A foundation for real recursive function theory

The class of recursive functions over the reals, denoted by REC(R), was introduced by Cristopher Moore in his seminal paper written in 1995. Since then many subsequent investigations brought new results: the class REC(R) was put in relation with the class of functions generated by the General Purpose Analogue Computer of Claude Shannon; classical digital computation was embedded in several ways into the new model of computation; restrictions of REC(R) were proved to represent different classes of recursive functions, e.g., recursive, primitive recursive and elementary functions, and structures such as the Ritchie and the Grzergorczyk hierarchies. The class of real recursive functions was then stratified in a natural way, and REC(R) and the analytic hierarchy were recently recognised as two faces of the same mathematical concept. In this new article, we bring a strong foundational support to the Real Recursive Function Theory, rooted in Mathematical Analysis, in a way that the reader can easily recognise both its intrinsic mathematical beauty and its extreme simplicity. The new paradigm is now robust and smooth enough to be taught. To achieve such a result some concepts had to change and some new results were added

Limits of Quantum Speed-Ups for Computational Geometry and Other Problems: Fine-Grained Complexity via Quantum Walks

Many computational problems are subject to a quantum speed-up: one might find that a problem having an Opn3q-time or Opn2q-time classic algorithm can be solved by a known Opn1.5q-time or Opnq-time quantum algorithm. The question naturally arises: how much quantum speed-up is possible? The area of fine-grained complexity allows us to prove optimal lower-bounds on the complexity of various computational problems, based on the conjectured hardness of certain natural, well-studied problems. This theory has recently been extended to the quantum setting, in two independent papers by Buhrman, Patro and Speelman [7], and by Aaronson, Chia, Lin, Wang, and Zhang [1]. In this paper, we further extend the theory of fine-grained complexity to the quantum setting. A fundamental conjecture in the classical setting states that the 3SUM problem cannot be solved by (classical) algorithms in time Opn2´εq, for any ε ą 0. We formulate an analogous conjecture, the Quantum-3SUM-Conjecture, which states that there exist no sublinear Opn1´εq-time quantum algorithms for the 3SUM problem. Based on the Quantum-3SUM-Conjecture, we show new lower-bounds on the time complexity of quantum algorithms for several computational problems. Most of our lower-bounds are optimal, in that they match known upper-bounds, and hence they imply tight limits on the quantum speedup that is possible for these problems. These results are proven by adapting to the quantum setting known classical fine-grained reductions from the 3SUM problem. This adaptation is not trivial, however, since the original classical reductions require pre-processing the input in various ways, e.g. by sorting it according to some order, and this pre-processing (provably) cannot be done in sublinear quantum time. We overcome this bottleneck by combining a quantum walk with a classical dynamic data-structure having a certain “history-independence” property. This type of construction has been used in the past to prove upper bounds, and here we use it for the first time as part of a reduction. This general proof strategy allows us to prove tight lower bounds on several computational-geometry problems, on Convolution-3SUM and on the 0-Edge-Weight-Triangle problem, conditional on the Quantum-3SUM-Conjecture. We believe this proof strategy will be useful in proving tight (conditional) lower-bounds, and limits on quantum speed-ups, for many other problems

Approximation algorithms and hardness of approximation for knapsack problems

We show various hardness of approximation algorithms for knapsack and related problems; in particular we will show that unless the Exponential-Time Hypothesis is false, then subset-sum cannot be approximated any better than with an FPTAS. We also give a simple new algorithm for approximating knapsack and subset-sum, that can be adapted to work for small space, or in small parallel time. Finally, we prove that knapsack can not be solved in Mulmuley's parallel PRAM model, even when the input is restricted to small bit-length

Reductions to the set of random strings:the resource-bounded case

This paper is motivated by a conjecture \cite{cie,adfht} that \BPP can be characterized in terms of polynomial-time nonadaptive reductions to the set of Kolmogorov-random strings. In this paper we show that an approach laid out in \cite{adfht} to settle this conjecture cannot succeed without significant alteration, but that it does bear fruit if we consider time-bounded Kolmogorov complexity instead. We show that if a set $A$ is reducible in polynomial time to the set of time-$t$-bounded Kolmogorov-random strings (for all large enough time bounds $t$), then $A$ is in \Ppoly, and that if in addition such a reduction exists for any universal Turing machine one uses in the definition of Kolmogorov complexity, then $A$ is in \PSPACE

Learning Weak Reductions to Sparse Sets

We study the consequences of NP having non-uniform polynomial size circuits of various types. We continue the work of Agrawal and Arvind~\cite{AA:96} who study the consequences of \SAT being many-one reducible to functions computable by non-uniform circuits consisting of a single weighted threshold gate. (\SAT \leq_m^p \LT). They claim that as a consequence \PTIME = \NP follows, but unfortunately their proof was incorrect. We take up this question and use results from computational learning theory to show that if \SAT \leq_m^p \LT then \PH = \PTIME^\NP. We furthermore show that if \SAT disjunctive truth-table (or majority truth-table) reduces to a sparse set then \SAT \leq_m^p \LT and hence a collapse of \PH to \PTIME^\NP also follows. Lastly we show several interesting consequences of \SAT \leq_{dtt}^p \SPARSE