Polynomial Calculus Research Articles

A Generalized Method for Proving Polynomial Calculus Degree Lower Bounds

We study the problem of obtaining lower bounds for polynomial calculus (PC) and polynomial calculus resolution (PCR) on proof degree, and hence by [Impagliazzo et al. ’99] also on proof size. [Alekhnovich and Razborov’03] established that if the clause-variable incidence graph of a conjunctive normal form (CNF) formula F is a good enough expander, then proving that F is unsatisfiable requires high PC/PCR degree. We further develop their techniques to show that if one can “cluster” clauses and variables in a way that “respects the structure” of the formula in a certain sense, then it is sufficient that the incidence graph of this clustered version is an expander. We also show how a weaker structural condition is sufficient to obtain lower bounds on width for the resolution proof system, and give a unified treatment that highlights similarities and differences between resolution and polynomial calculus (PC) lower bounds. As a corollary of our main technical theorem, we prove that the functional pigeonhole principle (FPHP) formulas require high PC/PCR degree when restricted to constant-degree expander graphs. This answers an open question in [Razborov’02], and also implies that the standard CNF encoding of the FPHP formulas require exponential proof size in polynomial calculus resolution (PCR). Thus, while onto-FPHP formulas are easy for polynomial calculus, as shown in [Riis’93], both FPHP and onto-PHP formulas are hard even when restricted to bounded-degree expanders.

Cutting Planes Width and the Complexity of Graph Isomorphism Refutations

The width complexity measure plays a central role in resolution and other propositional proof systems like Polynomial Calculus (under the name of degree). The study of width lower bounds is the most used method for proving size lower bounds, and it is known that for the mentioned proof systems, proofs with small width also imply the existence of proofs with small size. Not much has been studied, however, about the width parameter in the cutting planes (CP) proof system, a measure that was introduced by Dantchev and Martin in 2009 under the name of CP cutwidth. In this article, we study the width complexity of CP refutations of graph isomorphism formulas. For a pair of non-isomorphic graphs \(G\) and \(H\) , we show a direct connection between the Weisfeiler–Leman differentiation number \(\mathsf{WL}(G,H)\) of the graphs and the width of a CP refutation for the corresponding isomorphism formula \(\mathrm{Iso}(G,H)\) . In particular, we show that if \(\mathsf{WL}(G,H)\leq k\) , then there is a CP refutation of \(\mathrm{Iso}(G,H)\) with width \(k\) , and if \(\mathsf{WL}(G,H) \gt k\) , then there are no CP refutations of \(\mathrm{Iso}(G,H)\) with width \(k-2\) . Similar results are known for other proof systems, like Resolution, Sherali–Adams, or Polynomial Calculus. We also obtain polynomial-length CP refutations from our width bound for isomorphism formulas for graphs with constant Weisfeiler–Leman dimension. Furthermore, we notice that a length lower bound for refuting graph isomorphism formulas in the subsystem of tree-like cutting planes with polynomially bounded coefficients follows from known results.

Polynomial calculus for optimization

MaxSAT is the problem of finding an assignment satisfying the maximum number of clauses in a CNF formula. We consider a natural generalization of this problem to generic sets of polynomials and propose a weighted version of Polynomial Calculus to address this problem.Weighted Polynomial Calculus is a natural generalization of the systems MaxSAT-Resolution and weighted Resolution. Unlike such systems, weighted Polynomial Calculus manipulates polynomials with coefficients in a finite field and either weights in N or Z. We show the soundness and completeness of weighted Polynomial Calculus via an algorithmic procedure.Weighted Polynomial Calculus, with weights in N and coefficients in F2, is able to prove efficiently that Tseitin formulas on a connected graph are minimally unsatisfiable. Using weights in Z, it also proves efficiently that the Pigeonhole Principle is minimally unsatisfiable.

First-order reasoning and efficient semi-algebraic proofs

Semi-algebraic proof systems such as sum-of-squares (SoS) have attracted a lot of attention due to their relation to approximation algorithms: constant degree semi-algebraic proofs lead to conjecturally optimal polynomial-time approximation algorithms for important NP-hard optimization problems. Motivated by the need to allow a more streamlined and uniform framework for working with SoS proofs than the restrictive propositional level, we initiate a systematic first-order logical investigation into the kinds of reasoning possible in algebraic and semi-algebraic proof systems. Specifically, we develop first-order theories that capture in a precise manner constant degree algebraic and semi-algebraic proof systems: every statement of a certain form that is provable in our theories translates into a family of constant degree polynomial calculus or SoS refutations, respectively; and using a reflection principle, the converse also holds.This places algebraic and semi-algebraic proof systems in the established framework of bounded arithmetic, while providing theories corresponding to systems that vary quite substantially from the usual propositional-logic ones.We give examples of how our semi-algebraic theory proves statements such as the pigeonhole principle, we provide a separation between algebraic and semi-algebraic theories, and we describe initial attempts to go beyond these theories by introducing extensions that use the inequality symbol, identifying along the way which extensions lead outside the scope of constant degree SoS. Moreover, we prove new results for propositional proofs, and specifically extend Berkholz's dynamic-by-static simulation of polynomial calculus (PC) by SoS to PC with the radical rule.

On vanishing sums of roots of unity in polynomial calculus and sum-of-squares

We introduce a novel take on sum-of-squares that is ableto reason with complex numbers and still make use of polynomial inequalities.This proof system might be of independent interest since itallows to represent multivalued domains both with Boolean and Fourierencoding. We show degree and size lower bounds in this system for anatural generalization of knapsack: the vanishing sums of roots of unity.These lower bounds naturally apply to polynomial calculus as-well.

Space characterizations of complexity measures and size-space trade-offs in propositional proof systems

We identify two new clusters of proof complexity measures equal up to polynomial and log⁡n factors. The first cluster contains the logarithm of tree-like resolution size, regularized clause and monomial space, and clause space, ordinary and regularized, in regular and tree-like resolution. Consequently, separating clause or monomial space from the logarithm of tree-like resolution size is equivalent to showing strong trade-offs between clause space and length, and equivalent to showing super-critical trade-offs between clause space and depth. The second cluster contains width, Σ2 space (a generalization of clause space to depth 2 Frege systems), ordinary and regularized, and the logarithm of tree-like R(log⁡) size. As an application, we improve a known size-space trade-off for polynomial calculus with resolution. We further show a quadratic lower bound on tree-like resolution size for formulas refutable in clause space 4, and introduce a measure intermediate between depth and the logarithm of tree-like resolution size.

On Proof Complexity of Resolution over Polynomial Calculus

The proof system Res (PC d,R ) is a natural extension of the Resolution proof system that instead of disjunctions of literals operates with disjunctions of degree d multivariate polynomials over a ring R with Boolean variables. Proving super-polynomial lower bounds for the size of Res ( PC 1, R )-refutations of Conjunctive normal forms (CNFs) is one of the important problems in propositional proof complexity. The existence of such lower bounds is even open for Res ( PC 1,𝔽 ) when 𝔽 is a finite field, such as 𝔽 2 . In this article, we investigate Res ( PC d,R ) and tree-like Res ( PC d,R ) and prove size-width relations for them when R is a finite ring. As an application, we prove new lower bounds and reprove some known lower bounds for every finite field 𝔽 as follows: (1) We prove almost quadratic lower bounds for Res ( PC d ,𝔽)-refutations for every fixed d . The new lower bounds are for the following CNFs: (a) Mod q Tseitin formulas ( char (𝔽)≠ q ) and Flow formulas, (b) Random k -CNFs with linearly many clauses. (2) We also prove super-polynomial (more than n k for any fixed k ) and also exponential (2 nϵ for an ϵ > 0) lower bounds for tree-like Res ( PC d ,𝔽 )-refutations based on how big d is with respect to n for the following CNFs: (a) Mod q Tseitin formulas ( char (𝔽)≠ q ) and Flow formulas, (b) Random k -CNFs of suitable densities, (c) Pigeonhole principle and Counting mod q principle. The lower bounds for the dag-like systems are the first nontrivial lower bounds for these systems, including the case d =1. The lower bounds for the tree-like systems were known for the case d =1 (except for the Counting mod q principle, in which lower bounds for the case d > 1 were known too). Our lower bounds extend those results to the case where d > 1 and also give new proofs for the case d =1.

Practical algebraic calculus and Nullstellensatz with the checkers Pacheck and Pastèque and Nuss-Checker

AbstractAutomated reasoning techniques based on computer algebra have seen renewed interest in recent years and are for example heavily used in formal verification of arithmetic circuits. However, the verification process might contain errors. Generating and checking proof certificates is important to increase the trust in automated reasoning tools. For algebraic reasoning, two proof systems, Nullstellensatz and polynomial calculus, are available and are well-known in proof complexity. A Nullstellensatz proof captures whether a polynomial can be represented as a linear combination of a given set of polynomials by providing the co-factors of the linear combination. Proofs in polynomial calculus dynamically capture that a polynomial can be derived from a given set of polynomials using algebraic ideal theory. In this article we present the practical algebraic calculus as an instantiation of the polynomial calculus that can be checked efficiently. We further modify the practical algebraic calculus and gain LPAC (practical algebraic calculus + linear combinations) that includes linear combinations. In this way we are not only able to represent both Nullstellensatz and polynomial calculus proofs, but we are also able to blend both proof formats. Furthermore, we introduce extension rules to simulate essential rewriting techniques required in practice. For efficiency we also make use of indices for existing polynomials and include deletion rules too. We demonstrate the different proof formats on the use case of arithmetic circuit verification and discuss how these proofs can be produced as a by-product in formal verification. We present the proof checkers Pacheck, Pastèque, and Nuss-Checker. Pacheck checks proofs in practical algebraic calculus more efficiently than Pastèque, but the latter is formally verified using the proof assistant Isabelle/HOL. The tool Nuss-Checker is used to check proofs in the Nullstellensatz format.

Differential operator Dirac structures

Abstract As shown before, skew-adjoint linear differential operators, mapping efforts into flows, give rise to Dirac structures on a bounded spatial domain by a proper definition of boundary variables. In the present paper this is extended to pairs of linear differential operators defining a formally skew-adjoint relation between flows and efforts. Furthermore it is shown how the underlying repeated integration by parts operation is streamlined by the use of two-variable polynomial calculus. Dirac structures defined by formally skew adjoint operators together with differential operator effort constraints are treated within the same framework. Finally it is sketched how the approach can be also used for Lagrangian subspaces on bounded domains.

Пример полиномиального синтеза регулятора для неквадратного объекта с одним входом и двумя выходами

Polynomial methods for synthesizing controllers for automatic control systems with linear objects are becoming increasingly common. The synthesis of multichannel controllers is particularly difficult, which is caused by the need to use matrix polynomial calculus. However, this approach mainly considers objects with the number of inputs equal to the number of outputs. This is due to the convenience of solving a system of linear algebraic equations in matrix polynomial calculus. In this paper, we consider a polynomial method for synthesizing regulators for a non-square object, that is, one whose number of inputs is not equal to the number of outputs. The selected system contains not only a non-square object, but also a non-square controller.

Control of Zeros and Poles in Problems of Synthesis of Regulation Systems. Part I. Compensation Approach

A highly important place in the theory and practice of automatic systems occupy the problems of synthesis of automatic regulation systems (ARS) based on the requirements for the dynamic quality of regulation processes. For class of linear stationary ARS such requirements are imposed upon the type and parameters of its transition characteristic which is uniquely determined by its transfer function (TF). In this connection the setting of problem of ARS synthesis with desired TF, corresponding to the given amplification coefficient, zeros and poles of the synthesized system is regularity. The given problem is worth while to call as the control problem of zeros and poles of ARS. In the present paper consists into two parts, the questions of control of zeros and poles ARS, which are important for engineering applications are considered. A critical analysis of the known compensation and compensation-modal regulation schemes is given. And also the new circuit solutions combining the functionality possibilities of compensation and modal approaches are proposed. The first part of the article the effect of compensation of zeros and poles of control objects in ARS is analyzed. The understanding of this effect gives the representation of the system of canonical structure of R. Kalman, according to which the compensation of zeros and poles of object does not mean their physical liquidation. As a result of compensation, they become the poles of its unobservable and unmanageable parts, which will be tell upon the regulation processes in the conditions of disturbances of the state of control object. The given effect and its negative results are clearly detected in the classical method of constructing of controllers on a priori given (desired, reference) TF of closed ARS. The influence of the factor of non-minimal phase zeros on the dynamics of control systems is studied. The effect of negative ejection in the transition characteristic of the system is described and its quantitative assessment is given for the case of a single real right zero. In the second part of the article the well-known methods for ARS synthesis with desired TF, based on use of polynomial calculus apparatus are considered and analyzed. New compensatory modal methods which may be of interest in engineering applications are proposed.

Another look at degree lower bounds for polynomial calculus

Polynomial complexity is an algebraic proof system inspired by Gröbner bases which was introduced by Clegg, Edmonds and Impagliazzo. Alekhnovich and Razborov devised a method for proving degree lower bounds for polynomial calculus. We present an alternative account of their method, which also encompasses a generalization due to Galesi and Lauria.

Resolution over linear equations modulo two

We consider an extension of the resolution proof system that operates with disjunctions of linear equalities over F2; we denote this system by Res(⊕). It is well known that tree-like resolution is equivalent in behavior to DPLL algorithms for the Boolean satisfiability problem. Every DPLL algorithm splits the input problem into two by assigning two possible values to a variable; then it simplifies the two resulting formulas and makes two recursive calls. Tree-like Res(⊕)-proofs correspond to an extension of the DPLL paradigm, in which we can split by an arbitrary linear combination of variables modulo two. These algorithms quickly solve formulas that explicitly encode linear systems modulo two which were used for proving exponential lower bounds for conventional DPLL algorithms.We prove exponential lower bounds on the size of tree-like Res(⊕)-proofs. We also show that resolution and tree-like Res(⊕) do not simulate each other.We prove a space vs size tradeoff for Res(⊕)-proofs. We prove that Res(⊕) is implicationally complete and also that Res(⊕) is polynomially equivalent to its semantic version.We consider the proof system Res(⊕;⩽k) that is a restricted version of Res(⊕) operating with disjunctions of linear equalities such that at most k equalities depend on more than one variable. We simulate Res(⊕;⩽k) in the OBDD-based proof system with blowup 2k and in Polynomial Calculus Resolution with blowup 2nH(2k/n)poly(n), where n is the number of variables and H(p) is the binary entropy; the latter result implies exponential lower bounds on the size of Res(⊕;⩽δn)-proofs for some constant δ>0. Raz and Tzameret introduced the system R(lin) which operates with disjunctions of linear equalities with integer coefficients. We show that Res(⊕) can be p-simulated in R(lin).

ПОЛИНОМИАЛЬНЫЙ МЕТОД СИНТЕЗА МНОГОКАНАЛЬНЫХ СИСТЕМ ПОСРЕДСТВОМ ПЕРЕХОДА К МАТРИЧНОМУ ПОЛИНОМИАЛЬНОМУ ПРЕДСТАВЛЕНИЮ

Polynomial methods for synthesizing linear regulators for automatic control systems with linear objects, proposed by a number of authors, including Chen, Kailath, Gaiduk, and others, along with methods of synthesis in the state space, are becoming increasingly widespread. The synthesis of multichannel regulators caused by the need to use the matrix polynomial calculus is of a special difficulty, which is aggravated by a significant increase in the dimension of the matrices during the transition from polynomial matrices to numeric ones, in which Sylvester matrices are used. Herewith, it is necessary to take into account the requirements of controllability and observability, leading to the need to check for the presence of identical roots in polynomial matrices corresponding to the numerator and denominator of the object. This leads to the requirement of a relatively prime matrix polynomial fraction, which can be significantly weakened if it is possible to include in the desired characteristic matrix of the system some zeros and poles of the object located far to the left of the imaginary axis. In calculations using numerical matrices and, consequently, using Sylvester matrices, the latter degenerate due to the lowering of the rank, which complicates the calculations. The research continues to study polynomial synthesis of multichannel regulators based on the results obtained by Chen and other researchers and presents an algorithm for the synthesis of regulators, the feature of which is the possibility of introducing additional so-called free parameters that allow additional requirements for the automatic control system. The free parameters allow to obtain strictly proper regulators, along with the proper regulators.

Proof Complexity Meets Algebra

We analyze how the standard reductions between constraint satisfaction problems affect their proof complexity. We show that, for the most studied propositional, algebraic, and semialgebraic proof systems, the classical constructions of pp-interpretability, homomorphic equivalence, and addition of constants to a core preserve the proof complexity of the CSP. As a result, for those proof systems, the classes of constraint languages for which small unsatisfiability certificates exist can be characterized algebraically. We illustrate our results by a gap theorem saying that a constraint language either has resolution refutations of constant width or does not have bounded-depth Frege refutations of subexponential size. The former holds exactly for the widely studied class of constraint languages of bounded width. This class is also known to coincide with the class of languages with refutations of sublinear degree in Sums of Squares and Polynomial Calculus over the real field, for which we provide alternative proofs. We then ask for the existence of a natural proof system with good behavior with respect to reductions and simultaneously small-size refutations beyond bounded width. We give an example of such a proof system by showing that bounded-degree Lovász-Schrijver satisfies both requirements. Finally, building on the known lower bounds, we demonstrate the applicability of the method of reducibilities and construct new explicit hard instances of the graph three-coloring problem for all studied proof systems.

Circuit Complexity, Proof Complexity, and Polynomial Identity Testing

We introduce a new and natural algebraic proof system, whose complexity measure is essentially the algebraic circuit size of Nullstellensatz certificates. This enables us to exhibit close connections between effective Nullstellensatzë, proof complexity, and (algebraic) circuit complexity. In particular, we show that any super-polynomial lower bound on any Boolean tautology in our proof system implies that the permanent does not have polynomial-size algebraic circuits (VNP ≠ VP). We also show that super-polynomial lower bounds on the number of lines in Polynomial Calculus proofs imply the Permanent versus Determinant Conjecture. Note that there was no proof system prior to ours for which lower bounds on an arbitrary tautology implied any complexity class lower bound. Our proof system helps clarify the relationships between previous algebraic proof systems. In doing so, we highlight the importance of polynomial identity testing (PIT) in proof complexity. In particular, we use PIT to illuminate AC 0 [ p ]-Frege lower bounds, which have been open for nearly 30 years, with no satisfactory explanation as to their apparent difficulty. Finally, we explain the obstacles that must be overcome in any attempt to extend techniques from algebraic circuit complexity to prove lower bounds in proof complexity. Using the algebraic structure of our proof system, we propose a novel route to such lower bounds. Although such lower bounds remain elusive, this proposal should be contrasted with the difficulty of extending AC 0 [ p ] circuit lower bounds to AC 0 [ p ]-Frege lower bounds.

A Finite-Model-Theoretic View on Propositional Proof Complexity

We establish new, and surprisingly tight, connections between propositional proof complexity and finite model theory. Specifically, we show that the power of several propositional proof systems, such as Horn resolution, bounded-width resolution, and the polynomial calculus of bounded degree, can be characterised in a precise sense by variants of fixed-point logics that are of fundamental importance in descriptive complexity theory. Our main results are that Horn resolution has the same expressive power as least fixed-point logic, that bounded-width resolution captures existential least fixed-point logic, and that the polynomial calculus with bounded degree over the rationals solves precisely the problems definable in fixed-point logic with counting. By exploring these connections further, we establish finite-model-theoretic tools for proving lower bounds for the polynomial calculus over the rationals and over finite fields.

Gaussian distribution of LMOV numbers

Recent advances in knot polynomial calculus allowed us to obtain a huge variety of LMOV integers counting degeneracy of the BPS spectrum of topological theories on the resolved conifold and appearing in the genus expansion of the plethystic logarithm of the Ooguri–Vafa partition functions. Already the very first look at this data reveals that the LMOV numbers are randomly distributed in genus (!) and are very well parameterized by just three parameters depending on the representation, an integer and the knot. We present an accurate formulation and evidence in support of this new puzzling observation about the old puzzling quantities. It probably implies that the BPS states, counted by the LMOV numbers can actually be composites made from some still more elementary objects.

Space proof complexity for random 3-CNFs

We investigate the space complexity of refuting 3-CNFs in Resolution and algebraic systems. We prove that every Polynomial Calculus with Resolution refutation of a random 3-CNF φ in n variables requires, with high probability, Ω(n)distinct monomials to be kept simultaneously in memory. The same construction also proves that every Resolution refutation of φ requires, with high probability, Ω(n) clauses each of width Ω(n) to be kept at the same time in memory. This gives a Ω(n2) lower bound for the total space needed in Resolution to refute φ. These results are best possible (up to a constant factor) and answer questions about space complexity of 3-CNFs.The main technical innovation is a variant of Hall's Lemma. We show that in bipartite graphs with bipartition (L,R) and left-degree at most 3, L can be covered by certain families of disjoint paths, called VW-matchings, provided that L expands in R by a factor of (2−ϵ), for ϵ<15.

Tight Size-Degree Bounds for Sums-of-Squares Proofs

We exhibit families of 4-CNF formulas over n variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) d but require SOS proofs of size $${n^{\Omega{(d)}}}$$ for values of d = d(n) from constant all the way up to $${n^{\delta}}$$ for some universal constant $${\delta}$$ . This shows that the $${{n^{{\rm O}{(d)}}}}$$ running time obtained by using the Lasserre semidefinite programming relaxations to find degree-d SOS proofs is optimal up to constant factors in the exponent. We establish this result by combining NP-reductions expressible as low-degree SOS derivations with the idea of relativizing CNF formulas in Krajicek (Arch Math Log 43(4):427–441, 2004) and Dantchev & Riis (Proceedings of the 17th international workshop on computer science logic (CSL ’03), 2003) and then applying a restriction argument as in Atserias et al. (J Symb Log 80(2):450–476, 2015; ACM Trans Comput Log 17:19:1–19:30, 2016). This yields a generic method of amplifying SOS degree lower bounds to size lower bounds and also generalizes the approach used in Atserias et al. (2016) to obtain size lower bounds for the proof systems resolution, polynomial calculus, and Sherali–Adams from lower bounds on width, degree, and rank, respectively.