A.J. Wilkie. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Amer. Math. Soc. 9 (1996), 1051--1094. 34  Home/Search   Document Not in Database   Summary   Related Articles   Check

....submanifolds in a form suitable for constructing a bisimulation. Such partitions are called stratifications. Moreover, we show that relaxing the class of vector fields or sets in some naive ways leads to pathological situations. On the other hand, the concept of o minimal theories in logic [24,25,26] identifies classes of sets with good intersection properties suitable for the global study of trajectories of vector fields. The combination of techniques from both fields highlights the kind of properties of sets that play a central role in obtaining discrete abstractions. The outline of the ....

....in the extension of the theory of the real numbers by the exponential function, R, 1, exp) i.e. there is an additional symbol in the language for the exponential function) We denote this structure exp . While such theory does not admit elimination of quantifiers, it was shown in [25] that such theory is model complete, which in turns implies that it is o minimal. Another important extension is obtained as follows. Assume f is a real analytic function in a neighborhood of the cube [ 1, 1] Let f : be the function defined by f(x) f(x)ifx [ 1, 1] ....

A. J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted pfa#an functions and the exponential function, Journal of the American Mathematical Society 9 (1996), no. 4, 1051--1094.

....of R is a finite union of points and intervals (possibly unbounded) The class of o minimal theories is quite rich. Quantifier elimination implies that Lin(R)andOF(R) are o minimal. In addition, even though OF exp (R)does not admit elimination of quantifiers, such theory is indeed o minimal (see [16]) Another extension of OF(R) is obtained by adding to a symbol f for every function defined by f(x) f(x)ifx#[ 1,1] 0otherwise where f is a real analytic function in a neighborhood of the cube [ 1, 1] The resulting theory denoted OF an (R) is then an extension of ....

A. J. Wilkie. Model completeness results for expansions of the ordered field of real numbers by restricted pfa#an functions and the exponential function. Journal of the American Mathematical Society, 9(4):1051--1094, Oct 1996.

....definable by quantifier free formulas shows that the structure is o minimal. Tarski was also interested in extending this result to , where there is an additional symbol in the language for the exponential function. While this structure does not admit elimination of quantifiers, it was shown in [80] that this structure is o minimal. Another important extension is obtained as follows. Assume is a real analytic function in a neighborhood of the cube . Let be the function defined by if otherwise. We call such functions restricted analytic functions. These functions are useful to describe the ....

A. J. Wilkie, "Model completeness results for expansions of the ordered field of real numbers by restricted pfaffian functions and the exponential function," J. Amer. Math. Soc., vol. 9, no. 4, pp. 1051--1094, Oct. 1996.

....geometry. Hrushovski and Zilber in an earlier paper [33] had already proved a similar result for Zariski geometries , recovering algebraic geometry. Major results of the 90 s were Wilkie s proof of model completeness (and o minimality) of the real field equipped with the exponential function [66], and Hrushovski s proof of the Mordell Lang conjecture for function fields in all characteristics [32] Wilkie s ingenious proof made use of the general theory of o minimality. There is continuing work on finding richer o minimal expansions of the real field. Hrushovski s work was informed by ....

A.J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfa#an functions and the exponential function, Journal of the American Mathematical Society 9 (1996) 1051-1094.

....submanifolds in a form suitable for constructing a bisimulation. Such partitions are called stratifications. Moreover, we show that relaxing the class of vector fields or sets in some naive ways leads to pathological situations. On the other hand, the concept of o minimal theories in logic [26, 27, 28] identifies classes of sets with good intersection properties suitable for the global study of trajectories of vector fields. The combination of techniques from both fields highlights the kind of properties of sets that play a central role in obtaining discrete abstractions. The outline of the ....

....of the theory of the real numbers by the exponential function, R; Gamma; Theta; 0; 1; exp) i.e. there is an additional symbol in the language for the exponential function) We denote this structure by R exp . While such theory does not admit elimination of quantifiers, it was shown in [27] that such theory is model complete, which in turns implies that it is o minimal. Another important extension is obtained as follows. Assume f is a real analytic function in a neighborhood of the cube [ Gamma1; 1] n ae R n . Let f : R n R be the function defined by f(x) f(x) if x 2 ....

A. J. Wilkie. Model completeness results for expansions of the ordered field of real numbers by restricted pfaffian functions and the exponential function. Journal of the AMS, 9(4):1051--1094, Oct 1996.

....of the theory of the real numbers by the exponential function, R; Gamma; Theta; 0; 1; exp) i.e. there is an additional symbol in the language for the exponential function) We denote this structure by R exp . While such theory does not admit elimination of quantifiers, Wilkie showed in [18] that such theory is model complete, which in turns implies that it is o minimal. Another important extension is obtained as follows. Assume f is a (real )analytic function in a neighborhood of the cube [ Gamma1; 1] n ae R n . Let f : R n R be the function defined by f(x) f(x) if ....

A. J. Wilkie. Model completeness results for expansions of the ordered field of real numbers by restricted pfaffian functions and the exponential function. Journal of the AMS, 9(4):1051--1094, Oct 1996.

....has QE, and its first order theory is decidable, cf. 11] Polynomial Constraints: The real field hR; 0; 1; i is o minimal, has QE, and its first order theory is decidable. This follows from Tarski s theorem, cf. 11, 42] Exponential Constraints: hR; e x ; 0; 1; i is o minimal [48] but does not have QE [41] An example of a structure that is not o minimal is hN; i as the formula 9y(x = y y) defines the set of even numbers. Figure 1 provides examples of some often encountered structures and their standing with respect to o minimality and quantifier elimination. We ....

A.J. Wilkie. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Amer. Math. Soc. 9 (1996), 1051--1094. 32

....of the form fx j M j= x)g, where is a first order formula in the language of M, possibly supplemented with symbols for constants from M. All the structures on the reals we mentioned so far R lin , R, R exp are o minimal (the first two by Tarski s quantifier elimination, the last one by [39]) If M = hR; Omega i, we define M ; to be hR; Omega ; i. We often require that not just M but also M ; be ominimal. Note that this requirement is satisfied by R lin ; R and R exp . We also consider structures having finite VC dimension of definable families [3, 28] also known as ....

A.J. Wilkie. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Amer. Math. Soc. 9 (1996), 1051--1094.

....union of points and open intervals. Both R lin and R have quantifier elimination (by Fourier elimination [25] and Tarski s theorem [2, 3] respectively) which easily implies that they are o minimal. The exponential field hR; Delta; e x i is an example of a structure which is o minimal [24] but does not have quantifier elimination [22] For other o minimal structures on the reals, see [23] We shall deal with some structures on the integers. Of most interest to us is Z 0 = hZ; Gamma; 0; 1; j k ) k 1 i where n j k m iff n = m(mod k) This structure corresponds to constraints ....

A.J. Wilkie. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Amer. Math. Soc. 9 (1996), 1051--1094.

....of M. Classical examples of o minimal structures include the real field R and the real ordered group R lin (as an easy consequence of Tarski s quantifier elimination and the fundamental theorem of algebra) More recently, it was shown that the exponential field hR; Delta; e x i is o minimal [34]. Theorem 2 ( 8, 9] Let M be an o minimal structure admitting quantifier elimination. Then M admits the natural active collapse: FO(M;SC) FO act (M;SC) 2 Thus, the natural active collapse holds for FO Poly and FO Lin. The first result on the natural active collapse was the Hull Su theorem ....

A.J. Wilkie. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Amer. Math. Soc. 9 (1996), 1051--1094. 13

....elimination [48] it is decidable, and every formula (x) is equivalent to a Boolean combination of constraints of the form p(x) 0 or p(x) 0, where p is a polynomial, which implies o minimality. Some extensions of the real field are known to be o minimal, for example, hR; e x i [54]. Logics Since our goal is to develop a theory that can be used beyond the first order case, we consider a variety of logics here. Fix a structure M = hU ; Omega i. By L(SC; Omega Gamma we denote the language that consists of the schema predicates and the symbols in Omega Gamma By FO(SC ; M) ....

A.J. Wilkie. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Amer. Math. Soc. 9 (1996), 1051--1094. 35

....form fx j x ag, fx j x ag, fx j a x bg or closed, or half open half closed versions of these. If U = R, then examples of o minimal signatures include: ffl ( 0; 1) this follows from Tarski s quantifier elimination theorem. ffl ( e x ; 0) this follows from the work of [32], see also [31] ffl ( e x ; Gamma(x) 0) this was recently announced [30] 15 There are many other intersting examples of o minimal structures, see [31] For the rest of this section, we restrict our attention to signatures Omega that are o minimal. Our main result is Theorem 4 ....

A.J. Wilkie. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponentail function. To appear in Journal of the AMS.

....has QE, and its first order theory is decidable. Polynomial Constraints: The real field hR; 0; 1; i is o minimal, has QE, and its first order theory is decidable. This follows from Tarski s theorem, cf. 9, 36] Exponential Constraints: hR; e x ; 0; 1; i is ominimal, see [40]. If only quantifiers 8x 2 adom and 9x 2 adom are used in a query, it is called an active semantics (or active domain) query. This is the usual semantics for databases, and it will be the one used in most of the results in this paper. If quantification over the entire infinite universe is ....

A.J. Wilkie. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Amer. Math. Soc. 9 (1996), 1051--1094.

....structures on the reals we mentioned so far R lin , R, hR; i are o minimal (this is implied by quantifier elimination and the fundamental theorem of algebra, for the case of R. There are a number of known ominimal expansions of R, most notably, the exponential field hR; Delta; e x i [32]. A key property of o minimal structures is cell decomposition. A cell in R k is a subset homeomorphic to R k 0 , k 0 k (by convention, R 0 is a point) We now fix a structure M = hR; Omega i and define M cells by induction on dimension. An M cell in R 0 is just R 0 . M cells in R ....

A.J. Wilkie. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Amer. Math. Soc. 9 (1996), 1051--1094.

....structures on the reals we mentioned so far R lin , R, hR; i are o minimal (this is implied by quantifier elimination and the fundamental theorem of algebra, for the case of R. There are a number of known o minimal expansions of R, most notably, the exponential field hR; Delta; e x i [35]. A key property of o minimal structures is cell decomposition. A cell in R k is a subset homeomorphic to R k 0 , k 0 k (by convention, R 0 is a point) We now fix a structure M = hR; Omega i and define M cells by induction on dimension. An M cell in R 0 is just R 0 . M cells in R ....

A.J. Wilkie. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Amer. Math. Soc. 9 (1996), 1051--1094.

....This can be reformulated as the possibility of the quantifier simplification in the algebra of restricted analytic functions, i.e. functions y = f(x) x# R n ,y# R, such that, after embedding of R n in RP n and R in RP 1 , the graph of f is a subanalytic subset of RP n RP 1 . Wilkie [12] proved that unbounded quantifier simplification is possible for real semialgebraic expressions with the exponential function. Denef and van den Dries [1] found that quantifier elimination is possible in the algebra of all restricted analytic functions if we allow an additional operation of ....

A.J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfa#an functions and the exponential function, J. Amer. Math. Soc., v.9, p.1051-1094 (1996)

....of the form fx j M j= x)g, where is a first order formula in the language of M, possibly supplemented with symbols for constants from M. All the structures on the reals we mentioned so far R lin , R, R exp are o minimal (the first two by Tarski s quantifier elimination, the last one by [39]) If M = hR; Omega i, we define M ; to be hR; Omega ; i. We often require that not just M but also M ; be o minimal. We also consider structures having finite VC dimension of definable families [3, 28] also known as structures without the independence property [36] VC dimension, ....

A.J. Wilkie. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Amer. Math. Soc. 9 (1996), 1051--1094.

....quantifiers. For an o minimal structure with or without elimination of quantifiers we can obtain the above results with good time condition for some nondeterministic complexity classes (it may be necessary to go to PH) For example for R exp = R; Theta; 0; 1; exp) which is model complete [42], one can consider NP. 4.3 The Reals This section gather results that are specific to the field R of real numbers (in the sense that they do not apply to all other real closed fields) The following simple lemma will be useful. It is a straightforward consequence of the nested intervals property ....

....R n , a 2 X iff 9y n (y; ff) C n (a; y; ff) Thus, by elimination of quantifiers X 2 A M . A number of o minimal structures of interest are only model complete (i.e. every formula is equivalent to an existential formula) For instance, this is the case of the reals with exponentiation (see [42]) For this structure, the good question is NP = co NP and the above argument shows that NA=const = NA. We conclude this section with some applications of Theorem 4.19. The point is that some questions concerning the reals R or an arbitrary real closed field can be difficult to answer (due to ....

A. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted pfaffian functions and the exponential function, J. Amer. Math. Soc. 9 (1996) 1051-1094. 59

.... subset of R is a finite union of points and intervals (possibly unbounded) Examples of o minimal theories include (1) R = R; Gamma; Delta; 0; 1) 2) R sin = R; Gamma; Delta; sin j [ Gamma ; 0; 1) van den Dries, 1986) 3) R exp = R; Gamma; Delta; exp; 0; 1) (Wilkie, 1996), and (4) R sin;exp = R; Gamma; Delta; sin j [ Gamma ; exp; 0; 1) van den Dries and Miller, 1994) Based on this notion a new class of hybrid systems is defined. Definition 3. A hybrid system H = X; X 0 ; XF ; F; E; I ; G; R) is said to be o minimal if ffl XC = R n ffl for each ....

Wilkie, A. J. (1996). Model completeness results for expansions of the ordered field of real numbers by restricted pfaffian functions and the exponential function. Journal of the American Mathematical Society 9(4), 1051-- 1094.

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A.J. Wilkie. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Amer. Math. Soc. 9 (1996), 1051--1094. 34

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A. J. Wilkie. Model completeness results for expansions of the ordered field of real numbers by restricted pfaffian functions and the exponential function. Journal of the AMS, 9(4):1051--1094, Oct 1996.

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A. J. Wilkie. Model completeness results for expansions of the ordered field of real numbers by restricted pfaffian functions and the exponential function. Journal of the American Mathematical Society, 9(4):1051--1094, Oct 1996.

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A. J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Am. Math. Soc. 9, N 4, 1051--1094 (1996).

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A.J. Wilkie; Model completeness results for expansions of the ordered field of real numbers by restricted pfaffian functions and the exponential function; J. Am. Math. Soc., vol 9, Number 4, 1051-1094 (1996) Marcus Tressl, NWF-I Mathematik, 93040 Universitat Regensburg, Germany e-mail: marcus.tressl@mathematik.uni-regensburg.de

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A.J. Wilkie. Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function. J. Amer. Math. Soc. 9 (1996), 1051--1094.

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