Frameworks for logically classifying polynomial-time optimisation problems.

We show that a logical framework, based around a fragment of existential second-order logic formerly proposed by others so as to capture the class of polynomially-bounded P-optimisation problems, cannot hope to do so, under the assumption that P ≠ NP. We do this by exhibiting polynomially-bounded maximisation and minimisation problems that can be expressed in the framework but whose decision versions are NP-complete. We propose an alternative logical framework, based around inflationary fixed-point logic, and show that we can capture the above classes of optimisation problems. We use the inductive depth of an inflationary fixed-point as a means to describe the objective functions of the instances of our optimisation problems

Circuit complexity, proof complexity, and polynomial identity testing

We introduce a new algebraic proof system, which has tight connections to (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 is not equal to VP). As a corollary to the proof, we also show that super-polynomial lower bounds on the number of lines in Polynomial Calculus proofs (as opposed to the usual measure of number of monomials) imply the Permanent versus Determinant Conjecture. Note that, prior to our work, there was no proof system for which lower bounds on an arbitrary tautology implied any computational lower bound. Our proof system helps clarify the relationships between previous algebraic proof systems, and begins to shed light on why proof complexity lower bounds for various proof systems have been so much harder than lower bounds on the corresponding circuit classes. In doing so, we highlight the importance of polynomial identity testing (PIT) for understanding proof complexity. More specifically, we introduce certain propositional axioms satisfied by any Boolean circuit computing PIT. We use these PIT axioms to shed light on AC^0[p]-Frege lower bounds, which have been open for nearly 30 years, with no satisfactory explanation as to their apparent difficulty. We show that either: a) Proving super-polynomial lower bounds on AC^0[p]-Frege implies VNP does not have polynomial-size circuits of depth d - a notoriously open question for d at least 4 - thus explaining the difficulty of lower bounds on AC^0[p]-Frege, or b) AC^0[p]-Frege cannot efficiently prove the depth d PIT axioms, and hence we have a lower bound on AC^0[p]-Frege. Using the algebraic structure of our proof system, we propose a novel way to extend techniques from algebraic circuit complexity to prove lower bounds in proof complexity

Parallel Algorithm and Dynamic Exponent for Diffusion-limited Aggregation

A parallel algorithm for ``diffusion-limited aggregation'' (DLA) is described and analyzed from the perspective of computational complexity. The dynamic exponent z of the algorithm is defined with respect to the probabilistic parallel random-access machine (PRAM) model of parallel computation according to $T \sim L^{z}$, where L is the cluster size, T is the running time, and the algorithm uses a number of processors polynomial in L\@. It is argued that z=D-D_2/2, where D is the fractal dimension and D_2 is the second generalized dimension. Simulations of DLA are carried out to measure D_2 and to test scaling assumptions employed in the complexity analysis of the parallel algorithm. It is plausible that the parallel algorithm attains the minimum possible value of the dynamic exponent in which case z characterizes the intrinsic history dependence of DLA.Comment: 24 pages Revtex and 2 figures. A major improvement to the algorithm and smaller dynamic exponent in this versio

Circuit Lower Bounds, Help Functions, and the Remote Point Problem

We investigate the power of Algebraic Branching Programs (ABPs) augmented with help polynomials, and constant-depth Boolean circuits augmented with help functions. We relate the problem of proving explicit lower bounds in both these models to the Remote Point Problem (introduced by Alon, Panigrahy, and Yekhanin (RANDOM '09)). More precisely, proving lower bounds for ABPs with help polynomials is related to the Remote Point Problem w.r.t. the rank metric, and for constant-depth circuits with help functions it is related to the Remote Point Problem w.r.t. the Hamming metric. For algebraic branching programs with help polynomials with some degree restrictions we show exponential size lower bounds for explicit polynomials

A Measure of Space for Computing over the Reals

We propose a new complexity measure of space for the BSS model of computation. We define LOGSPACE\_W and PSPACE\_W complexity classes over the reals. We prove that LOGSPACE\_W is included in NC^2\_R and in P\_W, i.e. is small enough for being relevant. We prove that the Real Circuit Decision Problem is P\_R-complete under LOGSPACE\_W reductions, i.e. that LOGSPACE\_W is large enough for containing natural algorithms. We also prove that PSPACE\_W is included in PAR\_R