scholarly journals An Algebraic Model for Bounding Threshold Circuit Depth

1988 ◽  
Vol 17 (239) ◽  
Author(s):  
Joan Boyar ◽  
Gudmund Skovbjerg Frandsen ◽  
Carl Sturtivant
Keyword(s):  
Threshold Model ◽  
Algebraic Model ◽  
Constant Factor ◽  
Arbitrary Integer ◽  
Arithmetic Circuit ◽  
Threshold Circuit ◽  
Polynomial Size ◽  
Algebraic Laws ◽  
Threshold Circuits ◽  
Circuit Depth

We define a new structured and general model of computation: circuits using arbitrary fan- in arithmetic gates over the characteristic two finite fields (<strong>F</strong>_2n). These circuits have only one input and one output. We show how they correspond naturally to boolean computations with n inputs and n outputs. We show that if circuit sizes are polynomially related then the arithmetic circuit depth and the threshold circuit depth to compute a given function differ by at most a constant factor. We use threshold circuits that allow arbitrary integer weights; however, we show that when compared to the usual threshold model, the depth measure of this generalised model only differs by at most a constant factor (at polynomial size). The fan-in of our arithmetic model is also unbounded in the most generous sense: circuit size is measured as the number of Sum and ½ gates; there is no bound on the number of ''wires'' . We show that these results are provable for any ''reasonable'' correspondance between bit strings of n-bits and elements of <strong>F</strong>_ 2n. And, we find two distinct characterizations of ''reasonable''. Thus, we have shown that arbitrary fan-in arithmetic computations over <strong>F</strong>_ 2n constitute a precise abstraction of boolean threshold computations with the pleasant property that various algebraic laws have been recovered.

On the Computational Power of Threshold Circuits with Sparse Activity

2006 ◽  
Vol 18 (12) ◽  
pp. 2994-3008 ◽  
Author(s):  
Kei Uchizawa ◽  
Rodney Douglas ◽  
Wolfgang Maass
Keyword(s):  
Neural Circuits ◽  
Circuit Complexity ◽  
Complexity Measure ◽  
Computational Power ◽  
Complexity Measures ◽  
Threshold Circuit ◽  
Polynomial Size ◽  
Threshold Circuits ◽  
Size Threshold ◽  
The Brain

Circuits composed of threshold gates (McCulloch-Pitts neurons, or perceptrons) are simplified models of neural circuits with the advantage that they are theoretically more tractable than their biological counterparts. However, when such threshold circuits are designed to perform a specific computational task, they usually differ in one important respect from computations in the brain: they require very high activity. On average every second threshold gate fires (sets a 1 as output) during a computation. By contrast, the activity of neurons in the brain is much sparser, with only about 1% of neurons firing. This mismatch between threshold and neuronal circuits is due to the particular complexity measures (circuit size and circuit depth) that have been minimized in previous threshold circuit constructions. In this letter, we investigate a new complexity measure for threshold circuits, energy complexity, whose minimization yields computations with sparse activity. We prove that all computations by threshold circuits of polynomial size with entropy O(log n) can be restructured so that their energy complexity is reduced to a level near the entropy of circuit states. This entropy of circuit states is a novel circuit complexity measure, which is of interest not only in the context of threshold circuits but for circuit complexity in general. As an example of how this measure can be applied, we show that any polynomial size threshold circuit with entropy O(log n) can be simulated by a polynomial size threshold circuit of depth 3. Our results demonstrate that the structure of circuits that result from a minimization of their energy complexity is quite different from the structure that results from a minimization of previously considered complexity measures, and potentially closer to the structure of neural circuits in the nervous system. In particular, different pathways are activated in these circuits for different classes of inputs. This letter shows that such circuits with sparse activity have a surprisingly large computational power.

Some Results on Uniform Arithmetic Circuit Complexity

1991 ◽  
Vol 20 (343) ◽  
Author(s):  
Gudmund Skovbjerg Frandsen ◽  
Mark Valence ◽  
David Mix Barrington
Keyword(s):  
Arithmetic Function ◽  
Circuit Complexity ◽  
Arithmetic Circuits ◽  
Arithmetic Functions ◽  
Constant Depth ◽  
Arithmetic Circuit ◽  
Arithmetic Complexity ◽  
Polynomial Size ◽  
Threshold Circuits ◽  
Class Of Functions

We introduce a natural set of arithmetic expressions and define the complexity class AE to consist of all those arithmetic functions (over the fields F_(2)n) that are described by these expressions. We show that AE coincides with the class of functions that are computable with constant depth and polynomial size unbounded fan-in arithmetic circuits satisfying a natural uniformity constraint (DLOGTIME-uniformity). A 1-input and 1-output arithmetic function over the fields F_(2)n may be identified with an <em>n</em>-input and an n-output Boolean function when field elements are represented as bit strings. We prove that if some such representation is X-uniform (where X is P or DLOGTIME) then the arithmetic complexity of a function (measured with X-uniform unbounded fan-in arithmetic circuits) is identical to the Boolean complexity of this function (measured with X-uniform threshold circuits). We show the existence of a P-uniform representation and we give partial results concerning the existence of representations with more restrictive uniformity properties.

scholarly journals Uniform derandomization from pathetic lower bounds

2012 ◽  
Vol 370 (1971) ◽  
pp. 3512-3535 ◽  
Author(s):  
Eric Allender ◽  
V. Arvind ◽  
Rahul Santhanam ◽  
Fengming Wang
Keyword(s):  
Word Problem ◽  
Lower Bounds ◽  
Black Box ◽  
Arithmetic Circuits ◽  
List Type ◽  
Probabilistic Algorithms ◽  
Constant Depth ◽  
Subexponential Time ◽  
Polynomial Size ◽  
Threshold Circuits

The notion of probabilistic computation dates back at least to Turing, who also wrestled with the practical problems of how to implement probabilistic algorithms on machines with, at best, very limited access to randomness. A more recent line of research, known as derandomization, studies the extent to which randomness is superfluous. A recurring theme in the literature on derandomization is that probabilistic algorithms can be simulated quickly by deterministic algorithms, if one can obtain impressive (i.e. superpolynomial, or even nearly exponential) circuit size lower bounds for certain problems. In contrast to what is needed for derandomization, existing lower bounds seem rather pathetic. Here, we present two instances where ‘pathetic’ lower bounds of the form n 1+ ϵ would suffice to derandomize interesting classes of probabilistic algorithms. We show the following: — If the word problem over S 5 requires constant-depth threshold circuits of size n 1+ ϵ for some ϵ >0, then any language accepted by uniform polynomial size probabilistic threshold circuits can be solved in subexponential time (and, more strongly, can be accepted by a uniform family of deterministic constant-depth threshold circuits of subexponential size). — If there are no constant-depth arithmetic circuits of size n 1+ ϵ for the problem of multiplying a sequence of n  3×3 matrices, then, for every constant d , black-box identity testing for depth- d arithmetic circuits with bounded individual degree can be performed in subexponential time (and even by a uniform family of deterministic constant-depth AC 0 circuits of subexponential size).

scholarly journals Rewriting Environment for Arithmetic Circuit Verification

10.29007/rswk ◽  
2018 ◽  
Author(s):  
Cunxi Yu ◽  
Atif Yasin ◽  
Tiankai Su ◽  
Alan Mishchenko ◽  
Maciej Ciesielski
Keyword(s):  
Arithmetic Function ◽  
Software Tool ◽  
Logic Gates ◽  
Algebraic Model ◽  
Arithmetic Circuit ◽  
Circuit Verification ◽  
Functional Specification ◽  
Different Types ◽  
Polynomial Models ◽  
General Circuits

The paper describes a practical software tool for the verification of integer arithmetic circuits. It covers different types of integer multipliers, fused add-multiply circuits, and constant dividers - in general, circuits whose computation can be represented as a polynomial. The verification uses an algebraic model of the circuit and is accomplished by rewriting the polynomial of the binary encoding of the primary outputs (output signature), using the polynomial models of the logic gates, into a polynomial over the primary inputs (input signature). The resulting polynomial represents arithmetic function implemented by the circuit and hence can be used to extract functional specification from its gate-level implementation. The rewriting uses an efficient And-Inverter Graph (AIG) representation to enable extraction of the essential arithmetic components of the circuit. The tool is integrated with the popular ABC system. Its efficiency is illustrated with impressive results for integer multipliers, fused add-multiply circuits, and divide-by-constant circuits. The entire verification system is offered in an open source ABC environment together with an extensive set of benchmarks.

scholarly journals Pseudo-dimension of quantum circuits

2020 ◽  
Vol 2 (2) ◽  
Author(s):  
Matthias C. Caro ◽  
Ishaun Datta
Keyword(s):  
Probability Distributions ◽  
Expressive Power ◽  
Upper Bounds ◽  
Quantum Circuits ◽  
Polynomial Size ◽  
Probabilistic Concept ◽  
Circuit Depth ◽  
Concept Classes ◽  
Circuit Output ◽  
Output Probability

AbstractWe characterize the expressive power of quantum circuits with the pseudo-dimension, a measure of complexity for probabilistic concept classes. We prove pseudo-dimension bounds on the output probability distributions of quantum circuits; the upper bounds are polynomial in circuit depth and number of gates. Using these bounds, we exhibit a class of circuit output states out of which at least one has exponential gate complexity of state preparation, and moreover demonstrate that quantum circuits of known polynomial size and depth are PAC-learnable.

scholarly journals Counting, fanout and the complexity of quantum ACC

2002 ◽  
Vol 2 (1) ◽  
pp. 35-65
Author(s):  
F. Green ◽  
S. Homer ◽  
C. Moore ◽  
C. Pollett
Keyword(s):  
Complexity Classes ◽  
Constant Depth ◽  
Classical Counterpart ◽  
Classical Class ◽  
Polynomial Size ◽  
Constant Depth Circuits ◽  
Language Classes ◽  
Threshold Circuits ◽  
Quantum Analog ◽  
Quantum Complexity

We propose definitions of QAC^0, the quantum analog of the classical class AC^0 of constant-depth circuits with AND and OR gates of arbitrary fan-in, and QACC[q], the analog of the class ACC[q] where Mod_q gates are also allowed. We prove that parity or fanout allows us to construct quantum MOD_q gates in constant depth for any q, so QACC[2] = QACC. More generally, we show that for any q,p > 1, MOD_q is equivalent to MOD_p (up to constant depth and polynomial size). This implies that QAC^0 with unbounded fanout gates, denoted QACwf^0, is the same as QACC[q] and QACC for all q. Since \ACC[p] \ne ACC[q] whenever p and q are distinct primes, QACC[q] is strictly more powerful than its classical counterpart, as is QAC^0 when fanout is allowed. This adds to the growing list of quantum complexity classes which are provably more powerful than their classical counterparts. We also develop techniques for proving upper bounds for QACC in terms of related language classes. We define classes of languages closely related to QACC[2] and show that restricted versions of them can be simulated by polynomial-size circuits. With further restrictions, language classes related to QACC[2] operators can be simulated by classical threshold circuits of polynomial size and constant depth.

scholarly journals Constant-optimized quantum circuits for modular multiplication and exponentiation

2012 ◽  
Vol 12 (5&6) ◽  
pp. 361-394
Author(s):  
Igor L. Markov ◽  
Mehdi Saeedi
Keyword(s):  
Laboratory Experiments ◽  
Constant Factor ◽  
Quantum Circuits ◽  
Modular Exponentiation ◽  
Modular Multiplication ◽  
Worst Case ◽  
Gate Count ◽  
Shor's Algorithm ◽  
Circuit Depth ◽  
Additive Term

Reversible circuits for modular multiplication $Cx\%M$ with $x<M$ arise as components of modular exponentiation in Shor's quantum number-factoring algorithm. However, existing generic constructions focus on asymptotic gate count and circuit depth rather than actual values, producing fairly large circuits not optimized for specific $C$ and $M$ values. In this work, we develop such optimizations in a bottom-up fashion, starting with most convenient $C$ values. When zero-initialized ancilla registers are available, we reduce the search for compact circuits to a shortest-path problem. Some of our modular-multiplication circuits are asymptotically smaller than previous constructions, but worst-case bounds and average sizes remain $\Theta(n^2)$. In the context of modular exponentiation, we offer several constant-factor improvements, as well as an improvement by a constant additive term that is significant for few-qubit circuits arising in ongoing laboratory experiments with Shor's algorithm.

ENERGY-EFFICIENT THRESHOLD CIRCUITS COMPUTING MOD FUNCTIONS

2013 ◽  
Vol 24 (01) ◽  
pp. 15-29 ◽  
Author(s):  
AKIRA SUZUKI ◽  
KEI UCHIZAWA ◽  
XIAO ZHOU
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Energy Efficient ◽  
Extreme Case ◽  
Modulus Function ◽  
Threshold Circuit ◽  
Threshold Circuits

We prove that the modulus function MODm of n variables can be computed by a threshold circuit C of energy e and size s = O(e(n/m)1/(e − 1)) for any integer e ≥ 2, where the energy e is defined to be the maximum number of gates outputting "1" over all inputs to C, and the size s to be the number of gates in C. Our upper bound on the size s almost matches the known lower bound s = Ω(e(n/m)1/e). We also consider an extreme case where threshold circuits have energy 1, and prove that such circuits need at least 2(n − m)/2 gates to compute MODm of n variables.

Hyperstimulation of the immune system as a cause of autoimmune diseases

2020 ◽  
Vol 75 (3) ◽  
pp. 204-213
Author(s):  
Varvara A. Ryabkova ◽  
Leonid P. Churilov ◽  
Yehuda Shoenfeld
Keyword(s):  
Immune System ◽  
Autoimmune Diseases ◽  
Genetic Basis ◽  
Threshold Model ◽  
Subsequent Development ◽  
Threshold Effect ◽  
Successful Management ◽  
Polygenic Inheritance ◽  
Autoimmune Pathology ◽  
The Common

The pathogenesis of autoimmune diseases is very complex and multi-factorial. The concept of Mosaics of Autoimmunity was introduced to the scientific community 30 years ago by Y. Shoenfeld and D.A. Isenberg, and since then new tiles to the puzzle are continuously added. This concept specifies general pathological ideas about the multifactorial threshold model for polygenic inheritance with a threshold effect by the action of a number of external causal factors as applied to the field of autoimmunology. Among the external factors that can excessively stimulate the immune system, contributing to the development of autoimmune reactions, researchers are particularly interested in chemical substances, which are widely used in pharmacology and medicine. In this review we highlight the autoimmune dynamics i.e. a multistep pathogenesis of autoimmune diseases and the subsequent development of lymphoma in some cases. In this context several issues are addressed namely, genetic basis of autoimmunity; environmental immunostimulatory risk factors; gene/environmental interaction; pre-clinical autoimmunity with the presence of autoantibodies; and the mechanisms, underlying lymphomagenesis in autoimmune pathology. We believe that understanding the common model of the pathogenesis of autoimmune diseases is the first step to their successful management.

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