Computability and Continuity on the Real ... - Semantic Scholar
Computability and Continuity on the Real Arithmetic Hierarchy and the Power of Type-2 Nondeterminism Martin Ziegler Institut for Matematik og Datalogi, Syddansk Universitet, 5230 Odense M, Denmark [email protected]
Abstract. The sometimes so-called Main Theorem of Recursive Analysis implies that any computable real function is necessarily continuous. We consider three relaxations of this common notion of real computability for the purpose of treating also discontinuous functions f : R → R: – non-deterministic computation; – relativized computation, specifically given access to oracles like ∅ or ∅ ; – encoding input x ∈ R and/or output y = f (x) in weaker ways according to the Real Arithmetic Hierarchy. It turns out that, among these approaches, only the first one provides the required power.
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Motivation
What does it mean for a Turing Machine, capable of operating only on discrete objects, to compute a real number x: ρb,2 : To decide its binary expansion? ρCn : To compute a sequence (qn ) of rational numbers eventually converging to x? ρ : To compute a fast convergent sequence (qn ) ⊆ Q for x, i.e. with |x − qn | ≤ 2−n (in other words: to approximate x with error bounds)? ρ< : To approximate x from below, i.e., to compute (qn ) such that x = supn qn ? All these notions make sense in being closed under arithmetic operations like addition and multiplication. In fact they are well (known equivalent to variants) studied in literature1 ; e.g. [11], [2], [12], [13] in order. Now what does it mean for a Turing Machine M to compute a real function f : R → R? Most naturally it says that, upon input of x ∈ R given in one of 1
Supported by project 21-04-0303 of Statens Naturvidenskabelige Forskningsr˚ ad SNF. Their above names by Greek letters are taken from [13, Section 4.1].
S.B. Cooper, B. L¨ owe, and L. Torenvliet (Eds.): CiE 2005, LNCS 3526, pp. 562–571, 2005. c Springer-Verlag Berlin Heidelberg 2005
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the above ways, M outputs y = f (x) also in one (not necessarily the same) of the above ways. And, again, many possible combinations have already been investigated. For instance the standard notion of real function computation in Recursive Analysis [3, 6, 5, 13] refers (or is equivalent) to input and output given according to ρ. Here, the Main Theorem of Computable Analysis implies that any computable f will necessarily be continuous [13, Theorem 4.3.1]. We are interested in ways of lifting this restriction. A first approach might equip the Turing Machines under consideration with access to an oracle, say, for the Halting Problem ∅ or its iterated jumps ∅(d) in Kleene’s Arithmetic Hierarchy. However closer inspection in Section 3.1 reveals that this Main Theorem relies solely on information rather than recursion theoretic arguments and therefore requires continuity also for oracle-computable real functions with respect to input and output of form ρ. (For the special case of an ∅ –oracle, this had been observed in [4, Theorem 16].) A second idea changes the input and output representation for x and y = f (x) from ρ to a weaker form like, say, ρCn . This relates to the Arithmetic Hierarchy, too, however in a completely different way: Computing x in the sense of ρCn is equivalent to computing x in the sense of ρ [4, Theorem 9] relative (i.e., given access) to the Halting Problem ∅ . And, most promisingly, the Main Theorem [13, Corollary 3.2.12] which previously required continuity of computable real functions now does not apply any more since ρCn , in contrast to ρ, lacks the technical property of admissibility. It therefore came to quite a surprise when Brattka and Hertling established that any (ρCn → ρCn )–computable f (that is, with respect to input x and output f (x) encoded according to ρCn ) still satisfies continuity; see [13, Exercise 4.1.13d] or [2, Section 6]. In Section 3.2, we extend this result. Specifically it is proven that continuity ρ ≡ ρCn ρ is necessary for (ρ → ρ )–computability of f ; here, ρ . . . denote the first levels of an entire hierarchy of real number representations emerging naturally from the Real Arithmetic Hierarchy of Weihrauch and Zheng [14]. The class of (ρ → ρ)–computable functions f : R → R is well-known to admit an alternative characterization based on Caldwell’s and Pour-El’s famous Effective Weierstraß Theorem. Its extension based on relativization gives rise to a hierarchy of continuous functions f : R → R. The (ρ → ρ) –computable ones for example have been investigated in [4, Section 4]. Section 4 of the present work locates the class of (ρ → ρ )–computable functions among this hierarchy. Rather than endowing deterministic Turing Machines with oracles, in Section 5 we finally consider nondeterministic computation. Remarkably and in contrast to the classical (Type-1) theory, this significantly increases the principal capabilities. For example, discontinuous real functions now do become computable and so does conversion among the aforementioned representations ρCn and ρb,2 .
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Arithmetic Hierarchy of Reals
In [4], Ho observed an interesting relation between computability of a real number x in the respective senses of ρ and ρCn in terms of oracles: x = limn qn for an (eventually convergent) computable rational sequence (qn ) iff x admits a ∅ –computable fast convergent rational sequence, that is, a sequence (pm ) ⊆ Q recursive in ∅ with |x − pm | ≤ 2−m . This suggests to write ρ for ρCn ; and denoting by ∆1 R = Rc the set of reals computable in the sense of Recursive Analysis (that is with respect to ρ), it is therefore natural to write, in analogy to Kleenes classical Arithmetic Hierarchy, ∆2 R for the set of all x ∈ R computable with respect to ρ . Weihrauch and Zheng extended these considerations and obtained for instance [14, Corollary 7.3] the following characterization of ∆3 R: A real x ∈ R admits a fast convergent rational sequence recursive in ∅ iff x is computable in the sense of ρ defined as follows: ρ : x = limi limj qi,j for some computable rational sequence (qn ). Similarly, Σ1 R contains of all x ∈ R computable with respect to ρ< whereas Σ2 R includes all x computable in the sense of ρ< defined as follows: ρ< : x = supi inf j qi,j for some computable rational sequence (qn ). √ To Σ2 R belongs for instance the radius of convergence r = 1/ lim supn→∞ n an ∞ of a computable power series n=0 an xn [14, Theorem 6.2]. More generally we take from [14, Definition 7.1 and Corollary 7.3] the following Definition 1 (Real Arithmetic Hierarchy). Let d = 0, 1, 2, . . . (d) ρ< : Σd+1 R consists of all x ∈ R of the form x = supn1 inf n2 . . . Θnd+1 qn1 ,...,nd+1 for a computable rational sequence (qn ), where Θ= sup or Θ= inf depending on d’s parity; (d) ρ> : Πd+1 R similarly for x = inf n1 supn2 . . . ρ(d) : ∆d+1 R contains all x ∈ R of the form x = lim lim . . . lim qn1 ,...,nd n1 n2 nd for a computable rational sequence (qn ). (For levels beyond ω see [1]. . . ) The close analogy between the discrete and this real variant of the Arithmetic Hierarchy is expressed in [14] by a variety of elegant results like, e.g., Fact 2. a) x ∈ ∆d R iff deciding its binary expansion is in ∆d . b) x is computable with respect to ρ(d) iff there is a ∅(d) –computable fast convergent rational sequence for x. (d) c) x is computable with respect to ρ< iff x is the supremum of a ∅(d) –computable rational sequence. d) ∆d R = Σd R ∩ Πd R. e) Σd R ∪ Πd R ∆d+1 R. 2.1
Type-2 Theory of Effectivity
Specifying an encoding formalizes how to feed some general form of input like graphs or integers into a Turing Machine with fixed alphabet Σ. Already in the
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discrete case, the complexity of a problem usually depends heavily on the chosen encoding; e.g., numbers in unary versus binary. This dependence becomes even more important when dealing with objects from an continuum like the set of reals and their computability. While Recursive Analysis usually considers one particular encoding for R, the Type-2 Theory of Effectivity (TTE) due to Weihrauch provides (a convenient formal framework for studying and comparing) a variety of encodings for different universes. Formally speaking, a representation α for R ¯ ∈ dom(α) is a partial surjective mapping α :⊆ Σ ω → R; and an infinite string σ is regarded as a name for the real number x = α(¯ σ ). In this way, (α → β)– computing a real function f : R → R means to compute a transformation on ¯ for x = α(¯ σ ) gets infinite strings F :⊆ Σ ω → Σ ω such that any α–name σ transformed to a β–name τ¯ = F (¯ σ ) for f (x) = y, that is, satisfying β(¯ τ ) = y; cf. [13, Section 3]. Now observe that (the above characterization of) each level of the Real Arithmetic Hierarchy gives rise not only to a notion of computability for real numbers but also canonically to a representation for R; for instance let ρ : encode (arbitrary!) x ∈ R as a fast convergent rational sequence (qn ); ρ< : encode x ∈ R as supremum of a rational sequence: x = supn qn ; ρ : encode x ∈ R as limit of a rational sequence: x = limn qn ; ρ< : encode x ∈ R as (qn ) ⊆ Q with x = supi inf j qi,j ; ρ : encode x ∈ R as (qn ) ⊆ Q with x = limi limj qi,j . In fact the first three of them are known in TTE as ρ, ρ< , and ρCn , respectively [13, Section 4.1]. In general one obtains, similar to Definition 1, a hierarchy of real representations as follows: (0)
(0)
Definition 3. Let ρ(0) := ρ, ρ< := ρ< , ρ> := ρ> . Now fix 1 ≤ d ∈ N: A ρ(d) –name for x ∈ R is a (νQω –name for a) rational sequence (qn ) such that x = lim lim . . . lim qn1 ,...,nd . n1
n2
nd
(d)
A ρ< –name for x ∈ R is a (name for a) sequence (qn ) ⊆ Q such that x = supn1 inf n2 . . . Θnd+1 qn1 ,...,nd+1 . (d)
A ρ> –name for x ∈ R is a sequence (qn ) ⊆ Q such that x = inf n1 supn2 . . . Results from [14] about the Real Arithmetic Hierarchy are easily re-phrased in terms of these representations. However this leads only to non-uniform claims. Fact 2d) for example translates as follows: (d)
(d)
x is ρ(d) –computable iff it is both ρ< –computable and ρ> –computable. Closer inspection of the proofs in particular of Lemma 3.2 and Lemma 3.3 in [14] reveals them to hold even uniformly. More precisely we can prove ρ< ρ ≡ ρ< ρ> ρ< ρ ≡ . . . , Lemma 4. ρ ≡ ρ< ρ> where “ ” and “ ≡” denote computable reducibility and equivalence, respectively, of representations and “ ” their join, i.e., the least upper bound w.r.t. “ ”; see [13, Definition 2.3.2].
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Computability and Continuity
Recursive Analysis has established as folklore that any computable real function is continuous. More precisely, computability of a partial function from/to infinite strings f :⊆ Σ ω → Σ ω requires continuity with respect to the Cantor Topology τC [13, Theorem 2.2.3]; and this requirement carries over to functions f :⊆ A → B on other topological spaces (A, τA ) and (B, τB ) where input a ∈ A and output b = f (a) are encoded by respective admissible representations α and β. Roughly speaking, this property expresses that the mappings α :⊆ Σ ω → A and β :⊆ Σ ω → B satisfy a certain compatibility condition with respect to the topologies τA /τB and τC involved. For A = B = R, the (standard) representation ρ for example is admissible [13, Lemma 4.1.4.1], thus recovering the folklore claim. Now in order to treat and non-trivially investigate computability of discontinuous real functions f : R → R as well, there are basically two ways out: Either enhance the underlying Type-2 Machine model or resort to non-admissible representations. It turns out that for either choice, at least the straight-forward approaches fail: • extending Turing Machines with oracles as well as • considering weakened representations for R. 3.1
Type-2 Oracle Computation
Specifically concerning the first approach we make the following Observation 5. Let O ⊆ Σ ∗ be arbitrary. Replace in [13, Definition 2.1.1] the Turing Machine M by M O , that is, one with oracle access to O. Then Lemma 2.1.11, Theorem 2.1.12, Theorem 2.2.3, Corollary 3.2.12, and Theorem 4.3.1 in [13] still remain valid. In particular, the Main Theorem of Computable Analysis relativizes. Corollary 6. A partial function on infinite strings f :⊆ Σ ω → Σ ω is Cantorcontinuous iff it is computable relative to some oracle O. This is to be compared with Type-1 Theory (that is, computability on finite strings) where any function f :⊆ Σ ∗ → Σ ∗ becomes recursive in some appropriate O ⊆ Σ ∗ . Proof. If f is recursive in O, then it is also continuous by the relativized version of [13, Theorem 2.2.3]. Conversely let f be continuous; then a monotone total function h : Σ ∗ → Σ ∗ with hω = f according to [13, Definition 2.1.10.2] can be seen to exist. But being a classical Type-1 function, h is recursive in a certain oracle O ⊆ Σ ∗ . The relativization of [13, Lemma 2.1.11.2] then asserts also f to be computable in O.
While oracles thus do not increase the computational power of a Type-2 Machine sufficiently in order to handle also discontinuous functions, Section 5 reveals that nondeterminism does.
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Weaker Representations for Reals
Concerning the second approach, we are interested in relaxations of the standard representation ρ for single reals and their effect on the computability of function evaluation x → f (x). Since, with exception of ρ, none of the ones introduced in Definition 3 is admissible [13, Lemma 4.1.4, Example 4.1.14.1], chances are good for this problem to become computable even for discontinuous f : R → R. Example 7. Heaviside’s Function H : R → R,
x → 0 for x ≤ 0,
x → 1 for x > 0
is both (ρ< → ρ< )–computable and (ρ< → ρ< )–computable. Proof. Given (qn ) ⊆ Q with x = supn qn , compute pn := H(qn ). Then indeed, (pn ) ⊆ Q has supn pn = H(x): In case x ≤ 0, qn ≤ 0 and hence pn = 0 for all n; whereas in case x > 0, qn > 0 and hence pn = 1 for some n. Let x ∈ R be given by a rational double sequence (qi,j ) with x = supi inf j qi,j . Proceeding from qi,j to q˜i,j := max{q0,j , . . . , qi,j }, we assert inf j q˜i+1,j ≥ inf j q˜i,j . qi,j − 2−i ). Then in case x ≤ 0, it holds ∀i∃j : q˜i,j ≤ 2−i , Now compute pi,j := H(˜ i.e., pi,j = 0 and thus supi inf j pi,j = 0 = H(x). Similarly in case x > 0, there is some i0 such that inf j q˜i0 ,j > x/2 and thus inf j q˜i,j > x/2 for all i ≥ i0 . For i ≥ i0 with 2−i ≤ x/2, it follows pi,j = 1 ∀j and therefore supi inf j pi,j = 1 = H(x).
It turns out that the implication “(ρ< → ρ< ) ⇒ (ρ< → ρ< )” is not specific to H: Theorem 8. Consider f : R → R. a) If f is (ρ → ρ)–computable, then it is also (ρ → ρ )–computable. b) If f is (ρ → ρ< )–computable, then it is also (ρ → ρ< )–computable. c) If f is (ρ< → ρ< )–computable, then it is also (ρ< → ρ< )–computable. d) If f is (ρ → ρ )–computable, then it is also (ρ → ρ )–computable. Our proof exploits Fact 9 and Theorem 10 below. By the Main Theorem of Recursive Analysis, any (ρ → ρ)–computable real function is continuous. This claim has been extended in various ways. Fact 9. Consider f : R → R. a) If f is (ρ → ρ< )–computable, then it is lower semi continuous. b) If f is (ρ< → ρ< )–computable, then it is monotonically increasing. c) If f is (ρ → ρ )–computable, then it is continuous. The claims remain valid under oracle-supported computation. Proof. For a) see e.g. [7, Chapter 6.7]; c) is established in [2, Section 6]. These claims also generalize to the above hierarchy of representations: Theorem 10. Consider f : R → R. a) If f is (ρ → ρ )–computable, then it is continuous. b) If f is (ρ → ρ< )–computable, then it is lower semi-continuous. c) If f is (ρ< → ρ< )–computable, then it is monotonically increasing. The claims remain valid under oracle-supported computation.
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This allows us to strengthen Lemma 4: Corollary 11. ρ ≡ ρ< ρ> t ρ< t ρ ≡ ρ< ρ> t ρ< t ρ ≡ . . . where “ t” denotes continuous reducibility of representations [13, Def. 2.3.2]. Proof. The positive claims follow from Lemma 4 with Corollary 6. For a negative claim like for example “ρ< t ρ ” suppose the contrary. Then by Corollary 6, with the help of some appropriate oracle O, one can convert ρ< –names to ρ – names. As Heaviside’s Function H is (ρ → ρ< )–computable by Example 7 and Theorem 8b), composition with this conversion implies (ρ → ρ )–computability of H relative to O — contradicting Theorem 10a).
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Arithmetic Weierstraß Hierarchy
Recall that Weierstraß Approximation Theorem asserts any continuous real function f : [0, 1] → R to be the uniform limit f = ulimn Pn of a sequence of rational polynomials (Pn ) ⊆ Q[X]. Here, ‘ulim’ suggestively denotes uniform limits of continuous functions, that is, with sup |f (x) − Pn (x)| =: f − Pn → 0 as 0≤x≤1 n → ∞. For (ρ → ρ)–computable (and thus continuous) f , the well-known Effective Weierstraß Theorem yields even a computable and fast convergent such sequence (Pn ). Furthermore by allowing this sequence (Pn ) to be computable relative to ∅(d) , one naturally obtains, in analogy to the Real Arithmetic Hierarchy, a hierarchy of real functions f : [0, 1] → R with the effective case as lowest level. In fact its second level, too, has already been characterized namely by Ho. Lemma 12. a) A real function f : [0, 1]→ R is (ρ→ ρ)–computable if and only if it holds (ρ → ρ) ρ): There exists a computable sequence of (degrees and coefficients of ) rational polynomials (Pn ) ⊆ Q[X] such that f − Pn ≤ 2−n (1) b) To a real function f : [0, 1] → R, there exists a ∅ –computable sequence of polynomials (Pn ) satisfying Equation (1) if and only if it holds (ρ → ρ) : There is a computable sequence (Qm ) ⊆ Q[X] converging uniformly (although not necessarily ‘fast’) to f , that is, with f = ulimm→∞ Qm . c) To a real function f : [0, 1] → R, there exists a ∅ –computable sequence of polynomials (Pn ) satisfying Equation (1) if and only if it holds (ρ → ρ) : There is a computable sequence (Qm ) ⊆ Q[X] s.t. f = ulimi ulimj Qi,j . We emphasize the similarity to Fact 2b). Proof. a) See, e.g., [6, Section 0.7]. b) See [4, Theorem 16]. c) follows similarly from Shoenfield’s Limit Theorem. Recall that the (ρ → ρ )–computable functions are continuous by [2, Section 6] and thus applicable to the Weierstraß Theorem. The following result relates this class to relativized computation of Weierstraß approximations.
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Theorem 13. a) Let f : [0, 1] → R be (ρ → ρ) –computable in the sense of Lemma 12b). Then, f is (ρ → ρ )–computable. b) Let f : [0, 1] → R be (ρ → ρ )–computable. Then, f is (ρ → ρ) –computable. c) There is a (ρ → ρ )–computable but not (ρ → ρ) –computable f : [0, 1] → R. As opposed to d = 0, the hierarchies of (ρ(d) → ρ(d) )–computable functions on the one hand and of (ρ → ρ)(d) –computable functions on the other hand thus lie skewly to each other for d ≥ 1.
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Type-2 Nondeterminism
Observing that the proofs of the Main Theorem in Recursive Analysis as well as its generalizations to oracle- and weakened real computation in Theorem 10 crucially exploit the underlying Turing Machines to be deterministic, the question becomes evident whether nondeterminism might yield the additional power necessary for computing discontinuous real functions like Heaviside’s. In the discrete (i.e., Type-1) setting where any computation is required to terminate, the finitely many possible choices of a nondeterministic machine can of course be simulated by a deterministic one — however already here subject to the important condition that all paths of the nondeterministic computation indeed terminate, cf. [9]. In contrast, a Type-2 computation realizes a transformation from/to infinite strings and is therefore a generally non-terminating process. Therefore, nondeterminism here involves an infinite number of guesses which turns out cannot be simulated by a deterministic Type-2 machine. We also point out that nondeterminism has already before been revealed not only a useful but indeed the ¨chi most natural concept of computation on infinite strings. More precisely Bu extended Finite Automata from finite to infinite strings and proved that, here, nondeterministic ones are closed under complement [10] as opposed to deterministic ones. Since automata and Turing Machines constitute the bottom and top levels, respectively, of Chomsky’s Hierarchy of classical languages (Type-1 setting), we suggest that over infinite strings (Type-2 setting) both their respective counterparts, that is B¨ uchi Automata as well as Type-2 Machines, be considered nondeterministically. Definition 14. Let A and B be uncountable sets with respective representations α :⊆ Σ ω → A and β :⊆ Σ ω → B. A function f :⊆ A → B is called nondeterministically (α → β)–computable if some nondeterministic one-way Turing Machine, – upon input of any α–name for some a ∈ dom(f ), – has a computation which outputs a β–name for b = f (a) and – any infinite string output by some computation is a β–name for b = f (a). This definition is sensible insofar as it leads to closure under composition: Lemma 15. Let f :⊆ A → B be nondeterministically (α → β)–computable and g :⊆ B → C be nondeterministically (β → γ)–computable. Then, g ◦ f :⊆ A → C is nondeterministically (α → γ)–computable.
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A subtle point in Definition 14, computations leading only to finite output are semantically ignored. This enables algorithms to ‘withdraw’ a nondeterministic choice once its unfavourability is detected. In particular, one can nondeterministically convert forth and back among representations on the Real Arithmetic Hierarchy from Definition 1: Theorem 16. For each d = 0, 1, 2, . . ., the identity R x → x is nondeterministically (ρ(d+1) → ρ(d) )–computable. It is furthermore (ρ → ρb,2 )–computable. In particular and in striking contrast to usual Type-2 Theory, nondeterministic computability of real functions is largely independent of the representation under consideration. This notion includes discontinuous functions as, by Example 7, the Heaviside Function is nondeterministically computable in any reasonable sense. Proof (Theorem 16). Consider for simplicity the case d = 0. Let x ∈ R be given by a sequence (qn ) ⊆ Q eventually converging to x. There exists a strictly increasing sequence (nk ) ⊆ N such that (qnk ) converges fast to x, that is, satisfying ∀ ≤ k :
qnk ∈ [qn − 2− , qn + 2− ]
(2)
For each k ∈ N, the algorithm iteratively guesses nk > nk−1 , prints qnk , and checks Equation (2): If violated, abort having output only a finite string. For (ρ → ρb,2 )–computability, let x ∈ (0, 2) be given by a fast convergent sequence (qn ) ⊆ Q. We guess the leading digit b ∈ {0, 1} for x’s binary expansion b.∗; in case b = 0, check whether x > 1 — a ρ–semi decidable property — and if so, abort; similarly in case b = 1, abort if it turns out that x < 1. Otherwise (that is, proceeding while simultaneously continuing the above semi-decision process via dove-tailing) replace x by 2(x − b) and repeat guessing the next bit.
We point out that for non-unique binary expansion (i.e., for dyadic x), nondeterminism in the above (ρ → ρb,2 )–computation, in accordance with the third requirement of Definition 14, generates both possible expansions.
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Conclusion
Recursive Analysis is often criticized for being unable to (non-trivially) treat discontinuous functions. Strictly speaking, this reproach does not apply since for instance Zhong and Weihrauch do investigate the computability of generalized (and in particular of discontinuous) functions by suitable representations in [16]. However here, evaluating x → f (x) an L2 function or a distribution f at a point x ∈ R does not make sense already mathematically and therefore is not supported. Another reply to the aforementioned critics, Heaviside’s function — although discontinuous — is (ρ → ρ< )–computable; and this notion of function computability does allow for (lower) evaluation. On the other hand, it lacks closure under composition. Closure under composition holds for (ρ< → ρ< )–computability, but here closure under inversion f → −f fails.
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The present work investigates on sufficient and necessary conditions for some f : R → R to be (ρ(d) → ρ(d) )–computable. Being closed both under composition and inversion, respectively, these notions emerging from the Real Arithmetic Hierarchy are not applicable to the Main Theorem for d = 1, 2, . . . and might therefore include discontinuous functions. We extend the surprising result [2, Section 6] that (ρ → ρ )–computability does imply continuity to the case d ≥ 2. Due to the purely information-theoretic nature of the arguments employed by Brattka and Hertling, their result immediately relativizes, that is, even oracle support does not lift the continuity condition. So we looked for other ways of enhancing the underlying model. Replacing, in analogy to B¨ uchi Automata, deterministic Turing machines by nondeterministic ones turned out to do the job. While their practical realizability is admittedly questionable, the obtained notion of function computability benefits from its elegance and invariance under encodings of real numbers.
References 1. G. Barmpalias: “A Transfinite Hierarchy of Reals”, pp.163–172 in Mathematical Logic Quarterly vol.49(2) (2003). 2. V. Brattka, P. Hertling: “Topological Properties of Real Number Representations”, pp.241–257 in Theoretical Computer Science vol.284 (2002). 3. A. Grzegorczyk: “On the Definitions of Computable Real Continuous Functions”, pp.61–77 in Fundamenta Mathematicae 44 (1957). 4. C.-K. Ho: “Relatively Recursive Real Numbers and Real Functions”, pp.99–120 in Theoretical Computer Science vol.210 (1999). 5. Ker-I Ko: “Complexity Theory of Real Functions”, Birkh¨ auser (1991). 6. M.B. Pour-El, J.I. Richards: “Computability in Analysis and Physics”, Springer (1989). 7. J.F. Randolph: “Basic Real and Abstract Analysis”, Academic Press (1968). 8. R.I. Soare: “Recursively Enumerable Sets and Degrees”, Springer (1987). 9. E. Spaan, L. Torenvliet, P. van Emde Boas: “Nondeterminism, Fairness and a Fundamental Analogy”, pp.186–193 in The Bulletin of the European Association for Theoretical Computer Science (EATCS Bulletin) vol.37 (1989). 10. Thomas, W.: “Automata on Infinite Objects”, pp.133–191 in Handbook of Theoretical Computer Science, vol.B (Formal Models and Semantics), Elsevier (1990). 11. Turing, A.M.: “On Computable Numbers, with an Application to the Entscheidungsproblem”, pp.230–265 in Proc. London Math. Soc. vol.42(2) (1936). 12. Turing, A.M.: “On Computable Numbers, with an Application to the Entscheidungsproblem. A correction”, pp.544–546 in Proc. London Math. Soc. vol.43(2) (1937). 13. K. Weihrauch: “Computable Analysis”, Springer (2001). 14. X. Zheng, K. Weihrauch: “The Arithmetical Hierarchy of Real Numbers”, pp.51–65 in Mathematical Logic Quarterly vol.47 (2001). 15. X. Zheng: “Recursive Approximability of Real Numbers”, pp.131–156 in Mathematical Logic Quarterly vol.48 Supplement 1 (2002). 16. N. Zhong, K. Weihrauch: “Computability Theory of Generalized Functions”, pp.469–505 in J. ACM vol.50 (2003).
Abstract. The sometimes so-called Main Theorem of Recursive Analysis implies that any computable real function is necessarily continuous. We consider three relaxations of this common notion of real computability for the purpose of treating also discontinuous functions f : R → R: – non-deterministic computation; – relativized computation, specifically given access to oracles like ∅ or ∅ ; – encoding input x ∈ R and/or output y = f (x) in weaker ways according to the Real Arithmetic Hierarchy. It turns out that, among these approaches, only the first one provides the required power.
1
Motivation
What does it mean for a Turing Machine, capable of operating only on discrete objects, to compute a real number x: ρb,2 : To decide its binary expansion? ρCn : To compute a sequence (qn ) of rational numbers eventually converging to x? ρ : To compute a fast convergent sequence (qn ) ⊆ Q for x, i.e. with |x − qn | ≤ 2−n (in other words: to approximate x with error bounds)? ρ< : To approximate x from below, i.e., to compute (qn ) such that x = supn qn ? All these notions make sense in being closed under arithmetic operations like addition and multiplication. In fact they are well (known equivalent to variants) studied in literature1 ; e.g. [11], [2], [12], [13] in order. Now what does it mean for a Turing Machine M to compute a real function f : R → R? Most naturally it says that, upon input of x ∈ R given in one of 1
Supported by project 21-04-0303 of Statens Naturvidenskabelige Forskningsr˚ ad SNF. Their above names by Greek letters are taken from [13, Section 4.1].
S.B. Cooper, B. L¨ owe, and L. Torenvliet (Eds.): CiE 2005, LNCS 3526, pp. 562–571, 2005. c Springer-Verlag Berlin Heidelberg 2005
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the above ways, M outputs y = f (x) also in one (not necessarily the same) of the above ways. And, again, many possible combinations have already been investigated. For instance the standard notion of real function computation in Recursive Analysis [3, 6, 5, 13] refers (or is equivalent) to input and output given according to ρ. Here, the Main Theorem of Computable Analysis implies that any computable f will necessarily be continuous [13, Theorem 4.3.1]. We are interested in ways of lifting this restriction. A first approach might equip the Turing Machines under consideration with access to an oracle, say, for the Halting Problem ∅ or its iterated jumps ∅(d) in Kleene’s Arithmetic Hierarchy. However closer inspection in Section 3.1 reveals that this Main Theorem relies solely on information rather than recursion theoretic arguments and therefore requires continuity also for oracle-computable real functions with respect to input and output of form ρ. (For the special case of an ∅ –oracle, this had been observed in [4, Theorem 16].) A second idea changes the input and output representation for x and y = f (x) from ρ to a weaker form like, say, ρCn . This relates to the Arithmetic Hierarchy, too, however in a completely different way: Computing x in the sense of ρCn is equivalent to computing x in the sense of ρ [4, Theorem 9] relative (i.e., given access) to the Halting Problem ∅ . And, most promisingly, the Main Theorem [13, Corollary 3.2.12] which previously required continuity of computable real functions now does not apply any more since ρCn , in contrast to ρ, lacks the technical property of admissibility. It therefore came to quite a surprise when Brattka and Hertling established that any (ρCn → ρCn )–computable f (that is, with respect to input x and output f (x) encoded according to ρCn ) still satisfies continuity; see [13, Exercise 4.1.13d] or [2, Section 6]. In Section 3.2, we extend this result. Specifically it is proven that continuity ρ ≡ ρCn ρ is necessary for (ρ → ρ )–computability of f ; here, ρ . . . denote the first levels of an entire hierarchy of real number representations emerging naturally from the Real Arithmetic Hierarchy of Weihrauch and Zheng [14]. The class of (ρ → ρ)–computable functions f : R → R is well-known to admit an alternative characterization based on Caldwell’s and Pour-El’s famous Effective Weierstraß Theorem. Its extension based on relativization gives rise to a hierarchy of continuous functions f : R → R. The (ρ → ρ) –computable ones for example have been investigated in [4, Section 4]. Section 4 of the present work locates the class of (ρ → ρ )–computable functions among this hierarchy. Rather than endowing deterministic Turing Machines with oracles, in Section 5 we finally consider nondeterministic computation. Remarkably and in contrast to the classical (Type-1) theory, this significantly increases the principal capabilities. For example, discontinuous real functions now do become computable and so does conversion among the aforementioned representations ρCn and ρb,2 .
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In [4], Ho observed an interesting relation between computability of a real number x in the respective senses of ρ and ρCn in terms of oracles: x = limn qn for an (eventually convergent) computable rational sequence (qn ) iff x admits a ∅ –computable fast convergent rational sequence, that is, a sequence (pm ) ⊆ Q recursive in ∅ with |x − pm | ≤ 2−m . This suggests to write ρ for ρCn ; and denoting by ∆1 R = Rc the set of reals computable in the sense of Recursive Analysis (that is with respect to ρ), it is therefore natural to write, in analogy to Kleenes classical Arithmetic Hierarchy, ∆2 R for the set of all x ∈ R computable with respect to ρ . Weihrauch and Zheng extended these considerations and obtained for instance [14, Corollary 7.3] the following characterization of ∆3 R: A real x ∈ R admits a fast convergent rational sequence recursive in ∅ iff x is computable in the sense of ρ defined as follows: ρ : x = limi limj qi,j for some computable rational sequence (qn ). Similarly, Σ1 R contains of all x ∈ R computable with respect to ρ< whereas Σ2 R includes all x computable in the sense of ρ< defined as follows: ρ< : x = supi inf j qi,j for some computable rational sequence (qn ). √ To Σ2 R belongs for instance the radius of convergence r = 1/ lim supn→∞ n an ∞ of a computable power series n=0 an xn [14, Theorem 6.2]. More generally we take from [14, Definition 7.1 and Corollary 7.3] the following Definition 1 (Real Arithmetic Hierarchy). Let d = 0, 1, 2, . . . (d) ρ< : Σd+1 R consists of all x ∈ R of the form x = supn1 inf n2 . . . Θnd+1 qn1 ,...,nd+1 for a computable rational sequence (qn ), where Θ= sup or Θ= inf depending on d’s parity; (d) ρ> : Πd+1 R similarly for x = inf n1 supn2 . . . ρ(d) : ∆d+1 R contains all x ∈ R of the form x = lim lim . . . lim qn1 ,...,nd n1 n2 nd for a computable rational sequence (qn ). (For levels beyond ω see [1]. . . ) The close analogy between the discrete and this real variant of the Arithmetic Hierarchy is expressed in [14] by a variety of elegant results like, e.g., Fact 2. a) x ∈ ∆d R iff deciding its binary expansion is in ∆d . b) x is computable with respect to ρ(d) iff there is a ∅(d) –computable fast convergent rational sequence for x. (d) c) x is computable with respect to ρ< iff x is the supremum of a ∅(d) –computable rational sequence. d) ∆d R = Σd R ∩ Πd R. e) Σd R ∪ Πd R ∆d+1 R. 2.1
Type-2 Theory of Effectivity
Specifying an encoding formalizes how to feed some general form of input like graphs or integers into a Turing Machine with fixed alphabet Σ. Already in the
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discrete case, the complexity of a problem usually depends heavily on the chosen encoding; e.g., numbers in unary versus binary. This dependence becomes even more important when dealing with objects from an continuum like the set of reals and their computability. While Recursive Analysis usually considers one particular encoding for R, the Type-2 Theory of Effectivity (TTE) due to Weihrauch provides (a convenient formal framework for studying and comparing) a variety of encodings for different universes. Formally speaking, a representation α for R ¯ ∈ dom(α) is a partial surjective mapping α :⊆ Σ ω → R; and an infinite string σ is regarded as a name for the real number x = α(¯ σ ). In this way, (α → β)– computing a real function f : R → R means to compute a transformation on ¯ for x = α(¯ σ ) gets infinite strings F :⊆ Σ ω → Σ ω such that any α–name σ transformed to a β–name τ¯ = F (¯ σ ) for f (x) = y, that is, satisfying β(¯ τ ) = y; cf. [13, Section 3]. Now observe that (the above characterization of) each level of the Real Arithmetic Hierarchy gives rise not only to a notion of computability for real numbers but also canonically to a representation for R; for instance let ρ : encode (arbitrary!) x ∈ R as a fast convergent rational sequence (qn ); ρ< : encode x ∈ R as supremum of a rational sequence: x = supn qn ; ρ : encode x ∈ R as limit of a rational sequence: x = limn qn ; ρ< : encode x ∈ R as (qn ) ⊆ Q with x = supi inf j qi,j ; ρ : encode x ∈ R as (qn ) ⊆ Q with x = limi limj qi,j . In fact the first three of them are known in TTE as ρ, ρ< , and ρCn , respectively [13, Section 4.1]. In general one obtains, similar to Definition 1, a hierarchy of real representations as follows: (0)
(0)
Definition 3. Let ρ(0) := ρ, ρ< := ρ< , ρ> := ρ> . Now fix 1 ≤ d ∈ N: A ρ(d) –name for x ∈ R is a (νQω –name for a) rational sequence (qn ) such that x = lim lim . . . lim qn1 ,...,nd . n1
n2
nd
(d)
A ρ< –name for x ∈ R is a (name for a) sequence (qn ) ⊆ Q such that x = supn1 inf n2 . . . Θnd+1 qn1 ,...,nd+1 . (d)
A ρ> –name for x ∈ R is a sequence (qn ) ⊆ Q such that x = inf n1 supn2 . . . Results from [14] about the Real Arithmetic Hierarchy are easily re-phrased in terms of these representations. However this leads only to non-uniform claims. Fact 2d) for example translates as follows: (d)
(d)
x is ρ(d) –computable iff it is both ρ< –computable and ρ> –computable. Closer inspection of the proofs in particular of Lemma 3.2 and Lemma 3.3 in [14] reveals them to hold even uniformly. More precisely we can prove ρ< ρ ≡ ρ< ρ> ρ< ρ ≡ . . . , Lemma 4. ρ ≡ ρ< ρ> where “ ” and “ ≡” denote computable reducibility and equivalence, respectively, of representations and “ ” their join, i.e., the least upper bound w.r.t. “ ”; see [13, Definition 2.3.2].
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Computability and Continuity
Recursive Analysis has established as folklore that any computable real function is continuous. More precisely, computability of a partial function from/to infinite strings f :⊆ Σ ω → Σ ω requires continuity with respect to the Cantor Topology τC [13, Theorem 2.2.3]; and this requirement carries over to functions f :⊆ A → B on other topological spaces (A, τA ) and (B, τB ) where input a ∈ A and output b = f (a) are encoded by respective admissible representations α and β. Roughly speaking, this property expresses that the mappings α :⊆ Σ ω → A and β :⊆ Σ ω → B satisfy a certain compatibility condition with respect to the topologies τA /τB and τC involved. For A = B = R, the (standard) representation ρ for example is admissible [13, Lemma 4.1.4.1], thus recovering the folklore claim. Now in order to treat and non-trivially investigate computability of discontinuous real functions f : R → R as well, there are basically two ways out: Either enhance the underlying Type-2 Machine model or resort to non-admissible representations. It turns out that for either choice, at least the straight-forward approaches fail: • extending Turing Machines with oracles as well as • considering weakened representations for R. 3.1
Type-2 Oracle Computation
Specifically concerning the first approach we make the following Observation 5. Let O ⊆ Σ ∗ be arbitrary. Replace in [13, Definition 2.1.1] the Turing Machine M by M O , that is, one with oracle access to O. Then Lemma 2.1.11, Theorem 2.1.12, Theorem 2.2.3, Corollary 3.2.12, and Theorem 4.3.1 in [13] still remain valid. In particular, the Main Theorem of Computable Analysis relativizes. Corollary 6. A partial function on infinite strings f :⊆ Σ ω → Σ ω is Cantorcontinuous iff it is computable relative to some oracle O. This is to be compared with Type-1 Theory (that is, computability on finite strings) where any function f :⊆ Σ ∗ → Σ ∗ becomes recursive in some appropriate O ⊆ Σ ∗ . Proof. If f is recursive in O, then it is also continuous by the relativized version of [13, Theorem 2.2.3]. Conversely let f be continuous; then a monotone total function h : Σ ∗ → Σ ∗ with hω = f according to [13, Definition 2.1.10.2] can be seen to exist. But being a classical Type-1 function, h is recursive in a certain oracle O ⊆ Σ ∗ . The relativization of [13, Lemma 2.1.11.2] then asserts also f to be computable in O.
While oracles thus do not increase the computational power of a Type-2 Machine sufficiently in order to handle also discontinuous functions, Section 5 reveals that nondeterminism does.
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Weaker Representations for Reals
Concerning the second approach, we are interested in relaxations of the standard representation ρ for single reals and their effect on the computability of function evaluation x → f (x). Since, with exception of ρ, none of the ones introduced in Definition 3 is admissible [13, Lemma 4.1.4, Example 4.1.14.1], chances are good for this problem to become computable even for discontinuous f : R → R. Example 7. Heaviside’s Function H : R → R,
x → 0 for x ≤ 0,
x → 1 for x > 0
is both (ρ< → ρ< )–computable and (ρ< → ρ< )–computable. Proof. Given (qn ) ⊆ Q with x = supn qn , compute pn := H(qn ). Then indeed, (pn ) ⊆ Q has supn pn = H(x): In case x ≤ 0, qn ≤ 0 and hence pn = 0 for all n; whereas in case x > 0, qn > 0 and hence pn = 1 for some n. Let x ∈ R be given by a rational double sequence (qi,j ) with x = supi inf j qi,j . Proceeding from qi,j to q˜i,j := max{q0,j , . . . , qi,j }, we assert inf j q˜i+1,j ≥ inf j q˜i,j . qi,j − 2−i ). Then in case x ≤ 0, it holds ∀i∃j : q˜i,j ≤ 2−i , Now compute pi,j := H(˜ i.e., pi,j = 0 and thus supi inf j pi,j = 0 = H(x). Similarly in case x > 0, there is some i0 such that inf j q˜i0 ,j > x/2 and thus inf j q˜i,j > x/2 for all i ≥ i0 . For i ≥ i0 with 2−i ≤ x/2, it follows pi,j = 1 ∀j and therefore supi inf j pi,j = 1 = H(x).
It turns out that the implication “(ρ< → ρ< ) ⇒ (ρ< → ρ< )” is not specific to H: Theorem 8. Consider f : R → R. a) If f is (ρ → ρ)–computable, then it is also (ρ → ρ )–computable. b) If f is (ρ → ρ< )–computable, then it is also (ρ → ρ< )–computable. c) If f is (ρ< → ρ< )–computable, then it is also (ρ< → ρ< )–computable. d) If f is (ρ → ρ )–computable, then it is also (ρ → ρ )–computable. Our proof exploits Fact 9 and Theorem 10 below. By the Main Theorem of Recursive Analysis, any (ρ → ρ)–computable real function is continuous. This claim has been extended in various ways. Fact 9. Consider f : R → R. a) If f is (ρ → ρ< )–computable, then it is lower semi continuous. b) If f is (ρ< → ρ< )–computable, then it is monotonically increasing. c) If f is (ρ → ρ )–computable, then it is continuous. The claims remain valid under oracle-supported computation. Proof. For a) see e.g. [7, Chapter 6.7]; c) is established in [2, Section 6]. These claims also generalize to the above hierarchy of representations: Theorem 10. Consider f : R → R. a) If f is (ρ → ρ )–computable, then it is continuous. b) If f is (ρ → ρ< )–computable, then it is lower semi-continuous. c) If f is (ρ< → ρ< )–computable, then it is monotonically increasing. The claims remain valid under oracle-supported computation.
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This allows us to strengthen Lemma 4: Corollary 11. ρ ≡ ρ< ρ> t ρ< t ρ ≡ ρ< ρ> t ρ< t ρ ≡ . . . where “ t” denotes continuous reducibility of representations [13, Def. 2.3.2]. Proof. The positive claims follow from Lemma 4 with Corollary 6. For a negative claim like for example “ρ< t ρ ” suppose the contrary. Then by Corollary 6, with the help of some appropriate oracle O, one can convert ρ< –names to ρ – names. As Heaviside’s Function H is (ρ → ρ< )–computable by Example 7 and Theorem 8b), composition with this conversion implies (ρ → ρ )–computability of H relative to O — contradicting Theorem 10a).
4
Arithmetic Weierstraß Hierarchy
Recall that Weierstraß Approximation Theorem asserts any continuous real function f : [0, 1] → R to be the uniform limit f = ulimn Pn of a sequence of rational polynomials (Pn ) ⊆ Q[X]. Here, ‘ulim’ suggestively denotes uniform limits of continuous functions, that is, with sup |f (x) − Pn (x)| =: f − Pn → 0 as 0≤x≤1 n → ∞. For (ρ → ρ)–computable (and thus continuous) f , the well-known Effective Weierstraß Theorem yields even a computable and fast convergent such sequence (Pn ). Furthermore by allowing this sequence (Pn ) to be computable relative to ∅(d) , one naturally obtains, in analogy to the Real Arithmetic Hierarchy, a hierarchy of real functions f : [0, 1] → R with the effective case as lowest level. In fact its second level, too, has already been characterized namely by Ho. Lemma 12. a) A real function f : [0, 1]→ R is (ρ→ ρ)–computable if and only if it holds (ρ → ρ) ρ): There exists a computable sequence of (degrees and coefficients of ) rational polynomials (Pn ) ⊆ Q[X] such that f − Pn ≤ 2−n (1) b) To a real function f : [0, 1] → R, there exists a ∅ –computable sequence of polynomials (Pn ) satisfying Equation (1) if and only if it holds (ρ → ρ) : There is a computable sequence (Qm ) ⊆ Q[X] converging uniformly (although not necessarily ‘fast’) to f , that is, with f = ulimm→∞ Qm . c) To a real function f : [0, 1] → R, there exists a ∅ –computable sequence of polynomials (Pn ) satisfying Equation (1) if and only if it holds (ρ → ρ) : There is a computable sequence (Qm ) ⊆ Q[X] s.t. f = ulimi ulimj Qi,j . We emphasize the similarity to Fact 2b). Proof. a) See, e.g., [6, Section 0.7]. b) See [4, Theorem 16]. c) follows similarly from Shoenfield’s Limit Theorem. Recall that the (ρ → ρ )–computable functions are continuous by [2, Section 6] and thus applicable to the Weierstraß Theorem. The following result relates this class to relativized computation of Weierstraß approximations.
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Theorem 13. a) Let f : [0, 1] → R be (ρ → ρ) –computable in the sense of Lemma 12b). Then, f is (ρ → ρ )–computable. b) Let f : [0, 1] → R be (ρ → ρ )–computable. Then, f is (ρ → ρ) –computable. c) There is a (ρ → ρ )–computable but not (ρ → ρ) –computable f : [0, 1] → R. As opposed to d = 0, the hierarchies of (ρ(d) → ρ(d) )–computable functions on the one hand and of (ρ → ρ)(d) –computable functions on the other hand thus lie skewly to each other for d ≥ 1.
5
Type-2 Nondeterminism
Observing that the proofs of the Main Theorem in Recursive Analysis as well as its generalizations to oracle- and weakened real computation in Theorem 10 crucially exploit the underlying Turing Machines to be deterministic, the question becomes evident whether nondeterminism might yield the additional power necessary for computing discontinuous real functions like Heaviside’s. In the discrete (i.e., Type-1) setting where any computation is required to terminate, the finitely many possible choices of a nondeterministic machine can of course be simulated by a deterministic one — however already here subject to the important condition that all paths of the nondeterministic computation indeed terminate, cf. [9]. In contrast, a Type-2 computation realizes a transformation from/to infinite strings and is therefore a generally non-terminating process. Therefore, nondeterminism here involves an infinite number of guesses which turns out cannot be simulated by a deterministic Type-2 machine. We also point out that nondeterminism has already before been revealed not only a useful but indeed the ¨chi most natural concept of computation on infinite strings. More precisely Bu extended Finite Automata from finite to infinite strings and proved that, here, nondeterministic ones are closed under complement [10] as opposed to deterministic ones. Since automata and Turing Machines constitute the bottom and top levels, respectively, of Chomsky’s Hierarchy of classical languages (Type-1 setting), we suggest that over infinite strings (Type-2 setting) both their respective counterparts, that is B¨ uchi Automata as well as Type-2 Machines, be considered nondeterministically. Definition 14. Let A and B be uncountable sets with respective representations α :⊆ Σ ω → A and β :⊆ Σ ω → B. A function f :⊆ A → B is called nondeterministically (α → β)–computable if some nondeterministic one-way Turing Machine, – upon input of any α–name for some a ∈ dom(f ), – has a computation which outputs a β–name for b = f (a) and – any infinite string output by some computation is a β–name for b = f (a). This definition is sensible insofar as it leads to closure under composition: Lemma 15. Let f :⊆ A → B be nondeterministically (α → β)–computable and g :⊆ B → C be nondeterministically (β → γ)–computable. Then, g ◦ f :⊆ A → C is nondeterministically (α → γ)–computable.
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A subtle point in Definition 14, computations leading only to finite output are semantically ignored. This enables algorithms to ‘withdraw’ a nondeterministic choice once its unfavourability is detected. In particular, one can nondeterministically convert forth and back among representations on the Real Arithmetic Hierarchy from Definition 1: Theorem 16. For each d = 0, 1, 2, . . ., the identity R x → x is nondeterministically (ρ(d+1) → ρ(d) )–computable. It is furthermore (ρ → ρb,2 )–computable. In particular and in striking contrast to usual Type-2 Theory, nondeterministic computability of real functions is largely independent of the representation under consideration. This notion includes discontinuous functions as, by Example 7, the Heaviside Function is nondeterministically computable in any reasonable sense. Proof (Theorem 16). Consider for simplicity the case d = 0. Let x ∈ R be given by a sequence (qn ) ⊆ Q eventually converging to x. There exists a strictly increasing sequence (nk ) ⊆ N such that (qnk ) converges fast to x, that is, satisfying ∀ ≤ k :
qnk ∈ [qn − 2− , qn + 2− ]
(2)
For each k ∈ N, the algorithm iteratively guesses nk > nk−1 , prints qnk , and checks Equation (2): If violated, abort having output only a finite string. For (ρ → ρb,2 )–computability, let x ∈ (0, 2) be given by a fast convergent sequence (qn ) ⊆ Q. We guess the leading digit b ∈ {0, 1} for x’s binary expansion b.∗; in case b = 0, check whether x > 1 — a ρ–semi decidable property — and if so, abort; similarly in case b = 1, abort if it turns out that x < 1. Otherwise (that is, proceeding while simultaneously continuing the above semi-decision process via dove-tailing) replace x by 2(x − b) and repeat guessing the next bit.
We point out that for non-unique binary expansion (i.e., for dyadic x), nondeterminism in the above (ρ → ρb,2 )–computation, in accordance with the third requirement of Definition 14, generates both possible expansions.
6
Conclusion
Recursive Analysis is often criticized for being unable to (non-trivially) treat discontinuous functions. Strictly speaking, this reproach does not apply since for instance Zhong and Weihrauch do investigate the computability of generalized (and in particular of discontinuous) functions by suitable representations in [16]. However here, evaluating x → f (x) an L2 function or a distribution f at a point x ∈ R does not make sense already mathematically and therefore is not supported. Another reply to the aforementioned critics, Heaviside’s function — although discontinuous — is (ρ → ρ< )–computable; and this notion of function computability does allow for (lower) evaluation. On the other hand, it lacks closure under composition. Closure under composition holds for (ρ< → ρ< )–computability, but here closure under inversion f → −f fails.
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The present work investigates on sufficient and necessary conditions for some f : R → R to be (ρ(d) → ρ(d) )–computable. Being closed both under composition and inversion, respectively, these notions emerging from the Real Arithmetic Hierarchy are not applicable to the Main Theorem for d = 1, 2, . . . and might therefore include discontinuous functions. We extend the surprising result [2, Section 6] that (ρ → ρ )–computability does imply continuity to the case d ≥ 2. Due to the purely information-theoretic nature of the arguments employed by Brattka and Hertling, their result immediately relativizes, that is, even oracle support does not lift the continuity condition. So we looked for other ways of enhancing the underlying model. Replacing, in analogy to B¨ uchi Automata, deterministic Turing machines by nondeterministic ones turned out to do the job. While their practical realizability is admittedly questionable, the obtained notion of function computability benefits from its elegance and invariance under encodings of real numbers.
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