Computability on the Real Numbers

4. Computability on the Real Numbers Real numbers are the basic objects in analysis. For most non-mathematicians a real number is an infinite decimal fraction, for example π = 3•14159 ... Mathematicians prefer to define the real numbers axiomatically as follows: · (R, +, , 0, 1,<) is, up to isomorphism, the only Archimedean ordered field satisfying the axiom of continuity [Die60]. The set of real numbers can also be constructed in various ways, for example by means of Dedekind cuts or by completion of the (metric space of) rational numbers. We will neglect all foundational problems and assume that the real numbers form a well-defined set R with all the properties which are proved in analysis. We will denote the R real line topology, that is, the set of all open subsets of ,byτR. In Sect. 4.1 we introduce several representations of the real numbers, three of which (and the equivalent ones) will survive as useful. We introduce a rep- n n resentation ρ of R by generalizing the definition of the main representation ρ of the set R of real numbers. In Sect. 4.2 we discuss the computable real numbers. Sect. 4.3 is devoted to computable real functions. We show that many well known functions are computable, and we show that partial sum- mation of sequences is computable and that limit operator on sequences of real numbers is computable, if a modulus of convergence is given. We also prove a computability theorem for power series. Convention 4.0.1. We still assume that Σ is a fixed finite alphabet containing all the symbols we will need. 4.1 Various Representations of the Real Numbers According to the principles of TTE we introduce computability on R by nam- ∗ ⊆ → R ing systems. Since the set R is not countable, it has no notation ν : Σ (onto) but only representations. Most of its numerous representations have no applications. In this section we introduce three representations ρ, ρ< and ρ> of the set of real numbers (and some equivalent ones) which induce the most important computability concepts. We will also discuss some other representations which, for various reasons, are only of little interest in computable analysis. 86 4. Computability on the Real Numbers R Since the set Q of rational numbers is dense in , every real number has arbitrarily tight lower and arbitrarily tight upper rational bounds. Every real number x can be identified by the set { | ∈ } (a; b) a, b Q, a<x<b of all open intervals with rational endpoints containing x,bytheset | } {a∈Q a<x of all rational numbers smaller than x or by the set | } {a ∈ Q a>x of all rational numbers greater than x. According to the concept of standard admissible representations (Definitions 3.2.1, 3.2.2) a name of x will be a list of all open intervals with rational endpoints containing x, a list of all rational lower bounds of x or a list of all rational upper bounds of x, respectively. Convention 4.1.1. In the following we will abbreviate νQ(w)bywwhere ν Qis our standard notation of the rational numbers (Definition 3.1.2). First, we introduce a standard notation In of all rational n-dimensional cubes with edges parallel to the coordinate axes and rational vertices. Definition 4.1.2 (notation of rational cubes). Assume n ≥ 1. n 1. For (a1,...,an)∈ R define the (maximum) norm ||(a1,...,an)|| := max |a1|,...,|an| n and for x, y ∈ R define the (maximum) distance by d(x, y):=||x − y|| . (n) n ∈ Q } 2. Let Cb := {B(a, r) | a ∈ Q ,r ,r>0 be the set of open rational n| } balls (or cubes), where B(a, r):={x∈ R d(x, a) <r . 3. Define a notation In of the set Cb(n) by n I (ι(v1) ...ι(vn)ι(w)) := B((v1,...,vn),w). n 4. By I (w) we denote the closure of the cube In(w) . In particular, Cb(1) is the set of all open intervals with rational endpoints and I1(ι(v)ι(w)) is the open interval (v − w; v + w), Cb(2) is the set of all open squares with rational vertices (edges parallel to the coordinate axes) 2 and I (ι(v1)ι(v2)ι(w)) is the open square (v1 − w; v1 + w) × (v2 − w; v2 + w), Cb(3) is the set of all open cubes with rational vertices (edges parallel to the coordinate axes), etc. We introduce representations ρ, ρ< and ρ> as our standard representations of computable topological spaces as follows. 4.1 Various Representations of the Real Numbers 87 Definition 4.1.3 (the representations ρ, ρ< and ρ>). Define computable topological spaces (1) 1 • S= := (R, Cb , I ) , • ∞ S< := (R,σ<,ν<),ν<(w):=(w; ), • −∞ S> := (R,σ>,ν>),ν>(w):=( ; w) , and let ρ := δS= , ρ< := δS< and ρ> := δS> . (The sets σ< and σ> are defined implicitly.) Notice that S=, S< and S> are computable topological spaces (Definition 3.2.1), since the properties νQ(u)= 1 1 νQ(v)andI(u)=I (v) are decidable in (u, v). By Definition 3.2.2, ⇐⇒ { ∈ (1) 1 } ρ(p)=x J Cb | x ∈ J} = {I (w) | ι(w) p , or roughly speaking, iff p is a list of all J ∈ Cb(1) such that x ∈ J. Similarly, ρ<(p)=x,iffpis a list of all rational numbers a such that a<x,andρ>(p)=x,iffpis a list of all rational numbers a such that a>x. Fig. 4.1 shows some open intervals J ∈ Cb(1) with x ∈ J and some rational numbers a with a<y. - xy Fig. 4.1. Some open intervals J ∈ Cb(1) with x ∈ J and some rational numbers a with a<y The final topologies of the above representations can be characterized easily (Definition 3.1.3.2, Lemma 3.2.5.3): Lemma 4.1.4 (final topologies of ρ, ρ< and ρ>). 1. The final topology of ρ is the real line topology τR. { ∞ | ∈ R} 2. The final topology of ρ< is τρ< := (x; ) x . { −∞ | ∈ R} 3. The final topology of ρ> is τρ> := ( ; x) x . (1) Proof: Cb generates τR, σ< generates τρ< and σ> generates τρ> . Another important representation of the real numbers is the Cauchy representation. Since every real number is the limit of a Cauchy sequence of rational numbers, such sequences can be used as names of real numbers. However, the “naive Cauchy representation” which considers all converging sequences of rational numbers as names is not very useful (Example 4.1.14.1). 88 4. Computability on the Real Numbers For the Cauchy representation we consider merely the “rapidly converging” sequences of rational numbers. Definition 4.1.5 (Cauchy representation). Define the Cauchy repre- ω sentation ρC :⊆ Σ → R by ∈ there are words w0,w1... dom(ν Q) ρC(p)=x:⇐⇒ such that p = ι(w0)ι(w1)ι(w2) ..., −i |wi −wk|≤2 for i<k and x = limi→∞ wi . The representation ρ, the Cauchy representation ρC and many variants of them are equivalent: Lemma 4.1.6 (representations equivalent to ρ). The following representations of the real numbers are equivalent to the standard representation ω ρ :⊆ Σ → R: 1. the Cauchy representation ρC, 0 00 000 2. the representations ρC, ρC and ρC obtained by substituting −i −i −i |wi − wk| < 2 , |wi − x| < 2 or |wi − x|≤2 , −i respectively, for |wi − wk|≤2 in the definition of ρC, 3. 1 there are words u0,u1...∈dom(I ) such that p = ι(u0)ι(u1) ..., a ⇐⇒ ρ (p)=x: ∀ 1 ⊆ 1 1 −k ( k) I (uk+1) I (uk) and length(I (uk)) < 2 1 1 and {x} =I (u0)∩I (u1)∩... , 4. \ n o b 1 ρ (p)=x :⇐⇒ { x } = I ( v ) | ι ( v ) p . 0 00 000 The representations ρC, ρC and ρC are inessential modifications of the Cauchy representation, and ρa is a representation by strongly nested, rapidly converging sequences of open intervals. A ρb-name of x is a list of closed rational intervals for which x is the only common point. While a ρ-name of x is a list of all open rational intervals containing x, these characterizations show that it suffices to list merely “sufficiently many” of them. Proof: b a ρ ≤ ρ : Define representations δ1 and δ2 of R by δ1(p)=x,iff 1 1 1 1 p=ι(u0)ι(u1)..., I (uk+1) ⊆ I (uk )and{x}=I(u0)∩I(u1)∩... , and δ2(p)=x,iff 4.1 Various Representations of the Real Numbers 89 1 1 1 1 p = ι(u0)ι(u1) ..., I (uk+1) ⊆ I (uk )and{x}=I (u0)∩I (u1)∩... There is a computable function h :⊆ Σω → Σω such that \n o 1 1 | h(p)=ι(v0)ι(v1)... where I (vi)= I (v) ι(v) p<i+ip , ∈ 1 where ip is the smallest number k such that ι(v) p<k for some v dom(I ). b Then obviously, the function h translates ρ to δ1. × ∗ → ∗ There is a computable function g : N Σ Σ such that I1(w)=B(a, r)=⇒I1◦g(i, w)=B(a, (1 + 2−i) · r) . There is a computable function f :⊆ Σω → Σω such that f(ι(w0)ι(w1)ι(w2) ...)=ι◦g(0,w0)ι◦g(1,w1)ι◦g(2,w2)... Then f translates δ1 to δ2. It remains to select from any δ2-name a rapidly converging subsequence of intervals.

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