Computational speed-up by effective operators

1972 ◽  
Vol 37 (1) ◽  
pp. 55-68 ◽  
Author(s):  
Albert R. Meyer ◽  
Patrick C. Fischer
Keyword(s):  
Turing Machine ◽  
Computable Function ◽  
Turing Machines ◽  
Complexity Measures ◽  
Computational Speed ◽  
Partial Recursive Functions ◽  
Speed Up ◽  
Recursive Equations ◽  
Computable Functions ◽  
Effective Operators

The complexity of a computable function can be measured by considering the time or space required to compute its values. Particular notions of time and space arising from variants of Turing machines have been investigated by R. W. Ritchie [14], Hartmanis and Stearns [8], and Arbib and Blum [1], among others. General properties of such complexity measures have been characterized axiomatically by Rabin [12], Blum [2], Young [16], [17], and McCreight and Meyer [10].In this paper the speed-up and super-speed-up theorems of Blum [2] are generalized to speed-up by arbitrary total effective operators. The significance of such theorems is that one cannot equate the complexity of a computable function with the running time of its fastest program, for the simple reason that there are computable functions which in a very strong sense have no fastest programs.Let φi be the ith partial recursive function of one variable in a standard Gödel numbering of partial recursive functions. A family Φ0, Φ1, … of functions of one variable is called a Blum measure on computation providing(1) domain (φi) = domain (Φi), and(2) the predicate [Φi(x) = m] is recursive in i, x and m.Typical interpretations of Φi(x) are the number of steps required by the ith Turing machine (in a standard enumeration of Turing machines) to converge on input x, the space or number of tape squares required by the ith Turing machine to converge on input x (with the convention that Φi(x) is undefined even if the machine fails to halt in a finite loop), and the length of the shortest derivation of the value of φi(x) from the ith set of recursive equations.

scholarly journals Fractal Dimension versus Process Complexity

2016 ◽  
Vol 2016 ◽  
pp. 1-21 ◽  
Author(s):  
Joost J. Joosten ◽  
Fernando Soler-Toscano ◽  
Hector Zenil
Keyword(s):  
Fractal Dimension ◽  
Turing Machine ◽  
Time Complexity ◽  
Linear Time ◽  
Space Time ◽  
Polynomial Space ◽  
Turing Machines ◽  
Complexity Measures ◽  
Strong Relation ◽  
Process Complexity

We look at small Turing machines (TMs) that work with just two colors (alphabet symbols) and either two or three states. For any particular such machineτand any particular inputx, we consider what we call thespace-timediagram which is basically the collection of consecutive tape configurations of the computationτ(x). In our setting, it makes sense to define a fractal dimension for a Turing machine as the limiting fractal dimension for the corresponding space-time diagrams. It turns out that there is a very strong relation between the fractal dimension of a Turing machine of the above-specified type and its runtime complexity. In particular, a TM with three states and two colors runs in at most linear time, if and only if its dimension is 2, and its dimension is 1, if and only if it runs in superpolynomial time and it uses polynomial space. If a TM runs in timeO(xn), we have empirically verified that the corresponding dimension is(n+1)/n, a result that we can only partially prove. We find the results presented here remarkable because they relate two completely different complexity measures: the geometrical fractal dimension on one side versus the time complexity of a computation on the other side.

scholarly journals Accelerating simulations using REDCHEM_v0.0 for atmospheric chemistry mechanism reduction

2018 ◽  
Vol 11 (8) ◽  
pp. 3391-3407 ◽  
Author(s):  
Zacharias Marinou Nikolaou ◽  
Jyh-Yuan Chen ◽  
Yiannis Proestos ◽  
Jos Lelieveld ◽  
Rolf Sander
Keyword(s):  
Atmospheric Chemistry ◽  
Transport Model ◽  
A Priori ◽  
Chemical Mechanism ◽  
Climate Modelling ◽  
Mechanism Reduction ◽  
Computational Speed ◽  
Speed Up ◽  
Spatio Temporal ◽  
Skeletal Mechanism

Abstract. Chemical mechanism reduction is common practice in combustion research for accelerating numerical simulations; however, there have been limited applications of this practice in atmospheric chemistry. In this study, we employ a powerful reduction method in order to produce a skeletal mechanism of an atmospheric chemistry code that is commonly used in air quality and climate modelling. The skeletal mechanism is developed using input data from a model scenario. Its performance is then evaluated both a priori against the model scenario results and a posteriori by implementing the skeletal mechanism in a chemistry transport model, namely the Weather Research and Forecasting code with Chemistry. Preliminary results, indicate a substantial increase in computational speed-up for both cases, with a minimal loss of accuracy with regards to the simulated spatio-temporal mixing ratio of the target species, which was selected to be ozone.

Implementation of a Semi-Implicit Pressure-Based Multigrid Fluid Flow Algorithm on a Graphics Processing Unit

2009 ◽  
Author(s):  
Aaron F. Shinn ◽  
S. P. Vanka
Keyword(s):  
Stokes Equations ◽  
Graphics Processing Unit ◽  
Navier Stokes ◽  
Processing Unit ◽  
Navier Stokes Equations ◽  
Driven Cavity ◽  
Multigrid Algorithm ◽  
Computational Speed ◽  
Speed Up ◽  
Graphics Processing

A semi-implicit pressure based multigrid algorithm for solving the incompressible Navier-Stokes equations was implemented on a Graphics Processing Unit (GPU) using CUDA (Compute Unified Device Architecture). The multigrid method employed was the Full Approximation Scheme (FAS), which is used for solving nonlinear equations. This algorithm is applied to the 2D driven cavity problem and compared to the CPU version of the code (written in Fortran) to assess computational speed-up.

Eventually infinite time Turing machine degrees: infinite time decidable reals

2000 ◽  
Vol 65 (3) ◽  
pp. 1193-1203 ◽  
Author(s):  
P.D. Welch
Keyword(s):  
Turing Machine ◽  
Initial Segment ◽  
Upper Bounds ◽  
Turing Degrees ◽  
Turing Machines ◽  
Jump Operator ◽  
Infinite Time

AbstractWe characterise explicitly the decidable predicates on integers of Infinite Time Turing machines, in terms of admissibility theory and the constructible hierarchy. We do this by pinning down ζ, the least ordinal not the length of any eventual output of an Infinite Time Turing machine (halting or otherwise); using this the Infinite Time Turing Degrees are considered, and it is shown how the jump operator coincides with the production of mastercodes for the constructible hierarchy; further that the natural ordinals associated with the jump operator satisfy a Spector criterion, and correspond to the Lζ-stables. It also implies that the machines devised are “Σ2 Complete” amongst all such other possible machines. It is shown that least upper bounds of an “eventual jump” hierarchy exist on an initial segment.

scholarly journals Accelerated methods for direct computation of fusion alpha particle losses within, stellarator optimization

2020 ◽  
Vol 86 (2) ◽  
Author(s):  
Christopher G. Albert ◽  
Sergei V. Kasilov ◽  
Winfried Kernbichler
Keyword(s):  
Alpha Particle ◽  
Symplectic Integrators ◽  
Particle Loss ◽  
Early Classification ◽  
Statistical Computation ◽  
Computational Speed ◽  
Order Of Magnitude ◽  
Speed Up ◽  
Orbit Types ◽  
Optimization Loop

Accelerated statistical computation of collisionless fusion alpha particle losses in stellarator configurations is presented based on direct guiding-centre orbit tracing. The approach relies on the combination of recently developed symplectic integrators in canonicalized magnetic flux coordinates and early classification into regular and chaotic orbit types. Only chaotic orbits have to be traced up to the end, as their behaviour is unpredictable. An implementation of this technique is provided in the code SIMPLE (symplectic integration methods for particle loss estimation, Albert et al., 2020b, doi:10.5281/zenodo.3666820). Reliable results were obtained for an ensemble of 1000 orbits in a quasi-isodynamic, a quasi-helical and a quasi-axisymmetric configuration. Overall, a computational speed up of approximately one order of magnitude is achieved compared to direct integration via adaptive Runge–Kutta methods. This reduces run times to the range of typical magnetic equilibrium computations and makes direct alpha particle loss computation adequate for use within a stellarator optimization loop.

TWO TOPICS CONCERNING TWO-DIMENSIONAL AUTOMATA OPERATING IN PARALLEL

1992 ◽  
Vol 06 (02n03) ◽  
pp. 211-225 ◽  
Author(s):  
KATSUSHI INOUE ◽  
ITSUO SAKURAMOTO ◽  
MAKOTO SAKAMOTO ◽  
ITSUO TAKANAMI
Keyword(s):  
Turing Machine ◽  
Finite Automata ◽  
Space Complexity ◽  
Two Dimensional ◽  
Turing Machines ◽  
Small Space ◽  
Bottom Up ◽  
Constructible Function ◽  
Deterministic Turing Machine ◽  
Alternating Turing Machines

This paper deals with two topics concerning two-dimensional automata operating in parallel. We first investigate a relationship between the accepting powers of two-dimensional alternating finite automata (2-AFAs) and nondeterministic bottom-up pyramid cellular acceptors (NUPCAs), and show that Ω ( diameter × log diameter ) time is necessary for NUPCAs to simulate 2-AFAs. We then investigate space complexity of two-dimensional alternating Turing machines (2-ATMs) operating in small space, and show that if L (n) is a two-dimensionally space-constructible function such that lim n → ∞ L (n)/ loglog n > 1 and L (n) ≤ log n, and L′ (n) is a function satisfying L′ (n) =o (L(n)), then there exists a set accepted by some strongly L (n) space-bounded two-dimensional deterministic Turing machine, but not accepted by any weakly L′ (n) space-bounded 2-ATM, and thus there exists a rich space hierarchy for weakly S (n) space-bounded 2-ATMs with loglog n ≤ S (n) ≤ log n.

scholarly journals Revisiting the simulation of quantum Turing machines by quantum circuits

2019 ◽  
Vol 475 (2226) ◽  
pp. 20180767 ◽  
Author(s):  
Abel Molina ◽  
John Watrous
Keyword(s):  
Turing Machine ◽  
Polynomial Time ◽  
Computational Models ◽  
Circuit Complexity ◽  
Simulation Method ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Turing Machines ◽  
Generated Family ◽  
Quantum Turing Machine

Yao's 1995 publication ‘Quantum circuit complexity’ in Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science , pp. 352–361, proved that quantum Turing machines and quantum circuits are polynomially equivalent computational models: t ≥ n steps of a quantum Turing machine running on an input of length n can be simulated by a uniformly generated family of quantum circuits with size quadratic in t , and a polynomial-time uniformly generated family of quantum circuits can be simulated by a quantum Turing machine running in polynomial time. We revisit the simulation of quantum Turing machines with uniformly generated quantum circuits, which is the more challenging of the two simulation tasks, and present a variation on the simulation method employed by Yao together with an analysis of it. This analysis reveals that the simulation of quantum Turing machines can be performed by quantum circuits having depth linear in t , rather than quadratic depth, and can be extended to variants of quantum Turing machines, such as ones having multi-dimensional tapes. Our analysis is based on an extension of method described by Arright, Nesme and Werner in 2011 in Journal of Computer and System Sciences 77 , 372–378. ( doi:10.1016/j.jcss.2010.05.004 ), that allows for the localization of causal unitary evolutions.

scholarly journals Denjoy, Demuth and density

2014 ◽  
Vol 14 (01) ◽  
pp. 1450004 ◽  
Author(s):  
Laurent Bienvenu ◽  
Rupert Hölzl ◽  
Joseph S. Miller ◽  
André Nies
Keyword(s):  
Computable Function ◽  
Density Theorem ◽  
Positive Density ◽  
Random Real ◽  
Open Problems ◽  
Covering Problems ◽  
Closed Class ◽  
Random Reals ◽  
Upper And Lower Derivatives ◽  
Computable Functions

We consider effective versions of two classical theorems, the Lebesgue density theorem and the Denjoy–Young–Saks theorem. For the first, we show that a Martin-Löf random real z ∈ [0, 1] is Turing incomplete if and only if every effectively closed class 𝒞 ⊆ [0, 1] containing z has positive density at z. Under the stronger assumption that z is not LR-hard, we show that every such class has density one at z. These results have since been applied to solve two open problems on the interaction between the Turing degrees of Martin-Löf random reals and K-trivial sets: the noncupping and covering problems. We say that f : [0, 1] → ℝ satisfies the Denjoy alternative at z ∈ [0, 1] if either the derivative f′(z) exists, or the upper and lower derivatives at z are +∞ and -∞, respectively. The Denjoy–Young–Saks theorem states that every function f : [0, 1] → ℝ satisfies the Denjoy alternative at almost every z ∈ [0, 1]. We answer a question posed by Kučera in 2004 by showing that a real z is computably random if and only if every computable function f satisfies the Denjoy alternative at z. For Markov computable functions, which are only defined on computable reals, we can formulate the Denjoy alternative using pseudo-derivatives. Call a real zDA-random if every Markov computable function satisfies the Denjoy alternative at z. We considerably strengthen a result of Demuth (Comment. Math. Univ. Carolin.24(3) (1983) 391–406) by showing that every Turing incomplete Martin-Löf random real is DA-random. The proof involves the notion of nonporosity, a variant of density, which is the bridge between the two themes of this paper. We finish by showing that DA-randomness is incomparable with Martin-Löf randomness.

Two recursion theoretic characterizations of proof speed-ups

1989 ◽  
Vol 54 (2) ◽  
pp. 522-526 ◽  
Author(s):  
James S. Royer
Keyword(s):  
Recursive Function ◽  
Formal System ◽  
Complexity Measure ◽  
Incompleteness Theorem ◽  
Similar Treatment ◽  
Recursive Functions ◽  
Partial Recursive Functions ◽  
Speed Up

Smullyan in [Smu61] identified the recursion theoretic essence of incompleteness results such as Gödel's first incompleteness theorem and Rosser's theorem. Smullyan (improving upon [Kle50] and [Kle52]) showed that, for sufficiently complex theories, the collection of provable formulae and the collection of refutable formulae are effectively inseparable—where formulae and their Gödel numbers are identified. This paper gives a similar treatment for proof speed-up. We say that a formal system S1is speedable over another system S0on a set of formulaeAiff, for each recursive functionh, there is a formulaαinAsuch that the length of the shortest proof ofαin S0is larger thanhof the shortest proof ofαin S1. (Here we equate the length of a proof with something like the number of characters making it up,notits number of lines.) We characterize speedability in terms of the inseparability by r.e. sets of the collection of formulae which are provable in S1but unprovable in S0from the collectionA–where again formulae and their Gödel numbers are identified. We provide precise definitions of proof length, speedability and r.e. inseparability below.We follow the terminology and notation of [Rog87] with borrowings from [Soa87]. Below,ϕis an acceptable numbering of the partial recursive functions [Rog87] andΦa (Blum) complexity measure associated withϕ[Blu67], [DW83].

CLOSURE PROPERTY OF SPACE-BOUNDED TWO-DIMENSIONAL ALTERNATING TURING MACHINES, PUSHDOWN AUTOMATA, AND COUNTER AUTOMATA

2001 ◽  
Vol 15 (07) ◽  
pp. 1143-1165 ◽  
Author(s):  
TOKIO OKAZAKI ◽  
KATSUSHI INOUE ◽  
AKIRA ITO ◽  
YUE WANG
Keyword(s):  
Turing Machine ◽  
Nondecreasing Function ◽  
Closure Property ◽  
Two Dimensional ◽  
Turing Machines ◽  
Pushdown Automaton ◽  
Alternating Turing Machines ◽  
Positive Integers ◽  
Pushdown Automata

This paper investigates closure property of the classes of sets accepted by space-bounded two-dimensional alternating Turing machines (2-atm's) and space-bounded two-dimensional alternating pushdown automata (2-apda's), and space-bounded two-dimensional alternating counter automata (2-aca's). Let L(m, n): N2 → N (N denotes the set of all positive integers) be a function with two variables m (= the number of rows of input tapes) and n (= the number of columns of input tapes). We show that (i) for any function f(m) = o( log m) (resp. f(m) = o( log m/ log log m)) and any monotonic nondecreasing function g(n) space-constructible by a two-dimensional Turing machine (2-Tm) (resp. two-dimensional pushdown automaton (2-pda)), the class of sets accepted by L(m,n) space-bounded 2-atm's (2-apda's) is not closed under row catenation, row + or projection, and (ii) for any function f(m) = o(m/ log ) (resp. for any function f(m) such that log f(m) = o( log m)) and any monotonic nondecreasing function g(n) space-constructible by a two-dimensional counter automaton (2-ca), the class of sets accepted by L(m, n) space-bounded 2-aca's is not closed under row catenation, row + or projection, where L(m, n) = f(m) + g(n) (resp. L(m, n) = f(m) × g(n)).

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