Polynomial Equalities Research Articles

Synthesizing Invariants for Polynomial Programs by Semidefinite Programming

Constraint-solving-based program invariant synthesis takes a parametric invariant template and encodes the (inductive) invariant conditions into constraints. The problem of characterizing the set of all valid parameter assignments is referred to as the strong invariant synthesis problem , while the problem of finding a concrete valid parameter assignment is called the weak invariant synthesis problem . For both problems, the challenge lies in solving or reducing the encoded constraints, which are generally non-convex and lack efficient solvers. In this paper, we propose two novel algorithms for synthesizing invariants of polynomial programs using semidefinite programming (SDP): (1) The Cluster algorithm targets the strong invariant synthesis problem for polynomial invariant templates. Leveraging robust optimization techniques, it solves a series of SDP relaxations and yields a sequence of increasingly precise under-approximations of the set of valid parameter assignments. We prove the algorithm’s soundness, convergence, and weak completeness under a specific robustness assumption on templates. Moreover, the outputs can simplify the weak invariant synthesis problem. (2) The Mask algorithm addresses the weak invariant synthesis problem in scenarios where the aforementioned robustness assumption does not hold, rendering the Cluster algorithm ineffective. It identifies a specific subclass of invariant templates, termed masked templates, involving parameterized polynomial equalities and known inequalities. By applying variable substitution, the algorithm transforms constraints into an equivalent form amenable to SDP relaxations. Both algorithms have been implemented and demonstrated superior performance compared to state-of-the-art methods in our empirical evaluation.

MIMO Volterra polynomial equalizer for PDM ultrahigh-order QAM signals.

In this work, we proposed and experimentally demonstrated a multiple-input multiple-output (MIMO) Volterra polynomial equalizer (MVPE), in a probabilistic shaping (PS) polarization-division multiplexed (PDM) 4096-quadrature amplitude modulation (QAM) coherent optical transmission system. In comparison with the conventional single-input single-output (SISO) Volterra nonlinear equalizers (VNLEs), the proposed MVPE can equalize the in-phase (I) and quadrature (Q) components in both the X and Y polarizations simultaneously and effectively alleviate nonlinear distortions. In the PS-PDM-4096-QAM experiment, the normalized generalized mutual information (NGMI) reached the threshold of DVB-S2 low-density parity-check (LDPC) codes with a 20% overhead. Compared to the SISO VNLE scheme, the MVPE scheme improves receiver sensitivity by approximately 2.8 dB while maintaining nearly similar computational complexity.

Uni/multi variate polynomial embeddings for zkSNARKs

A zero-knowledge proof is a cryptographic primitive that enables a prover to convince a verifier the validity of a mathematical statement (an NP statement) without revealing any secret inputs to the verifier. A special case, called zero-knowledge Succinct Non-interactive ARgument of Knowledge (zkSNARK) is particularly designed for arithmetic circuit proof systems which have important applications in blockchain privacy. The major computations in this type of zkSNARK proofs with post-quantum security are polynomial evaluations and Lagrange interpolations over finite fields. Given a sequence over a finite field, in the field of coding and sequences research, we understand that there are two representations of the sequence, one is a univariate polynomial and the other, a multivariate polynomial. This is exactly what is done in those zero-knowledge proof systems to transform the proof of a R1CS relation to evaluate uni/multi variate polynomials at some random points in the finite field. In this paper, we present a comparative analysis on how to convert a rank 1 constrained satisfiability (R1CS) system (more general than a circuit system) into a polynomial equality and provide analysis on the concrete complexities of provers, proof sizes and verifiers. We use two concrete zkSNARK schemes, i.e., Polaris, univariate polynomial encodings and Spartan, multivariate polynomial encodings, as examples to show our analysis. Secondly, we propose to select interpolating sets as subfields instead of affine spaces of a large field for Lagrange interpolation. This new method has improved the performance of R1CS encodings largely. We comment that post-quantum secure zkSNARKs yield post-quantum digital signatures with security only depending on symmetric-key schemes. Some open problems are proposed at the end of the paper.

Polynomial Equivalence of Complexity Geometries

This paper proves the polynomial equivalence of a broad class of definitions of quantum computational complexity. We study right-invariant metrics on the unitary group—often called `complexity geometries' following the definition of quantum complexity proposed by Nielsen—and delineate the equivalence class of metrics that have the same computational power as quantum circuits. Within this universality class, any unitary that can be reached in one metric can be approximated in any other metric in the class with a slowdown that is at-worst polynomial in the length and number of qubits and inverse-polynomial in the permitted error. We describe the equivalence classes for two different kinds of error we might tolerate: Killing-distance error, and operator-norm error. All metrics in both equivalence classes are shown to have exponential diameter; all metrics in the operator-norm equivalence class are also shown to give an alternative definition of the quantum complexity class BQP. My results extend those of Nielsen et al., who in 2006 proved that one particular metric is polynomially equivalent to quantum circuits. The Nielsen et al. metric is incredibly highly curved. I show that the greatly enlarged equivalence class established in this paper also includes metrics that have modest curvature. I argue that the modest curvature makes these metrics more amenable to the tools of differential geometry, and therefore makes them more promising starting points for Nielsen's program of using differential geometry to prove complexity lowerbounds. In a previous paper my collaborators and I—inspired by the UV/IR decoupling that happens in the phenomenon of renormalization—conjectured that high- dimensional metrics that look very different at short scales will often nevertheless give rise at long scales to the same emergent effective geometry. The results of this paper provide evidence for those conjectures, since many complexity metrics that have radically different penalty factors and therefore radically different short- distance properties are shown to belong to the same long-distance equivalence class.

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We study the cone of completely positive (cp) matrices for the first interesting case n=5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n = 5$$\\end{document}. This is a semialgebraic set for which the polynomial equalities and inequlities that define its boundary can be derived. We characterize the different loci of this boundary and we examine the two open sets with cp-rank 5 or 6. A numerical algorithm is presented that is fast and able to compute the cp-factorization even for matrices in the boundary. With our results, many new example cases can be produced and several insightful numerical experiments are performed that illustrate the difficulty of the cp-factorization problem.

Products, Polynomials and Differential Equations in the Stream Calculus

We study connections among polynomials, differential equations, and streams over a field 𝕂, in terms of algebra and coalgebra. We first introduce the class of (F,G) - products on streams, those where the stream derivative of a product can be expressed as a polynomial function of the streams and their derivatives. Our first result is that, for every (F,G) -product, there is a canonical way to construct a transition function on polynomials such that the resulting unique final coalgebra morphism from polynomials into streams is the (unique) commutative 𝕂-algebra homomorphism—and vice versa. This implies that one can algebraically reason on streams via their polynomial representation. We apply this result to obtain an algebraic-geometric decision algorithm for polynomial stream equivalence, for an underlying generic (F,G) -product. Finally, we extend this algorithm to solve a more general problem: finding all valid polynomial equalities that fit in a user specified polynomial template.

Simultaneous stoquasticity

Stoquastic Hamiltonians play a role in the computational complexity of the local Hamiltonian problem as well as the study of classical simulability. In particular, stoquastic Hamiltonians can be straightforwardly simulated using Monte Carlo techniques. We address the question of whether two or more Hamiltonians may be made simultaneously stoquastic via a unitary transformation. This question has important implications for the complexity of simulating quantum annealing where quantum advantage is related to the stoquasticity of the Hamiltonians involved in the anneal. We find that for almost all problems no such unitary exists and show that the problem of determining the existence of such a unitary is equivalent to identifying if there is a solution to a system of polynomial (in)equalities in the matrix elements of the initial and transformed Hamiltonians. Solving such a system of equations is NP-hard. We highlight a geometric understanding of this problem in terms of a collection of generalized Bloch vectors.

Isomorphism testing of read-once functions and polynomials

In this paper, we study the isomorphism testing problem of formulas in the Boolean and arithmetic settings. We show that isomorphism testing of read-once formulas is complete for log-space. In contrast, we observe that the problem becomes polynomial-time equivalent to the graph isomorphism problem when the input formulas can be represented as OR of two or more read-once formulas.We address the polynomial isomorphism problem, a special case of polynomial equivalence problem which in turn is important from a cryptographic perspective [Patarin EUROCRYPT '96, and Kayal SODA '11]. As our main result, we propose a deterministic polynomial time canonization scheme for polynomials computed by read-once arithmetic formulas. In contrast, we show that when the arithmetic formula is allowed to read a variable twice, this problem is at least as hard as the graph isomorphism problem.

Error bounds for the solution sets of generalized polynomial complementarity problems

ABSTRACT In this paper, several error bounds for the solution sets of the generalized polynomial complementarity problems (GPCPs) with explicit exponents are given. As the solution set of a GPCP is the solution set of a system of polynomial equalities and inequalities, the state-of-art results in error bounds for polynomial systems can be applied directly. Starting from this, a much better error bound result for the solution set of a GPCP based on exploring the intrinsic sparsity via tensor decomposition is established.

Low-complexity equalizer with a hybrid decision scheme for 50 Gb/s/λ PAM4-PON using a low-cost 10 G receiver.

In this Letter, we experimentally demonstrate a 50Gb/s/λ four-level pulse amplitude modulation-based passive optical network system with a 10 G class receiver. A memory polynomial equalizer (MPE) combined with a decision feedback equalizer (DFE) is applied to eliminate channel distortions in the system. To further improve the performance of the MPE-DFE, for the first time, to the best of our knowledge, a low-complexity hybrid decision scheme (HDS) is proposed, which consists of single-symbol decision (SSD) and multi-symbol decision (MSD). The SSD is exactly the conventional hard decision based on minimum Euclidean distance, whereas MSD is based on a simplified maximum likelihood detection principle with M-algorithm. In terms of complexity, MSD requires 19.1% more multiplications than SSD, but the symbol number of MSD only accounts for less than 20% of the total signal frame when the received optical power is greater than -27dBm. Experimental results show that the proposed MPE-DFE with HDS achieves a 0.7 dB and 1.3 dB sensitivity gain compared with conventional SSD, and up to 35.4 dB and 31.4 dB link power budget, regarding the forward error correction threshold of 10-2 and 10-3, respectively.

Nested covariance determinants and restricted trek separation in Gaussian graphical models

Directed graphical models specify noisy functional relationships among a collection of random variables. In the Gaussian case, each such model corresponds to a semi-algebraic set of positive definite covariance matrices. The set is given via a parametrization, and much work has gone into obtaining an implicit description in terms of polynomial (in)equalities. Implicit descriptions shed light on problems such as parameter identification, model equivalence and constraint-based statistical inference. For models given by directed acyclic graphs, which represent settings where all relevant variables are observed, there is a complete theory: All conditional independence relations can be found via graphical $d$-separation and are sufficient for an implicit description. The situation is far more complicated, however, when some of the variables are hidden (or in other words, unobserved or latent). We consider models associated to mixed graphs that capture the effects of hidden variables through correlated error terms. The notion of trek separation explains when the covariance matrix in such a model has submatrices of low rank and generalizes $d$-separation. However, in many cases, such as the infamous Verma graph, the polynomials defining the graphical model are not determinantal, and hence cannot be explained by $d$-separation or trek-separation. In this paper, we show that these constraints often correspond to the vanishing of nested determinants and can be graphically explained by the (more general) notion of restricted trek separation.

Orthogonal tensor decomposition and orbit closures from a linear algebraic perspective

We study orthogonal decompositions of symmetric and ordinary tensors using methods from linear algebra. For the field of real numbers we show that the sets of decomposable tensors can be defined by equations of degree 2. This gives a new proof of some of the results of Robeva and Boralevi et al. Orthogonal decompositions over the field of complex numbers had not been studied previously; we give an explicit description of the set of decomposable tensors using polynomial equalities and inequalities, and we begin a study of their closures. The main open problem that arises from this work is to obtain a complete description of the closures. This question is akin to that of characterizing border rank of tensors in algebraic complexity. We give partial results using in particular a connection with approximate simultaneous diagonalization (the so-called ASD property).

C-band 56-Gb/s PAM4 transmission over 80-km SSMF with electrical equalization at receiver.

In this paper, a memory polynomial equalizer combined with decision feedback equalizer (MPE-DFE) is proposed to eliminate channel distortions for intensity modulation and direct detection (IM/DD) systems. Compared with traditional feedforward equalizer and decision feedback equalizer (FFE-DFE), the proposed MPE-DFE introduces extra square terms and cubic terms to jointly equalize chromatic dispersion and nonlinear distortions. We demonstrated a C-band 56-Gb/s four-level pulse-amplitude modulation (PAM4) system over 80-km standard single mode fiber (SSMF) transmission. Experimental results show that the proposed MPE-DFE achieved up to 6.2 dB higher SNR than traditional FFE-DFE. Moreover, the achieved bit error ratio (BER) with MPE-DFE reaches 3.1 × 10-3, which is below 7% feedforward error correction (FEC) threshold of 3.8 × 10-3. To the best of our knowledge, we achieved a record transmission distance for C-band 56-Gb/s PAM4 signal with only electrical equalization at the receiver.

Polynomial Equivalence of the Problems “Predicate Formulas Isomorphism and Graph Isomorphism”

The problem of isomorphism checking of two elementary conjunctions of predicate formulas is considered in this work. Such a problem appears while solving some Artificial Intelligence problems, admitting formalization by means of predicate calculus language. The exact definition of the concept of isomorphism of such formulas is given in this paper. However, isomorphic elementary conjunctions of predicate formulas are formulas that, with some substitution of variables instead of their arguments, coincide with the accuracy of the order of writing literals. Problems are described that, when solved, mean the necessity of testing formulas for isomorphism arises. Polynomial equivalence of this problem with the Graph Isomorphism (GI) problem is proved.

Polynomial equivalence of the problems predicate formulas isomorphism and graph isomorphism

Polynomial equivalence of the problems predicate formulas isomorphism and graph isomorphism

On the Complexity of the Vertex 3-Coloring Problem for the Hereditary Graph Classes With Forbidden Subgraphs of Small Size

The 3-coloring problem for a given graph consists in verifying whether it is possible to divide the vertex set of the graph into three subsets of pairwise nonadjacent vertices. A complete complexity classification is known for this problem for the hereditary classes defined by triples of forbidden induced subgraphs, each on at most 5 vertices. In this article, the quadruples of forbidden induced subgraphs is under consideration, each on atmost 5 vertices. For all but three corresponding hereditary classes, the computational status of the 3-coloring problem is determined. Considering two of the remaining three classes, we prove their polynomial equivalence and polynomial reducibility to the third class.

Error Bounds for the Solution Sets of Quadratic Complementarity Problems

In this article, two types of fractional local error bounds for quadratic complementarity problems are established, one is based on the natural residual function and the other on the standard violation measure of the polynomial equalities and inequalities. These fractional local error bounds are given with explicit exponents. A fractional local error bound with an explicit exponent via the natural residual function is new in the tensor/polynomial complementarity problems literature. The other fractional local error bounds take into account the sparsity structures, from both the algebraic and the geometric perspectives, of the third-order tensor in a quadratic complementarity problem. They also have explicit exponents, which improve the literature significantly.

Discovery of statistical equivalence classes using computer algebra

Discrete statistical models supported on labeled event trees can be specified using so-called interpolating polynomials which are generalizations of generating functions. These admit a nested representation which is a notion formalized in this paper. A new algorithm exploits the primary decomposition of monomial ideals associated with an interpolating polynomial to quickly compute all nested representations of that polynomial. It hereby determines an important subclass of all trees representing the same statistical model. To illustrate this method we analyze the full polynomial equivalence class of a staged tree representing the best fitting model inferred from a real-world dataset.

Service system design with immobile servers, stochastic demand and concave-cost capacity selection

The service system design problem is a location-allocation problem with service quality considerations that is often modeled as a network of M/M/1 queues to minimize facility setup, customer access, and waiting costs. Traditionally, capacity decisions are either ignored or modeled as a selection among discrete capacity levels. In this work, we study the general continuous capacity case and account for economies-of-scale in its cost through an increasing concave function. We focus on the special square-root case that has been shown to model capacity in terms of the number of servers needed under Poisson arrivals and exponential service times. The problem is formulated as a mixed-integer nonlinear program with concave and convex terms in the objective function. Two novel resolution approaches are proposed: In the first, the problem is reformulated as a mixed-integer quadratic program with fourth-degree polynomial equality constraints. These constraints and the quadratic objective function are approximated using piecewise-linear segments. In the second, we use Lagrangian relaxation to decompose the problem and reformulate the subproblems as second-order cone programs that are solved at multiple utilization levels. The Lagrangian multipliers are updated using a cutting-plane method and a feasible solution is obtained by solving the corresponding set-covering formulation. The solution approaches are tested and compared. The linearization approach provides high quality solutions within short computational times for small instances and lower accuracy; whereas the Lagrangian approach scales well as size increases.

64-Gb/s SSB-PAM4 Transmission Over 120-km Dispersion-Uncompensated SSMF With Blind Nonlinear Equalization, Adaptive Noise-Whitening Postfilter and MLSD

A novel low-complexity digital signal processing solution is presented to compensate the impairments of limited bandwidth and nonlinearity. It is based on the equalizer, adaptive noise-whitening postfilter, and maximum likelihood sequence detection (MLSD). The system performance loss is induced by higher noise power at the signal band edges after the equalizer can be mitigated by the postfilter and MLSD. Given the above structure, it is possible to enrich the equalizer by providing nonlinear compensation capability. A memory polynomial equalizer (MPE), instead of Volterra equalizer (VE), is applied as the equalizer to compensate the nonlinearity impairments in order to make a tradeoff between complexity and performance. For the adaption of MPE, the blindly adaptive multistep-size decision-directed least-mean-square algorithm is selected. By using a dual-drive Mach–Zehnder and direct detection, 64-Gb/s single-sideband 4-ary pulse amplitude modulation (SSB-PAM4) transmission over 120-km dispersion-uncompensated standard single-mode fiber (SSMF) with 10-dB bandwidth roughly 13.5 GHz is experimentally demonstrated. Given a 20% overhead soft-decision forward-error correction with bit error rate threshold of $2\times 10^{{-2}}$ , the dispersion-uncompensated SSMF transmission distance can be significantly increased from 40 to 120 km with the proposed receiver side solution. The optical signal noise ratio performances for different fiber reach and receiver structure are investigated. Furthermore, we compare the computational complexity of MPE with VE and conclude that MPE can substantially reduce computation complexity with negligible performance loss.