Pudlák, Pavel

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Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaePavel PudlákA definition of exponentiation by a bounded arithmetical formulaCommentationes Mathematicae Universitatis Carolinae, Vol. 24 (1983), No. 4, 667--671Persistent URL: http://dml.cz/dmlcz/106264Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1983Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czCOMMENTATIONES MATHEMATICAE UNIVERSITAHS CAROLINAE 24,4 (1983) A DEFINITION OF EXPONENTIATION BY A BOUNDED ARITHMETICAL FORMULA Pavel PUDLAK Abstract: A new definition of exponentiation by a bounded arithmetical formula is presented. Key words: Exponentiationf bounded arithmetical formula. Classification: 03-?30 It was shown by Godel that the relation z - X* can be de-fined using only + and • . In complexity theory and also for in-vestigations of weak fragments of arithmetic it is very useful to have definitions by bounded arithmetical formulae, i.e. arit-hmetical formulae in which every quantification is of the form 3 x £ y or Vx_ryf where y is some free variable of the formula. Once we have a definition of exponentiation by a bounded formu-la, we can easily construct such definitions for a variety of other concepts. The first definition of z « x^ by a bounded formula was found by Bennet H 3 , another formula was construc-ted by Paris, see [21. Since so far there is no known simple definition of exponentiation, by a bounded formula, it seems reasonable to look for alternative definitions- Another such definition in prepented here* Let F(x), x|y be the obvious bounded formulae defining "x is a prime", "x divides yu resp. We shall also use the ab-breviation {WX cp(x) for a bounded formula gp(x) &. V y < x i<p(y). It is enough to define a a r 7 only for x prime since then we obtain a general definition by (x% Vpfk(P(x)&pk|x~* p ^ U ) . Also we can restrict ourselves to sufficiently large exponents, since for a fixed exponent the definition is trivial, fhe for-mula is constructed in three steps? 1. PP(p,a) -*—v ^ Yq-£a (q l a —> q » 1 v p | q ) 2 . E(p f k f a f b) <----> d f ^ b ( P P ( p , a ) & P P ( p > b ) f c . ( 3 u ^ b ) ( b » (a -1) . [ (a -1)^ t t+k]+D) 3 . Exp (p f k f e) -*—» ^ ( 3 r a f n f a f b f c f d , £% ) ( e » b«o & &k » n + m 2 &n.*2.m<a - 1 & E(p f m f a f c)&E(p f m f p.a f a . c ) &. E ( p f n f a f d ) & E ( p f n f p * a t b * d ) ) . Theorem. For p prime, k > 9 and e a r b i t r a r y , Exp(p fk fe) i f f p k «e . Proof: Let p be a prime number. 1. Clearly PP(pfa) iff a is a power of p. 2. If a and b are some powers of p then b is a power of a iff a-l|b-1f just imagine the p-adic representations of af bf a-1, b-1. Therefore if E(pfkfafb) then b is a power of a* We shall show that for k< a-1f E(pfkfafb) iff ak » b. By the remark above it is enough to consider only b s which are powers of a. Let b » am. Using binomial expansion of ((a-1)+1)m we can represent b uniauely as - 668 -b - (a-1) [ (a-1) u + n.3 + 1 f with n < a ~ U Moreover we have n-=rm mod (a-1) t Hence a i s the smallest power of a such that ak m (a-1) I(a-1) u+k] + 1. 3. Suppose Exp(pfkfe) holds true. Then using 2 we can replace the last four clauses by am - cf (p.a)m a a,cf &n * d, (p*a)n - b.d. 2 The first two of them imply a - pmf hence c - pm ; the last two imply b - pnf Hence e * b.o - p n . pra2 - p n + m 2 - pk, 2 low suppose p - e, Thenf clearly, e » pn . pm for some mf n such that k » n + m and n_?2m. To obtain Exp(pfkfe) one needs only to check that 2m < a - 1f where a - pmt which is true if n»>3t hence if k>9. The largest of the quantified numbers is 2 a «n _mn * _2m ^ A2 r\ x? T% a » a * p f p ^ e .» Q. £.. I). The proof above was done in the standard model of arithme-tic. If we use such a formula in a weak fragment of arithmetic, we would like to be able to prove that it defines exponentiati-on there. But what does it mean? A natural formalization of this requirement is that the theory proves the inductive conditions: 1 m X°f Z « X^—> X • Z » x7 • It has been conjectured that for some of the definitions of ex-ponentiation by a bounded formula the conditions are provable using only bounded induction. The formula presented here may seem not to have this property, since in the critical part of the proof we referred to p-adic representation and binomial ex-pansion. However, this was done only for the sake of simplicity of the proof, and we conjecture that our formula is a right one. - 669 -It would be a really tedious work to check it formally, the-refore we consider the inductive conditions only for the for-mula E(p9k9a9b). First we need the following lemma. Lemma. c> 1 &c|a.b —• B BIB >1 & s|c &(s|avslb)3 is pro-vable using only bounded induction. Proofs Let A(a9b9c) denote the formula above. Using in-duction we prove Vn B(n) where B(n) denotes the following bounded formula Va9b9o ta.b+c^n— .»A(a9b fc)3. B(0) is trivial. Assume B(n) and let a.b+c^n+19 o 1 9 cla.b. 1. If c^a then c|(a-c).b< a.b9 hence by the induction assump-tion there exists s>1 9 sic such that al(a-c) or sib. Thus also s|a or sib. 2. The case of o^ b is symmetrical. 3« Suppose o a and c>b« Let d be such that c.d • a.b9 then d*-c. By the induction assumption there exists s > 19 s Id such that s'la or s'lb. Because of symmetry we can investigate only the case of s'la. Let a 9 d be such that a .a' » a9 ft*.s'* d. Then c.d* » a .b<a.b9 and we c *n use the induction assumption again to obtain s>1 9 sic such that els' or sib, whence sla or sib , Q.E.D. Proposition. P(p) &E(p9k9a9b) —> E(p9k+1 ,a,a.b) Is pro-vable using only bounded induction. Proofs Assume the antecedent of the implication, i.e. ?J(p,a)9 PP(p9b) aid b is minimal such that b - (a-i). Ua-O.u+kD + 19 for some u. Then a.b -= (a-1). [{a-1).(a.u+k)+k+1.l +1 f hen^t »«b has the required form. If o > 1 and cla.b then, by - 670 -Lemma, for some s > 1 , sic &(s(a velb). Sinoe PP(pfa) and P-?(pt*)» *• a*ve pis, hence pic. Thus we have proved PP(pfa.b). It remains to prove the minima-lity of a.b . Suppose c<a.b has these properties. Then o£a and since PP(p,c) we have alo, (consider g.c.d. of a and c). Let o • a.d, thus (*) a.d m (a-1). [(a-1).v+k+1] +1 m a2.v+a.(k+1-2v)+v-kf for some v. Hence v-k * t.a for some t. If we substitute v • m t.a+k in (*) and divide it by af we get, after some computa-tion, d - (a-1). C(a-1).t+k] +1. Also PP(pfd) can be shown easily. But a.d -= c«ra.b, so d<b. Thus b is not minimal - contradiction. This proves that a.b is minimal. Q.E.D. R e f e r e n c e s 11] BENNETT, J.H.: On Spectra, Ph.D. dissertation, Princeton University 1962. 12] GAIPMAN, H. and DIBHTRACOPOLOUS C : Fragments of Peano's arithmetic and the MRDP Theorem, in Logic and Al-gorithmic, Geneve 1982, 187-206. Matematicky ustav ?SA7, fitnd 25, Praha 1, Czechoslovakia. (Oblatum 20.9. 1983) o?i 

A definition of exponentiation by a bounded arithmetical formula

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A definition of exponentiation by a bounded arithmetical formula

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Institute of Mathematics AS CR, v. v. i.