Hardness vs Randomness for Bounded Depth Arithmetic Circuits

In this paper, we study the question of hardness-randomness tradeoffs for bounded depth arithmetic circuits. We show that if there is a family of explicit polynomials {f_n}, where f_n is of degree O(log^2n/log^2 log n) in n variables such that f_n cannot be computed by a depth Delta arithmetic circuits of size poly(n), then there is a deterministic sub-exponential time algorithm for polynomial identity testing of arithmetic circuits of depth Delta-5. This is incomparable to a beautiful result of Dvir et al.[SICOMP, 2009], where they showed that super-polynomial lower bounds for depth Delta circuits for any explicit family of polynomials (of potentially high degree) implies sub-exponential time deterministic PIT for depth Delta-5 circuits of bounded individual degree. Thus, we remove the "bounded individual degree" condition in the work of Dvir et al. at the cost of strengthening the hardness assumption to hold for polynomials of low degree. The key technical ingredient of our proof is the following property of roots of polynomials computable by a bounded depth arithmetic circuit : if f(x_1, x_2, ..., x_n) and P(x_1, x_2, ..., x_n, y) are polynomials of degree d and r respectively, such that P can be computed by a circuit of size s and depth Delta and P(x_1, x_2, ..., x_n, f) equiv 0, then, f can be computed by a circuit of size poly(n, s, r, d^{O(sqrt{d})}) and depth Delta + 3. In comparison, Dvir et al. showed that f can be computed by a circuit of depth Delta + 3 and size poly(n, s, r, d^{t}), where t is the degree of P in y. Thus, the size upper bound in the work of Dvir et al. is non-trivial when t is small but d could be large, whereas our size upper bound is non-trivial when d is small, but t could be large

Interpolation in Valiant's theory

We investigate the following question: if a polynomial can be evaluated at rational points by a polynomial-time boolean algorithm, does it have a polynomial-size arithmetic circuit? We argue that this question is certainly difficult. Answering it negatively would indeed imply that the constant-free versions of the algebraic complexity classes VP and VNP defined by Valiant are different. Answering this question positively would imply a transfer theorem from boolean to algebraic complexity. Our proof method relies on Lagrange interpolation and on recent results connecting the (boolean) counting hierarchy to algebraic complexity classes. As a byproduct we obtain two additional results: (i) The constant-free, degree-unbounded version of Valiant's hypothesis that VP and VNP differ implies the degree-bounded version. This result was previously known to hold for fields of positive characteristic only. (ii) If exponential sums of easy to compute polynomials can be computed efficiently, then the same is true of exponential products. We point out an application of this result to the P=NP problem in the Blum-Shub-Smale model of computation over the field of complex numbers.Comment: 13 page

Subclasses of Presburger Arithmetic and the Weak EXP Hierarchy

It is shown that for any fixed $i>0$, the $\Sigma_{i+1}$-fragment of Presburger arithmetic, i.e., its restriction to $i+1$ quantifier alternations beginning with an existential quantifier, is complete for $\mathsf{\Sigma}^{\mathsf{EXP}}_{i}$, the $i$-th level of the weak EXP hierarchy, an analogue to the polynomial-time hierarchy residing between $\mathsf{NEXP}$ and $\mathsf{EXPSPACE}$. This result completes the computational complexity landscape for Presburger arithmetic, a line of research which dates back to the seminal work by Fischer & Rabin in 1974. Moreover, we apply some of the techniques developed in the proof of the lower bound in order to establish bounds on sets of naturals definable in the $\Sigma_1$-fragment of Presburger arithmetic: given a $\Sigma_1$-formula $\Phi(x)$, it is shown that the set of non-negative solutions is an ultimately periodic set whose period is at most doubly-exponential and that this bound is tight.Comment: 10 pages, 2 figure

Making proofs without Modus Ponens: An introduction to the combinatorics and complexity of cut elimination

This paper is intended to provide an introduction to cut elimination which is accessible to a broad mathematical audience. Gentzen's cut elimination theorem is not as well known as it deserves to be, and it is tied to a lot of interesting mathematical structure. In particular we try to indicate some dynamical and combinatorial aspects of cut elimination, as well as its connections to complexity theory. We discuss two concrete examples where one can see the structure of short proofs with cuts, one concerning feasible numbers and the other concerning "bounded mean oscillation" from real analysis

Unbounded arithmetic

When comparing the different axiomatizations of bounded arithmetic and Peano arithmetic, it becomes clear that there are similarities between the fragments of these theories. In particular, it is tempting to draw an analogy between the hierarchies of bounded arithmetic and Peano arithmetic. However, one cannot deny that there are essential and deeply rooted differences and the most one can claim is a weak analogy between these hierarchies. The following quote by Kaye expresses this argument in an elegant way: "Many authors have emphasized the analogies between the fragments Sigma(b)(n)-IND of IDelta0+(Vx)(xlog x) exists) and the fragments ISigma(n) of Peano arithmetic. Sometimes this is helpful, but often one feels that the bounded hierarchy of theories is of a rather different nature and new techniques must be developed to answer the key questions concerning them." In this paper, we propose a (conjectured) hierarchy for Peano arithmetic which is much closer to that of bounded arithmetic than the existing one. In light of this close relation, techniques developed to establish properties of the new hierarchy should carry over naturally to the (conjectured) hierarchy of bounded arithmetic. As the famous P vs. NP problem is related to the collapse of the hierarchy of bounded arithmetic, the new hierarchy may prove particularly useful in solving this famous problem

Progress on Polynomial Identity Testing - II

We survey the area of algebraic complexity theory; with the focus being on the problem of polynomial identity testing (PIT). We discuss the key ideas that have gone into the results of the last few years.Comment: 17 pages, 1 figure, surve

On the role of entanglement in quantum computational speed-up

For any quantum algorithm operating on pure states we prove that the presence of multi-partite entanglement, with a number of parties that increases unboundedly with input size, is necessary if the quantum algorithm is to offer an exponential speed-up over classical computation. Furthermore we prove that the algorithm can be classically efficiently simulated to within a prescribed tolerance \eta even if a suitably small amount of global entanglement (depending on \eta) is present. We explicitly identify the occurrence of increasing multi-partite entanglement in Shor's algorithm. Our results do not apply to quantum algorithms operating on mixed states in general and we discuss the suggestion that an exponential computational speed-up might be possible with mixed states in the total absence of entanglement. Finally, despite the essential role of entanglement for pure state algorithms, we argue that it is nevertheless misleading to view entanglement as a key resource for quantum computational power.Comment: Main proofs simplified. A few further explanatory remarks added. 22 pages, plain late