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Conference in Bedlewo (Poland) 21th of july – 27th of july 2012 In memoriam of professors

Conference in Bedlewo (Poland) 21th of july – 27th of july 2012 In memoriam of professors Henryk Kotlarski and Zygmunt Ratajczyk. Nézondet ‘s p -destinies for theories with two and three quantifiers. The five missing numbers.

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Conference in Bedlewo (Poland) 21th of july – 27th of july 2012 In memoriam of professors

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  1. Conference in Bedlewo (Poland) 21th of july – 27th of july 2012 In memoriam of professors Henryk Kotlarski and Zygmunt Ratajczyk • Nézondet ‘s p-destinies for theories • with two and three quantifiers The five missing numbers Denis RICHARD, professeur émérite à l’Université d’Auvergne

  2. The notion of p-destiny is due to Francis Nézondet In his thesis (1997). Annie Château (2000) gave the best written survey of this thesis in 2000 and also wrote the first programm in CAML deciding Th2(N,S,⊥). A group including YIN Jilei (Fudan DaXue), GUILLAUME Marcel(Université Blaise Pascal) and DR (Université d’Auvergne) attacked in 2005 the problem of constructing the essential exhaustive tranverse destiny of Th3(N,S,⊥) which would provide an algorithm of decision and an axiomatisation of this theory. Therefore, there are 5 remaining cases. In each case it has to be construct a number or to prove it does not exist.

  3. 0. What are p-destinies ? 0.1 The (trivial) 2-destiny Th2(N,S,P). A p-destiny is a tree of height p of possibilities in a theory with p quantifiers and a language L of finitely many relations. Suppose p=2 and N is the set of nodes of the tree and take the predicates < and P for primeness. We intend to obtain a description of all possible (up to isomorphism) situations which can exist with two variables.

  4. Working with the tree of root 0 and denoting by a circle the prime numbers, we get the exhaustive 2-destiny of 0 inTh2(N,<,P) Which , up to isomorphism, produce the following essential 2-destiny of 0 in Th (N,<,P)

  5. Of course, the same is possible for any integer, giving the exhaustive simplified 2-destiny of Th2(N,<,P) :

  6. Just keeping one representative for each class of isomorphic trees, we can construct the so-call exhaustive essential 2-destiny ( also called essential tranverse 2- destiny) of Th2(N,<,P)

  7. With the essential tranverse 2-destiny, we have An algorithm of decision for Th2(N,<,P) : for Q1x1 Q2x2 F(x1, x2) The variable x1 takes its value(s) satisfying F(x1, x2) on the roots of the trees (all values if Q1 is universal and at least one value if Q1 is existential). The same for x2 but at the depth 2. The same essential tranverse 2-destiny provides an (trivial) axiomatisation (more than 22 axioms) of Th2(N,<,P) ∃x(∃y⏋(y>x)∧ ⏋(x>y))) ∃x(∃y⏋(P(y)∧ (x<y))) ∃x(∃y(P(y)∧ (x<y))) ∃x( ∀ y(⏋(y<x))) And so and …

  8. 0.2. Some definitions about p-destinies • Def. 0.2.1 (destiny). A p-destinyis a regulartree(X,P,r) of heightpwith a structuration mapc : • X is the set of nodes of the tree • Pis the fatherhood relation • Ris the root of the tree • cmaps a L-structure denotedc(n) to a noden, withdomain the branche [r,n] sothat if n1 and n2 are on a same branch, n1 being an ancestor of n2 thenc(n1) is a substructure of c(n2) • Def. 0.2.2 (M-completetree of heightp). We callM-completetree of heightpwithrootx the regulartree(LIST(M,p,x), P, (x)), where: • (LIST(M,p,x) is the set of lists of elements of M withlenghtat • mostp reaching the rootx • the root (x) is the listwith a unique element x • Pis the binary relation defined on the set of lists by x1P x2iff x1 • is the list x2withoutits first element.

  9. Def. 0.2.3 (induced canonical structuration) Let Mbe a L-structure and xϵ M. Wedefine a structuration mapc over (LIST(M,p,x),P,(x)) by the following condition: For everypredicateR of L witharityk , for all familiesx1 = (y1,…), x2 = (y2,…), …,(xk = (yk,…) of lists with respective first elements y1,y2,…,ykbelonging to the support of a same branch having z as a leaf of the tree (LIST(M,p,x),P,(x)) , we have the followingequivalence: c(z) satisfies R(x1, x2, …,xk) iff M satisfies R(y1, y2, …,yk) Def. 0.2.4 (exhaustive p-destiny) Let Mbe a L-structure p an integer and xϵ M. Wedefine the exhaustive p-destiny of x as being the p-destiny : (LIST(M,p,x),P,(x),c) where (LIST(M,p,x),P,(x)) is the M-completetreewithheightp and root x, and c is the structuration canonical mapinduced on thistree by M.

  10. Working with the tree of root 0 and denoting by a circle the prime numbers, we get the exhaustive 2-destiny of 0 in (N,<,P)

  11. A stalk(pétiolein french) is a subtree of a destinyequippedwith the structuration induced by the structuration of the wholedestiny Def. 0.2.5 (stalkisomorphism). An isomorphism of stalkbetweentwosubtrees (resp. twodestinies) is a bijection preserving the tree structure (i.e. the fatherhood relation) and being un isomorphism for the structuration map. Def. 0.2.6 (superimposition (or simplification) of rankk ) Given a p-destiny(A,P,r,c), wesay(A’,P,’r,’c’) is a superimposition of rankk ( 0<k<p ) of thisdestiny, iff for everynodex, we have chosen a reprentative for everyisomorphism class of stalks for the stalkshaving as roots the sons of x

  12. simplification of rank2 for destiniesassociated to the congruence modulo 3 (a cross meansthere no equalitymod 3)

  13. simplification of rank2 for destinies associated to the congruence modulo 3 (a cross meansthere no equalitymod 3)

  14. Def. 0.2.7 (simplication of rankkandgeneral simplification) Given a family of superimpositions over the set of p-destinies of a fixed language Sj , Sj ,…, Sp-1, each with a rank corresponding to its index, we call simplification at the k rank, the map Sj o Sj … o Sp-1 . We call general simplification a simplification at first rank. . Def. 0.2.8 (essential destiny). We call an essential p-destinyor an essential tranversep-destinyor a tranversep-destinyany range by a generalsimplication of ap-destiny 14

  15. Def. 0.2.8 (essential destiny). An essential p-destiny is the range of a general (of rankp) simplification of a p-destiny Transverse essential 2-destiny for the congruence modulo3

  16. 0.3 some reasons of becoming attached to p-destinies • This the best possibility I know for effectivelywriting a programmdeciding a theorywithfixednumber of quantifiers and finitelymany relations (FRT), • Destinies are good tools to test the truth in FRT to computers • Destiniesprovide an axiomatization for FRT, • Transform problems into the research of finitely many elements • determined by conditions, • In the case of arithmeticaltheories, construction of destiniesturns • intofindingintegersrealizingsome conditions or does not exist. • Destinies lead to the number-theoritical key-questions of a theory , and focus on the actual expressive power of a FRT • Use the powerfultechnic of FRAISSE-EHRENFURT back-and- • - forth in the steps of simplification • Could be useful in data basis

  17. Interlude : Why to choose relations S and ⊥ ?? Alan WOODS (1981) and DR (1982) gave independent proofs of undecidability of Th(N, S, ⊥) and Th(N, =, S, ⊥). Many interesting problems in number theory emerge from the thesis of Alan Woods . One is now known as Erdös-Woods conjecture: The following are equivalent: (i) z=xyis (S, ⊥)- definable in N (ii) z = x + y is (S, ⊥)- definable in N (iii) x ⩽ y is (S, ⊥)- definable in N (iv) x = y is (S, ⊥)- definable in N (v) (Erdös-Woods Conjecture) There is some k >0 such that every natural number x is determined uniquely by the sequence S0, S1,…, Skof sets of (distinct) prime numbers definedby Si= {p / p divides x + i} (vi) For any fixed prime p, the map n→pn is (S, ⊥) - definable in N.

  18. Importance of EWC. Let us call Supp(a) the set of prime (distinct) divisors of a. Consider the following conjectures of Number Theory : Oesterlé-Masser’s conjecture. (Also called a-b-c conjecture) For all (a,b) Î(N* )2 , there exists an effectively computable constant C such that : Supp((a + b)ab) > C[ (a+ b)/ gcd (a,b) ] 1-ε(a) with ε(a) tends to 0 when a tends to infinity. Hall’s conjecture . Suppose x3 ≠ y2, for an effectively computable constant C, we have │ x3 − y2│> [C. Max (x3 , y2)] 1/ 6 Hall and Schinzel ‘s conjectureSuppose xm ≠ yn, for an effectively computable constant C, we have │ xm− yn│> [Max (x3 , y2)] C M. Langevin results in [1988] the three previous conjectures are false in case EWC fails

  19. Moreover every conjecture which is false if Hall’s conjecture is false, becomes false in turn if EWC is. This is the case for Lang-Waldschmidt‘s conjecture(which provides lower bounds of linear forms of logarithms) and for Vojta ‘s conjectureabout abelian varieties. Obviously a positive solution to one of this conjecture would give a positive solution to EWC . More precisely Hall and Schinzel‘s conjecture implies that if a is sufficiently large, then a is determined by So ,S1 , …, S20 .

  20. 1) An algorithm of decision for Th2(N,S,⊥) The root x can be or not coprime with x cannot be in a relation of succession with x; for instance: For a leaf k, there are relations of k to k and of k to x, : so that we have a vector with 3 componenets as shown below ; the arrow from left to right indicates x=S(k) and from right to left that k=S(x).

  21. Construction of the exhaustive essential destiny of Th2(N,<,P) Destinies of 0,1, 2 and 3 are straightforward:

  22. When n>3, the essential 2-destiny of n is as follows:

  23. Now, it is easy to produce the essential exhaustive transverse destiny of Th2(N,S,⊥) which provides an algorithm of decision together with an axiomatisation of this theory:

  24. 2) Results and questions on Th3(N,S,⊥) As usual, le case with 3 is much more difficult than with 2 We denote Supp(n) the set of prime divisors of n. Below for 3-destiny, n is the root of the tree, k is a son of x and u a son of k 2.2) The nodes k and n are distant (S is not involved) Then, there are 5 basis cases associated to u which are : u = 0 is never coprime with an integer u = 1 is coprime with any integer uis not coprime to itself and coprime with k and n u is not coprime to itself and coprime with k but not with n u is not coprime to itself and coprime with n but not with k

  25. Therefore, there are at most 5 branches (n,k,u) for u far off from k and n for given k and n. The number of branches depends on four relative positions of Supp(k) and Supp(n) as follows: Supp(k) ⊂ Supp(n) and Supp(k) ≠ Supp(n) Supp(n) ⊂ Supp(k) and Supp(n) ≠ Supp(k) Supp(n) = Supp(k) Supp(n) \ Supp(k) ≠ ⍉ and Supp(k) \Supp(k) ≠ ⍉ In this situation 2.2, the 3-destinies are easy to construct (cz14)

  26. 2.3) The node u is close to k or to n when k and n are distant (S is involved) Because u ϵ { k-1, k+1, n-1, n+1} and k-1, k+1 have no successor link with n (similarly n-1, n+1 have no successor relation with k), it remains to discuss thecoprimeness relationship between k-1, k+1 (resp. n-1, n+1) with n (resp. k). For each case, there are 2 possibilities so that the number of possible cases is 16, giving the following set of configurations of branches (n,k,u) for u close to n or to k

  27. Counting. For each (n,k) with n far off k, there are 5 possibilities for the branches (n,k,u) when u is distant from k and from n, and there are 16 possibilities for the branches (n,k,u) when u is close to k or to n, so that we have 80 possible stalks (pétiole in french) for n distant from k. But each of these branches can or cannot appear in the destiny of a given n, so that, a priori , there are 280 trees in the essential tranverse destiny of Th3(N,S,⊥) Fortunately, we can enormously reduce the number of possible trees.

  28. 2.4) Branches (n,k) when k is close to n (S is involved) We begin by n>3(so that n-2 ≠0 and n-2 ≠1) . The discussion is rather technical depending on the parity of n, and on the combinatorial of Supp(n) and Supp(n-2) introducing the cases n=4, n=6 , all numbers of the form 2(2α -1) and 2(2α +1) After cleaning (n=4, …) the key-cases are the following : ⏋(n-3 ⊥ n) which appears iff n = 0 (mod 3) ; ⏋(n+1 ⊥ n-2) which appears iff n = 2 (mod 3); (n - 3 ⊥ n) and (n+1 ⊥ n-2) which appears iff n = 1 (mod 3), So that congruences modulo 3 appear in the construction This is the first time, we observe Th3(N,S,⊥)depends of exponents into the primary decomposition of integers and the following cases appear

  29. Stalk (n,n-2) : we know exactly when the cases a, b, c, d and e appear or not (case k=n-2). There are 9 possible cases (excepting n=4 and n=5

  30. Stalk (n,n-1) : There are only 2 possible cases : • (n is even and ⏋ (n-2 ⊥ n) (case a) or (n+1 ⊥ n-1)(case b) • (n is odd and (n-2 ⊥ n) (case a) or ⏋(n+1 ⊥ n-1) (case b)

  31. Stalk (n,n) : This is the unique possible case

  32. Stalk (n,n+1) : It is similar to the case k=n-1 • (n is even and (n-1⊥ n+1) (case a) or⏋(n+2 ⊥ n)(case b) • (n is odd and⏋(n-1 ⊥ n+1) (case a) or (n+2 ⊥ n) (case b)

  33. Stalk (n,n+2) : It is similar to the case k=n-2

  34. (Pathological) cases of Stalk(n,0) : here Supp(n) Supp(k) Stalk(n,1) : 2 possibilities according the parity of n Stalk(n,2) : Case a appears iff n=2α ; case b appears iff n is odd; Casec appears iff n=0(mod3); case d and e appear iff n is odd

  35. 2.3 Counting all possibilities Counting the cases when k is close to n and the destinies of 0,1,2, we find 14 cases Since 80 configurations was a majoration of the cases for k distant from n, we have at most 14×280 possible destinies. What is huge But in fact one can reduce to 646 cases allowing an exhaustive inspection by a computer : Annie château did it and proved the number of 3-destinies is at most 72.

  36. 2.4 –Five open number theory questions necessary to construct an essential transverse destiny of Th3(N, S, ⊥). In fact , surprisingly, S and ⊥ do express properties of the exponents of the prime factors of integers. It turns out that two sets are the keys of our construction: X = {nϵN /∃kϵN (Supp(k) ⫋Supp(n) ∧(k-1⊥ n)∧ ⏋(k+1⊥ n)} Y={nϵN /∃kϵN (Supp(k) ⫋Supp(n) ∧⏋(k-1⊥ n)∧ ⏋(k+1⊥ n)} This construction depends on the intersections of { 2 n +1 / n ϵ N } and { 2 n -1 / n ϵ N } with X and Y This means an algorithm of decision through an essential transverse 3-destiny will need to solve some questions in number theory

  37. The work around these sets consists to find , for every form of n we discuss previously , • a witness in X, or to prove there is no such integer • a witness outside of Y, or to prove there is no such integer • a witness in Y\X , or to prove there is no such integer. Prop 1. (Characterization of Y) Y is the set of naturalintegershavingtwo distinct prime factorsp and qsuchthatOrd(p,q) (the order of the element q within the group (Z/pZ)* ) iseven Prop 2. (Characterization of X∩(2N+1)). A positive oddintegerbelongs to X iffthereis one of its prime divisors p, and twointegers b and c with disjoint supports included in Supp(n) \ {p} suchthatv2(Ord(b,p) > v2(Ord(c,p) More precisely, on the 9 questions sufficient to achieve the 3-destiny, 5 remains open:

  38. Sketch of the proofs • Finite groups theory • Quadratic reciprocity law • Aurifeuille’s decomposition of numbers of the form exp(2, ij2α ) + 1. The construction of the essential 3-destiny of Th3(N, S, ⊥) depends on the answer (by yes or no) to nine questions

  39. 1) Is there a = (2n - 1) ∧│Supp (a)│= 2∧⏋(a inY) Answer :YES with a = 2227 - 1 or a = 2269 - 1 2) Is there a = (2n - 1) ∧│Supp (a)│= 2∧⏋(ain Y) Answer : YES with a = 2(2269 - 1) 3) Is there a= (2n - 1) ∧│Supp (a)│⩾ 3 ∧⏋(ain Y) OPEN 4) Is there a=(2n + 1 ∧ neven∧│Supp (a)│= 2 ∧⏋(a inY) OPEN 5)= Is there a = (2n + 1) ∧ n even ∧│Supp (a)│ ⩾ 3 ∧⏋(a inY) OPEN

  40. 6)= Is there a = (2n - 1) ∧ n odd∧│Supp (a)│⩾ 3 ∧ (a inY\X) Answer: YES 225 -1 = 31.601.1801 7)= Is there a = (2n - 1) ∧ neven ∧│Supp (a)│ ⩾ 3 ∧⏋(a inX) Answer : NO 8)= Is there a =(2n + 1) ∧ n odd ∧│Supp (a)│ ⩾ 3 ∧⏋(a inX) Answer : OPEN 9)= Is there a =(2n + 1) ∧ neven∧│Supp (a)│ ⩾ 3 ∧⏋(a in Y\X) Answer : OPEN

  41. Conclusion. To – day , we know 67 treesof this 3-destiny (unpublished) which is made of at most 72 trees

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