Countably categorical almost sure theories Ove Ahlman, Uppsala University ove@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 , R 1 , ..., R k ). 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 , R 1 , ..., R k ). For each n ∈ ◆ let K n be a finite set of finite structures and µ n a probability measure on K n . If ϕ is a formula let µ n ( ϕ ) = µ n ( {N ∈ K n : N | = ϕ } ) K = � ∞ n =1 K n has a convergence law if for each first order formula ϕ , lim n →∞ µ n ( ϕ ) converges. Countably categorical almost sure theories Ove Ahlman, Uppsala University
K 3 K 1 K 2 a a a a a c c µ 1 (Α) = 1 b b b b µ (A) = 1/6 µ (B) = 1/6 µ (Α) = 1/3 µ (Β) = 2/3 3 3 2 2 a a c c b b µ (C) = 1/3 µ (D) = 1/3 3 3 If we let ϕ be the formula ∃ x ∃ y ( xEy ) then µ 1 ( ϕ ) = 0 µ 2 ( ϕ ) = 2 / 3 µ 3 ( ϕ ) = 5 / 6 lim n →∞ µ n ( 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 n →∞ µ n ( ϕ ) = 1 lim or n →∞ µ n ( ϕ ) = 0 lim then K has 0 − 1 law. Countably categorical almost sure theories Ove Ahlman, Uppsala University
0-1 laws If for each formula ϕ n →∞ µ n ( ϕ ) = 1 lim or n →∞ µ n ( ϕ ) = 0 lim then K has 0 − 1 law. Let K n consisting of all structures with universe { 1 , ..., n } (over a 1 fixed vocabulary) with µ n ( N ) = | K n | . 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 1 Let K consist of all partial orders and let µ n ( M ) = | K n | . Compton (1988): K has a 0 − 1 law. Countably categorical almost sure theories Ove Ahlman, Uppsala University
More 0 − 1 Laws 1 Let K consist of all partial orders and let µ n ( M ) = | K n | . Compton (1988): K has a 0 − 1 law. Let K consist of all graphs but let µ n 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 1 Let K consist of all partial orders and let µ n ( M ) = | K n | . Compton (1988): K has a 0 − 1 law. Let K consist of all graphs but let µ n 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 µ n a certain, edge depending, probability measure. Haber and Krivelevich (2010) proved that K n has a 0 − 1 law. Countably categorical almost sure theories Ove Ahlman, Uppsala University
More 0 − 1 Laws 1 Let K consist of all partial orders and let µ n ( M ) = | K n | . Compton (1988): K has a 0 − 1 law. Let K consist of all graphs but let µ n 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 µ n a certain, edge depending, probability measure. Haber and Krivelevich (2010) proved that K n 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 1 uniform (the normal | K n | ) 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 ϕ k (for graphs) if: A , B ⊆ N , A ∩ B = ∅ , | A ∪ B | ≤ k ⇒ ∃ z : aEz and ¬ bEz for each a ∈ A , b ∈ B z A B Countably categorical almost sure theories Ove Ahlman, Uppsala University
Fagins method of proving 0 − 1 laws N satisfies the k-extension property ϕ k (for graphs) if: A , B ⊆ N , A ∩ B = ∅ , | A ∪ B | ≤ k ⇒ ∃ z : aEz and ¬ bEz for each a ∈ A , b ∈ B z A B If K consist of all structures, then lim n →∞ µ n ( ϕ k ) = 1. We say that ϕ k 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 ϕ k (for graphs) if: A , B ⊆ N , A ∩ B = ∅ , | A ∪ B | ≤ k ⇒ ∃ z : aEz and ¬ bEz for each a ∈ A , b ∈ B z A B If K consist of all structures, then lim n →∞ µ n ( ϕ k ) = 1. We say that ϕ k is an almost sure property. T K = { ϕ : lim n →∞ µ n ( 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 ϕ k (for graphs) if: A , B ⊆ N , A ∩ B = ∅ , | A ∪ B | ≤ k ⇒ ∃ z : aEz and ¬ bEz for each a ∈ A , b ∈ B z A B If K consist of all structures, then lim n →∞ µ n ( ϕ k ) = 1. We say that ϕ k is an almost sure property. T K = { ϕ : lim n →∞ µ n ( ϕ ) = 1 } is called the almost sure theory. Note: T K is complete iff K has a 0 − 1 law. Countably categorical almost sure theories Ove Ahlman, Uppsala University
Let κ ≥ ℵ 0 . For κ -categorical theories completeness is equivalent with not having any finite models. Theorem T K is ℵ 0 − categorical. Hence this will prove that K has a 0 − 1 law. Countably categorical almost sure theories Ove Ahlman, Uppsala University
Let κ ≥ ℵ 0 . For κ -categorical theories completeness is equivalent with not having any finite models. Theorem T K is ℵ 0 − categorical. Hence this will prove that K has a 0 − 1 law. Proof. Take N , M | = T K . Build partial isomorphisms back and forth by using the extension properties to help. M N iso A B 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 T K is ℵ 0 − categorical iff K almost surely satisfies all extension properties z A B Extension properties may be very complicated. Countably categorical almost sure theories Ove Ahlman, Uppsala University
M eq is constructed from a structure M by for each ∅− definable r − ary equivalence relation E : ◮ Add unique element e ∈ M eq − M for each E − equivalence class. ◮ Add new unary relation symbol P E such that e represents an E − equivalence class iff M eq | = P E ( e ) ◮ Add a r + 1-ary relation symbol R E ( y , ¯ a ∈ M is x ) such that ¯ in the equivalence class of e iff M eq | = R E ( e , ¯ a ). Countably categorical almost sure theories Ove Ahlman, Uppsala University
M eq is constructed from a structure M by for each ∅− definable r − ary equivalence relation E : ◮ Add unique element e ∈ M eq − M for each E − equivalence class. ◮ Add new unary relation symbol P E such that e represents an E − equivalence class iff M eq | = P E ( e ) ◮ Add a r + 1-ary relation symbol R E ( y , ¯ a ∈ M is x ) such that ¯ in the equivalence class of e iff M eq | = R E ( e , ¯ a ). 2 1 1 M 3 2 E 1 1 2 E 2 1 2 3 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 = { E 1 , ..., E n } is a finite set of ∅− definable equivalence relations then let K E be K where we add the M eq structure for only the equivalence relations in E to each N ∈ K . Countably categorical almost sure theories Ove Ahlman, Uppsala University
If E = { E 1 , ..., E n } is a finite set of ∅− definable equivalence relations then let K E be K where we add the M eq structure for only the equivalence relations in E to each N ∈ K . Theorem Let K be a set of finite relational structures with almost sure theory T K , then K has a 0 − 1 law and T K is ω − categorical. iff K E has a 0 − 1 law and T K E is ω − categorical. Countably categorical almost sure theories Ove Ahlman, Uppsala University
If E = { E 1 , ..., E n } is a finite set of ∅− definable equivalence relations then let K E be K where we add the M eq structure for only the equivalence relations in E to each N ∈ K . Theorem Let K be a set of finite relational structures with almost sure theory T K , then K has a 0 − 1 law and T K is ω − categorical. iff K E has a 0 − 1 law and T K E 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
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