Real recursive functions and Baire classes 1. Introduction - MIMUW

Fundamenta Informaticae XXI (2001) 1001–1016

1001

IOS Press

Real recursive functions and Baire classes Jerzy Mycka Institute of Mathematics M. Curie-Sklodowska University pl. M. Curie-Sklodowskiej 1, 20-031 Lublin, Poland [email protected]

Abstract. Recursive functions over the reals [6] have been considered, first as a model of analog computation, and second to obtain analog characterizations of classical computational complexity classes [2]. However, one of the operators introduced in the seminal paper by Cris Moore (in 1996), the minimalization operator, creates some difficulties: (a) although differential recursion (the analog counterpart of classical recurrence) is, in some extent, directly implementable in the General Purpose Analog Computer of Claude Shannon, analog minimalization is far from physical realizability, and (b) analog minimalization was borrowed from classical recursion theory and does not fit well the analytic realm of analog computation. In this paper we use the most natural operator captured from Analysis - the operator of taking a limit - instead of the minimalization with respect to the equivalance of these operators given in [8]. In this context the natural question about coincidence between real recursive functions and Baire classes arises. To solve this problem the limit hierarchy of real recursive funcions is introduced. Also relations between Baire classes, effective Baire classes and the limit hierachy are studied.

Keywords: theory of computation, real recursive functions, hierarchy of real functions

1.

Introduction

The classical theory of computation deals with functions on enumerable (especially natural) domains. The fundamental notion on this field is the notion of a (partial) recursive function. During past years many mathematicians have been interested in creating the analogous models of computation on real numbers (see, for example, Grzegorczyk[5], Blum, Shub, Smale[1]). Address for correspondence: J. Mycka, Institute of Mathematics, UMCS, pl. M. Curie-Sklodowskiej 1,20-031 Lublin, Poland

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Some existing models of analog computation have the strong connection with the mathematical analysis and its tools. The best example is Shannon’s General Purpose Analog Computer [12]. The General Purpose Analog Computer (GPAC) is a computer whose computation evolves in continuous time. The outputs are generated from the inputs by means of a dependence defined by a finite directed graph (not necessarily acyclic) where each node is one of the following boxes - integrator: a two-input, one-output unit with a setting for initialR condition, if the inputs are unary functions u, v, then the output is the t Riemann-Stieljes integral λt. t0 u(x)dv(x) + a, where a and t0 are real constants defined by the initial settings of the integrator; constant multiplier: a one-input, one-output unit associated with a real number, if u is the input of a constant multiplier associated with the real number k, then the output is ku; adder: a two-input, one-output unit, if u and v are the inputs, then the output is u + v; multiplier: a twoinput, one-output unit, if u and v are the inputs, then the output is uv; constant function: a zero-input, one-output unit, the value of the output is always 1. Rubel in his papers [10, 11] extended this model by an introduction of new boxes to define Extended Analog Computer. This model is similar to the GPAC, but it allows, in addition, other types of units, e.g. units that solve boundary value problems (here we allow several independent variables because Rubel is not seeking any equivalence with existing models) and infinite limits. The EAC permits the operations of ordinary analysis, except the unrestricted taking of limits. To avoid too many generated functions in this model, limits have some restrictions: indices can go only through some subsets of real numbers. The new units add an extended computational power relatively to the GPAC. For example, the EAC can solve the Dirichlet problem for Laplace’s equation in the disk and can generate the Γ function (it is known that the GPAC cannot solve these problems [10]). It is not known if there exists a physical version of the EAC. The new approach was given by Moore. In the work [6] he defined a set of functions on the reals (called R-recursive functions) analogous to the classical recursive functions on the natural numbers. His model has a continuous time of computation (a continuous integration instead of a discrete recursion). The great importance in Moore’s model has the zero-finding operation µ (as a ’strong uncomputable’ operation) which is used to construct µ-hierarchy of R-recursive functions. However, the operation of infinte limits (including limes inferior and limes superior) is a natural operation on functions. The main field for the limit operation is in the mathematical analysis (for the real functions). But also the case of natural functions is considered in mathematics and computer science (for example the important Shoenfield’s Limit Lemma). In this paper we use the limit operation as an ’ideal’ component of computing systems (after a suggestion found in [6] and the idea of Rubel [11]). It was shown [8] that the zero-finding operator µ can be replaced by the operation of infinite limits. It allows us to define a limit hierarchy and relate it to the µ-hierarchy. In this situation we can analyse a complexity (a difficulty) of real recursive functions by their position in the µ-hierarchy. Here we can compare the definitions of the same functions with infinite limits or the minimalization operators. In this context infinite limits are also useful tool to find the connection which exists between the µ-hierarchies and Baire classes of real functions. The additional impact is given by the fact that real recursive functions defined without limits and a minimalization operator are compatible with GPAC-computable functions [3]. Hence the upper classes of the µ-hierarchy and the limit hierarchy should be carefully analysed. In this paper we show connections between the limit hierarchy and Baire classes and effective Baire classes. This can lead to identify some problems of analog computation with descriptive set theory. Moreover, the problem of noncollapsing character of the hierarchy of real recursive functions is solved

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in this way.

2.

Preliminaries

We start with the definition of some class of real functions called R-recursive functions given by Moore (see [6]). Definition 2.1. The set of R-recursive functions is generated from the constants 0, 1, −1 by the operations: 1. composition: h(¯ x) = f (g(¯ x)); 2. differential recursion: h(¯ x, 0) = f (¯ x), ∂y h(¯ x, y) = g(¯ x, y, h(¯ x, y)) (the equivalent formulation can be given by integrals: Ry h(¯ x, y) = f (¯ x) + 0 g(¯ x, y 0 , h(¯ x, y 0 ))dy 0 ); 3. minimalization (µ-recursion) h(¯ x) = µy f (¯ x, y) = inf{y : f (¯ x, y) = 0}, where infimum chooses the number y with the smallest absolute value and for two y with the same absolute value the negative one. 4. vector-valued functions can be defined by defining their components. This definition needs some comments. A solution of a differential equation need not be unique or can diverge, hence, we assume that if h is defined by differential recursion then h is defined only where a finite and unique solution exists. Therefore the set of R-recursive functions includes also partial functions. The above definition raises few problems. First, to define the natural operation of functions iteration we need minimalization operator. Contrary, in the case of natural functions an iteration operator is equivalent to a recursion operation. Also a differential recursion has some non-standard properties. The solution of such differential equations can be a nonanalytic function (like |x|). This is the result of unbounded variant of an integration used in the differential recursion. The next problem is connected with minimalization. This operator breaks continuity of functions. The need of control of such behaviour is evident. Moore proposes the η-function for this purpose, but the construction fails, because he uses an assumption that f (x) · 0 = 0 even when f (x) is undefined or reaches infinity. Because of the mentioned difficulties let us propose another definition in which we use rather vector than scalar functions and infinite limits instead of the µ-operator. The below definition is given after [9]. Definition 2.2. The set of real recursive vectors is generated from the real recursive scalars 0, 1, −1 and the real recursive projections Ini (x1 , . . . , xn ) = xi , 1 ≤ i ≤ n, n > 0, by the operators: 1. composition: if f is a real recursive vector with n k-ary components and g is a real recursive vector with k m-ary components, then the vector with n m-ary components (1 ≤ i ≤ n) λx1 . . . xm .fi (g1 (x1 , . . . , xm ), . . . , gk (x1 , . . . , xm )) is real recursive.

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2. differential recursion: if f is a real recursive vector with n k-ary components and g is a real recursive vector with n k + n + 1-ary components, then the vector h of n k + 1-ary components which is the solution of the Cauchy problem for 1 ≤ i ≤ n hi (x1 , . . . , xk , 0) = fi (x1 , . . . , xk ), ∂y hi (x1 , . . . , xk , y) = gi (x1 , . . . , xk , y, h1 (x1 , . . . , xk , y), . . . , hn (x1 , . . . , xk , y)) is real recursive whenever h and its derivative are continuous in y on the largest interval containing 0 in which a unique solution exists except for a countable set of isolated points of discontinuity (of its derivative) where only one analytical continuation exists. 3. infinite limits: if f is a real recursive vector with n k +1-ary components, then the vectors h, h0 , h00 with n k-ary components (1 ≤ i ≤ n) hi (x1 , . . . , xk ) = lim fi (x1 , . . . , xk , y), y→∞

h0i (x1 , . . . , xk ) = lim inf fi (x1 , . . . , xk , y), y→∞

h00i (x1 , . . . , xk ) = lim sup fi (x1 , . . . , xk , y), y→∞

are real recursive, whenever these limits are defined for all 1 ≤ i ≤ n. 4. Arbitrary real recursive vectors can be defined by assembling scalar real recursive components. 5. If f is a real recursive vector, than each of its components is a real recursive scalar. Let us discuss carefully the definition. For differential recursion we restrict a domain to an interval of continuity. For given functions from the class C k this operator gives the result (a defined function) in the same class C k . This eliminates a possibility of defining such functions as λx.|x|. The bounded integration in differential recursion prevents us from creating nonanalytical functions by this operation and gets the equivalence of the set of real recursive functions defined without limits and GPAC-computable functions [3]. It is proved in [8] that the replacing of the µ-recursion operator (like in [6]) by infinite limits is permitted. The replacement of the minimalization operator by infinite limits helps also to avoid problems with the definition of the η-function. This change is made to express the notion of real recursive function in the way closer to the field of mathematical analysis and descriptive set theory. Especially, we should remember that Baire classes are created by some kind of infinite limits. Of course, the third kind of infinite limits h(¯ x) = limy→∞ g(¯ x, y) can be omitted without any consequence in the power of the system of real recursive functions. We introduced it only to avoid artificial troubles in definitions of particular real recursive functions. Let us recall here that such functions as +, ×, −, exp, sin, cos, λx. x1 , /, ln, λxy.xy and the Kronecker δ function, the signum function, absolute value, the Heaviside Θ function, the binary maximum max, the square-wave function s and the floor function are all real recursive too (see [9]). Also Bessel functions of the first kind, Euler’s Γ-function and Riemann’s ζ-function are real recursive.

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Hierarchy

The infinite limits are most important in generating the complicated real recursive functions. This fact suggests creating a hierarchy which is built with respect to the number of uses of limits in the definition of a given f . However, we treat two distinguished functions in the special manner. Let us recall the Heaviside Θ function equal to 1 if x ≥ 0, otherwise 0. With this very natural function Θ it is sufficient to define the iteration operator without more complex operators like limits or minimalization [3]. This is why we can place Θ on the zero level of our hierarchy, although it can be defined as Θ(x) = (sgn(x) + 1 2 y δ(x) + 1)/2, where δ(x) = lim inf y→∞ ( 1+x 2 ) , sgn(x) = lim inf y→∞ 1+exp(−xy) − 1. The floor function is important, when the change between real and natural domain is implemented. To avoid unnecessery complexity we use this function as the function of the zero level too. The precise description of a construction of the floor function is given by bxc = w(x)p(2x) + w(x − 21 )(1 − p(2x)), where p(x) = s(x)(1 − δ(sin( (x−1)π ))) and w(0) = 0, ∂x w(x) = 4 sin2 2πxΘ(− sin 2πx). 2 The above remarks lead us to the following construction of η-hierarchy of real recursive functions (see [9]). We start with the notion of syntactic n-ary descriptions of real recursive vectors. Let us introduce some kind of symbols called basics descriptors for all basic real recursive functions. The combination of such descriptions for given real recursive functions will form a new description of another function. Let us have: ijk is a k-ary description for projection Ikj for all 1 ≤ j ≤ k; 1k , ¯1k , 0k are k-ary descriptions for constants 1, −1, 0 used with k variables, Θ is a descriptor of the function λx.Θ(x) and f l is a descriptor of λx.bxc. We must add also operator symbols (descriptors) for all introduced operators: dr - for a differential recursion, c - for a composition, l, ls, li for a respective kind of limits (lim, lim sup, lim inf). Definition 3.1. The collection of descriptors of real recursive vectors is inductively defined as follows: • ijn is a n-ary description of Inj , 1 ≤ j ≤ n ∈ N ; • 1n is a n-ary description of f (x1 , . . . , xn ) = 1, for all (x1 , . . . , xn ) ∈ Rn , n ∈ N ; • ¯1n is a n-ary description of f (x1 , . . . , xn ) = −1, for all (x1 , . . . , xn ) ∈ Rn , n ∈ N ; • 0n is a n-ary description of f (x1 , . . . , xn ) = 0, for all (x1 , . . . , xn ) ∈ Rn , n ∈ N ; • Θ is an unary descriptor of the function λx.Θ(x); • f l is an unary descriptor of λx.bxc; • if hhi = hh1 , . . . , hm i is a k-ary description of the real recursive vector h and hgi = hg1 , . . . , gk i is a n-ary description of the real recursive vector g, then c(hhi, hgi) is a n-ary description of the composition of h ang g; • if hhi = hh1 , . . . , hn i is a k-ary description of the real recursive vector h and hgi = hg1 , . . . , gn i is a k +n+1-ary description of the real recursive vector g, then dr(hhi, hgi) is a k +1-ary description of the solution of the Cauchy Problem for h, g (if such a solution exists);

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• if hhi = hh1 , . . . , hm i is a n + 1-ary description of the real recursive vector h, then l(hhi), li(hhi), ls(hhi) is a n-ary description of an appropriate infinite limit (respectively lim, lim inf, lim sup) of h (if such limits exist); • if hf1 i, . . . , hfm i are n-ary descriptions of real recursive k-ary scalars f1 , . . . , fm , then hf i = hf1 , . . . , fm i is a k-ary description of the real recursive vector f = (f1 , . . . , fm ); • if hf i = hf1 , . . . , fm i is a k-ary description of the real recursive vector f , then each hfi i, 1 ≤ i ≤ m, is a k-ary description of the real recursive scalar fi . Now we can find the η-number for a description of some function f . Definition 3.2. For a given n-ary description s of a vector f let Eik (s) (the η-number with respect to i-th variable of the k-component) be defined as follows: 1. Ei1 (0n ) = Ei1 (1n ) = Ei1 (¯1n ) = E11 (Θ) = E11 (f l) = 0; 2. Eim (c(hhi, hgi)) = max1≤j≤k (Ejm (hhi) + Eij (hgj i)), where h is a n components k-ary vector ang g is a k-components m-ary vector; 3. for a differential recursion we distinguish two cases: • i ≤ k: Eij (dr(hf i, hgi)) = max(Ei1 (hf1 i), . . . , Ei1 (hfn i), Ei1 (hg1 i), . . . , Ei1 (hgn i), 1 (hg i), . . . , E 1 (hg i)) Ek+1 n 1 k+1 • i = k + 1: Eij (dr(hf i, hgi)) = 1 1 (hgn i))) (hg1 i), . . . , Ek+m+1 max0≤m≤n (max(Ek+m+1 where f is a n components k-ary vector and g is a n components k + n + 1-ary vector; k (hhi)) + 1, where h is a k 4. Eik (l(hhi)) = Eik (li(hhi)) = Eik (ls(hhi)) = max(Eik (hhi), En+1 components n + 1-ary vector.

For the n-ary description s of m components we can define now E(hhi) = maxk maxi Eik (hhi) for 1 ≤ i ≤ n, 1 ≤ k ≤ m. Now we can deal with the η-number for a real recursive functions. Definition 3.3. For a given real recursive function f , let η(f ) be defined as the minimum of E(hf i) for all possible descriptions of the function f . We are ready to conclude with a definition of η-hierarchy as a family of Hj = {f : η(f ) ≤ j}. It will be comfortable to think about the η-hierarchy as the measure of the difficulty of real recursive functions. Let us explain that we give explicitly the operator f (¯ x) = limy→∞ g(¯ x, y) in the above definitions to avoid its construction by other operators, which would efect in higher class of a complexity of a function f. Of course (compare with [9]) the standard arithmetical functions (e.g. +, ×, −, exp, λx. x1 ) are in H0 , because they can be defined by a differential recursion. Let us try to classify a few simple and useful functions.

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Lemma 3.4. The Kronecker δ function, the signum function, absolute value, maximum max(x, y), the square-wave function s are in H0 . Proof: Let us present the following constructions: δ(x) = Θ(x)Θ(−x), sgn(x) = Θ(x) − Θ(−x), |x| = sgn(x)x, max(x, y) = Θ(x − y)x + (1 − Θ(x − y))y, s(x) = Θ(sin(πx)). u t In the work [4] we can find the following result (given in a different formulation), which compares a classical computability notion to our hierarchy. Lemma 3.5. If the function f : N k → N is a primitive recursive function then there exists such a function F : Rk → R that (∀¯ n ∈ N k )f (¯ n) = F (¯ n) is in H0 . Let us recall (see [9]) that for a given real recursive function F : R → R we can define the real recursive function G(n, x) = F n (x) in the following method, f, g : R → R: f (0) = g(0) = x, 2 cos2 (

πt )∂t g(t) = (F (f (t)) − f (t))Θ2 (sin πt), 2

sin2 (πt)r(t)∂t f (t) = σ(g(t) − f (t))Θ2 (− sin πt), R1 where Θ2 (x) = x2 Θ(x), σ = 0 Θ2 (sin πx)dx, r(0) = 0, ∂x r(x) = 2Θ2 (sin πx) − σ. The n-th iteration G(n, x) of the function F satisfies the following equation: G(n, x) = f (2n) = g(2n) for natural n. Now we can add the observation connecting the natural partial recursive functions with the η-hierarchy. Lemma 3.6. If f : N k → N is a partial recursive function, then there exists such the function F : Rk → R that F (¯ n) = f (¯ n) or both are undefined and F ∈ H1 . Proof: It is sufficient to use Kleene normal form of f : f (¯ n) = u(µz v(¯ n, z)), where u, v are primitive recursive functions in the natural domain, µ is the minimalization operator on the natural numbers. From the previous lemma we have such functions U, V in H0 that f (¯ n) = U (µz V (¯ n, z)). The only problem is to present theQnatural minimalization operator by limits. By an iteration we can simply get the function V1 (¯ n, z) = zi=0 V (¯ n, i) which is equal to zero for z ≥ µz V (¯ n, z), Then we can construct by an iteration such the function   z = 0,  0 V2 (¯ n, z) = V2 (¯ n, z − 1) V1 (¯ n, z) = 0, z ≥ 1,   V2 (¯ n, z − 1) + 1 otherwise. This function has the obvious property µz V (¯ n, z)) = limx→∞ V2 (¯ n, bxc). Because V1 , V2 are both in H0 , hence µz V (¯ n, z)) is in H1 . u t

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It is an important task to control the domain of defined real recursive function. In the case of a definition by infinite limits we should consider three cases. For h(¯ x) = lim inf y→∞ g(¯ x, y) we can use the 1 equivalence (lim supy→∞ g(¯ x, y) = −∞) ∨ (lim supy→∞ g(¯ x, y) = +∞) ≡ lim inf y→∞ |g(¯x,y)|+1 =0 to check whether h(¯ x) is defined. The similiar equivalence is sufficient for h(¯ x) = lim supy→∞ g(¯ x, y): 1 (lim inf y→∞ g(¯ x, y) = +∞) ∨ (lim inf y→∞ g(¯ x, y) = −∞) ≡ lim supy→∞ |g(¯x,y)|+1 = 0. We will summarize and extend these considerations in the following lemma. Lemma 3.7. Let g : Rn+1 → R be a total function. For the functions h1 (¯ x) = lim inf g(¯ x, y), y→∞

h2 (¯ x) = lim sup g(¯ x, y), y→∞

h(¯ x) = lim g(¯ x, y), y→∞

there exist everywhere defined functions ηgs , ηgi , ηg : Rn → R such that h1 (¯ x) is defined iff ηgi (¯ x) 6= 0, s h2 (¯ x) is defined iff ηg (¯ x) 6= 0, h(¯ x) is defined iff ηg (¯ x) 6= 0. Proof: The ηgs , ηgi are given in the discussion before this lemma.

t u

Now let us finish by the proper construction of ηg (¯ x, y). In this case two conditions are needed: both x, y) are defined and they are equal to each other. lim inf y→∞ g(¯ x, y) and lim supy→∞ g(¯ Let us define hi (¯ x) = lim inf (ηgi (¯ x, y)) · g(¯ x, y), y→∞

hs (¯ x) = lim inf (ηgs (¯ x, y)) · g(¯ x, y). y→∞

Then if lim inf y→∞ g(¯ x, y) is defined we have hi (¯ x) = lim inf y→∞ g(¯ x, y), otherwise hi (¯ x) = 0. The s analogous property holds for h (¯ x). Now it is sufficient to write: ηg (¯ x) = ηgs (¯ x) · ηgi (¯ x) · δ(hi (¯ x) − hs (¯ x)). From the proof of the above lemma we can get an interesting corollary. Corollary 3.8. If the function g : Rn → R is in the class Hi , i ≥ 0 then ηg , ηgs , ηgi which control the domain of the respective functions: limy→∞ g(¯ x, y), lim supy→∞ g(¯ x, y), lim inf y→∞ g(¯ x, y) are in the class Hi+1 .

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Baire classes and effective Baire classes

We recall some standard notions of descriptive set theory after [7]. Let us start with the definition of Borel sets hierarchy. Definition 4.1. Borel classes of finite order Σ0n , n ≥ 1 are defined by the induction: Σ01 = all open sets in Rk × N l , k, l ≥ 0; Σ0n+1 = {P ⊂ Rk × N l , k, l ≥ 0 : (∃P 0 )P 0 ∈ Σ0n , P 0 ⊂ Rk × N l+1 , P (¯ x, n ¯ ) ≡ (∃k ∈ N )¬P 0 (¯ x, n ¯ , k)}. Usually two more kinds of classes are added for all n ≥ 1. Π0n = {P : ¬P ∈ Σ0n }, ∆0n = Σ0n ∩ Π0n . Each set contained in any Borel class is called a Borel set. We can use a countable basis for the topology of some product space Rk × N l : B0 , . . . , Bn , . . .. With each Bi there are associated two values: the center ci of Bi and the radius ri . Now we are ready to give the definition of Σ0n -measurable functions. Definition 4.2. The function f : Rk × N l → Rm × N n is Σ0n -measurable if for all s ∈ N we have: f −1 (Bs ) ∈ Σ0n . Another notion important for us is a notion of Baire classes for real functions. Definition 4.3. A function f : Rk × N l → Rm × N n is of Baire class 0 if it is continuous, and it is of Baire class t if it is not of Baire class s < t and there exists a sequence f0 , f1 , . . . , fi , . . . (where fi : Rk × N l → Rm × N n ) such that each fi , i ∈ N is of Baire class p < t and for all x ¯, n ¯ ∈ Rk × N l we have f (¯ x, n ¯ ) = limi→∞ fi (¯ x, n ¯ ) or f (¯ x, n ¯ ) is undefined where the above limit does not exist. These two notions given above (namely Baire classes and Borel sets) are linked. The type of such a connection is formulated in the below important theorem (see [7]). Theorem 4.4. If the function f : Rk × N l → Rm × N n is of Baire class t then the function f is Σ0t+1 measurable. If the function f : Rk × N l → Rm × N n is Σ0t+1 -measurable then it is of Baire class s such that s ≤ t . There is an efective refinement of the above notions. The set G ∈ Rk × N l is called semirecursive S if it can be presented as t Bu(t) , where u : N → N is a natural recursive function. If we use the semirecursive sets as the fundament to the similar construction as Borel sets, then we obtain the following arithmetical hierarchy. Definition 4.5. Arithmetical classes of finite order Σ0n , n ≥ 1 are defined by the induction: Σ01 = all semirecursive sets in Rk × N l , k, l ≥ 0; Σ0n+1 = {P ⊂ Rk × N l , k, l ≥ 0 : (∃P 0 )P 0 ∈ Σ0n , P 0 ⊂ Rk × N l+1 , P (¯ x, n ¯ ) ≡ (∃k ∈ N )¬P 0 (¯ x, n ¯ , k)}.

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Analogously, like in the case of Borel classes we usually use also two more types of arithmetical sets: Π0n = {P : ¬P ∈ Σ0n }, n ≥ 0; ∆0n = Σ0n ∩ Π0n , n ≥ 0. Every set contained in any level of the arithmetical hierarchy is called an arithmetical set. Let us point out the difference in the notation. While Borel sets are denoted by the boldface case of Greek letters, for the arithmetical hierarchy we use a standard font. With the arithmetical hierarchy we may introduce the new kind of functions. First we associate with each function f : Rk × N l → Rm × N n the neighborhood diagram Gf ⊂ Rk × N l+1 such that Gf (¯ x, n ¯ , k) ≡ f (¯ x, n ¯ ) ∈ Bk . Definition 4.6. The function f : Rk × N l → Rm × N n is Σ0t -recursive if Gf ∈ Σ0t . For t = 1 Σ0t -recursive functions are simply called recursive functions. It is important to rememeber that here the notion of recursiveness is extended to real realm (not only for natural domains). We have also a possibility of an effective analog of Baire classes. The below definition is formulated for the case of domains which are the product of real and natural sets. Definition 4.7. A function f : Rk × N l → Rm × N n is of effective Baire class 0 if it is recursive, and it is of effective Baire class t + 1 if it is not of Baire class s < t + 1 and there exists a function g : Rk × N l+1 → Rm × N n such that g is of Baire class t and for all (¯ x, n ¯ ) ∈ Rk × N l we have f (¯ x, n ¯ ) = limi→∞ g(¯ x, n ¯ , i) or f (¯ x, n ¯ ) is undefined iff limi→∞ g(¯ x, n ¯ , i) does not exist. Here we have the following result connecting effective Baire classes and Σ0n -recursive functions. Theorem 4.8. A function f : Rk × N l → Rm × N n is of effective Baire class p ≤ t iff the function f is Σ0t+1 -recursive. Let us observe the obvious difference between Σ0t -recursiveness and Σ0t -measurability which lies in the degree of effective computability. For example the Σ01 -measurable functions are exactly the same as continuous functions without any information about a method of establishing a result of a given function. On the other side we have Σ01 -recursive functions. In this case we can effectively compute arbitrarily good approximations of a result: we can look for such a neighborhood which has sufficiently small radius and contains a value of this function.

5.

η-hierarchy and effective Baire classes

Let us start with an analysis of the proper place of recursive functions (i.e. Σ01 -recursive functions) in the η-hierarchy. For this purpose we use effective Baire classes as a tool. Lemma 5.1. Let f : Rn → R be in the effective Baire class 0. Then f belongs to H4 .

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Proof: At the beginning we use the theorem of associated functions (see [7]): f : Rn → (0, 1) is Σ01 -recursive iff f ∗ : Rn × N → N is Σ01 -recursive, where the value of the function f ∗ (¯ x, n) is established as the n-th number of a decimal expansion of f (¯ x). By a convention we consider only expansions without infinite number of zeros. In the next step we use the theorem on functions graph (also from [7]) to get an ∗ equivalent problem: for the function f ∗ the graph Gf ∗

Gf (¯ x, n, m) = {(¯ x, n, m) : f ∗ (¯ x, n) = m} should be of the class Σ01 . By the definition of the class Σ01 now S we can reduce this condition to the ∗ following expression: the graph Gf ⊂ Rn × N 2 has the form n Bh(n) , where Bi , i ≥ 0 is a countable basis of topology of Rn × N 2 , h : N → N is some natural recursive function. Hence, if the function f : Rn → (0, 1) is recursive then there exists some natural recursive function S ∗ h, such that Gf = n Bh(n) . Now let us start with the observation that all sets from Bi , i ≥ 0 are (as open intervals) also Rrecursive sets, so there exists the function ( 1 (¯ x, m, n) ∈ Bi χ(i, x ¯, m, n) = . 0 (¯ x, m, n) 6∈ Bi Let (rj )j≥0 be an enumeration of rational numbers and by (i)k , k = 1, 2, 3 we mean the primitive recursive decoding of natural numbers, where i is obtained as the encoded triple of natural numbers i = h(i)1 , (i)2 , (i)3 i. We can use the basis Bi , i ≥ 0 given in the standard way: Bi is an open ball with 1 the center r(i)0 and the radius (i)(i)2 +1 . In such case all natural functions used to establish the interval Bi are primitive recursive functions and can be simulated (see [3]) by real recursive functions with an iteration and Θ functions. It means that χ is in H (0 . 1 (∃k)χ(h(k), x ¯, m, n) = 1, Now we should consider χGf ∗ (¯ x, m, n) = because each natural 0 (∀k)χ(h(k), x ¯, m, n) = 0, recursive function is at most in H1 , so χ(h(k), x ¯, m, n) is in the class H1 . We change the form of this definition:  P x, m, n) 6= 0 and   0 η ki=0 χ(h(i),¯x,m,n) (¯ Pk χGf ∗ (¯ x, m, n) = ¯, m, n) = 0, limk→∞ i=0 χ(h(i), x   1 otherwise, where ηg is a characteristic function of a convergence of g in infinity. The sum of χ(h(i), x ¯, m, n) can be defined from Θ by an iteration operation, so we have its class equal to H1 . Consequently, χGf ∗ is in the class H2 . From χGf ∗ (¯ x, m, n) is possible to get the function f ∗ by the equation: f ∗ (¯ x, m) = µn χGf ∗ (¯ x, m, n). However, the µ-operator on the natural numbers can be replaced by the infinite limit operation, hence f∗ Pn is in H3 . Finally we have f (¯ x) = limn→∞ i=1 f ∗ (¯ x, i)10−i and we get f ∈ H4 . The last step will extend the proof to the case of function f with values from R but not only from the interval (0, 1). For this purpose we can use such function g : R → (0, 1), which is always defined, one-to-one, and has properties: g is recursive and its inverse g −1 is in H0 . Then for given f we can finish

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the proof with h(¯ x) = g(f (¯ x)) and f (¯ x) = g −1 (h(¯ x)). From many appropriate functions g we can take 2 x2 for example such g(x) that g(x) = 2x2 +2 for x > 0 and g(x) = 2xx2 +2 + 12 for x ≤ 0. t u Now we can extend the above result to the theorem about any function f from some effective Baire class and find the level of this function in the η-hierarchy. Theorem 5.2. Let f : Rn → R be in effective Baire class n. Then f belongs to Hn+4 . Proof: We will proceed in the inductive way. For n = 0 we can use the results from the previous lemma. Let us assume an inductive hypothesis: if f : Rn → R is in effective Baire class n, then f belongs to Hn+4 . Now we consider the function g : Rk → R, such that g is in effective Baire class n + 1. Then g(¯ x) = limm→∞ G(¯ x, m), where G is at most the effective class n. By the inductive hypothesis we have also G ∈ Hn+4 . So for g(¯ x) = limy→∞ G(¯ x, byc) we get g ∈ Hn+4+1 . t u Of course these results can be expressed in terms of Σ0n -recursive functions. Then we can obtain from the above theorem and Theorem 4.8 the following fact. Corollary 5.3. If the function f : Rn → R is Σ0n -recursive function then it belongs to Hn+3 . This fact can us resolve the problem of the noncollapsing character of the η-hierarchy. We know that each class of all Σ0n -recursive functions is a subset of the level Hn+3 of η-hierarchy. Because the hierarchy of classes of Σ0n -recursive functions, n ≥ 1, is infinite we can conclude this section with the important corollary from the above consideration. Theorem 5.4. The η-hierarchy is an infinite hierarchy (it has a noncollapsing character).

6.

Modified η-hierachy and Baire classes

In this section we use the slightly modificated definition of η-hierarchy. Now we would like to take into account the computational complexity of the distinguished functions Θ, λx.bxc. Meanwhile, this decision results in avoiding technical problems with noncontinuous functions in comparison to the class of Σ01 -measurable functions. Let us describe precisely the new hierarchy. For a given real recursive expression s(¯ x), let Ei0k (s) be defined like in the in Definition 3.2 with only this change that we omit the functions Θ(x), bxc on the zero level (so only Ei01 (0n ), Ei01 (1n ), Ei01 (¯1n ) are equal to 0). Let us use the rest of definitions from Section 3 unchanged except use of a prime notation as a sign of the modification. Then we get the modified η-hierarchy with the levels Hk0 , k ≥ 0. At the beginning we can establish the proper levels of the mentioned functions in the modified ηhierarchy. For this purpose we cite result from [9] which can be verified by equations given at the beginning of Section 3. Lemma 6.1. The function Θ(x) and the floor function bxc are in the class H10 of the modified ηhierarchy.

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Before we give the general results we can analyse these two special functions in the context of their position in the hierarchy of Σ0i -measurable functions. Let us start with Θ(x). It is simple to observe that   ∅ 0, 1 6∈ Bs     h0, ∞) 1 ∈ Bs , 0 6∈ Bs Θ−1 (Bs ) =  (−∞, 0) 0 ∈ Bs , 1 6∈ Bs     (−∞, ∞) 0, 1 ∈ B s so Θ is ∆02 -measurable Σ02 -measurable. We can get the S with respect to the set h0, ∞) and consequently −1 0 equation bBs c = n∈Bs hn, n + 1). This means that bxc is Σ3 -measurable. With these observations we can analyse the whole modified η-hierarchy. Let us start with the class H00 . Lemma 6.2. Let f : Rn → R be a real recursive function from the class H00 . Then a function f is Σ01 -measurable. Proof: Let us recall that in the class H00 we have the constants 0, 1, −1 and all other functions generated from the given ones by the operators of composition and differential recursion, where the defined function and its derivative are continuous on the largest interval containing 0 in which a unique solution exists except for a countable set of isolated points of discontinuity (of its derivative) where only one analytical continuation exists. Because the operators of composition and differential recursion do not change the continuity of given functions, therefore all function in H00 are continuous in their domain, hence they are Σ01 -measurable. t u Now let us consider the following problem. Let f, g be in the class Hi0 , i > 0. We can define a new function h by a differential recursion: h(¯ x, 0) = f (¯ x), ∂y h(¯ x, y) = g(¯ x, y, h(¯ x, y)) with properties given in the definition of real recursive functions. This function will be in the same class Hi0 , i > 0. But the important point is a position of h in the hierarchy of Σ0k -measurable, k > 1, functions for functions f, g which belong to Σ0n -measurable functions for some n ∈ N . From the definition of the operator of differential recursion which assume continuity of a defined function we get the below conclusion. Lemma 6.3. Let a function h be defined by differential recursion from f, g, which are Σ0n -measurable functions. Then h is Σ01 -measurable function. Because the class of Σ0n -measurable function is contained in the class of Σ0n+1 -measurable functions, hence from the above lemma we get the corollary that all classes of Σ0n -measurable functions for a given n are closed under the operation of differential recursion. The other important forms of functions definitions can be given by the operations of supremum and infimum. They are useful to express the notion of infinite limit (especially limes superior and limes inferior) in their terms. Lemma 6.4. Let functions h1 , h2 be defined as h1 (¯ x) = supy f (¯ x, y), h2 (¯ x) = inf y f (¯ x, y), where f is 0 0 Σn -measurable function. Then h1 , h2 are Σn+4 -measurable functions.

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Proof:

(

1 f (¯ x, y) ≤ z, The function χf (¯ x, y, z) is a Σ0n 0 f (¯ x, y) > z. ( 1 (∀y)χf (¯ x, y, z) = 1, f measurable function for the same function f . Then the function χub (¯ x, z) = f 0 (∃y)χ (¯ x, y, z) = 0, can be constructed as Let us start with the definition χf (¯ x, y, z) =

χfub (¯ x, z)

= lim lim lim

a→∞ b→∞ n→∞

n Y i=0

χf (¯ x, a + i

b−a , z). n

It means that χfub is Σ0n+3 -measurable. The equation χfub (¯ x, z) = 1 holds if z is the upper bound for f (¯ x, y), where y is going through the set of real numbers. It is simple to observe that ( 0 z < supy f (¯ x, y) f χub (¯ x, z) = 1 z ≥ supy f (¯ x, y) Then by some technical steps we can build from χfub the supermum with the use of the following auxiliary functions   w ≤ supy f (¯ x, y) − 12 , Z w− 1  0 2 f t(¯ x, w) = x, z)dz = χub (¯ x ∈ (0, 1) w ∈ (supy f (¯ x, y) − 12 , supy f (¯ x, y) + 12 ),  w− 12  1 w ≥ supy f (¯ x, y) + 12 ; ( 0 t(¯ x, w) = 0 ∨ t(¯ x, w) = 1, 0 t (¯ x, w) = 1 otherwise. x, y) − 21 , supy f (¯ With such a definition we have t0 (¯ x, w) = 1 iff w is in the interval (supy f (¯ x, y) + 12 ). The final step is given by the construction of supy f (¯ x, y) as Ra 0 R −a 0 x, y) lima→∞ 0 wt (¯ x, w)dw+lima→∞ 0 wt (¯ x, w)dw. From the above we can conclude that supy f (¯ is Σ0n+4 -measurable and in the analogous way we can obtain the same result for inf y f (¯ x, y). t u Now we can take a look at the whole modified η-hierarchy. The general problem to discuss is the relation between infinite limits and the particular levels Hi0 , i ≥ 0 and the Σ0n -measurable functions. Theorem 6.5. Let a function f : Rn → R be a real recursive function, which belongs to the class Hi0 , i ≥ 0. Then f is Σ08i+1 -measurable function. Proof: In this proof we will proceed with an induction. Because for i = 0 the thesis is true by Lemma 6.2, we have to deal only with i > 0. 0 . By the induction hypothesis the Let us start with the case f (¯ x) = limy→∞ g(¯ x, y), where g ∈ Hi−1 function g is Σ08(i−1)+1 -measurable. But all the functions gi (¯ x) = g(¯ x, i), i ≥ 0 are then also Σ08(i−1)+1 measurable. By the fact f (¯ x) = limi→∞ gi (¯ x) and Theorem 4.4 we have f is Σ06(i−1)+2 -measurable and, consequently, Σ08i+1 -measurable.

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1015

0 . Now let us consider two remaining cases. Let f (¯ x) = lim inf y→∞ g(¯ x, y), where g ∈ Hi−1 0 Then from the induction hypothesis the function g is Σ8(i−1)+1 -measurable. However we know that lim inf y→∞ g(¯ x, y) = supz inf y>z g(¯ x, y) into an equivalent form (x, y). We change inf y>z g(¯ g(¯ x, y) y > z, inf y g t (¯ x, y, z), where g t (¯ x, y, z) = for some t from a domain of g. In this moment g(¯ x, t) y ≤ z; we can write f (¯ x) = sup inf g t (¯ x, y, z). z

y

Because g t is Σ08(i−1)+1 -measurable then (by Lemma 6.4) we get from the above construction the function f in the class of Σ08(i−1)+1+4+4 -measurable functions. The case f (¯ x) = lim supy→∞ g(¯ x, y) can be handled in the same manner. The situation that f is defined by composition or differential recursion can be ommited as a result of our previous consideration, because the classes Σ0i -measurable functions, i ≥ 1 are closed under these operations. t u

7.

Final remarks

In the final part of his paper [6], Moore considers the possibility of taking limits and questioned himself about the consequences of an introduction of infinite limit in the realm of analog computations. The proof that minimalization can be expressed in terms of infinite limits is presented in [8] as one of such consequences. This paper gives some links between descriptive set theory and the set of real recursive functions. Particulary we have discovered a relationship between hierarchies of real recursive functions and Baire classes. Also the problem of noncollapsing character of the η-hierarchy is solved. However, we are interested in more precise description of mutual dependencies of these two fields. Especially, the parametrization and uniformization theorems should be studied in this area. The other interesting questions are connected with the role of inifinite limits in an analog computation. In some situations limits and differential recursion can be interchangeable (for example ex can be defined by both the above mentioned methods). We would like to identify sufficient conditions for such a replacement theorem, and to find a fragment of the language that can be expressed by differential recursion. Because limits are also a tool used in the Rubel’s Extended Analog Computer (EAC), we would like to know in which way his hierarchy of EAC-computable functions is connected with the η-hierarchy.

References [1] Blum, L., Shub, M., Smale, S.: On a theory of computation and complexity over the real numbers: NPcompletness, recursive functions and universal machines, Bull. Amer. Math. Soc., 21, 1989, 1–46. [2] Campagnolo, M.: The complexity of real recursive functions, Unconventional Models of Computation, UMC 2002 (C. Calude, M. Dinneen, F. Peper, Eds.), LNCS, LNCS, Springer-Verlag, 2002, 1–14. [3] Campagnolo, M., Moore, C., Costa, J.: Iteration, inequalities, and differentiability in analog computers, Journal of Complexity, 16(4), 2000, 642–660.

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[4] Campagnolo, M., Moore, C., Costa, J.: An analog characterization of the Grzegorczyk hierarchy, Journal of Complexity, 18(4), 2002, 977–1000. [5] Grzegorczyk, A.: On the definition of computable real continuous functions, Fund. Math., 44, 1957, 61–71. [6] Moore, C.: Recursion theory on the reals and continuous-time computation, Theoretical Computer Science, 162, 1996, 23–44. [7] Moschovakis, Y.: Descriptive Set Theory, North-Holland, 1980. [8] Mycka, J.: µ-Recursion and infinite limits, Theoretical Computer Science, 302, 2003, 123–133. [9] Mycka, J., Costa, J.: Real recursive functions and their hierarchy, Journal of Complexity, in print. [10] Rubel, L.: Some mathematical limitations of the General-Purpose Analog Computer, Advances in Applied Mathematics, 9, 1988, 22–34. [11] Rubel, L.: The Extended Analog Computer, Advances in Applied Mathematics, 14, 1993, 39–50. [12] Shannon, C.: Mathematical theory of the differential analyser, J. Math. Phys. MIT, 20, 1941, 337–354.