ON THE EXPANSION (N; +; 2x) OF PRESBURGER ARITHMETIC FRANC¸OISE POINT1 1. Introduction. This is based on a preprint ([9]) which appeared in the Proceedings of the fourth Easter Conference on model theory, Gross Ko¨ris, 1986, 17-34, Seminarberichte 86, Humboldt University, Berlin, where, with G. Cherlin, we gave a detailed proof of a result of Alexei L. Semenov that the theory of (N; +; 2x) is decidable and admits quantifier elimination in an expansion of the language containing the Presburger congruence predicates and a logarithmic function. Expansions of Presburger arithmetic have been (and are still) extensively studied (see, for instance [5]). Let us give a quick review on the expansions of (N; +;P2), where P2 is the set of powers of 2. J. Richard B¨uchi showed that this expansion is decidable using the fact that the definable subsets are recognizable by a finite 2-automaton (and Kleene's theorem that the empty problem for finite automata is decidable). (In his article, a stronger result is claimed, namely that T h!(N;S), the weak monadic second- order theory of N with the successor function S, is bi-interpretable with T h(N; +;P2), which is incorrect, as later pointed out by R. McNaughton ([19])). In his review, McNaughton suggested to replace the predicate P2 by the binary predicate 2(x; y) interpreted by "x is a power of 2 and appears in the binary ex- pansion of y". It is easily seen that this predicate is inter-definable with the unary function V2(y) sending y to the highest power of 2 dividing it. Since then, sev- eral proofs of the fact that T h(N; +;V2) is bi-interpretable with T h!(N;S) and that Def(N; +;V2) are exactly the 2-recognizable sets (in powers of N) appeared (see [6], [7]), where Def(N; +;V2) are the definable sets in the structure (N; +;V2). A.L. Semenov exhibited a family of 2-recognizable subsets which are not definable in (N; +;P2) (see [24] Corollary 4 page 418). Another way to show that this last theory has less expressive power than T h(N; +;V2) is to use a result of C. Elgot and M. Rabin ([16] Theorem 2) that if g is a function from P2 to P2 with the property that g skips at least one value, namely that 8n > 1 8m (m > n ! (9y 2 P2 g(m) > y > g(n))), x then T h(N; +;V2; n ! g(n)) is undecidable and so T h(N; +;V2; 2 ) is undecidable (another proof was given by G. Cherlin (see [9]). Consequences are that neither x x the graph of 2 is definable in (N; +;V2), nor the graph of V2 in (N; +; 2 ) and that x T h(N; +;P2) has less expressive power than T h(N; +; 2 ). Which unary predicate can we add to the structure (N; +;V2) and retain decidabil- ity? Let us mention two kinds of results. On one hand, R. Villemaire showed that Date: July 23, 2010. 1Senior Research Associate at the "Fonds National de la Recherche Scientifique”. 1 2 FRANC¸OISE POINT1 T h(N; +;V2;V3) is undecidable ([26]) and this has been strengthened by A. B`eswho proved that T h(N; +;V2;P3) is undecidable ([4]), and on the other hand, P. Bate- man, C. Jockusch and A. Woods showed that, under the linear Schintzel hypothesis, n T h(N; +;V2; (2 )n2P ), where P denotes the set of prime numbers, is decidable ([2]). Then, one may ask what are the complexities of the definable sets of these struc- tures? In an analogous way to his proof of the model-completeness of the theory of (R; +; :; λ2) ([13]), where λ2 is a unary function sending a strictly positive real number r to the biggest power of 2 smaller than r, L. van den Dries ([14]) gave a universal _ : axiomatisation T of T h(N; +;P2) in the language f+; −; ≤; 0; 1; n ; n 2 !; λ2;P2g and showed that T was model-complete (and so it admits quantifier elimination). Both proofs were model-theoretic, using a description of 1-extensions. Recently, J. Avigad and Y. Yin gave an effective proof of van den Dries' quantifier elimination result for (R; +; :; λ2) (see [1]). R. Villemaire ([26]) showed that the quantifier complexity of Def(N; +;V2) has no more than three alternations of quantifiers : 989, by showing that any subset of Nn which is recognizable by a finite 2-automaton is definable in (N; +;V2). Finally, let us say a word on other expansions of Presburger arithmetic. H. Put- nam showed that (N; +;C), where C is a unary predicate for the set of squares, is undecidable, then J.R. B¨uchi extended that result to (N; +;R), where R is a unary predicate which is ultimately the image of a polynomial of degree bigger than or equal to 2 ([8]). On the other hand he showed that T h(N; +;R) is decidable whenever it is ultimately periodic. A.L. Sem¨enov ([25]) described a class of unary predicates R (effectively) sparse and a class of functions f : N ! N,(effectively)compatible with addition for which one x gets analogous (decidability and) definability results as for (N; +;P2) and (N; +; 2 ) (see section 4). Revisiting a result of A. Muchnik ([21]), C. Michaux and R. Villemaire showed that whenever one has a subset R0 of Nm, for some m, which is not already de- finable in Presburger arithmetic, then one can find a subset R of N such that R 2 Def(N; +;R0) − Def(N; +) (see [20]). A. B`es showed showed that there ex- ~ ~ ists R := (rn)n2! ⊂ N with R 2 Def(N; +;R) and such that rn+1 − rn ≥ n ([4]). To a subset of N, one can associate a numeration system. In [23], we gave necessary conditions on the numeration system under which R = (rn)n2N and fR : n ! rn, ful- filled Sem¨enov requirements and so for which T h(N; +;R) (respectively T h(N; +; fR)) was model-complete and/or decidable. Note that there are unary predicates R such that T h(N; +;R) is decidable and n model-complete, but T h(N; +;VR) is undecidable. (For instance, take R := (2 + n)n2!.) 2. Axiomatisation of (N; +; 2x) Let N∗ := N − f0g. _ : ∗ x Definition 2.1. Let L := f+; −; ≤; 0; 1; n ; n 2 N ; 2 ; `2(x)g: Let x; y; z 2 N and n 2 N∗, let us define (1) x−_ y = 0 if y > x, and x−_ y = x − y if y ≤ x. x ON THE EXPANSION (N; +; 2 ) OF PRESBURGER ARITHMETIC 3 x (2) n = y iff x = n:y + z, where 0 ≤ z < n. y y+1 (3) If x 6= 0; y 6= 0, then (`2(x) = y iff 2 ≤ x < 2 ) and `2(0) = 0. `2(x) (4) If x 6= 0, then (P2(x) iff x = 2 ). (5) If x 6= 0; y 6= 0, then (λ2(x) = y iff (y ≤ x < 2:y and P2(y))) and λ2(0) = 0). _ x ∗ We abbreviate x− d :d = m, where 0 ≤ m ≤ d − 1 and d 2 N , by the congruence _ : ∗ condition: x ≡d m. Let LP := f+; −; ≤; 0; 1; n ; n 2 N g: Let TP res be the LP -theory of Presburger arithmetic: (1) 8x 8y 8z ((x + y) + z = x + (y + z)), (2) 8x (x + 0 = 0 + x = x), (3) 8x 8y 8z (x + z = y + z ! x = y), (4) 8x 8y (x + y = y + x), (5) 8x 8y (x ≤ y $ 9u (x + u = y)), (6) 8x 8y (x ≤ y or x ≤ y), (7) 8x (x ≥ 0 & x 6= 0 ! x ≥ 1) & 0 6= 1, W ∗ (8) 8x 9y ( 0≤k<n x = n:y + k), for each n 2 N ; x W ∗ (9) 8x 8y( n = y $ 0≤k<n x = n:y + k), for each n 2 N ; (10) 8x 8y 8z (x−_ y = z $ ((x ≥ y & x = y + z) or (x ≤ y & z = 0))). Let φ be the Euler function, namely φ(m) for natural number m is the number of natural numbers coprime to m and less than or equal to m. We will include in our φ(m) axioms a special case of Euler's theorem, namely that if m is odd, then 2 ≡m 1. In the following, we will use 2:x as an abbreviation for x + x. Let Texp be the following L [ fλ2g-theory (1) TP res (2) 8x (x 6= 0 ! (λ2(x) ≤ x < 2.λ2(x))), λ2(0) = 0, (3) 8x 8y (0 < x ≤ y ! `2(x) ≤ `2(y)), (4) `2(0) = 0, `2(1) = 0, (5) 8x (x ≥ 1 ! `2(2:x) = `2(x) + 1), `2(x) (6) 8x (x ≥ 1 ! 2 = λ2(x)), x (7) 8x (`2(2 ) = x), (8) 8x (2x+1 = 2x + 2x), (9) 8 (x ≥ 1 ! 2x−_ 1 ≥ x), x ∗ (10) 8x (x ≡φ(m) 0 ! 2 ≡m 1) for every odd natural number m 2 N . x We will show that Texp axiomatizes the theory of (N; +; 2 ); this will be a conse- quence of the quantifier elimination (q.e.) result for this theory. Indeed, (N; +; 2x) is a model of Texp and it embeds in any model of that theory, so in particular it will be a prime model of Texp. Before proving the q.e. result, we list a series of properties that hold in any model of Texp.