Defining a Path Beyond First Order Model Theory

Monica VanDieren, Ph.D.

2022 European Computer Science Logic Conference

Logic Mentoring Workshop

20 Years of Tameness +

20 Years of Tameness +

Honors Program

Agenda

• Model Theory Background 
  • First Order Logic
  • Ehrenfeucht-Fraïssé Games
  • Morley's Categoricity Theorem
• Multiple Paths Beyond First Order
  • Non-first order logics
  • Classification Theory for Abstract Elementary Classes
  • Limit Models

• Verification
• Database Theory
• Computational
  Complexity
• 0-1 Laws

Model Theory

Computer Science

Image not drawn to scale

• Verification
• Database Theory
• Computational
  Complexity
• 0-1 Laws

Model Theory

Computer Science

Image not drawn to scale

?

your ideas

Fundamental Ideas in
First Order Model Theory

• Gödel's Completeness Theorem
• Compactness Theorem
• Löwenheim-Skolem Theorems
• Definability 
• Ehrenfeucht-Fraïssé Constructions
• Morley's Categoricity Theorem
• Independence relations (e.g. forking, dividing, splitting)

Fundamental Ideas in
First Order Model Theory

• Gödel's Completeness Theorem
• Compactness Theorem
• Löwenheim-Skolem Theorems
• Definability 
• Ehrenfeucht-Fraïssé Constructions
• Morley's Categoricity Theorem
• Independence relations (e.g. forking, dividing, splitting)

• Ehrenfeucht-Fraïssé Constructions
• Morley's Categoricity Theorem
• Independence relations (e.g. forking, dividing, splitting)

Ehrenfeucht-Fraïssé Games

• Computational complexity (Is NP=coNP?)

• Helps to capture the expressive power of first order logic

Morley's Categoricity Theorem

• Marks the beginning of Classification Theory
• Foundations for independence, rank functions, geometric model theory

Graphs

Signature:

Formulas:

First Order Logic (FOL)

Graphs

Signature:

Formulas:

First Order Logic (FOL)

Graphs

Signature:

Formulas:

First Order Logic (FOL)

Models (triangle free graphs)

Graphs

Signature:

Formulas:

Rings

Signature

Formulas (axioms of rings)

Models (your favorite ring)

First Order Logic (FOL)

Models (triangle free graphs)

Theory of the Random Graph

Signature:

Theory (collection of formulas)

Erdős–Rényi Graph

Theory of the Random Graph

Signature:

Theory (collection of formulas)

Erdős–Rényi Graph

Signature:

Theory (collection of formulas)

Theory of the Random Graph

Erdős–Rényi Graph

Signature:

Theory (collection of formulas)

Theory of the Random Graph

For     and     disjoint, finite subsets of vertices, there is a 

   that is connected to  everything in   
and z is connected to
nothing in   

A

B

z

A

B

z

Erdős–Rényi Graph

Existence of a Countable Random Graph

Existence of a Countable Random Graph

Existence of a Countable Random Graph

Recipe: Establish a countable set of vertices and enumerate all pairs.  For each pair, flip a coin to determine whether or not to connect them with an edge.

Uniqueness of Countable Random Graph

Ehrenfeucht-Fraïssé Game (back-and-forth isomorphism)

Player I: picks a point in \(N\) on even turns and \(M\) on odd turns
Player II: extend a partial isomorphism from previous round (\(f_i\)) from \(N\) to \(M\) using Player I's point

Player I wins if Player II can't play.
If Player II wins, there is an isomorphism between \(N\) and \(M\)

\(N\)

\(M\)

Round 1 - Player I

Player I: picks a point in \(N\) on odd turns and \(M\) on even turns
Player II: uses Player I's point to extend a partial isomorphism from previous round from \(N\) to \(M\)

Player I wins if II can't play. If Player II wins, there is an isomorphism between \(N\) and \(M\)

\(N\)

\(M\)

Round 1 - Player II

Player I: picks a point in \(N\) on odd turns and \(M\) on even turns
Player II: uses Player I's point to extend a partial isomorphism from previous round from \(N\) to \(M\)

Player I wins if II can't play. If Player II wins, there is an isomorphism between \(N\) and \(M\)

\(N\)

\(M\)

\(f_1\)

Player I: picks a point in \(N\) on odd turns and \(M\) on even turns
Player II: uses Player I's point to extend a partial isomorphism from previous round from \(N\) to \(M\)

Player I wins if II can't play. If Player II wins, there is an isomorphism between \(N\) and \(M\)

\(N\)

\(M\)

Round 2 - Player I pickes a connected vertex

\(f_1\)

Player I: picks a point in \(N\) on odd turns and \(M\) on even turns
Player II: uses Player I's point to extend a partial isomorphism from previous round from \(N\) to \(M\)

Player I wins if II can't play. If Player II wins, there is an isomorphism between \(N\) and \(M\)

\(N\)

\(M\)

\(x_1\)

\(y_1\)

Round 2 - Player II responds

\(f_1\)

Player I: picks a point in \(N\) on odd turns and \(M\) on even turns
Player II: uses Player I's point to extend a partial isomorphism from previous round from \(N\) to \(M\)

Player I wins if II can't play. If Player II wins, there is an isomorphism between \(N\) and \(M\)

\(N\)

\(M\)

\(x_1\)

\(y_1\)

\(z_1\)

Round 2 - Player II responds

\(f_1\)

\(f_2\)

Player I: picks a point in \(N\) on odd turns and \(M\) on even turns
Player II: uses Player I's point to extend a partial isomorphism from previous round from \(N\) to \(M\)

Player I wins if II can't play. If Player II wins, there is an isomorphism between \(N\) and \(M\)

\(N\)

\(M\)

Round 2 - Player I picks disconnected vertex

\(f_1\)

Player I: picks a point in \(N\) on odd turns and \(M\) on even turns
Player II: uses Player I's point to extend a partial isomorphism from previous round from \(N\) to \(M\)

Player I wins if II can't play. If Player II wins, there is an isomorphism between \(N\) and \(M\)

\(N\)

\(M\)

\(x_1\)

\(y_1\)

Round 2 - Player II responds

\(f_1\)

Player I: picks a point in \(N\) on odd turns and \(M\) on even turns
Player II: uses Player I's point to extend a partial isomorphism from previous round from \(N\) to \(M\)

Player I wins if II can't play. If Player II wins, there is an isomorphism between \(N\) and \(M\)

\(N\)

\(M\)

\(x_1\)

\(y_1\)

\(z_1\)

Round 2 - Player II responds

\(f_1\)

\(f_2\)

Player I: picks a point in \(N\) on odd turns and \(M\) on even turns
Player II: uses Player I's point to extend a partial isomorphism from previous round from \(N\) to \(M\)

Player I wins if II can't play. If Player II wins, there is an isomorphism between \(N\) and \(M\)

\(N\)

\(M\)

Round \(i+1\)

Player I: picks a point in \(N\) on odd turns and \(M\) on even turns
Player II: uses Player I's point to extend a partial isomorphism from previous round from \(N\) to \(M\)

Player I wins if II can't play. If Player II wins, there is an isomorphism between \(N\) and \(M\)

\(N\)

\(M\)

Round \(i+1\) 

\(f_i\)

Player I: picks a point in \(N\) on odd turns and \(M\) on even turns
Player II: uses Player I's point to extend a partial isomorphism from previous round from \(N\) to \(M\)

Player I wins if II can't play. If Player II wins, there is an isomorphism between \(N\) and \(M\)

\(N\)

\(M\)

Round \(i+1\) - Player 1 picks a vertex

\(f_i\)

Player I: picks a point in \(N\) on odd turns and \(M\) on even turns
Player II: uses Player I's point to extend a partial isomorphism from previous round from \(N\) to \(M\)

Player I wins if II can't play. If Player II wins, there is an isomorphism between \(N\) and \(M\)

\(N\)

\(M\)

Round \(i+1\) - Player 2 responds

\(x_1\)

\(x_2\)

\(x_n\)

\(y_1\)

\(y_2\)

\(y_m\)

\(y_3\)

\(y_4\)

\(f_i\)

Player I: picks a point in \(N\) on odd turns and \(M\) on even turns
Player II: uses Player I's point to extend a partial isomorphism from previous round from \(N\) to \(M\)

Player I wins if II can't play. If Player II wins, there is an isomorphism between \(N\) and \(M\)

\(N\)

\(M\)

Round \(i+1\) - Player 2 responds

\(x_1\)

\(x_2\)

\(x_n\)

\(y_1\)

\(y_2\)

\(y_m\)

\(y_3\)

\(y_4\)

\(z\)

\(f_i\)

Player I: picks a point in \(N\) on odd turns and \(M\) on even turns
Player II: uses Player I's point to extend a partial isomorphism from previous round from \(N\) to \(M\)

Player I wins if II can't play. If Player II wins, there is an isomorphism between \(N\) and \(M\)

\(N\)

\(M\)

Round \(i+1\) - Player 2 has a winning strategy!

\(f_i\)

\(f_{i+1}\)

Categoricity

Defn: If      is a class of models,      is categorical in    if there is exactly one model in     of cardinality     up to isomorphism.

Categoricity

Defn: If      is a class of models,      is categorical in    if there is exactly one model in     of cardinality     up to isomorphism.

Fact:  The class of models of the theory of the random graph is categorical in \(\aleph_0\).

Categoricity

Defn: If      is a class of models,      is categorical in    if there is exactly one model in     of cardinality     up to isomorphism.

Fact:  The class of models of the theory of the random graph is categorical in \(\aleph_0\).

Fact:  The class of models of the theory of algebraically closed fields of characterstic 0 is categorical in every uncountable cardinality.

Morley's Categoricity Theorem

Theorem (Morley 1965) Let      be the class of models of a complete first order theory with countable signature.  If 

is categorical in some uncountable cardinality, then      is categorical in all uncountable cardinalities.

Classifying First Order (countable) Theories

All complete first order theories of countable signature

Classifying First Order (countable) Theories

All complete first order theories of countable signature

Theory of random graph

Theory of multicolored directed graphs omitting directed cycles 

Theory of free groups on n>1
generators 

Theory of differentially closed fields of characteristic 0 

Theory of algebraically closed fields of characteristic 0 

Classifying First Order (countable) Theories

All complete first order theories of countable signature

Theory of random graph

uncountably categorical

Theory of multicolored directed graphs omitting directed cycles 

Theory of free groups on n>1
generators 

Theory of differentially closed fields of characteristic 0 

Theory of algebraically closed fields of characteristic 0 (Steinitz, 1910)

Classifying First Order (countable) Theories

All complete first order theories of countable signature

Theory of random graph

uncountably categorical

superstable

stable

simple

NSOP

Theory of multicolored directed graphs omitting directed cycles (Shelah, 1996)

Theory of random graph

Theory of free groups on n>1
generators (Sela, 2006)

Theory of differentially closed fields of characteristic 0 (Blum, 1968)

Theory of algebraically closed fields of characteristic 0 (Steinitz, 1910)

Classifying First Order (countable) Theories

All complete first order theories of countable signature

uncountably categorical

superstable

stable

simple

NSOP

Classifying First Order (countable) Theories

All complete first order theories of countable signature

uncountably categorical

superstable

stable

simple

NSOP

Classifying First Order (countable) Theories

All complete first order theories of countable signature

Increasing levels of structure

(e.g. independence relations)

Can we develop a similar classification for non-first order theories?

?

Why consider non-FOL?

• Limited expressive power
  • Queries such as disconnectivity and k-colorability of finite graphs are not FO-definable
  • Mathematical concepts such as pure-embeddings for modules are not FO-definable

2nd Order

Generalized quantifiers

Infinitary Logics

Non-exhaustive list

2nd Order

Generalized quantifiers

Infinitary Logics

Non-exhaustive list

2nd order

Generalized quantifiers

Infinitary Logics

First order plus extra quantifiers such as

Non-exhaustive list

2nd order

Generalized quantifiers

Infinitary Logics

First order plus extra quantifiers such as

Non-exhaustive list

2nd order

Generalized quantifiers

Infinitary Logics

First order plus extra quantifiers such as

Non-exhaustive list

Forget the Logic:

Tame AECs

Tame Abstract Elementary Classes

Alternative semantic approach to studying non-FO theories

Fix a signature and consider a class of models      in this signature along with a partial ordering on the models, which induces embeddings between models.  Assume this class satisfies several natural properties:

A1. Closure under isomorphisms

A2. Extends submodel relation

A3. Coherence

A4. Löwenheim Skolem

AP, JEP, Tameness

A5. Union Conditions

This implies the existence of a monster model, \(\frak{C}\).

Tame Abstract Elementary Classes

Alternative semantic approach to studying non-FO theories

Fix a signature and consider a class of models      in this signature along with a partial ordering on the models, which induces embeddings between models.  Assume this class satisfies several natural properties.

F.O. Model Theory Ideas

• Gödel's Completeness Theorem
• Compactness Theorem
• Löwenheim-Skolem Theorems
• Definability 
• Ehrenfeucht-Fraïssé Constructions
• Morley's Categoricity Theorem
• Independence relations 

Tame Abstract Elementary Classes

Alternative semantic approach to studying non-FO theories

Fix a signature and consider a class of models \(\mathcal{K}\) in this signature along with a partial ordering on the models, which induces embeddings between models.  Assume this class satisfies several natural properties.

F.O. Model Theory Ideas

• Gödel's Completeness Theorem
• Compactness Theorem
• Löwenheim-Skolem Theorems
• Definability 
• Ehrenfeucht-Fraïssé Constructions
• Morley's Categoricity Theorem
• Independence relations 

Tame Abstract Elementary Classes

Alternative semantic approach to studying non-FO theories

Fix a signature and consider a class of models \(\mathcal{K}\) in this signature along with a partial ordering on the models, which induces embeddings between models.  Assume this class satisfies several natural properties.

F.O. Model Theory Ideas

• Gödel's Completeness Theorem
• Compactness Theorem
• Löwenheim-Skolem Theorems
• Definability 
• Ehrenfeucht-Fraïssé Constructions
• Morley's Categoricity Theorem
• Independence relations 

Tame Abstract Elementary Classes

Alternative semantic approach to studying non-FO theories

Fix a signature and consider a class of models \(\mathcal{K}\) in this signature along with a partial ordering on the models, which induces embeddings between models.  Assume this class satisfies several natural properties.

F.O. Model Theory Ideas

• Gödel's Completeness Theorem
• Compactness Theorem
• Löwenheim-Skolem Theorems
• Definability 
• Ehrenfeucht-Fraïssé Constructions
• Morley's Categoricity Theorem
• Independence relations 

Tame Abstract Elementary Classes

Alternative semantic approach to studying non-FO theories

Fix a signature and consider a class of models \(\mathcal{K}\) in this signature along with a partial ordering on the models, which induces embeddings between models.  Assume this class satisfies several natural properties.

F.O. Model Theory Ideas

• Gödel's Completeness Theorem
• Compactness Theorem
• Löwenheim-Skolem Theorems
• Definability
• Ehrenfeucht-Fraïssé Constructions
• Morley's Categoricity Theorem
• Independence relations 

Tame Abstract Elementary Classes

Alternative semantic approach to studying non-FO theories

Fix a signature and consider a class of models      in this signature along with a partial ordering on the models, which induces embeddings between models.  Assume this class satisfies several natural properties:

F.O. Model Theory Ideas

• Gödel's Completeness Theorem
• Compactness Theorem
• Löwenheim-Skolem Theorems
• Definability
• Ehrenfeucht-Fraïssé Constructions
• Morley's Categoricity Theorem
• Independence relations 

Tame Abstract Elementary Classes

Alternative semantic approach to studying non-FO theories

Fix a signature and consider a class of models      in this signature along with a partial ordering on the models, which induces embeddings between models.  Assume this class satisfies several natural properties:

F.O. Model Theory Ideas

• Gödel's Completeness Theorem
• Compactness Theorem
• Löwenheim-Skolem Theorems
• Definability
• Ehrenfeucht-Fraïssé Constructions
• Morley's Categoricity Theorem
• Independence relations 

Tame Abstract Elementary Classes

Alternative semantic approach to studying non-FO theories

Fix a signature and consider a class of models      in this signature along with a partial ordering on the models, which induces embeddings between models.  Assume this class satisfies several natural properties:

F.O. Model Theory Ideas

• Gödel's Completeness Theorem
• Compactness Theorem
• Löwenheim-Skolem Theorems
• Definability
• Ehrenfeucht-Fraïssé Constructions
• Morley's Categoricity Theorem
• Independence relations

First Order Theories

\(L_{\omega_1,\omega}\) Theories

\(L_{\kappa^+,\omega}\) Theories

Abstract Elementary Classes

Tame

Incomplete Map of Non-FO Classes of Models

2

Tame AECs are "Everywhere"

• \(Mod(\psi)\) where \(\psi\in L_{\kappa,\omega}\) with \(\kappa\) strongly compact (Makkai-Shelah)
   
• Homogeneous Classes
   
• Finitary Classes
   
• Quasi-minimal class axiomatizing Schanuel's Conjecture (Zilber)
   
• Excellent classes (Kolesnikov-Grossberg)
   
• Universal classes (Boney)
   
• All AECs are tame iff there is class many almost strongly compact cardinals (Boney, Boney-Unger)

Can we develop a classification for
Tame AECs?

?

Can we develop a classification for
Tame AECs?

Yes! And...

categorical in high enough     

superstable

stable

Shelah's Categoricity Conjecture holds in many instances (Shelah, Grossberg-V., Vasey, ...)

New Ideas Arise:
Limit Models

\(M_0\)

\(M_i\)

\(M_{i+1}\)

. . .

. . .

\(\bigcup_{i<\theta}M_{i}\)

21

Definition (Shelah). If there exists a \(\prec\)-increasing sequence \(\langle M_i\in\mathcal{K}\mid i<\theta\rangle\) of models of cardinality \(\mu\) so that for each \(i<\theta\), \(M_{i+1}\) is universal over \(M_i\), then \(\bigcup_{i<\theta}M_i\) is a \((\mu,\theta)\)-limit model.

New Ideas Arise:
Limit Models

\(M_0\)

\(M_i\)

\(M_{i+1}\)

. . .

. . .

\(\bigcup_{i<\theta}M_{i}\)

21

\(M'\)

Definition (Shelah). If there exists a \(\prec\)-increasing sequence \(\langle M_i\in\mathcal{K}\mid i<\theta\rangle\) of models of cardinality \(\mu\) so that for each \(i<\theta\), \(M_{i+1}\) is universal over \(M_i\), then \(\bigcup_{i<\theta}M_i\) is a \((\mu,\theta)\)-limit model.

New Ideas Arise:
Limit Models

\(M_0\)

\(M_i\)

\(M_{i+1}\)

. . .

. . .

\(\bigcup_{i<\theta}M_{i}\)

21

\(f(M')\)

\(M'\)

\(f\)

Definition (Shelah). If there exists a \(\prec\)-increasing sequence \(\langle M_i\in\mathcal{K}\mid i<\theta\rangle\) of models of cardinality \(\mu\) so that for each \(i<\theta\), \(M_{i+1}\) is universal over \(M_i\), then \(\bigcup_{i<\theta}M_i\) is a \((\mu,\theta)\)-limit model.

New Ideas Arise:
Limit Models

\(M_0\)

\(M_i\)

\(M_{i+1}\)

. . .

. . .

Definition (Shelah). If there exists a \(\prec\)-increasing sequence \(\langle M_i\in\mathcal{K}\mid i<\theta\rangle\) of models of cardinality \(\mu\) so that for each \(i<\theta\), \(M_{i+1}\) is universal over \(M_i\), then \(\bigcup_{i<\theta}M_i\) is a \((\mu,\theta)\)-limit model.

\(\bigcup_{i<\theta}M_{i}\)

21

\(f(M')\)

\(M'\)

\(f\)

\(f\restriction M_i=id_{M_i}\)

\(M=M^\alpha_0\)

\(M^\alpha_{i+1}\)

\(M^\alpha_{i}\)

\(M^\alpha=\bigcup_{i<\alpha} M^\alpha_i\)

Uniqueness question:  Under what non-trivial conditions can we conclude that if \(M^\alpha\) is a \((\mu,\alpha)\)-limit model over \(M\) and \(M^\theta\) is a \((\mu,\theta)\)-limit model over \(M\), then there exists \(f:M^\alpha\cong M^\theta\) with \(f\restriction M=id_M\)?

\(M=M^\alpha_0\)

\(M^\alpha_{i+1}\)

\(M^\alpha_{i}\)

\(M^\alpha=\bigcup_{i<\alpha} M^\alpha_i\)

Uniqueness question:  Under what non-trivial conditions can we conclude that if \(M^\alpha\) is a \((\mu,\alpha)\)-limit model over \(M\) and \(M^\theta\) is a \((\mu,\theta)\)-limit model over \(M\), then there exists \(f:M^\alpha\cong M^\theta\) with \(f\restriction M=id_M\)?

\(M=M^\alpha_0\)

\(M^\alpha_{i+1}\)

\(M^\alpha_{i}\)

\(M^\alpha=\bigcup_{i<\alpha} M^\alpha_i\)

Uniqueness question:  Under what non-trivial conditions can we conclude that if \(M^\alpha\) is a \((\mu,\alpha)\)-limit model over \(M\) and \(M^\theta\) is a \((\mu,\theta)\)-limit model over \(M\), then there exists \(f:M^\alpha\cong M^\theta\) with \(f\restriction M=id_M\)?

\(M^\theta_{i+1}\)

. . .

. . .

\(\bigcup_{i<\theta}M^\theta_{i}\)

\(M=M^\alpha_0\)

f(\(M^\alpha)\)

Uniqueness question:  Under what non-trivial conditions can we conclude that if \(M^\alpha\) is a \((\mu,\alpha)\)-limit model over \(M\) and \(M^\theta\) is a \((\mu,\theta)\)-limit model over \(M\), then there exists \(f:M^\alpha\cong M^\theta\) with \(f\restriction M=id_M\)?

\(M^\theta_{i+1}\)

. . .

. . .

\(\bigcup_{i<\theta}M^\theta_{i}\)

\(M^\alpha_{i+1}\)

\(M^\alpha_{i}\)

\(M^\alpha=\bigcup_{i<\alpha} M^\alpha_i\)

\(M=M^\alpha_0\)

\(M^\alpha_{i+1}\)

\(M^\alpha_{i}\)

\(M^\alpha=\bigcup_{i<\alpha} M^\alpha_i\)

Uniqueness question:  Under what non-trivial conditions can we conclude that if \(M^\alpha\) is a \((\mu,\alpha)\)-limit model over \(M\) and \(M^\theta\) is a \((\mu,\theta)\)-limit model over \(M\), then there exists \(f:M^\alpha\cong M^\theta\) with \(f\restriction M=id_M\)?

\(M^\theta_{i+1}\)

. . .

. . .

\(\bigcup_{i<\theta}M^\theta_{i}\)

Case \(cf(\alpha)=cf(\theta)\): Back and forth construction produces \(f\).

Case \(cf(\alpha)\neq cf(\theta)\): Answer is related to "superstability" (V. 2006, 2016a, 2016b, Grossberg-Boney-Vasey-V, ...)