K. Ko and H. Friedman. Computational complexity of real functions. Theoretical Computer Science, 20:323-352, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of the event S. Define u n = fx j jxj = ng) For each n, let (x) be the conditional probability of x in fx j jxj = ng. That is, n (x) x) u n , if u n 0, and n (x) 0 for x 2 fx j jxj = ng, if u n = 0. A function from S to [0; 1] is computable in polynomial time [KF82] if there is a polynomial time bounded transducer M such that for every string x and every positive integer n, j(x) Gamma M(x;1 )j n . Consistent with Levin s hypothesis that natural distributions are computable in polynomial time, we restrict our attention entirely to such ....

K. Ko and H. Friedman. Computational complexity of real functions. Theoretical Computer Science, 20:323--352, 1982.

....n, #(n, k) is also a polynomial. # Corollary 4.4 Let f # C[0, 1] be polynomial time computable, then i) Under(# f ) # # P 2 [B f ] and, 12 ii) p n ) n#N # # P 2 [B f ] Remark 4. 5 If we want to take into account the complexity of integration (oracle B f ) the best result is given in [KF82]. If f # C[0, 1] is polynomial time computable then the real number # 1 0 f(x) dx is PSPACER . ....

K.-I. Ko and H. Friedman. Computational complexity of real functions. Theoretical Computer Science, 20:323--352, 1982.

....event S. Define u n = fx j jxj = ng) For each n, let 0 n (x) be the conditional probability of x in fx j jxj = ng. That is, 0 n (x) 0 (x) u n , if u n 0, and 0 n (x) 0 for x 2 fx j jxj = ng, if 3 u n = 0. A function from S to [0; 1] is computable in polynomial time [KF82] if there is a polynomial time bounded transducer M such that for every string x and every positive integer n, j(x) M(x;1 n )j 1 2 n . Consistent with Levin s hypothesis that natural distributions are computable in polynomial time, we restrict our attention entirely to such distributions. ....

K. Ko and H. Friedman. Computational complexity of real functions. Theoretical Computer Science, 20:323--352, 1982.

....independently proposed (Godel, Church, or Turing) they turned out to be equivalent. In the case of computability over the reals the situation is essentially different from the classical one, since some nonequivalent notions have been proposed among others by Moschovakis [10] Blum [3] Freedman [6], Ershov [5] Pour El [14] and others. This was apparently for the reason that different authors took notice of different aspects of computational processes over abstract structure in particularly over the reals. So in this case it is desirable to find some more general and more Inst. of ....

H. Freedman and K. Ko, Computational complexity of real functions, Theoret. Comput. Sci. , 1982, v. 20, pages 323--352.

....functions are presented. 1 Introduction In the recent time, attention to the problems of computability over uncountable structures, particularly over the reals, is constantly raised. The theories proposed by Barwise [1] Scott [18] Ershov [7] Grzegorczyk [9] Moschovakis [15] Freedman [8] got further development in the works of Blum, Shub, Smail [4] Poul El, Richards [16] Edalat, Sunderhauf [6] Stoltenberg Hansen, Tucker [19] Korovina, Kudinov [13] and others. This work continues the investigation of the approach to computability proposed in [13] Developing our approach we ....

H. Freedman and K. Ko, Computational complexity of real functions, Theoret. Comput. Sci. , v. 20, 1982, pages 323--352.

....definability theory. In the case of computability over the reals the situation is essentially different from the classical one, where computability over the natural numbers is studied. Some notions of computability over the reals have been proposed by Grzegorczyk [10] Moschovakis [14] Freedman [8], Ershov [7] Blum [4] Pour El, Richards [18] Edalat [6] and others. Most of them are not equivalent since different authors took notice of different aspects of computational processes Inst. of Math. University pr. 4, Novosibirsk, Russia, email: rita ssc.nsu.ru y Inst. of Math. University ....

H. Freedman and K. Ko, Computational complexity of real functions, Theoret. Comput. Sci. , 1982, v. 20, pages 323--352.

....of recursive analysis, i.e. a real or p adic number is viewed as infinite object, and is understood as the limit of finite objects, namely rational numbers. This approach for the case of real numbers has been extensively developed by various authors. A partial list of related work follows: [1, 2, 7, 8, 6, 10, 13, 15, 16, 17, 18, 14, 20]. The main issue in the present work is to study the analogous problems in the case of p adic numbers and verify which ones carry over and which ones fail and the reason for the failures. The present approach to the nature of a real or a p adic number should be contrasted with the ....

....derivative f # can be calculated by a primitive recursive procedure. A point of interest here is that the analogous closure property for the polynomially time computable real numbers (real closed) is proved in a di#erent manner than the one for the p adic case by H. Friedman and Ker I Ko [6]. The statement for the polynomially time computable real numbers depends on the complexity of the roots of analytic functions. In the case of real numbers an upper bound for the complexity of roots of analytic polynomially time computable functions is proved. From this upper bound we have that ....

H. Friedman and K. Ko. Computational complexity of real functions. Theoretical Computer Science, 20:323--352, 1982.

....and continuity of functions. There are also many undecidability and complexity results. Also in this line of research there is a plethora of different approaches. A key point of difference among these approaches is the notion of computability used. Turing machines are used in Ko and Friedman n [12], 13] Turing [32] Wiedmer [36] A theory of computability of Baire spaces called type 2 recursion theory is used to study computability on real number in Kreitz and Weihrauch [14] Finally approximation spaces are introduced and used in Lacombe [15] Martin Lof [16] D. Scott [26] 26] ....

K. Ko and Friedmann, "Computational Complexity of Real Functions." Theoret. Comput. Sci. 20 (1982) 323-352. 78

....to semantics via continuity spaces but leaving the interested reader to fill in the gaps in the literature (q.v. 12, 11, 13] 2 The Lazy Reals One of the most widely used representation of the reals in both practical and theoretical computer science is the signed digit representation (q.v. [4, 14, 18, 26, 16]) The reason comes from its simplicity, computability of the arithmetic operations, and easy implementation in lazy functional languages. Here a real number is represented by an infinite list or string. The reals represented in this way are usually called Lazy Reals since they can be implemented ....

K. I. Ko and H. Friedman. Computational complexity of real functions. Theoretical Computer Science, 20, 1982.

....of such real valued functions Approximable in Probabilistic Polynomial time will be denoted as APP, while the corresponding deterministic class as AP. It is worth pointing out that our definition of APP is different from the definition of the class of efficiently computable real functions f in [KF82, Ko91], where f is required to be approximable to within ffl in probabilistic time poly(n; 1= log ffl) It is not hard to see that BPP corresponds exactly to the subclass of Boolean functions of APP. It is also easy to see that the problem of computing the acceptance probability of a given Boolean ....

K. Ko and H. Friedman. Computational complexity of real functions. Theoretical Computer Science, 20:323--352, 1982.

....depends on a notion of bounding computational resources (in this case running time) in a general computational model in some natural way. Questions about feasibility arise when dealing with type 2 functionals as well, for example, in the study of reducibilities ( 9] computable analysis ([7]) and descriptive set theory ( 10] Also, in [3] feasibility for functions of arbitrary finite type is presented as a means of interpreting systems of feasibly constructive arithmetic. Constable s paper [2] appears to be the first to consider the notion of feasibility for type 2 functionals. ....

H. Friedman and K. Ko. Computational Complexity of Real Functions. Theoretical Computer Science 20 (1982), 323-352.

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K. Ko and H. Friedman. Computational complexity of real functions. Theoretical Computer Science, 20:323-352, 1982.

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K.-I. Ko. Computational Complexity of Real Functions. Birkhauser Boston, 1991.

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K.-I. Ko and H. Friedman. Computational complexity of real functions. Theoretical Computer Science, 20:323--352, 1982.

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K.-I Ko. Computational Complexity of Real Functions. Birkhauser, 1991.

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K. Ko and H. Friedman. Computational complexity of real functions. Theoretical Computer Science, 20:323--352, 1982.

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Ko, K.-I, and H. Friedman, On the computational complexity of real functions, Theoretical Computer Science 20 (1982), 323--352.

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K.--I. Ko, H. Friedman, Computational complexity of real functions. Theor. Comput. Sc. 20, 1982, 323--352

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Ko, Ker-I, and H. Friedman, On the computational complexity of real functions, Theor. Comput. Sci., 20 (1982), 323-352.

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Ko, Ker-I, and H. Friedman, On the computational complexity of real functions, Theor. Comput. Sci., 20 (1982), 323-352.

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K. Ko and H. Friedman. Computational complexity of real functions. Theorical Computer Science, pages 20:323--352, 1982.

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H. Friedman and K. Ko, Computational complexity of real functions, Theoret. Comput. Sci. , v. 20, 1982, pages 323-352.

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H. Freedman and K. Ko, Computational complexity of real functions, Theoret. Comput. Sci. , v. 20, 1982, pages 323-352.

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K. Ko and H. Friedman. Computational complexity of real functions. Theorical Computer Science, pages 20:323--352, 1982.

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H. Freedman and K. Ko, Computational complexity of real functions, Theoret. Comput. Sci. , v. 20, 1982, pages 323--352.

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