Literature for
"Limits of finite structures (Spring 2014)
Obviously, we were not able to
get to all topics listed below; a lot of the
references served as pointers
to topics that are relevant but were not discussed.
We may return to some of them in future.
Pseudofinitiness
J. Vaananen,
Pseudo-finite model theory, Matematica Contemporanea,
vol 24, 2003, pp.169-183.
Non-standard analysis and Loeb's measure
I.Davis,
An introduction to nonstandard analysis
I.Goldbring,
Nonstandard Analysis
(Lecture notes, UCLA Summer School in Logic, 2009 & 2012).
W.Henson,
Foundations of nonstandard analysis:
A gentle introduction to nonstandard extensions
, lecture notes.
P.A.Loeb,
Conversion from nonstandard to standard measure spaces and applications in probability theory,
Transactions of the American Mathematical Society, 211, (197), pp.113–122.
T.Tao,
Ultraproducts as a bridge between discrete and continuous analysis, an exposition based on a talk
at the workshop “Neo-Classical methods in discrete analysis“
(the Simons institute for the theory of computing), 2013.
Amalgamation
W.Hodges, A shorter model theory, CUP,
(1997).
(for general amalgamation)
E. Hrushovski. A new strongly minimal set. Annals of Pure and Applied Logic, 52:147–166, 1993.
F.Wagner's presentation of Hrushovski's amalgamation, 2006.
M.Ziegler,
An exposition of Hrushovski's new strongly minimal set, 2011.
Model-theoretic forcing
W.Hodges, Building models by games, Dover Publ., (2006).
T.Zhang's presentation, 2002.
Limits of graphs
G.Elek and B.Szegedy,
Limits of Hypergraphs, Removal and Regularity Lemmas. A Non-standard Approach,
(2007),
Paper in the ArXiv.
L.Lovasz and B.Szegedy,
Limits of dense graph sequences, (2004),
Paper in the ArXiv.
J.Nesetril and P.Ossona De Mendez, A Model Theory
Approach to Structural Limits,
Commentationes Mathematicae Universitatis Carolinae 53, 4 (2012) 581-603.
Paper in the ArXiv.
Asymptotic and robust classes
R.Elwes,
Asymptotic classes of finite structures,
J. of Symbolic Logic, 72(2), (2007), pp.418-438.
R.Elwes and D.Macpherson,
A Survey Of Asymptotic Classes and Measurable Structures,
in Model Theory with Applications to Algebra and Analysis Volume 2 (Cambridge University Press, 2008).
A. Macintyre, L. v.d. Dries,
Z. Chatzidakis,
Definable sets over finite fields,
Journal für die reine und angewandte Mathematik (Crelles Journal). Issue 427, (1992), pp.107–136.
D.Macpherson and C.Steinhorn, De finability in
classes of finite structures, in: Javier Esparza, C. Michaux,
and C. Steinhorn, editors, Finite and Algorithmic
Model Theory, London Mathematical Society Lecture Note
Series (No. 379), pages 140–176. Cambridge University Press, 2011.
Paper.
Smoothly approximable structures
G. Cherlin, E. Hrushovski,
Finite structures with few types,
Annals of Mathematics Studies No. 152,
Princeton University Press, Princeton, 2003
Nonstandard finite structures
in proof complexity
M.Ajtai,
Generalizations of the compactness theorem and Gödel's completeness theorem for nonstandard finite structures,
in:
Proc. of the 4th international conference on Theory and applications of models of computation,
Springer-Verlag, (2007), pp.13-33.
M.Garlik,
Ajtai's Completeness Theorem for Nonstandard Finite Structures,
Archive for Math.Logic, to app.
J.Krajicek,
Uniform families of polynomial equations over a finite field and structures admitting an Euler characteristic of definable sets,
Proc. London Mathematical Society,
3(81), (2000), pp.257-284.
J.Krajicek,
Forcing with random variables and proof complexity,
London Mathematical Society Lecture Note Series, No.382, Cambridge University Press, (2011).
Appendix (available online from
the draft) presents the ultraproduct construction
of an Aleph_1-saturated model of true arithmetic.
J.Krajicek and
T. Scanlon,
Combinatorics with definable sets: Euler characteristics and Grothendieck rings,
Bulletin of Symbolic Logic, 3(3), (2000), pp.311-330.