Notes on Space Complexity of Integration of Computable Real

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1.1 CF computable real numbers and functions Cauchy functions in the model defined in [1] are functions binary converging to real numbers. A function φ : N → D (here D is the set of dyadic rational numbers) is said to binary converge to real number x if − |φ(n) − x|≤ 2 n for all n ∈ N; CFx denotes the set of all functions binary converging to x. Real number x is said to be a CF computable real number if CFx contains a computable function φ. Real function f on interval [a, b] is said to be a CF computable function on interval [a, b] if there exists a function-oracle Turing machine M such that for all x ∈ [a, b] and for all φ ∈ CFx function ψ computed by M with oracle φ is in CFf(x). The input of functions φ and ψ is 0n (0 repeated n times) when a number or a function is evaluated to precision 2−n. Definition 1. [1] Function f :[a, b] → R is said to be computable in time t(n) real function on interval [a, b] if for all computable real numbers x ∈ [a, b] function ψ ∈ CFf(x) (ψ is from the definition of CF computable real function) is computable in time t(n). Definition 2. [1] Function f :[a, b] → R is said to be computable in space s(n) real function on interval [a, b] if for all computable real numbers x ∈ [a, b] function ψ ∈ CFf(x) (ψ is from the definition of CF computable real function) is computable in space s(n). FP denotes the class of string functions computable in polynomial time on Turing machines, FLINSPACE denotes the class of string functions computable in linear space on Turing machines, and FEXPTIME denotes the class of string functions computable in exponential time on Turing machines. According to these notations, polynomial-time computable real functions are said to be FP computable real functions, linear-space computable real functions are said to be FLINSPACE computable real functions, and exponential-time computable real functions are said to be FEXPTIME computable real functions. The set of FP computable real functions on interval [a, b] is denoted by FPC[a,b], the set of FLINSPACE computable real functions on interval [a, b] is denoted by FLINSPACEC[a,b], and the set of FEXPTIME computable real functions on interval [a, b] is denoted by FEXPTIMEC[a,b]. 2 The set of the C [0, 1] real functions from class FPC[a,b] is denoted by FPC2[a,b], and the set of ∞ the C [0, 1] real functions from class FPC[a,b] is denoted by FPC∞[a,b]. The same definitions are for other complexity classes.

1.2 Integration of FP computable real functions The main results from [1] regarding integration of FP computable real functions are the following. Theorem 1. [1, 5.33] The following are equivalent: x a) Let f be in FPC[0,1]. Then, the function g(x)= 0 f(t)dt is polynomial-time computable. x ∞ b) Let f be in FPC [0,1]. Then, the function g(x)=R 0 f(t)dt is polynomial-time computable. R 2 c) FP = #P. It means if FP 6= #P then the integral of a polynomial-time computable real function f is not necessarily polynomial-time computable even if f is known to be infinitely differentiable. But if f is polynomial-time computable and is analytic on [0, 1] then the integral of f must be computable in polynomial time. Some additional results regarding the time complexity of integration of computable real functions can be found in [6]. For example, the computation of the volume of a one-dimensional convex set K is #P-complete if K is represented by a polynomial-time computable function defining its boundary.

2 Upper bound of the time complexity of integration

Let f be a linear-space computable C2[0, 1] real function on interval [0, 1]. Let’s consider the composite trapezoidal rule [7] for function f on interval [a, x] for a < x ≤ b:

x g(x)= f(t)dt = Za k−1 h = f(t )+ 2 · f(ti) + f(tk) − 2 0 i=1 ! ! 3 X (b − a) ′′ − f (ξ) 12 · k2

x−a wherein k is a natural number, h = k , t0 = a, ti = a + i · h, tk = b, and a<ξ

x g(x)= f(t)dt = Z0 k−1 h = f(t )+ 2 · f(ti) + f(tk) − (1) 2 0 i=1 ! ! X 1 ′′ − f (ξ) 12 · k2 x wherein h = , t0 = 0, ti = i · h, tk = x, and 0 <ξ

∗ |∆(g; x)| = |g (x) − g(x)| = k−1 h ∗ ∗ ∗ = f (t )+ 2 · f (ti) + f (tk) − 2 0 i=1 ! ! X k−1 h − f(t )+ 2 · f(ti) + f(tk) + 2 0 i=1 ! ! X 1 ′′ + f (ξ) ≤ 12 · k2 k−1 h ∗ ∗ ≤ |f (t ) − f(t )| + 2 ·|f (ti) − f(ti)| + 2 0 0 i=1 ! X ∗ 1 ′′ + |f (tn) − f(tk)| + f (ξ). 12 · k2

3 So, we have: ∗ |∆(g; x)| = |g (x) − g(x)|≤ h 1 2(k − 1) 1 1 ≤ + + + C ≤ 2 2m 2m 2m 12 · k2 1 1 k 1 1 1 ≤ + C = + C ; k 2m 12 · k2 1 2m k2 2 ′′ here the fact that f is bounded above if f ∈ C2[0, 1] is taken into account. If we take m = 2n and n k = C22 then

1 1 −n |∆(g; x)| = 2n + 2 2n C2 < 2 . 2 (C2) 2 ∗ It means it is suﬃcient to compute in a loop for i ∈ [0,k] approximations g (x) of function g on n ∗ interval [0, 1] using formula (1) wherein k = C22 and f (xi) are the approximations of function f −m at points xi, i ∈ [0..k], to precision 2 wherein m =2n. So, the following theorems holds. Theorem 2. If f is a linear-space computable C2[0, 1] real function on interval [0, 1] then real function x g(x)= 0 f(t)dt is a linear-space computable real function on interval [0, 1]. 2 TheR same holds for class FDSPACE(n ). Theorem 3. If f is a FDSPACE(n2) computable C2[0, 1] real function on interval [0, 1] then real x 2 function g(x)= 0 f(t)dt is a FDSPACE(n ) computable real function on interval [0, 1]. Theorem 4. If f is a polynomial-time and linear-space computable C2[0, 1] real function on interval R x [0, 1] then real function g(x) = 0 f(t)dt is an exponential-time and linear-space computable real function on interval [0, 1]. R

2 3 Function from FDSPACE(n )C[a,b] that not in FPC[a,b] if FP =6 #P

∞ Let’s consider fucntion f(x)= n=1 fn(x) from proof (d) ⇒ (e) of theorem 5.32 [1, p.186]. Some of the properties of function f are as follows: ∞ P 1) f ∈ C [0, 1]; 1 2) if 0 f ∈ FPC[0,1] then FP = #P; 2 2 3) f ∈R FDSPACE(n )C[a,b] if one takes set B ( [1, theorem 5.32, p.184]) such that B ∈ DSPACE(n ); 1 2 4) 0 f ∈ FDSPACE(n )C[a,b] according to theorem 3. Therefore,R the following theorem holds 2 Theorem 5. FPC[0,1] 6= FDSPACE(n )C[a,b] if FP 6= #P.

4 Conclusion

x In the present paper, it is shown that real function g(x) = 0 f(t)dt is a linear-space computable real function on interval [0, 1] whenever f is a linear-space computable C2[0, 1] real function on R interval [0, 1]. This result diﬀers from the result from [1] regarding the time complexity of integration of polynomial-time computable real functions in the sense that integration of a polynomial-time computable real function may be not polynomial-time computable (if f is not analytic and FP 6= #P), but integration of linear-space computable C2 real functions is always linear-space computable. Regarding further investigations, it is interesting to derive results regarding the space complexity of other operators, for example, the space complexity of diﬀerentiation of computable functions.

4 References

[1] KoK. Complexity Theory of Real Functions. Boston: Birkhauser, 1991. 309 p. [2] AberthO. Computable calculus. Academic Press, 2001. 192 p. [3] KushnerB.A. Lectures on Constructive Mathematical Analysis. American Mathematical Society, 1984. 346 p. [4] WeihrauchK. Computable analysis. New York: Springer, 2000. 285 p. [5] DuD., KoK. Theory of Computational Complexity. New York: John Wiley & Sons, 2000. 491 p. [6] DuD., KoK. Computational complexity of integration and diﬀerentiation on convex functions // System Sci. and Math. Sci 2, pp. 70–79. [7] BerezinI.S., ZhidkovN.P. Numeric analysis. M.:Fiz-Mat Lit, 1962, Vol.1. 464 p.