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53
Definable sets in ordered structures
 Bull. Amer. Math. Soc. (N.S
, 1984
"... Abstract. This paper introduces and begins the study of a wellbehaved class of linearly ordered structures, the ^minimal structures. The definition of this class and the corresponding class of theories, the strongly ©minimal theories, is made in analogy with the notions from stability theory of m ..."
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Abstract. This paper introduces and begins the study of a wellbehaved class of linearly ordered structures, the ^minimal structures. The definition of this class and the corresponding class of theories, the strongly ©minimal theories, is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories. Theorems 2.1 and 2.3, respectively, provide characterizations of Cminimal ordered groups and rings. Several other simple results are collected in §3. The primary tool in the analysis of ¿¡minimal structures is a strong analogue of "forking symmetry, " given by Theorem 4.2. This result states that any (parametrically) definable unary function in an (5minimal structure is piecewise either constant or an orderpreserving or reversing bijection of intervals. The results that follow include the existence and uniqueness of prime models over sets (Theorem 5.1) and a characterization of all N0categorical ¿¡¡minimal structures (Theorem 6.1). 1. Introduction. The
Stability theory, Permutations of Indiscernibles, and Embedded Finite Models
 Trans. Amer. Math. Soc
, 2000
"... Abstract. We show that the expressive power of firstorder logic over finite models embedded in a model M is determined by stabilitytheoretic properties of M. In particular, we show that if M is stable, then every class of finite structures that can be defined by embedding the structures in M, can ..."
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Abstract. We show that the expressive power of firstorder logic over finite models embedded in a model M is determined by stabilitytheoretic properties of M. In particular, we show that if M is stable, then every class of finite structures that can be defined by embedding the structures in M, can be defined in pure firstorder logic. We also show that if M does not have the independence property, then any class of finite structures that can be defined by embedding the structures in M, can be defined in firstorder logic over a dense linear order. This extends known results on the definability of classes of finite structures and ordered finite structures in the setting of embedded finite models. These results depend on several results in infinite model theory. Let I be a set of indiscernibles in a model M and suppose (M,I) is elementarily equivalent to (M1,I1) whereM1 is I1  +saturated. If M is stable and (M,I) is saturated, then every permutation of I extends to an automorphism of M and the theory of (M,I) isstable. LetI be a sequence of <indiscernibles in a model M, which does not have the independence property, and suppose (M,I) is elementarily equivalent to (M1,I1) where(I1,<) is a complete dense linear order and M1 is I1  +saturated. Then (M, I)types over I are orderdefinable and if (M, I) isℵ1saturated, every order preserving permutation of I can be extended to a backandforth system. 1.
Model theory of differentiable fields
 Lecture Notes in Logic 5
, 1996
"... Abstract. This article surveys the model theory of differentially closed fields, an interesting setting where one can use modeltheoretic methods to obtain algebraic information. The article concludes with one example showing how this information can be used in diophantine applications. A differenti ..."
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Cited by 13 (1 self)
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Abstract. This article surveys the model theory of differentially closed fields, an interesting setting where one can use modeltheoretic methods to obtain algebraic information. The article concludes with one example showing how this information can be used in diophantine applications. A differential field is a field K equipped with a derivation δ: K → K; recall that this means that, for x, y ∈ K, we have δ(x + y) = δ(x) + δ(y) and δ(xy) = x δ(y) + yδ(x). Roughly speaking, such a field is called differentially closed when it contains enough solutions of ordinary differential equations. This setting allows one to use modeltheoretic methods, and particularly dimensiontheoretic ideas, to obtain interesting algebraic information. In this lecture I give a survey of the model theory of differentially closed fields, concluding with an example — Hrushovski’s proof of the Mordell–Lang conjecture in characteristic zero — showing how modeltheoretic methods in this area can be used in diophantine applications. I will not give the proofs of the main theorems. Most of the material in Sections 1–3 can be found in [Marker
A primer of simple theories
 Archive Math. Logic
"... Abstract. We present a selfcontained exposition of the basic aspects of simple theories while developing the fundamentals of forking calculus. We expound also the deeper aspects of S. Shelah’s 1980 paper Simple unstable theories. The concept of weak dividing has been replaced with that of forking. ..."
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Abstract. We present a selfcontained exposition of the basic aspects of simple theories while developing the fundamentals of forking calculus. We expound also the deeper aspects of S. Shelah’s 1980 paper Simple unstable theories. The concept of weak dividing has been replaced with that of forking. The exposition is from a contemporary perspective and takes into account contributions due to S. Buechler, E. Hrushovski, B. Kim, O. Lessmann, S. Shelah and A. Pillay.
On Counting Generalized Colorings
, 2008
"... The notion of graph polynomials definable in Monadic Second Order Logic, MSOL, was introduced in [Mak04]. It was shown that the Tutte polynomial and its generalization, as well as the matching polynomial, the cover polynomial and the various interlace polynomials fall into this category. In this pap ..."
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Cited by 5 (5 self)
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The notion of graph polynomials definable in Monadic Second Order Logic, MSOL, was introduced in [Mak04]. It was shown that the Tutte polynomial and its generalization, as well as the matching polynomial, the cover polynomial and the various interlace polynomials fall into this category. In this paper we present a framework of graph polynomials based on counting functions of generalized colorings. We show that this class encompasses the examples of graph polynomials from the literature. Furthermore, we extend the definition of graph polynomials definable in MSOL to allow definability in full second order, SOL. Finally, we show that the SOLdefinable graph polynomials extended with a combinatorial counting function are exactly the counting functions of generalized colorings definable in SOL.