Near-Optimal Complexity Bounds for Fragments of the Skolem Problem

Given a linear recurrence sequence (LRS), specified using the initial conditions and the recurrence relation, the Skolem problem asks if zero ever occurs in the infinite sequence generated by the LRS. Despite active research over last few decades, its decidability is known only for a few restricted subclasses, by either restricting the order of the LRS (upto 4) or by restricting the structure of the LRS (e.g., roots of its characteristic polynomial). In this paper, we identify a subclass of LRS of arbitrary order for which the Skolem problem is easy, namely LRS all of whose characteristic roots are (possibly complex) roots of real algebraic numbers, i.e., roots satisfying x^d = r for r real algebraic. We show that for this subclass, the Skolem problem can be solved in NP^RP. As a byproduct, we implicitly obtain effective bounds on the zero set of the LRS for this subclass. While prior works in this area often exploit deep results from algebraic and transcendental number theory to get such effective results, our techniques are primarily algorithmic and use linear algebra and Galois theory. We also complement our upper bounds with a NP lower bound for the Skolem problem via a new direct reduction from 3-CNF-SAT, matching the best known lower bounds

Complexity of Restricted Variants of Skolem and Related Problems

Given a linear recurrence sequence (LRS), the Skolem problem, asks whether it ever becomes zero. The decidability of this problem has been open for several decades. Currently decidability is known only for LRS of order upto 4. For arbitrary orders (i.e., number of terms the n-th depends on), the only known complexity result is NP-hardness by a result of Blondel and Portier from 2002. In this paper, we give a different proof of this hardness result, which is arguably simpler and pinpoints the source of hardness. To demonstrate this, we identify a subclass of LRS for which the Skolem problem is in fact NP-complete. We show the generic nature of our lower-bound technique by adapting it to show stronger lower bounds of a related problem that encompasses many known decision problems on linear recurrent sequences

Imperfect Gaps in Gap-ETH and PCPs

We study the role of perfect completeness in probabilistically checkable proof systems (PCPs) and give a way to transform a PCP with imperfect completeness to one with perfect completeness, when the initial gap is a constant. We show that PCP_{c,s}[r,q] subseteq PCP_{1,s\u27}[r+O(1),q+O(r)] for c-s=Omega(1) which in turn implies that one can convert imperfect completeness to perfect in linear-sized PCPs for NP with a O(log n) additive loss in the query complexity q. We show our result by constructing a "robust circuit" using threshold gates. These results are a gap amplification procedure for PCPs, (when completeness is not 1) analogous to questions studied in parallel repetition [Anup Rao, 2011] and pseudorandomness [David Gillman, 1998] and might be of independent interest. We also investigate the time-complexity of approximating perfectly satisfiable instances of 3SAT versus those with imperfect completeness. We show that the Gap-ETH conjecture without perfect completeness is equivalent to Gap-ETH with perfect completeness, i.e. MAX 3SAT(1-epsilon,1-delta), delta > epsilon has 2^{o(n)} algorithms if and only if MAX 3SAT(1,1-delta) has 2^{o(n)} algorithms. We also relate the time complexities of these two problems in a more fine-grained way to show that T_2(n) <= T_1(n(log log n)^{O(1)}), where T_1(n),T_2(n) denote the randomized time-complexity of approximating MAX 3SAT with perfect and imperfect completeness respectively

On Oracles and Algorithmic Methods for Proving Lower Bounds

This paper studies the interaction of oracles with algorithmic approaches to proving circuit complexity lower bounds, establishing new results on two different kinds of questions. 1) We revisit some prominent open questions in circuit lower bounds, and provide a clean way of viewing them as circuit upper bound questions. Let Missing-String be the (total) search problem of producing a string that does not appear in a given list L containing M bit-strings of length N, where M < 2?. We show in a generic way how algorithms and uniform circuits (from restricted classes) for Missing-String imply complexity lower bounds (and in some cases, the converse holds as well). We give a local algorithm for Missing-String, which can compute any desired output bit making very few probes into the input, when the number of strings M is small enough. We apply this to prove a new nearly-optimal (up to oracles) time hierarchy theorem with advice. We show that the problem of constructing restricted uniform circuits for Missing-String is essentially equivalent to constructing functions without small non-uniform circuits, in a relativizing way. For example, we prove that small uniform depth-3 circuits for Missing-String would imply exponential circuit lower bounds for ?? EXP, and depth-3 lower bounds for Missing-String would imply non-trivial circuits (relative to an oracle) for ?? EXP problems. Both conclusions are longstanding open problems in circuit complexity. 2) It has been known since Impagliazzo, Kabanets, and Wigderson [JCSS 2002] that generic derandomizations improving subexponentially over exhaustive search would imply lower bounds such as NEXP ? ? ?/poly. Williams [SICOMP 2013] showed that Circuit-SAT algorithms running barely faster than exhaustive search would imply similar lower bounds. The known proofs of such results do not relativize (they use techniques from interactive proofs/PCPs). However, it has remained open whether there is an oracle under which the generic implications from circuit-analysis algorithms to circuit lower bounds fail. Building on an oracle of Fortnow, we construct an oracle relative to which the circuit approximation probability problem (CAPP) is in ?, yet EXP^{NP} has polynomial-size circuits. We construct an oracle relative to which SAT can be solved in "half-exponential" time, yet exponential time (EXP) has polynomial-size circuits. Improving EXP to NEXP would give an oracle relative to which ?? ? has "half-exponential" size circuits, which is open. (Recall it is known that ?? ? is not in "sub-half-exponential" size, and the proof relativizes.) Moreover, the running time of the SAT algorithm cannot be improved: relative to all oracles, if SAT is in "sub-half-exponential" time then EXP does not have polynomial-size circuits

Polyolefin elastomer blends as an alternative to poly(vinyl chloride) flooring

Poly(vinyl chloride) (PVC) flooring products in the form of flexible sheets and tiles dominate the resilient flooring market. These flooring products are produced by calendering compounds based on vinyl chloride-vinyl acetate copolymer filled with calcium carbonate, essentially with plasticizers and other additives. Plasticizers have become matter of concern from indoor air quality (IAQ) point of view as they are suspected to be the one of the sources of volatile organic compound (VOC) emissions, considered harmful to human health. In the research work reported here, polyolefin elastomers (POE) filled with calcium carbonate filler are prepared and their mechanical properties are evaluated as an alternative to poly(vinyl chloride), PVC matrix for flooring products. POEs are olefin copolymers synthesized using metallocene technology and does not require plasticizers for processing, yet can provide similar flexibility to the finished product as that of plasticized PVC. Also, POE can incorporate relatively higher amounts of post consumer polyethylene (PC-PE) while maintaining adequate properties. The POE used in this study is Engage-8440, which is an ethylene-octene copolymer. Two types of PC-PE are studied in this research: the first one is a 50-50 mixture of linear low density polyethylene (LLDPE) and low density polyethylene (LDPE) and the second is a high density polyethylene (HDPE). The mechanical and thermal properties, as well as the technique is used to study the difference in thermal properties of the two recycled resins. Further, heat and UV stabilizers are added and the tensile properties of POE/PC-PE blend composites are tested for accelerated weathering effects. As a result of the present study, an alternative flooring material is attempted with appreciable properties, such as Young's modulus, tensile strength at break and yield point, elongation at break, impact, hardness and static load limit values

On Provable Copyright Protection for Generative Models

There is a growing concern that learned conditional generative models may output samples that are substantially similar to some copyrighted data $C$ that was in their training set. We give a formal definition of $\textit{near access-freeness (NAF)}$ and prove bounds on the probability that a model satisfying this definition outputs a sample similar to $C$, even if $C$ is included in its training set. Roughly speaking, a generative model $p$ is \textit{k-NAF} if for every potentially copyrighted data $C$, the output of $p$ diverges by at most $k$-bits from the output of a model $q$ that \textit{did not access C at all}. We also give generative model learning algorithms, which efficiently modify the original generative model learning algorithm in a black box manner, that output generative models with strong bounds on the probability of sampling protected content. Furthermore, we provide promising experiments for both language (transformers) and image (diffusion) generative models, showing minimal degradation in output quality while ensuring strong protections against sampling protected content.Comment: Accepted at ICML 202

On Privileged and Convergent Bases in Neural Network Representations

In this study, we investigate whether the representations learned by neural networks possess a privileged and convergent basis. Specifically, we examine the significance of feature directions represented by individual neurons. First, we establish that arbitrary rotations of neural representations cannot be inverted (unlike linear networks), indicating that they do not exhibit complete rotational invariance. Subsequently, we explore the possibility of multiple bases achieving identical performance. To do this, we compare the bases of networks trained with the same parameters but with varying random initializations. Our study reveals two findings: (1) Even in wide networks such as WideResNets, neural networks do not converge to a unique basis; (2) Basis correlation increases significantly when a few early layers of the network are frozen identically. Furthermore, we analyze Linear Mode Connectivity, which has been studied as a measure of basis correlation. Our findings give evidence that while Linear Mode Connectivity improves with increased network width, this improvement is not due to an increase in basis correlation.Comment: In the Workshop on High-dimensional Learning Dynamics at ICML 202