We investigate the unprovability of NP$\not\subseteq$P/poly in various fragments of arithmetic. The unprovability is usually obtained by showing hardness of propositional formulas encoding superpolynomial circuit lower bounds. Firstly, we discuss few relevant techniques and known theorems. Namely, natural proofs, feasible interpolation, KPT theorem, iterability, gadget generators etc. Then we prove some original results. We show the unprovability of superpolynomial circuit lower bounds for systems admitting certain forms of feasible interpolation (modulo a hardness assumption) and for systems roughly described as tree-like Frege systems working with formulas using only a small fraction of variables of the statement that is supposed to be proved. These results are obtained by proving the hardness of the Nisan-Wigderson generators in corresponding proof systems.

We study the problem of proving lower bounds for depth four arithmetic circuits. Depth four circuits have been receiving much attraction when it comes to recent circuit lower bound results, as a result of the series of results culminating in the fact that strong enough lower bounds for depth four circuits will imply super-polynomial lower bounds for general arithmetic circuits, and hence solve one of the most central open problems in algebraic complexity i.e a separation between the VP and VNP classes. However despite several efforts, even for general arithmetic circuits, the best known lower bound is Omega(N log N) by Baur and Strassen (1983), where N is the number of input variables. In the case of arithmetic formulas, Kalorkoti (1985) proved a lower bound that is quadratic in the number of input variables, which has not seen any improvement since then. The situation for depth three arithmetic circuits was also similar for many years, until a recent result by Kayal et. al. (2016) achieved an almost cubic lower bound that improved over the previous best quadratic bound by Shpilka and Wigderson (1999). As the main contribution of this thesis, we prove an Omega(N^1.5) lower bound on the size of a depth four circuit, for an explicit multilinear N-variate polynomial in VNP with degree d = Theta(sqrt(N)). Our approach offers a potential route to proving a super-quadratic lower bound for depth four circuits. Taking cue from the numerous successful results recently, we use the technique of the shifted partial derivatives measure to achieve the said lower bound. Particularly, we use the Dimension of Projected Shifted Partials (DPSP) measure which has been previously used in recent depth four results. Coming to the choice of the hard polynomial, we again follow the status quo and use a variant of the Nisan-Wigderson (NW) polynomial family that has proved to be very helpful over the past few years in arithmetic circuit complexity. Finally, we do a careful analysis of Shoup-Smolensky (1997) and Raz (2010) and compare their techniques to ours. We conclude that our result can potentially be used as a starting point, and techniques similar to Kayal et. al. (2016) can likely be used to strengthen our lower bound to Omega(N^2.5), for general depth four arithmetic circuits. However, unlike depth three circuits, proving a super-quadratic lower bound for depth four circuits remains a prevalent open problem for many years. Previous work like Shoup-Smolensky and Raz implied super-linear lower bounds. To the best of our knowledge, the previous best known lower bound for general depth four circuits is Omega(N^1.33).

Understanding the structure and complexity of a polynomial family is a fundamental problem of arithmetic circuit complexity. There are various approaches like studying the lower bounds, which deals with nding the smallest circuit required to compute a polynomial, studying the orbit and stabilizer of a polynomial with respect to an invertible transformation etc to do this. We have a rich understanding of some of the well known polynomial families like determinant, permanent, IMM etc. In this thesis we study some of the structural properties of the polyno-mial family called the Nisan-Wigderson polynomial family. This polynomial family is inspired from a well known combinatorial design called Nisan-Wigderson design and is recently used to prove strong lower bounds on some restricted classes of arithmetic circuits ([KSS14],[KLSS14], [KST16]). But unlike determinant, permanent, IMM etc, our understanding of the Nisan-Wigderson polynomial family is inadequate. For example we do not know if this polynomial family is in VP or VNP complete or VNP-intermediate assuming VP 6= VNP, nor do we have an understanding of the complexity of its equivalence test. We hope that the knowledge of some of the inherent properties of Nisan-Wigderson polynomial like group of symmetries and Lie algebra would provide us some insights in this regard. A matrix A 2 GLn(F) is called a symmetry of an n-variate polynomial f if f(A x) = f(x): The set of symmetries of f forms a subgroup of GLn(F), which is also known as group of symmetries of f, denoted Gf . A vector space is attached to Gf to get the complete understanding of the symmetries of f. This vector space is known as the Lie algebra of group of symmetries of f (or Lie algebra of f), represented as gf . Lie algebra of f contributes some elements of Gf , known as continuous symmetries of f. Lie algebra has also been instrumental in designing e cient randomized equivalence tests for some polynomial families like determinant, permanent, IMM etc ([Kay12], [KNST17]). In this work we completely characterize the Lie algebra of the Nisan-Wigderson polynomial family. We show that gNW contains diagonal matrices of a speci c type. The knowledge of gNW not only helps us to completely gure out the continuous symmetries of the Nisan-Wigderson polynomial family, but also gives some crucial insights into the other symmetries of Nisan-Wigderson polynomial (i.e. the discrete symmetries). Thereafter using the Hessian matrix of the Nisan-Wigderson polynomial and the concept of evaluation dimension, we are able to almost completely identify the structure of GNW . In particular we prove that any A 2 GNW is a product of diagonal and permutation matrices of certain kind that we call block-permuted permutation matrix. Finally, we give explicit examples of nontrivial block-permuted permutation matrices using the automorphisms of nite eld that establishes the richness of the discrete symmetries of the Nisan-Wigderson polynomial family.