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Unformatted text preview: ON THE EXPANSION ( N , + , 2 x ) OF PRESBURGER ARITHMETIC FRANC ¸OISE POINT 1 1. Introduction. This is based on a preprint () which appeared in the Proceedings of the fourth Easter Conference on model theory, Gross K ¨ o 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 , + , 2 x ) 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 ). Let us give a quick review on the expansions of ( N , + ,P 2 ), where P 2 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 Th ω ( N ,S ), the weak monadic second- order theory of N with the successor function S , is bi-interpretable with Th ( N , + ,P 2 ), which is incorrect, as later pointed out by R. McNaughton ()). In his review, McNaughton suggested to replace the predicate P 2 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 V 2 ( y ) sending y to the highest power of 2 dividing it. Since then, sev- eral proofs of the fact that Th ( N , + ,V 2 ) is bi-interpretable with Th ω ( N ,S ) and that Def ( N , + ,V 2 ) are exactly the 2-recognizable sets (in powers of N ) appeared (see , ), where Def ( N , + ,V 2 ) are the definable sets in the structure ( N , + ,V 2 ). A.L. Semenov exhibited a family of 2-recognizable subsets which are not definable in ( N , + ,P 2 ) (see  Corollary 4 page 418). Another way to show that this last theory has less expressive power than Th ( N , + ,V 2 ) is to use a result of C. Elgot and M. Rabin ( Theorem 2) that if g is a function from P 2 to P 2 with the property that g skips at least one value , namely that ∀ n > 1 ∀ m ( m > n → ( ∃ y ∈ P 2 g ( m ) > y > g ( n ))), then Th ( N , + ,V 2 ,n → g ( n )) is undecidable and so Th ( N , + ,V 2 , 2 x ) is undecidable (another proof was given by G. Cherlin (see ). Consequences are that neither the graph of 2 x is definable in ( N , + ,V 2 ), nor the graph of V 2 in ( N , + , 2 x ) and that Th ( N , + ,P 2 ) has less expressive power than Th ( N , + , 2 x ). Which unary predicate can we add to the structure ( N , + ,V 2 ) and retain decidabil- ity? Let us mention two kinds of results. On one hand, R. Villemaire showed that Date : July 23, 2010....
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.
- Fall '08