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Countably categorical almost sure theories

Ove Ahlman, Uppsala University

• ve@math.uu.se

Countably categorical almost sure theories Ove Ahlman, Uppsala University

Introduction

A finite graph G = (G, E) is a finite set G with a binary “edge” relation E. Generalized to finite relational first order structures M = (M, R1, ..., Rk). ◆

Countably categorical almost sure theories Ove Ahlman, Uppsala University

Introduction

A finite graph G = (G, E) is a finite set G with a binary “edge” relation E. Generalized to finite relational first order structures M = (M, R1, ..., Rk). For each n ∈ ◆ let Kn be a finite set of finite structures and µn a probability measure on Kn. If ϕ is a formula let µn(ϕ) = µn({N ∈ Kn : N | = ϕ}) K = ∞

n=1 Kn has a convergence law if for each first order formula

ϕ, limn→∞ µn(ϕ) converges.

Countably categorical almost sure theories Ove Ahlman, Uppsala University

K1 µ 1(Α) = 1 K2

b a

2

µ (Β) = 2/3

a b

2

µ (Α) = 1/3 K3

a b c a b c a b c a b c

3

µ (B) = 1/6

3

µ (C) = 1/3

3

µ (A) = 1/6

3

µ (D) = 1/3

a

If we let ϕ be the formula ∃x∃y(xEy) then µ1(ϕ) = 0 µ2(ϕ) = 2/3 µ3(ϕ) = 5/6 limn→∞ µn(ϕ) converges if the sequence 0, 2/3, 5/6, ... converges.

Countably categorical almost sure theories Ove Ahlman, Uppsala University

0-1 laws

If for each formula ϕ lim

n→∞ µn(ϕ) = 1

• r

lim

n→∞ µn(ϕ) = 0

then K has 0 − 1 law.

Countably categorical almost sure theories Ove Ahlman, Uppsala University

0-1 laws

If for each formula ϕ lim

n→∞ µn(ϕ) = 1

• r

lim

n→∞ µn(ϕ) = 0

then K has 0 − 1 law. Let Kn consisting of all structures with universe {1, ..., n} (over a fixed vocabulary) with µn(N) =

1 |Kn|. Fagin (1976) and

independently Glebksii et. al.(1969) proved that this K has a 0 − 1 law.

Countably categorical almost sure theories Ove Ahlman, Uppsala University

More 0 − 1 Laws

Let K consist of all partial orders and let µn(M) =

1 |Kn|. Compton

(1988): K has a 0 − 1 law.

Countably categorical almost sure theories Ove Ahlman, Uppsala University

More 0 − 1 Laws

Let K consist of all partial orders and let µn(M) =

1 |Kn|. Compton

(1988): K has a 0 − 1 law. Let K consist of all graphs but let µn give high probability to sparse (few edges) graphs. Shelah and Spencer (1988) showed that K has a 0 − 1 law

Countably categorical almost sure theories Ove Ahlman, Uppsala University

More 0 − 1 Laws

Let K consist of all partial orders and let µn(M) =

1 |Kn|. Compton

(1988): K has a 0 − 1 law. Let K consist of all graphs but let µn give high probability to sparse (few edges) graphs. Shelah and Spencer (1988) showed that K has a 0 − 1 law Let K consist of all d−regular graphs and µn a certain, edge depending, probability measure. Haber and Krivelevich (2010) proved that Kn has a 0 − 1 law.

Countably categorical almost sure theories Ove Ahlman, Uppsala University

More 0 − 1 Laws

Let K consist of all partial orders and let µn(M) =

1 |Kn|. Compton

(1988): K has a 0 − 1 law. Let K consist of all graphs but let µn give high probability to sparse (few edges) graphs. Shelah and Spencer (1988) showed that K has a 0 − 1 law Let K consist of all d−regular graphs and µn a certain, edge depending, probability measure. Haber and Krivelevich (2010) proved that Kn has a 0 − 1 law. Let K consist of all l−coloured structures with a vectorspace

• pregeometry. Koponen (2012) proved a 0 − 1 law for K under both

uniform (the normal

1 |Kn|) and dimension conditional measure.

Countably categorical almost sure theories Ove Ahlman, Uppsala University

Fagins method of proving 0 − 1 laws

N satisfies the k-extension property ϕk (for graphs) if: A, B ⊆ N, A ∩ B = ∅, |A ∪ B| ≤ k ⇒ ∃z : aEz and ¬bEz for each a ∈ A, b ∈ B

z B A

Countably categorical almost sure theories Ove Ahlman, Uppsala University

Fagins method of proving 0 − 1 laws

N satisfies the k-extension property ϕk (for graphs) if: A, B ⊆ N, A ∩ B = ∅, |A ∪ B| ≤ k ⇒ ∃z : aEz and ¬bEz for each a ∈ A, b ∈ B

z B A

If K consist of all structures, then limn→∞ µn(ϕk) = 1. We say that ϕk is an almost sure property.

Countably categorical almost sure theories Ove Ahlman, Uppsala University

Fagins method of proving 0 − 1 laws

N satisfies the k-extension property ϕk (for graphs) if: A, B ⊆ N, A ∩ B = ∅, |A ∪ B| ≤ k ⇒ ∃z : aEz and ¬bEz for each a ∈ A, b ∈ B

z B A

If K consist of all structures, then limn→∞ µn(ϕk) = 1. We say that ϕk is an almost sure property. TK = {ϕ : lim

n→∞ µn(ϕ) = 1}

is called the almost sure theory.

Countably categorical almost sure theories Ove Ahlman, Uppsala University

Fagins method of proving 0 − 1 laws

N satisfies the k-extension property ϕk (for graphs) if: A, B ⊆ N, A ∩ B = ∅, |A ∪ B| ≤ k ⇒ ∃z : aEz and ¬bEz for each a ∈ A, b ∈ B

z B A

If K consist of all structures, then limn→∞ µn(ϕk) = 1. We say that ϕk is an almost sure property. TK = {ϕ : lim

n→∞ µn(ϕ) = 1}

is called the almost sure theory. Note: TK is complete iff K has a 0 − 1 law.

Countably categorical almost sure theories Ove Ahlman, Uppsala University

Let κ ≥ ℵ0. For κ-categorical theories completeness is equivalent with not having any finite models.

Theorem

TK is ℵ0−categorical. Hence this will prove that K has a 0 − 1 law.

Countably categorical almost sure theories Ove Ahlman, Uppsala University

Let κ ≥ ℵ0. For κ-categorical theories completeness is equivalent with not having any finite models.

Theorem

TK is ℵ0−categorical. Hence this will prove that K has a 0 − 1 law.

Proof.

Take N, M | = TK. Build partial isomorphisms back and forth by using the extension properties to help.

A B M N iso Countably categorical almost sure theories Ove Ahlman, Uppsala University

The proof method with extension properties has been used in multiple articles proving 0 − 1 laws. In general we get the following

Countably categorical almost sure theories Ove Ahlman, Uppsala University

The proof method with extension properties has been used in multiple articles proving 0 − 1 laws. In general we get the following

Theorem

K has a 0 − 1 law and TK is ℵ0−categorical iff K almost surely satisfies all extension properties

z B A

Extension properties may be very complicated.

Countably categorical almost sure theories Ove Ahlman, Uppsala University

Meq is constructed from a structure M by for each ∅−definable r−ary equivalence relation E:

◮ Add unique element e ∈ Meq − M for each E−equivalence

class.

◮ Add new unary relation symbol PE such that e represents an

E−equivalence class iff Meq | = PE(e)

◮ Add a r + 1-ary relation symbol RE(y, ¯

x) such that ¯ a ∈ M is in the equivalence class of e iff Meq | = RE(e, ¯ a).

Countably categorical almost sure theories Ove Ahlman, Uppsala University

Meq is constructed from a structure M by for each ∅−definable r−ary equivalence relation E:

◮ Add unique element e ∈ Meq − M for each E−equivalence

class.

◮ Add new unary relation symbol PE such that e represents an

E−equivalence class iff Meq | = PE(e)

◮ Add a r + 1-ary relation symbol RE(y, ¯

x) such that ¯ a ∈ M is in the equivalence class of e iff Meq | = RE(e, ¯ a).

E 1

1 2

E 2

1 2 3

M

1 2 3 1 2

Could be thought of as an “Anti-quotient”. A very important structure in infinite model theory.

Countably categorical almost sure theories Ove Ahlman, Uppsala University

If E = {E1, ..., En} is a finite set of ∅−definable equivalence relations then let KE be K where we add the Meq structure for

• nly the equivalence relations in E to each N ∈ K.

Countably categorical almost sure theories Ove Ahlman, Uppsala University

If E = {E1, ..., En} is a finite set of ∅−definable equivalence relations then let KE be K where we add the Meq structure for

• nly the equivalence relations in E to each N ∈ K.

Theorem

Let K be a set of finite relational structures with almost sure theory TK, then K has a 0 − 1 law and TK is ω−categorical. iff KE has a 0 − 1 law and TKE is ω−categorical.

Countably categorical almost sure theories Ove Ahlman, Uppsala University

If E = {E1, ..., En} is a finite set of ∅−definable equivalence relations then let KE be K where we add the Meq structure for

• nly the equivalence relations in E to each N ∈ K.

Theorem

Let K be a set of finite relational structures with almost sure theory TK, then K has a 0 − 1 law and TK is ω−categorical. iff KE has a 0 − 1 law and TKE is ω−categorical. Proof: An application of the previous theorem.

Countably categorical almost sure theories Ove Ahlman, Uppsala University

Strongly minimal countably categorical almost sure theories

A theory T is strongly minimal if for each M | = T, formula ϕ(x, ¯ y) and ¯ a ∈ M. ϕ(M, ¯ a) = {b ∈ M : M | = ϕ(b, ¯ a)} or ¬ϕ(M, ¯ a) is finite.

Countably categorical almost sure theories Ove Ahlman, Uppsala University

Theorem

Assume K has a 0 − 1 law and N ∈ Kn implies |N| = n. Then TK is strongly minimal and ω−categorical ⇔ There exists m ∈ ◆ such that lim

n→∞

µn({M ∈ Kn : there is X ⊆ M, |X| ≤ m, SymX(M) ≤ Aut(M)}) = 1

X .....

Countably categorical almost sure theories Ove Ahlman, Uppsala University

Questions?

• ve@math.uu.se

www.math.uu.se/∼ove

• O. Ahlman, Countably categorical almost sure theories, Preprint (2014)
• 1. K.J. Compton, The computational complexity of asymptotic problems I: partial
• rders, Inform. and comput. 78 (1988), 108-123.
• 2. R. Fagin, Probabilities on finite model theory, J. Symbolic Logic 41 (1976), no.

1, 55-58.

• 3. Y. V. Glebskii. D. I . Kogan, M.I. Liogonkii, V.A. Talanov, Volume and fraction
• f Satisfiability of formulas of the lower predicate calculus, Kibernetyka Vol. 2

(1969) 17-27.

• 4. S. Haber, M. Krivelevich, The logic of random regular graphs, Journal of

combinatorics, Volume 1 (2010) 389-440.

• 5. V. Koponen, Asymptotic probabilities of extension properties and random

l-colourable structures, Annals of Pure and Applied Logic, Vol. 163 (2012) 391-438.

• 6. J. Spencer, S. Shelah, Zero-one laws for sparse random graphs, Journal of the

american mathematical society, Volume 1 (1988) 97-115.

Countably categorical almost sure theories Ove Ahlman, Uppsala University