of
symmetric functions. We show that the logarithm of the spectral norm is of the
same order of magnitude as where ,
and and are the smallest integers less than such that
or is constant for all with . We mention some applications to the decision tree and communication
complexity of symmetric functions
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