Disproving the Single Level Conjecture

It is well known that among the three classes of the PRAM models, namely, CRCW, CREW, and EREW, the CRCW models are the weakest, in the sense that, they permit concurrent read/write by processors. Accordingly, algorithms on the CRCW model mainly concentrate on the core computations without much ado about data access. Consequently, this model, at least in principle, allows for the design of the fastest algorithm for a problem. It is intriguing to ask how fast prefixes can be computed on the CRCW models. Since CRCW models are equivalent to the unbounded fan-in circuits (refer to Chapter 2), the task of developing the fastest algorithms for the prefix problems is pursued in the context of the unbounded fan-in circuits. Recall from Chapter 2, that while the standard measures, such as, size and depth are still used to quantify the goodness of unbounded fanin circuits, the size of the circuit is measured by the total number of edges incident on all of its operation nodes, instead of by the number of operations nodes. It turns out that the size and depth of unbounded fanin circuits for computing prefixes, depends critically on the structure of the underlying semigroup from which the input elements are drawn. The principal result of this concluding Chapter may be stated as follows: There exists unbounded fan-in parallel prefix circuits of constant depth and polynomial size if, and only if, the underlying semigroup is group free. The proof of this result involves a very clever synthesis of a number of ideas drawn from different directions — structure of group free semigroups, their relations to a special class of regular sets, called non-counting regular sets, the relation of this latter class of regular sets to yet another class of regular sets defined by star-free regular expressions, and the design of a special class of finite state deterministic automata called RS machines that accept star-free regular expressions. In this context, it is convenient to define the notion of small circuits as the class of circuits with constant depth and polynomial size.

Quadratization of symmetric pseudo-Boolean functions

The theory of spin-wave resonance in gradient ferromagnetic films with magnetic parameters varying in space described by both concave and convex quadratic functions is developed. Gradient structures such as a potential well, a potential barrier, and a monotonic change in potential between the film surfaces for both quadratic functions are considered. The waveforms of oscillations mn(z), the laws of the dependence of discrete frequencies ωn, and relative susceptibilities χn/χ10 of spin-wave resonances on the resonance number n are studied. It is shown that the law ωn∝n for n<nc, where nc is the resonance level near the upper edge of the gradient inhomogeneity, which is well known for a parabolic potential well, is also valid for the potential barrier and for the monotonic change in potential, if these structures are formed by a concave quadratic function. It is shown that the law ωn∝(n−1/2)1/2, which we numerically derived and approximated by the analytical formula, is valid for all three structures formed by a convex quadratic function. It is shown that the magnetic susceptibility χn of spin-wave resonances for n<nc is much greater than the susceptibility of resonances in a uniform film. An experimental study of both laws ωn(n) and χn(n) would allow one to determine the type of quadratic function that formed the gradient structure and the form of this structure. The possibility of creating gradient films with different laws ωn(n) and the high magnitude of the high-frequency magnetic susceptibility χn(n) at n<nc make these metamaterials promising for practical applications.

Quadratic response functions are derived and implemented for a vibrational configuration interaction state. Combined electronic and vibrational quadratic response functions are derived using Born-Oppenheimer vibronic product wave functions. Computational tractable expressions are derived for determining the total quadratic response contribution as a sum of contributions involving both electronic and vibrational linear and quadratic response functions. In the general frequency-dependent case this includes a new and more troublesome type of electronic linear response function. Pilot calculations for the FH, H(2)O, CH(2)O, and pyrrole molecules demonstrate the importance of vibrational contributions for accurate comparison to experiment and that the vibrational contributions in some cases can be very large. The calculation of transition properties between vibrational states is combined with sum-over-states expressions for analysis purposes. On the basis of this some simple analysis methods are suggested. Also, a preliminary study of the effect of finite lifetimes on quadratic response functions is presented.

The present paper extends previous findings already published where the relationships among total porosity, bulk density, compressive strength, and thermal conductivity were analysed. The paper brings more accuracy into some original mathematical relationships and takes into account two additional physical properties (pore volume and the median pore radius). It has been found that the relationships between physical properties and a dose of any admixture added to brick clay could be better described using a quadratic function. The relationships between pairs of physical characteristics can be described by three types of mathematical function (linear, exponential, and power). The amount of admixture used, the firing temperature, and type of clay affect the value of the coefficients but not the type of functions.

Design and simulation of Full-Subtractor based on Quantum-Dot cellular automata technology

We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in mathsf {Quasi}text {-}mathsf {NP} = mathsf {NTIME}[n^{(log n)^{O(1)}}] and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes mathcal { C}, by showing that mathcal { C} admits non-trivial satisfiability and/or # SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a non-trivial # SAT algorithm for a circuit class {mathcal C}. Say that a symmetric Boolean function f(x1,…,xn) is sparse if it outputs 1 on O(1) values of {sum }_{i} x_{i}. We show that for every sparse f, and for all “typical” mathcal { C}, faster # SAT algorithms for mathcal { C} circuits imply lower bounds against the circuit class f circ mathcal { C}, which may be stronger than mathcal { C} itself. In particular: # SAT algorithms for nk-size mathcal { C}-circuits running in 2n/nk time (for all k) imply NEXP does not have (f circ mathcal { C})-circuits of polynomial size.# SAT algorithms for 2^{n^{{varepsilon }}}-size mathcal { C}-circuits running in 2^{n-n^{{varepsilon }}} time (for some ε > 0) imply Quasi-NP does not have (f circ mathcal { C})-circuits of polynomial size.Applying # SAT algorithms from the literature, one immediate corollary of our results is that Quasi-NP does not have EMAJ ∘ ACC0 ∘ THR circuits of polynomial size, where EMAJ is the “exact majority” function, improving previous lower bounds against ACC0 [Williams JACM’14] and ACC0 ∘THR [Williams STOC’14], [Murray-Williams STOC’18]. This is the first nontrivial lower bound against such a circuit class.

We exhibit supercritical trade-off for monotone circuits, showing that there are functions computable by small circuits for which any small circuit must have depth superlinear or even super-polynomial in the number of variables, far exceeding the linear worst-case upper bound. We obtain similar trade-offs in proof complexity, where we establish the first size-depth trade-offs for cutting planes and resolution that are truly supercritical, i.e., in terms of formula size rather than number of variables, and also show supercritical trade-offs between width and size for treelike resolution. Our results build on a new supercritical width-depth trade-off for resolution, obtained by refining and strengthening the compression scheme for the cop-robber game in [Grohe, Lichter, Neuen & Schweitzer 2023]. This yields robust supercritical trade-offs for dimension versus iteration number in the Weisfeiler-Leman algorithm, which also translate into trade-offs between number of variables and quantifier depth in first-order logic. Our other results follow from improved lifting theorems that might be of independent interest.

In recent years, the size and complexity of digital circuits have grown drastically. Consequently, bugs may appear in a circuit in different phases of design and synthesis. If they remain undetected, they propagate to the physical chip and cause huge financial loss for the designers and manufacturers. Formal verification is an important task after design to ensure the correctness of a circuit. Recently, polynomial formal verification has gained special attention. If we prove that the space and time complexity of a verification method is polynomial, we can ensure that it is always scalable. Many digital circuits which are used in different applications have a tree-like structure. It has been proven that if a tree-like circuit is made of basic logic gates (AND, OR, NOT, NAND, and NOR), it can be verified polynomially using BDDs. However, these proofs cannot be extended to the general tree-like circuits containing XOR gates. In this paper, we propose a method to prove the correctness of a general tree-like circuit in polynomial time.

In recent years there has been a flurry of activity proving lower bounds for homogeneous depth-4 arithmetic circuits, which has brought us very close to statements that are known to imply VP != VNP. It is a big question to go beyond homogeneity, and in this paper we make progress towards this by considering depth-4 circuits of low algebraic rank, which are a natural extension of homogeneous depth-4 arithmetic circuits. A depth-4 circuit is a representation of an N-variate, degree n polynomial P as P = sum_{i=1}^T Q_{i1} * Q_{i2} * ... * Q_{it} where the Q_{ij} are given by their monomial expansion. Homogeneity adds the constraint that for every i in [T], sum_{j} degree(Q_{ij}) = n. We study an extension where, for every i in [T], the algebraic rank of the set of polynomials {Q_{i1}, Q_{i2}, ... ,Q_{it}} is at most some parameter k. We call this the class of spnew circuits. Already for k=n, these circuits are a strong generalization of the class of homogeneous depth-4 circuits, where in particular t<=n (and hence k<=n). We study lower bounds and polynomial identity tests for such circuits and prove the following results. 1. Lower bounds: We give an explicit family of polynomials {P_n} of degree n in N = n^{O(1)} variables in VNP, such that any spnewn circuit computing P_n has size at least exp{(Omega(sqrt(n)*log(N)))}. This strengthens and unifies two lines of work: it generalizes the recent exponential lower bounds for homogeneous depth-4 circuits [KLSS14, KS-full] as well as the Jacobian based lower bounds of Agrawal et al. which worked for spnew circuits in the restricted setting where T * k <= n. 2. Hitting sets: Let spnewbounded be the class of spnew circuits with bottom fan-in at most d. We show that if d and k are at most poly(log(N)), then there is an explicit hitting set for spnewbounded circuits of size quasipolynomial in N and the size of the circuit. This strengthens a result of Forbes which showed such quasipolynomial sized hitting sets in the setting where d and t are at most poly(log(N)). A key technical ingredient of the proofs is a result which states that over any field of characteristic zero (or sufficiently large characteristic), upto a translation, every polynomial in a set of algebraically dependent polynomials can be written as a function of the polynomials in the transcendence basis. We believe this may be of independent interest. We combine this with shifted partial derivative based methods to obtain our final results.

Understanding the structure–function relationship in a neuronal network is one of the major challenges in neuroscience research. Despite increasing researches at circuit connectivity and neural network structure, their structure-based biological interpretability remains unclear. Based on the attractor theory, here we develop an analytical framework that links neural circuit structures and their functions together through fixed point attractor in Caenorhabditis elegans. In this framework, we successfully established the structural condition for the emergence of multiple fixed points in C. elegans connectome. Then we construct a finite state machine to explain how functions related to bistable phenomena at the neural activity and behavioral levels are encoded. By applying the proposed framework to the command circuit in C. elegans, we provide a circuit level interpretation for the forward-reverse switching behaviors. Interestingly, network properties of the command circuit and first layer amphid interneuron circuit can also be inferred from their functions in this framework. Our research indicates the reliability of the fixed point attractor bridging circuit structure and functions, suggesting its potential applicability to more complex neuronal circuits in other species.

Since the advent of the computer, many circuit analysis programs have been developed for the purpose of eliminating much of the expensive breadboarding associated with a design. However, a large class of circuits cannot be analyzed by the programs and hence many circuits are still being evaluated only by breadboard methods. A much larger class of circuits could be analyzed by the programs if the limitations on circuit size, input, and output were removed. Described in this paper is a new non-linear d-c circuit analysis program called TESS-DC, which was designed specifically to analyze a much larger class of d-c problems. Included in this paper is the description as well as an example of the program's input language. The program's capabilities are presented and a number of flow diagrams provided to illustrate the operation of the program.

As we have pointed out in the preceding chapters, in deterministic global optimization it is of utmost importance to take advantage of the particular mathematical structure of the problem under study. There are in general two aspects of nonconvexity deserving special attention. First, the rank of nonconvexity, i.e. roughly speaking, the number of nonconvex variables. In Chapter 7 we have discussed decomposition methods for handling low rank nonconvex problems. The second aspect is the degree of nonconvexity, i.e. the extent to which the variables are nonconvex. This last Chapter is devoted to nonconvex optimization problems which involve only linear or quadratic functions, i.e. in a sense functions with lowest degree of nonconvexity.KeywordsQuadratic ProgramConvex EnvelopeNonconvex FunctionNonconvex Quadratic ProgramBilinear ProgramThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Location‐only and use‐availability data: analysis methods converge

We investigate about a lower bound on the size of a Boolean circuit that computes the clique function with a limited number of negation gates. To derive strong lower bounds on the size of such a circuit we develop a new approach by combining the three approaches: the restriction applied for constant depth circuits[Has], the approximation method applied for monotone circuits[Raz2] and boundary covering developed in the present paper. Based on the approach the following statement is established: If a circuit C with at most ⌈(1/6) log log m⌋ negation gates detects cliques of size \((\log m)^{3(\log {\mathbf{ }}m)^{{\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}} }\) in a graph with m vertices, then C contains at least \(2^{({\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 5$}})(\log {\mathbf{ }}m)^{(\log {\mathbf{ }}m)^{{\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}} } }\) gates. In addition, we present a general relationship between negation-limited circuit size and monotone circuit size of an arbitrary monotone function.