Algebraic Independence over Positive Characteristic: New Criterion and Applications to Locally Low Algebraic Rank Circuits

The motivation for this work comes from two problems--test algebraic independence of arithmetic circuits over a field of small characteristic, and generalize the structural property of algebraic dependence used by (Kumar, Saraf CCC\u2716) to arbitrary fields. It is known that in the case of zero, or large characteristic, using a classical criterion based on the Jacobian, we get a randomized poly-time algorithm to test algebraic independence. Over small characteristic, the Jacobian criterion fails and there is no subexponential time algorithm known. This problem could well be conjectured to be in RP, but the current best algorithm puts it in NP^#P (Mittmann, Saxena, Scheiblechner Trans.AMS\u2714). Currently, even the case of two bivariate circuits over F_2 is open. We come up with a natural generalization of Jacobian criterion, that works over all characteristic. The new criterion is efficient if the underlying inseparable degree is promised to be a constant. This is a modest step towards the open question of fast independence testing, over finite fields, posed in (Dvir, Gabizon, Wigderson FOCS\u2707). In a set of linearly dependent polynomials, any polynomial can be written as a linear combination of the polynomials forming a basis. The analogous property for algebraic dependence is false, but a property approximately in that spirit is named as ``functional dependence\u27\u27 in (Kumar, Saraf CCC\u2716) and proved for zero or large characteristic. We show that functional dependence holds for arbitrary fields, thereby answering the open questions in (Kumar, Saraf CCC\u2716). Following them we use the functional dependence lemma to prove the first exponential lower bound for locally low algebraic rank circuits for arbitrary fields (a model that strongly generalizes homogeneous depth-4 circuits). We also recover their quasipoly-time hitting-set for such models, for fields of characteristic smaller than the ones known before. Our results show that approximate functional dependence is indeed a more fundamental concept than the Jacobian as it is field independent. We achieve the former by first picking a ``good\u27\u27 transcendence basis, then translating the circuits by new variables, and finally approximating them by truncating higher degree monomials. We give a tight analysis of the ``degree\u27\u27 of approximation needed in the criterion. To get the locally low algebraic rank circuit applications we follow the known shifted partial derivative based methods

Algebraic Dependencies and PSPACE Algorithms in Approximative Complexity

Testing whether a set $\mathbf{f}$ of polynomials has an algebraic dependence is a basic problem with several applications. The polynomials are given as algebraic circuits. Algebraic independence testing question is wide open over finite fields (Dvir, Gabizon, Wigderson, FOCS'07). The best complexity known is NP$^{\#\rm P}$ (Mittmann, Saxena, Scheiblechner, Trans.AMS'14). In this work we put the problem in AM $\cap$ coAM. In particular, dependence testing is unlikely to be NP-hard and joins the league of problems of "intermediate" complexity, eg. graph isomorphism & integer factoring. Our proof method is algebro-geometric-- estimating the size of the image/preimage of the polynomial map $\mathbf{f}$ over the finite field. A gap in this size is utilized in the AM protocols. Next, we study the open question of testing whether every annihilator of $\mathbf{f}$ has zero constant term (Kayal, CCC'09). We give a geometric characterization using Zariski closure of the image of $\mathbf{f}$; introducing a new problem called approximate polynomials satisfiability (APS). We show that APS is NP-hard and, using projective algebraic-geometry ideas, we put APS in PSPACE (prior best was EXPSPACE via Grobner basis computation). As an unexpected application of this to approximative complexity theory we get-- Over any field, hitting-set for $\overline{\rm VP}$ can be designed in PSPACE. This solves an open problem posed in (Mulmuley, FOCS'12, J.AMS 2017); greatly mitigating the GCT Chasm (exponentially in terms of space complexity)

The Means/Side-Effect Distinction in Moral Cognition: A Meta-Analysis

Experimental research suggests that people draw a moral distinction between bad outcomes brought about as a means versus a side effect (or byproduct). Such findings have informed multiple psychological and philosophical debates about moral cognition, including its computational structure, its sensitivity to the famous Doctrine of Double Effect, its reliability, and its status as a universal and innate mental module akin to universal grammar. But some studies have failed to replicate the means/byproduct effect especially in the absence of other factors, such as personal contact. So we aimed to determine how robust the means/byproduct effect is by conducting a meta-analysis of both published and unpublished studies (k = 101; 24,058 participants). We found that while there is an overall small difference between moral judgments of means and byproducts (standardized mean difference = 0.87, 95% CI 0.67 – 1.06; standardized mean change = 0.57, 95% CI 0.44 – 0.69; log odds ratio = 1.59, 95% CI 1.15 – 2.02), the mean effect size is primarily moderated by whether the outcome is brought about by personal contact, which typically involves the use of personal force

Factorization of Polynomials Given by Arithmetic Branching Programs

Given a multivariate polynomial computed by an arithmetic branching program (ABP) of size s, we show that all its factors can be computed by arithmetic branching programs of size poly(s). Kaltofen gave a similar result for polynomials computed by arithmetic circuits. The previously known best upper bound for ABP-factors was poly(s^(log s))