Cardinalities of definable sets in finite structures Dugald - PowerPoint PPT Presentation

Cardinalities of definable sets in finite structures Dugald Macpherson University of Leeds July 4, 2017 (joint work with Anscombe, Steinhorn, Wolf) Dugald Macpherson (University of Leeds) Finite structures July 4, 2017 1 / 18

CDM Theorem Theorem. [Chatzidakis, van den Dries and Macintyre 1992] Let ϕ ( x 1 , . . . , x n ; y 1 , . . . , y m ) be a formula in the language of rings. Then there is a positive constant C and finitely many pairs ( d i , µ i ) (1 ≤ i ≤ K ), with d i ∈ { 0 , 1 , . . . , n } and µ i ∈ Q > 0 a positive rational number such that for each a ∈ F m q , if the set ϕ ( F n finite field F q , where q is a prime power, and each ¯ q , ¯ a ) is nonempty, then � − µ i q d i � � < Cq d i − ( 1 / 2 ) � �� F n �� � � ϕ q , ¯ a for some i ≤ K . Moreover, for each pair ( d i , µ i ) , there is a formula � F m � ψ i ( y 1 , . . . , y m ) in the language of rings such that ψ i consists of those q a ∈ F m ¯ q for which the corresponding inequality with ( µ i , d i ) holds. Dugald Macpherson (University of Leeds) Finite structures July 4, 2017 2 / 18

The CDM theorem was turned into a definition of an abstract model-theoretic framework in work of M+Steinhorn, Elwes, Ryten,... Dugald Macpherson (University of Leeds) Finite structures July 4, 2017 3 / 18

The CDM theorem was turned into a definition of an abstract model-theoretic framework in work of M+Steinhorn, Elwes, Ryten,... A 1-dimensional asymptotic class is essentially a class of finite structures satisfying the conclusion of the theorem (e.g. finite fields). Dugald Macpherson (University of Leeds) Finite structures July 4, 2017 3 / 18

The CDM theorem was turned into a definition of an abstract model-theoretic framework in work of M+Steinhorn, Elwes, Ryten,... A 1-dimensional asymptotic class is essentially a class of finite structures satisfying the conclusion of the theorem (e.g. finite fields). Elwes (2007): notion of N -dimensional asymptotic class (e.g. class of groups SL 2 ( q ) is a 3-dimensional asymptotic class). Dugald Macpherson (University of Leeds) Finite structures July 4, 2017 3 / 18

The CDM theorem was turned into a definition of an abstract model-theoretic framework in work of M+Steinhorn, Elwes, Ryten,... A 1-dimensional asymptotic class is essentially a class of finite structures satisfying the conclusion of the theorem (e.g. finite fields). Elwes (2007): notion of N -dimensional asymptotic class (e.g. class of groups SL 2 ( q ) is a 3-dimensional asymptotic class). Ryten (PhD thesis 2007): For any fixed Lie type τ , the class of all finite simple groups of type τ is an asymptotic class. Dugald Macpherson (University of Leeds) Finite structures July 4, 2017 3 / 18

The CDM theorem was turned into a definition of an abstract model-theoretic framework in work of M+Steinhorn, Elwes, Ryten,... A 1-dimensional asymptotic class is essentially a class of finite structures satisfying the conclusion of the theorem (e.g. finite fields). Elwes (2007): notion of N -dimensional asymptotic class (e.g. class of groups SL 2 ( q ) is a 3-dimensional asymptotic class). Ryten (PhD thesis 2007): For any fixed Lie type τ , the class of all finite simple groups of type τ is an asymptotic class. Corresponding notion of measurable infinite structure (M + Steinhorn) (measurable implies supersimple, finite SU rank) – e.g. pseudofinite fields. Dugald Macpherson (University of Leeds) Finite structures July 4, 2017 3 / 18

The CDM theorem was turned into a definition of an abstract model-theoretic framework in work of M+Steinhorn, Elwes, Ryten,... A 1-dimensional asymptotic class is essentially a class of finite structures satisfying the conclusion of the theorem (e.g. finite fields). Elwes (2007): notion of N -dimensional asymptotic class (e.g. class of groups SL 2 ( q ) is a 3-dimensional asymptotic class). Ryten (PhD thesis 2007): For any fixed Lie type τ , the class of all finite simple groups of type τ is an asymptotic class. Corresponding notion of measurable infinite structure (M + Steinhorn) (measurable implies supersimple, finite SU rank) – e.g. pseudofinite fields. Fact: Any ultraproduct of an asymptotic class is measurable. Dugald Macpherson (University of Leeds) Finite structures July 4, 2017 3 / 18

We aim to broaden this framework, e.g. allow parts of the structure (sorts? coordinatising geometries?) to vary independently, not require that ultraproducts have finite rank, or even have simple theory, not be specific about the form of the functions giving approximate cardinalities (no longer just of form µ q d ). Dugald Macpherson (University of Leeds) Finite structures July 4, 2017 4 / 18

We aim to broaden this framework, e.g. allow parts of the structure (sorts? coordinatising geometries?) to vary independently, not require that ultraproducts have finite rank, or even have simple theory, not be specific about the form of the functions giving approximate cardinalities (no longer just of form µ q d ). Possible examples to keep in mind: Dugald Macpherson (University of Leeds) Finite structures July 4, 2017 4 / 18

We aim to broaden this framework, e.g. allow parts of the structure (sorts? coordinatising geometries?) to vary independently, not require that ultraproducts have finite rank, or even have simple theory, not be specific about the form of the functions giving approximate cardinalities (no longer just of form µ q d ). Possible examples to keep in mind: Pairs ( V , F q ) (2-sorted language), V a finite-dim vector space over F q . Dugald Macpherson (University of Leeds) Finite structures July 4, 2017 4 / 18

We aim to broaden this framework, e.g. allow parts of the structure (sorts? coordinatising geometries?) to vary independently, not require that ultraproducts have finite rank, or even have simple theory, not be specific about the form of the functions giving approximate cardinalities (no longer just of form µ q d ). Possible examples to keep in mind: Pairs ( V , F q ) (2-sorted language), V a finite-dim vector space over F q . Disjoint unions of complete graphs all of same size ( n copies of K m ) Dugald Macpherson (University of Leeds) Finite structures July 4, 2017 4 / 18

We aim to broaden this framework, e.g. allow parts of the structure (sorts? coordinatising geometries?) to vary independently, not require that ultraproducts have finite rank, or even have simple theory, not be specific about the form of the functions giving approximate cardinalities (no longer just of form µ q d ). Possible examples to keep in mind: Pairs ( V , F q ) (2-sorted language), V a finite-dim vector space over F q . Disjoint unions of complete graphs all of same size ( n copies of K m ) Finite abelian groups Dugald Macpherson (University of Leeds) Finite structures July 4, 2017 4 / 18

We aim to broaden this framework, e.g. allow parts of the structure (sorts? coordinatising geometries?) to vary independently, not require that ultraproducts have finite rank, or even have simple theory, not be specific about the form of the functions giving approximate cardinalities (no longer just of form µ q d ). Possible examples to keep in mind: Pairs ( V , F q ) (2-sorted language), V a finite-dim vector space over F q . Disjoint unions of complete graphs all of same size ( n copies of K m ) Finite abelian groups Finite graphs of bounded degree Dugald Macpherson (University of Leeds) Finite structures July 4, 2017 4 / 18

For a class C of finite L -structures and a tuple ¯ y of variables, let ( C , ¯ y ) be the � � M ∈ C , ¯ a ∈ M | ¯ y | � � ( M , ¯ a ) of pairs consisting of a structure in C and a set ¯ y -tuple from that structure (‘pointed structures in C ’). Dugald Macpherson (University of Leeds) Finite structures July 4, 2017 5 / 18

For a class C of finite L -structures and a tuple ¯ y of variables, let ( C , ¯ y ) be the � � M ∈ C , ¯ a ∈ M | ¯ y | � � ( M , ¯ a ) of pairs consisting of a structure in C and a set ¯ y -tuple from that structure (‘pointed structures in C ’). A finite partition Φ of ( C , ¯ y ) , is ∅ -definable if for each P ∈ Φ there exists an L -formula φ P (¯ y ) without parameters such that � ¯ � ( M , ¯ b ∈ M | ¯ y | � � φ P ( M ) = b ) ∈ P , for each M ∈ C . Dugald Macpherson (University of Leeds) Finite structures July 4, 2017 5 / 18

Definition of R -m.a.c. Let R be any set of functions C − → R ≥ 0 . A class C of finite L -structures is an R -multidimensional asymptotic class ( R -m.a.c. ) if for every formula φ (¯ x ;¯ y ) there is a finite ∅ -definable partition Φ of ( C , ¯ y ) and a set H Φ := { h P ∈ R | P ∈ Φ } ⊂ R such that for each P ∈ Φ , x ; ¯ � = o ( h P ( M )) � � � | φ (¯ b ) | − h P ( M ) (1) for ( M , ¯ b ) ∈ P as | M | − → ∞ . Dugald Macpherson (University of Leeds) Finite structures July 4, 2017 6 / 18

Definition of R -m.a.c. Let R be any set of functions C − → R ≥ 0 . A class C of finite L -structures is an R -multidimensional asymptotic class ( R -m.a.c. ) if for every formula φ (¯ x ;¯ y ) there is a finite ∅ -definable partition Φ of ( C , ¯ y ) and a set H Φ := { h P ∈ R | P ∈ Φ } ⊂ R such that for each P ∈ Φ , x ; ¯ � = o ( h P ( M )) � � � | φ (¯ b ) | − h P ( M ) (1) for ( M , ¯ b ) ∈ P as | M | − → ∞ . R - m.e.c. (multidimensional exact class) if above we have x , ¯ | φ (¯ b ) | = h P ( M ) . Dugald Macpherson (University of Leeds) Finite structures July 4, 2017 6 / 18