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

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

Study on Relationship Between Individual Work Value and Work Performance of Civil Servants-Based on the Research in China

Civil servants are the key power to government’s development and social demands. However, present large amount of researches on public HR management mainly study from macro fields such as policy, education and team setup, the study of civil servants’ inner factors and outer work performance is comparatively much less, while the demonstrative study is the least. This paper proposes research hypothesis of civil servants’ work value and work performance on the basis of literature review; with the design of questionnaire, statistics analysis and research, finds the influencing factors and reasons why individual work value affects work performance; gets the specific influencing degree the parameters affect work performance. On the basis of demonstrative study, this paper proposes applicable methods and suggestions for civil servants’ work value on individual work performance management in administration departments, suggestions especially for HR management, screening and selection, stimulating, training and development.Key words: Civil servant; Work value; Work performanc

Catalytic space: non-determinism and hierarchy

Catalytic computation, defined by Buhrman, Cleve, Koucký, Loff and Speelman (STOC 2014), is a space-bounded computation where in addition to our working memory we have an exponentially larger auxiliary memory which is full; the auxiliary memory may be used throughout the computation, but it must be restored to its initial content by the end of the computation. Motivated by the surprising power of this model, we set out to study the non-deterministic version of catalytic computation. We establish that non-deterministic catalytic log-space is contained in ZPP, which is the same bound known for its deterministic counterpart, and we prove that non-deterministic catalytic space is closed under complement (under a standard derandomization assumption). Furthermore, we establish hierarchy theorems for non-deterministic and deterministic catalytic computation