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Some thoughts on hypercomputation

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N.C.A. da Costa *, F.A. Doria

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Institute for Advanced Studies, University of Sa˜o Paulo, Av. Prof. Luciano Gualberto, Trav. J 374, 05655-010 Sa˜o Paulo, SP, Brazil

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6 Abstract

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7 We first show that the Halting Function (the noncomputable function that solves the Halting Problem) has explicit 8 expressions in the language of calculus. Out of that fact we elaborate on the possible meaning of hypercomputation theory 9 within the setting of formal mathematical theories. 10  2005 Published by Elsevier Inc. 11 12 1. Introduction

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The concept of a Turing machine is a formal, abstract, concept that arises out of a basic idea: any computing device should be able to calculate a (computable) function by splitting it into small operational ‘‘bricks,’’ or elementary operations, whose manifold combinations would then give us all possible computable functions. Turing machines [18] are usually introduced via their tables, that specify elementary operations like moving the head to the left or to the right, erasing or printing a symbol on the tapeÕs square under the head, and so on. However it is more convenient for us to give an alternative definition for the concept in this paper, and we will use the (fully equivalent) partial recursive function picture to stress that we are dealing with a formal construction. Let pU(k, x0, x1, x2, . . . , xk) be an universal Diophantine polynomial [8,21] which we suppose to be fixed. We define the partial recursive function {e} of Go¨del number e that acts on natural number m as its input and has natural number n as its output as

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24 Definition 1.1

½fegðmÞ ¼ n $ Def ½9x0 ; . . . ; xk 2 xpU ðhe; m; ni; x0 ; x1 ; . . . ; xk Þ ¼ 0. 27 (x is the set of natural numbers, hx, y, zi = hx, hy, zii and h i is the usual pairing function; for the compu28 tation of e and the construction of pU see [8,21,18].) 29 Partial recursive function {e} is given by the preceding definition. Of course there is a relation between (the 30 abstract objects) Turing machines and concrete objects of our real world such as computers (which can be best q *

N.C.A. da Costa is supported in part by CNPq, Philosophy Section. Corresponding author. E-mail addresses: [email protected] (N.C.A. da Costa), [email protected], [email protected] (F.A. Doria).

0096-3003/$ - see front matter  2005 Published by Elsevier Inc. doi:10.1016/j.amc.2005.09.073

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31 seen as realizations of finite automata) but we restrict our attention to the mathematical object characterized 32 above. 33 We can also put I e ¼ fn 2 x: for some m 2 x; n ¼ fegðmÞg.

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36 The Ie are the recursively enumerable sets. Clearly for all e, Ie  x. As there is only a countably infinity of 37 such Ie, we may consider sets E so that E 62 {Ie : e 2 x}. 38 1.1. Exploring extensions

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51 Which is the class of the sets Ej given by n 2 Ej M $m(n = fj(m), where fj is explicitly given by an expression with real-valued and real-defined polynomials, sines, exponentials, j. . .j, plus infinite sums, derivatives an integrals—that is, operations to be found in the toolbox of classical elementary analysis? That class includes, as we will see, explicit expressions within the language of classical elementary analysis for characteristic functions of subsets of x in the complete arithmetic degrees 0, 0 0 , 000 , . . .. Now we may ask a further question: is there a physical, real-world device that would actually compute those characteristic functions? Scarpellini [19,20] wondered about its realizability some 40 years ago. Of course we will probably never be able to fully make a concrete counterpart of the ideal objects we will discuss in the next section, as much as we cannot fully realize in the concrete world a counterpart of a general Turing machine, with its potentially unbounded memory. Yet a question remains: how far can we go in the concrete realizations of the formal hypercomputing objects we introduce here?

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52 53

Can we extend those ideas? Say: let us define some set of functions fj ; j 2 I, from x to x, where I is some indexing set. Can we find a set of reasonably intuitive, more involved, elementary operations, so that each fj splits up into those elementary operations? Besides the trivial answer (take the fj as the ‘‘bricks’’ one is looking for, the question we ask here is in fact—can we usefully generalize the concept of computability? We must restrict our quest, as the above question is too general. We may for instance just consider arithmetical subsets of the natural numbers [18]. Our characterization of the recursively enumerable sets Ie uses a polynomial over the natural numbers, that is, sums and products of natural numbers. What happens if we allow for series? What happens if we somehow extend our admissible operations to encompass a few simple real-defined and real-valued functions such as again polynomials, sines, exponentials? If we allow for operations like the positive value jxj of x, derivatives or integrals? More precisely

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63 2. Expressions for the Halting Function and beyond 64

The present section follows published results by the authors [3,4,6,7].

65 2.1. An informal discussion

66 We start from a very informal discussion; one must always keep in mind that it is here given as a rather naı¨ 67 ve but nevertheless suggestive, starting point. 68 69 70 71 72 73 74 75

Remark 2.1. Go to the blackboard; pick up a piece of chalk, draw a circle on the blackboard. Then mark a point within the circle, and another one outside it. Join both by a continuous line. You will immediately see that this line crosses the circle. This is the intuition behind the Jordan Curve Theorem, a notoriously tricky theorem when we have to prove it. Place your 6-month old child (or grandchild. . .) within a childÕs playpen. After a few minutes he or she will get bored and start crying, asking to be taken out of the pen, whose boundaries restrain the babyÕs movements. So, the essential content of the Jordan Curve Theorem—shall we say intuitively? naı¨vely?—appears to be known even to a baby, while again very few adults can fathom its proof.

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Take elementary Euclidean plane geometry, and consider two open-ended line segments within a given square. Can we check whether the two segments do intersect? That again appears to be a trivial question—but the corresponding mathematical problem is (in a sense made precise below) algorithmically unsolvable: namely, given the equations for a straight line and for a curve described by elementary functions with rational coefficients and p, we cannot decide in general whether the curve and the straight line do intersect. A simple map then transfers that result to the interior of a square.

84 2.2. A Turing machine with an analog oracle 85

We will consider in this section an ideal device composed of

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82 Those examples suggest, in a naı¨ve way, that perhaps our brain operates with nonrecursive processes. If 83 that is the case, hypercomputation theories are not just abstract exercises.

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86 • A Turing machine with an oracle. 87 • The oracle is an analog oracle that decides, given two smooth curve segments without endpoints (homeo88 morphic to an open segment) within a square on the plane, whether they have a common point. 89 90 The oracle above described amounts as we will see to an ideal device that settles the Halting Function. We 91 will elaborate on that assertion. The key idea is, there are infinitely many expressions for the Halting Function, 92 the function that settles the Halting Problem, within some simple formal languages that extend arithmetic. 93 (For a review of our results with full references see [6]. The whole constructions will appear in [7].) 94 We have to move to a rather technical presentation now. 95 2.3. RichardsonÕs map

Refer to our definition of a Turing machine (Definition 1.1).

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97 Proposition 2.2. If {e}(a) = b, for natural numbers a, b, then we can algorithmically construct a polynomial pe 98 over the natural numbers so that [{e}(a) = b] M [$x1, x2, . . . , xk 2 xpe(a, b, x1, x2, . . . , xk) = 0]. 99

For the proof see [8]. Follows:

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100 Proposition 2.3. a 2 Re, where Re is a recursively enumerable set, if and only if there are e and p so that 101 $x1, x2, . . . , xk 2 x[pe(a, x1, x2, . . . , xk) = 0]. 102 Proposition 2.3 is the main content of the solution given by Matyasevich, Davis and Robinson for HilbertÕs 103 10th Problem—a set of Diophantine equations with parameter a, pe(a, x1, . . .) = 0 has solutions if and only if 104 a 2 Re, Re a recursively enumerable set. This result implies the following well-known negative result: in the 105 general case, there is no algorithm to decide whether a Diophantine equation has roots or not. 106 2.4. RichardsonÕs map, multidimensional version 107 Let x be the set of natural numbers, as above, and let A be the real-defined and real-valued algebra of 108 polynomials, trigonometric functions, and exponentials, closed under sum, product and function composition; 109 we add to A the number p and close. 110 Proposition 2.4 (RichardsonÕs Map, I). There is an injection jP : P ! A, where P denotes the algebra of 111 x-defined and x-valued polynomials in a finite number of variables, and A is the algebra of functions described 112 above, such that: 113 1. jP is constructive, that is, given the expression for p in arithmetic, there is an effective procedure so that we can 114 obtain the corresponding expression for F = jP(p) in A. 115 2. jP is 1–1.

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116 3. For ~ x ¼ ðx1 ; . . . ; xn Þ, 9~ x 2 xn pðm;~ xÞ ¼ 0 if and only if 9~ x 2 Rn F ðm;~ xÞ ¼ 0 if and only if 9~ x 2 Rn F ðm;~ xÞ 6 1, 117 for p 2 P and F 2 A. 118 4. The injection jP is proper. 119

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120 The crucial property is given in step 3.: it allows us to translate the existence of roots for Diophantine equa121 tions into roots of the corresponding transformed real-defined and real-valued function, with some extras. 122 Notice that this construction can be fully formalized within Zermelo–Fraenkel set theory ZF (with or without 123 the Axiom of Choice); ZFC is the full set theory plus the Axiom of Choice. 124 Remark 2.5. The map from Diophantine polynomials into the algebra of elementary real-valued and real125 defined functions is given by the following construction. Given " # i¼n X 4 2 4 f ðm; x1 ; . . . ; xn Þ ¼ ðn þ 1Þ p2 ðm; x1 ; . . . ; xn Þ þ ðsin pxi Þk i ðm; x1 ; . . . ; xn Þ ; i¼1

k i ðm; x1 ; . . . ; xn Þ >

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128 where p(m, x1, . . .) is a Diophantine polynomial and (for p extended over the reals); ki satisfies o 2 ðp ðm; x1 þ Di ; . . . ; xn þ Dn ÞÞ; oxi

131 with jDij 6 1; an expression for ki can be explicitly constructed—see the references. Then put

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F ðm; x1 ; . . . ; xn Þ ¼ f ðm; x21 ; . . . ; x2n Þ

135 2.5. RichardsonÕs map: one-dimensional version

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134 and we have our desired jP transform.

138 Remark 2.6. We define: • hðxÞ ¼ x sin x. • gðxÞ ¼ x sin x3 .

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136 We can obtain a one-dimensional version of the preceding results. Here we only sketch the main ideas, leav137 ing the details for the references.

141 Given F(m, x1, . . . , xn), we make the following substitutions: 142 143 144 145 146 147

• • • • • •

x1 = h(x). x2 = h  g(x). x .. 3 = h  g  g(x). . xn1 = h  g   g(x). Here g is composed n  2 times. And xn = g  g   g(x). Here g is composed n times.

148 149 The result of those substitutions is G(m, x), a function parametrized by m and defined over R with values in 150 R, where as usual m 2 x. 151

Now put Lðm; xÞ ¼ Gðm; xÞ  12. Then

152 Proposition 2.7 (RichardsonÕs Map, II). Let A1 be the algebra defined above and restricted to a single real 153 variable x. Then there is a map j0 : P ! A1 such that 154 1. j 0 is constructive. 155 2. j 0 is 1  1.

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156 j0 ðP Þ  A1 is proper. 158 3. The inclusion n 157 4. 9~ x 2 x pðm;~ xÞ ¼ 0 if and only if $x 2 R L(m, x) = 0 if and only if $x 2 RG(m, x) 6 1. 159 Remark 2.8. The preceding theorem implies the claim made at the opening of this section: given two smooth curves on the plane, there is no general algorithm to decide whether they cross or not; a compactification that maps those curves within the interior of a square leads to our claim—there is no general algorithm to check whether any two curve segments without endpoints within a square on the plane do cross or not.

168 169 170 171

Remark 2.9. LetÕs go back to the Turing machine picture for a moment. Let Mm(a)# mean: ‘‘Turing machine of Go¨del number m stops over input a and gives some output.’’ (We can also take {m}(a)# to mean that {m} is defined at a). Similarly Mm(a)" means, ‘‘Turing machine of Go¨del number m enters an infinite loop over input a’’. (Or {m}(a)" means that {m} is undefined at a). Then we can define the Halting Function h

2.6. The Halting Function

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• h(m, a) = 1 if and only if Mm(a)#. • h(m, a) = 0 if and only if Mm(a)".

174 h(m, a) is the Halting Function for Mm over input a.

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175 Since the Halting Problem is not algorithmically solvable, we know that h is not a Turing-computable func176 tion. The idea is to obtain an explicit expression for h in the language of classical analysis. 177 Then, if r is the sign function, r(±x) = ±1 and r(0) = 0.

hðn; qÞ ¼ rðGn;q Þ; Z þ1 2 C n;q ðxÞex dx; Gn;q ¼ 1

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178 Proposition 2.10 (The Halting Function). The Halting Function h(n, q) is explicitly given by

C m;q ðxÞ ¼ jF m;q ðxÞ  1j  ðF m;q ðxÞ  1Þ. F n;q ðxÞ ¼ jP pn;q .

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181 Here pn,q is the two-parameter universal Diophantine polynomial pðhn; qi; x1 ; x2 ; . . . ; xr Þ

184 and jP is as in Proposition 2.4. 185 186 187 188 189 190 191 192 193

Remark 2.11. Notice that there is also an expression for the Halting Function within a theory that includes arithmetic plus some definition for infinite series. Let p(n, x) be a one-parameter universal polynomial; x abbreviates x1, . . . , xp. Then either p2(n, x) P 1, for all x 2 xp, or there are x in xp such that p2(n, x) = 0 sometimes. As r(x) when restricted to x is primitive recursive, we may define a function w(n, x) = 1  rp2(n, x) such that • Either for all x 2 xp, w(n, x) = 0. • Or there are x 2 xp so that w(n, x) = 1 sometimes. Thus the Halting Function can be represented as " # X wðn; xÞ hðnÞ ¼ r ; sq ðxÞ! sq ðxÞ

196 where sq(x) denotes the positive integer given out of x by the pairing function s: if sq maps q-tuples of positive 197 integers onto single positive integers, sq+1 = s(x, sq(x)). 198 The infinite sum makes the difference.

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200 199 201 2.7. Undecidability and incompleteness 202 Definition 2.12. A theory T that includes formalized arithmetic is arithmetically sound if T has a model where 203 arithmetic is standard.

207 Proposition 2.13. If T is arithmetically sound, then

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204 T may be, say, ZFC (or adequate extensions), or a fragment of it that includes the main tools of classical 205 elementary analysis. Let P(x) be a one-variable formula in the language of T so that there are adequately 206 defined terms h, h 0 , defined for example by using the description symbol, with T ‘ P(h) and T ‘ :P ðh0 Þ. Then

208 1. There is a term h so that neither T 0:P ðhÞ nor T 0 P(h), but N  P(h), where N makes T arithmetically sound. 209 2. There is an infinite denumerable set of defined terms hm, m 2 x, such that there is no general decision procedure 211 210 to ascertain, for an arbitrary m, whether P(hm) or :P ðhm Þ is provable in T. 212 For the proof see [4,6,7]. However we can go beyond that.

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214 2.8. Characteristic functions for higher complete arithmetic degrees

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215 The idea behind the next theorem is simple: we construct a Turing machine with an oracle given by h(m) [3], and 216 out of the modified Diophantine polynomial we construct corresponding versions of the h function. We thus get a 217 h00 function that is a characteristic function for a set of complete degree 000 , and so on. The general result is

Given those h(p), we can even go beyond them and obtain explicit expressions for a characteristic function h for a subset in degree 0x. This concludes the claim made in the introduction. A final remark. Consider Proposition 2.7. From what we have presented here, it is easily seen that our informal assertion made at the beginning of this section, that an oracle that settles whether given two smooth curves with open endpoints on the plane, they have a common point, also settles something that cannot be algorithmically decided within the language of classical analysis. Even a simple question like—does an equation f(x) = 0 built out of elementary functions have roots?—is algorithmically unsolvable. (x)

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227 2.9. A conjecture

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218 Proposition 2.14. For all p 2 x expressions h(p)(m) can be explicitly constructed for characteristic functions of 219 sets in the complete degrees 0(p).

228 The idea implicit in the preceding results is: can we use elementary functions in classical analysis as building 229 bricks for an useful, implementable, extended computation theory? We therefore formulate the following 230 admittedly risky conjecture: 232

231 Every mathematical operation up to the level of elementary calculus can be ‘‘usefully’’ implemented and computed by some computing device, at least in an approximate way.

233 234 235 236 237 238 239 240 241

(The assumption that lies in the background of the conjecture is: mathematics somehow mirrors what goes on in the world, and therefore mathematical procedures can be simulated by some concrete process. The gist of our conjecture is: how close to truly useful can we get our ‘‘useful’’?) If that holds, we are done. This idea is an old one; to our knowledge it goes back to Scarpellini in 1962 [19,20], and has been explored in several ways (cf. [14]) before our 1990 work (published in 1991 [2,22]). Can it ever be implemented? The whole point has to do with the imprecisions of analog computers—but anyway when we see two lines on the blackboard, we immediately see whether they cross or not. Is that a pointer towards the feasibility of our oracle? We may of course weaken it and say Every mathematical operation up to the level of elementary calculus can be physically implemented in an approximate but useful way and computed by some computing device in order to give some nontrivial information.

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The qualifications weaken it and make it more plausible. Say, how about the physical implementation of an analog device that computes instances of h(m) up to some finite but large m? For we are only interested (in practical situations) with finite, even if large, specific instances of the halting problem. In order to get an affirmative answer for the Halting Problem—Mm(n) halts—one only needs a computation that performs a finite number of steps. Would an actual, concrete, analog oracle be able to give us the negative answers?

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251 3. Further remarks

There is a well-known relation between undecidability and incompleteness. The Davis [8] version of Go¨delÕs first incompleteness theorem (out of an idea that goes back to Post in 1944) proves that theorem out of the nonexistence of an algorithm to settle HilbertÕs 10th Problem. Can we turn the tables, and look at undecidability from the viewpoint of incompleteness? Let us consider the following question: can we algorithmically check whether the usual formal sentence Consis(PA) that asserts the consistency of Peano Arithmetic is true? Of course that cannot be done within Peano Arithmetic itself. However consider the following question:

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Suppose that we give, for each natural number n, a finite set Sn. Suppose that, again for each n, we hand out an algorithm to compute the elements of Sn. Consider the function F(n) = maxSn + 1. Is it a total function? Can we compute it, for an arbitrary value?

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Most mathematicians would say that, once we have a recipe for the computation of any Sn, then we can compute its maximum value, and add one and get a value for F(n). Is that so, always? (On this discussion see [16].) As it is well known, the function usually noted F0 cannot be proved to be total within Peano Arithmetic, even if it is naı¨vely and intuitively total, and can be given an explicit algorithm (it is of very large computational complexity). The theory PA þ ½F0 is total is very strong, as it proves Consis(PA). There is a beautiful algorithmic implementation of a related computation by Kunen [15]. Kunen uses primitive recursive arithmetic (but for a single step) to build a program that proves the Paris–Harrington theorem. His construction is clear cut, and we can actually see where is the step that corresponds to GentzenÕs transfinite induction, a step that can be informally implemented in KunenÕs algorithm for the proof of Paris–Harrington, but which does not fit within Peano Arithmetic (please see the reference for details). Kunen notices in his paper [15] that one could in fact extend his algorithm to one that proves Consis(PA).

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274 3.1. The Turing–Feferman theorem: very brief remarks 275 Can we generalize that construction? One possible, even if controversial, road to follow would be some276 thing like the Turing–Feferman theorem. Notice that each failed instance—negative answer—of the Halting 277 Problem can be formalized as a P1 sentence: there is a Diophantine polynomial p so that, given partial recur278 sive function {m} and input n, {m}(n) is undefined (does not halt, in the Turing machine picture) if and only if 8x1 ; x2 ; . . . ; xk pðhm; ni; x1 ; . . . ; xk Þ > 0 281 282 283 284 285 286

is true of the standard integers. The present discussion is informal and sketchy. However we believe that it gives an idea of the subject matter. The Turing–Feferman theorem is an old conundrum (for references and a summary see [1,12,13]). There are several technical difficulties and subtleties to be considered, but, roughly, the idea of the theorem (in TuringÕs version) is that there is a sequence of theories that include PA ([13, p. 381]), T ¼ T 0 ; T 1 ; T 2 ; . . . ; T x ; T xþ1 ;

289 (Here, T1 = T0 + Consis(T0), and in general Tj+1 = Tj + Consis(Tj), where Consis(Tj) is the usual sentence 290 that asserts the consistency of theory Tj)—so that for some k, Tk proves a given P1 sentence /. Therefore, 291 with some machinery we can show that all instances of the Halting Problem, each one given by a P1 sentence,

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will be decided within that sequence of theories by some specific Tk; see [1]. (Actually, PA itself proves infinitely many P1 sentences which correspond to instances of the Halting Problem.) For the sake of progressions as above we will just replace PA for T. The question is: can we turn this result into some reasonable procedure, that will end up by deciding each particular instance of the Halting Problem, even if the full procedure has a possibly unbounded computational complexity? We ask that question in the light of the following example. Let us for a moment go back to theories T which include arithmetic, have a recursively enumerable set of theorems, and are arithmetically sound; for the formal definition of R1-soundness as a reflection principle in the sense of Feferman see [1,12,13]. Then

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301 Definition 3.1. A bounding total recursive function FT over T is a recursive function which is diagonal over all 302 T-provably total recursive functions. 303

The following results are well-known [1]:

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Proposition 3.2 306 1. T ‘ [FT is total] M [T is R1-sound]. 308 307 2. T ‘ [T is R1-sound] ! Consis(T). 309

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310 Assertion 1 can be given an elegant proof [1]; for Peano Arithmetic, there is a proof in [17]. Assertion 2 in 311 Proposition 3.2 means that T cannot prove that FT is total. However it is naı¨vely obvious that FT is total, or, 312 more precisely, it holds of the standard model for arithmetic that FT is total. 313 Now the sequence of theories:

316 implies, due to Proposition 3.2, the sequence, T 0 ; T 1 ¼ T 0 þ ConsisðT 0 Þ; . . . ; T xþ1 .

328 329

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Put T0 as PA, Peano Arithmetic. Suppose that we are satisfied that PA is consistent—actually, it is enough to naı¨vely check KunenÕs [15] algorithm for Paris–Harrington, and deduce Consis(PA). Follows the consistency of any finite segment of that sequence. In order to decide a particular negative instance of the Halting Problem, say, is {m}(n) undefined?—one must go up in that sequence just a finite number of steps. Can we algorithmically deal with it, even if only in an informal way, through some informal procedure? A reasonable kind of (extra, hyper) computational procedure can perhaps be implemented if we get an affirmative answer to the following question:

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T 00 ¼ T ; T 01 ¼ T 00 þ ½T 00 is R1 -sound; T 02 ; . . . ; T 0x ; T 0xþ1

327 Can we construct a recursively enumerable sequence T of strictly increasing total recursive functions so that, given any strictly increasing total recursive function G there is a function Fj in the sequence so that Fj dominates G?

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Given some theory T 0j in the sequence T 00 ; T 01 ; . . . ; T j ; . . . ;

333 there will always be some function Fj 2 T so that Fj dominates all provably total recursive functions in 334 T 0j þ ½T 0j is R1 -sound. Therefore T 0j þ ½Fj is total proves Consis(Tj) and will therefore prove all P1 sentences 335 that are proved by Tj+1 = Tj + Consis(Tj), including specific instances of the Halting Problem. 336 We can wonder whether one could use them to implement some kind of hypercomputational procedure. 337 338 339 340 341

Remark 3.3. We wish to stress the following point: the present discussion cannot be formalized within any theory T as characterized above: one that is consistent, includes PA, is arithmetically sound, and has a recursively enumerable set of theorems, since there is a bound to the growth speed of total recursive functions in those theories [1,5]. The same is true even if we substitute ZFC, Zermelo–Fraenkel set theory plus the Axiom of Choice, for PA.

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342 The argument we have just sketched has a strong appeal, but it depends on the construction of a feasible T. 344 343 For a rigorous criticism see [13]. 345 4. What do we lose with hypercomputation?

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One usually considers the difficulties concerning hypercomputation from the side of its practical implementation. But let us consider a different side of the question here: if we can plug into a Turing machine an oracle that settles the Halting Problem, we can decide the truth of the whole arithmetic hierarchy. That means: we verify the truth in the standard model of sentences in the arithmetic hierarchy. More precisely: if you have a device that settles the Halting Problem [2,3,6], you so to say reach degree 0 0 of the arithmetic hierarchy. Given knowledge of subsets of x in that degree as an oracle, you can reach 000 , and so on [6,7]. Therefore hypercomputation allows us to decide sentences along the arithmetic hierarchy—with respect to the standard model of arithmetic. This may be seen as impoverishing mathematics. Untrue theories, such as ZFC þ :ConsisðZFCÞ, where Consis(ZFC) is the usual formal sentence that asserts the consistency of ZFC, are apparently outside the scope of hypercomputation, essentially because they need a nonstandard kind of arithmetic in their models. However one may object that a theory like ZFC þ :ConsisðZFCÞ is too abstract and too far from everyday considerations. But we can give here an example of a theory that refers to a concrete problem and that may require nonstandard models of arithmetic for its interpretation. It is quite simple: there is a Diophantine equation p(m0, x1, . . . , xk) = 0 so that it is true of the standard model for arithmetic that it has no solutions, while that fact [8] can neither be proved nor disproved within ZFC. Yet there is a model for ZFC with nonstandard arithmetic where that equation does have solutions. That means: there will be a corresponding Turing machine that does not stop over m0 for the standard integers, but does stop over m0 for nonstandard integers. Can we give some concrete meaning for that result?

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365 4.1. A note on nonstandard models and P vs. NP

• Say, in the work by DeMillo and Lipton [10,11] where these authors prove the consistency of P = NP with fragments of arithmetic with the help of, among other tools, nonstandard models for the theories they handle. • This turns out to be the case of our recent paper [5], where we have that (see the reference) if ZFC + what we have called the exotic formulation [P = NP]F for the P = NP hypothesis, is an x-consistent theory, then ZFC + [P = NP] is consistent.

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366 We notice that the importance of nonstandard models for the P vs. NP question and for computer science 367 in general has already been pointed out in several opportunities

We will have to consider nonstandard models to make sure that the x-consistency hypothesis holds; see on that the sketch in [6, Remark 2, p. 22]. Can we give some concrete, say, ‘‘computably useful,’’ meaning for those models? (Quotation marks required here, of course). These papers give a nontrivial example of the importance of nonstandard models in the mathematics of computer science. So, the point is: we may get a lot about the standard model of arithmetic out of hypercomputation, but we may possibly lose many interesting results that depend on dealing with nonstandard models for arithmetic.

382 5. Uncited reference 383

[9]

384 Acknowledgements 385 The authors wish to thank the Institute for Advanced Studies at the University of Sa˜o Paulo, as well as its 386 Director Professor J. Steiner for support of this ongoing research project. F.A.D. also thanks the Graduate

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Studies Program at the School of Communications, Federal University at Rio de Janeiro. Finally, portions of this work were done during the COBERA March 2005 Workshop at Galway, Ireland; we wish to thank Professor Vela Velupillai for the stimulating and fruitful environment he so kindly sponsored at that meeting. We finally heartily thank E. Bir who uncovered for us several hard-to-find bibliographical items.

391 References

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[1] L. Beklemishev, Provability and reflection, Lecture Notes for ESSLLI’ 97 (1997). [2] N.C.A. da Costa, F.A. Doria, Undecidability and incompleteness in classical mechanics, Int. J. Theor. Phys. 30 (1991) 1041–1073. [3] N.C.A. da Costa, F.A. Doria, Suppes predicates and the construction of unsolvable problems in the axiomatized sciences, in: P. Humphreys (Ed.), Patrick Suppes, Scientific Philosopher, II, Kluwer, 1994, pp. 151–191. [4] N.C.A. da Costa, F.A. Doria, Variations on an original theme, in: J. Casti, A. Karlqvist (Eds.), Boundaries and Barriers, AddisonWesley, 1996. [5] N.C.A. da Costa, F.A. Doria, Consequences of an exotic formulation for P = NP, Applied Mathematics and Computation 145 (2003) 655–665, also Erratum, Applied Mathematics and Computation, in press. [6] N.C.A. da Costa, F.A. Doria, Computing the future, in: K. Vela Velupillai (Ed.), Computability, Complexity and Constructivity in Economic Analysis, Blackwell, 2005. [7] N.C.A. da Costa, F.A. Doria, Metamathematics of Science, in press. [8] M. Davis, Hilberts Tenth Problem is unsolvable, Computability and Unsolvability, Dover, 1982. [9] M. Davis, Introduction, Applied Mathematics and Computation, this issue. [10] R.A. DeMillo, R.J. Lipton, Some connections between computational complexity theory and mathematical logic, Proc. 12th Ann. ACM Symp. Theory Comput. (1979) 153–159. [11] R.A. DeMillo, R.J. Lipton, The consistency of P = NP and related problems with fragments of number theory, Proc. 12th Ann. ACM Symp. Theory Comput. (1980) 45–57. [12] S. Feferman, Transfinite recursive progressions of axiomatic theories, J. Symbolic Logic 27 (1962) 259–316. [13] T. Franzen, Transfinite progressions: a second look at completeness, Bull. Symbolic Logic 10 (2004) 367–389. [14] G. Kreisel, A notion of mechanistic theory, in: P. Suppes (Ed.), Logic and Probability in Quantum Mechanics, D. Reidel, 1976. [15] K. Kunen, A Ramsey theorem in Boyer–Moore logic, J. Automated Reasoning 15 (1995) 217. [16] G. Longo, On the proofs of some formally unprovable propositions and prototype proofs in type theory. Lecture Notes in Computer Science, 2277, Springer, 2002, p. 160. [17] J. Paris, L. Harrington, A mathematical incompleteness in Peano Arithmetic, in: J. Barwise (Ed.), Handbook of Mathematical Logic, Springer, 1989. [18] H. Rogers Jr., Theory of recursive functions and of effective computability, reprint, MIT Press, 1992. [19] B. Scarpellini, Two undecidable problems of analysis, Minds Mach. 13 (2003) 49–77. [20] B. Scarpellini, Comments to Ôtwo undecidable problems of analysisÕ, Minds Mach. 13 (2003) 79–85. [21] C. Smory´nski, Logical Number Theory, I, Springer, 1991. [22] I. Stewart, Deciding the undecidable, Nature 352 (1991) 664–665.

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Some thoughts on hypercomputation q

18 However it is more convenient for us to give an alternative definition for the ... q N.C.A. da Costa is supported in part by CNPq, Philosophy Section. ..... 247 practical situations) with finite, even if large, specific instances of the halting problem.

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