Differential recursion

Kawamura, Akitoshi

doi:10.1145/1507244.1507252

Cited by 7 publications

(3 citation statements)

References 36 publications

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“…The formula was inserted into the chip as on analog data processor from the sensor. 9,10,11 A conversion was performed to compare output bit sensor with the number of bacteria in the preparat. This conversion was performed by making a plot between two variables so that regression equation which would be used in the chip should be obtained.…”

Section: Resultsmentioning

confidence: 99%

Tuberculosis Counter (Tc) as the Equipment to Measure the Level of Tb in Sputum

Purwanda

Fibriawan

Sasmito

et al. 2016

IJTID

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Tuberculosis (TB) is an infectious disease caused by Mycobacterium tuberculosis. This disease is the third killer disease after cardiovascular diseases and respiratory diseases, and is also he number one killer disease in a group of infectious diseases. This is partly due to the late handling and a non real time detection, both of which will inhibit the therapy which yields a large number of microorganisms in the body

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Section: Resultsmentioning

confidence: 99%

Tuberculosis Counter (Tc) as the Equipment to Measure the Level of Tb in Sputum

Purwanda

Fibriawan

Sasmito

et al. 2016

IJTID

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“…7 Related results and open problems 7.1 Other results on differential equations Table 7.1 summarizes what is known about the computability and complexity of the initial value problem (1.1) in our sense. Computability of other aspects of the solution is discussed by Cenzer & Remmel (2004), Graça et al (2008) and Kawamura (2009a). Edalat & Pattinson (2004) give a domain-theoretic account for the problem.…”

Section: Proof Of Theorem 55mentioning

confidence: 99%

Lipschitz Continuous Ordinary Differential Equations are Polynomial-Space Complete

Kawamura

2009

2009 24th Annual IEEE Conference on Computational Complexity

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ABSTRACT. In answer to Ko's question raised in 1983, we show that an initial value problem given by a polynomial-time computable, Lipschitz continuous function can have a polynomial-space complete solution. The key insight is simple: the Lipschitz condition means that the feedback in the differential equation is weak. We define a class of polynomial-space computation tableaux with equally restricted feedback, and show that they are still polynomial-space complete. The same technique also settles Ko's two later questions on Volterra integral equations.Let gW OE0; 1 R ! R be a continuous function and consider the initial value problemThe Picard-Lindelöf (or Cauchy-Lipschitz) Theorem [15] states 1 that this equation has a unique solution hW OE0; 1 ! R if g is Lipschitz continuous (along its second argument), i. e., (2) jg.t; y 0 / g.t; y 1 /j Ä Z jy 0 y 1 j; t 2 OE0; 1; y 0 ; y 1 2 R for some constant Z independent of y 0 , y 1 and t . We are interested in the computational complexity of the solution h under this condition. Our model of computation of real functions is reviewed in Section 1. It is adopted from computable analysis and is thus consistent with the conventional notion of computability (see Section 5.3 for other perspectives on the "complexity" of similar problems). We formulate our main result in Section 2: the solution of the above equation can be polynomialspace complete, even if g is polynomial-time computable. This was open since 1983 [18]. The proof will be given in Section 3. The main idea is to regard the differential equation with the Lipschitz condition as a polynomial-space computation tableau with certain restrictions. In Section 4, we use the same techniques solve several other problems, including the ones about Volterra integral equations [20]. Section 5 discusses related results and problems.Key words and phrases. computable analysis; computational complexity; exponential space; initial value problem; Lipschitz condition; ordinary differential equations; Picard-Lindelöf Theorem; polynomial space; Volterra integral equations.This work was supported in part by the Nakajima Foundation and by the Natural Sciences and Engineering Research Council of Canada.1 There are several variants of the theorem; here is a proof sketch for ours. Let C be the set of all continuous real functions on OE0; 1. A solution of (1) is a fixed point of the operator T W C ! C defined byThis solution exists and is unique by the Banach fixed point theorem, because T is a contraction with respect to the distance d given by d.h 0 ; h 1 / D max t2OE0;1 exp. 2Zt /jh 0 .t/ h 1 .t/j for h 0 , h 1 2 C . To compute a real function f , the machine should output an approximation of f .t / to given precision 2 m by consulting the oracle for approximations of t to any precision 2 n it desires (left). An alternative picture (right) is that the machine converts any stream of improving approximations of t to a stream of improving approximations of f .t/.

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“…Such a hierarchy has been proven to be proper; its first level contains the discrete primitive recursive functions, the second level contains the partial recursive functions, and the following level contains the solution to the halting problem. Recently, A. Kawamura [19] tried to redefine MRec exploring several ways to remove the ambiguities in the original definition, in particular the behavior of the operators on partial functions. Further work on MRec is mentioned below when discussing the relationships with recursive analysis.…”

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confidence: 99%

A survey of recursive analysis and Moore’s notion of real computation

Gomaa

2011

Nat Comput

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The theory of analog computation aims at modeling computational systems that evolve in a continuous manner. Unlike the situation with the discrete setting there is no unified theory of analog computation. There are several proposed theories, some of them seem quite orthogonal. Some theories can be considered as generalizations of the Turing machine theory and classical recursion theory. Among such are recursive analysis and Moore's class of recursive real functions. Recursive analysis was introduced by A. Turing [1936], A. Grzegorczyk [1955], and D. Lacombe [1955]. Real computation in this context is viewed as effective (in the sense of Turing machine theory) convergence of sequences of rational numbers. In 1996 Moore introduced a function algebra that captures his notion of real computation; it consists of some basic functions and their closure under composition, integration and zerofinding. Though this class is inherently unphysical, much work have been directed at stratifying, restricting, and comparing it with other theories of real computation such as recursive analysis and the GP AC. In this article we give a detailed exposition of recursive analysis and Moore's class and the relationships between them.

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Differential recursion

Cited by 7 publications

References 36 publications

Tuberculosis Counter (Tc) as the Equipment to Measure the Level of Tb in Sputum

Tuberculosis Counter (Tc) as the Equipment to Measure the Level of Tb in Sputum

Lipschitz Continuous Ordinary Differential Equations are Polynomial-Space Complete

A survey of recursive analysis and Moore’s notion of real computation

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