Properly ergodic random structures

Transcription

1 Properly ergodic random structures Alex Kruckman University of California, Berkeley June 11, 2015 Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

2 Properly ergodic random structures Joint work with Nate Ackerman, Cameron Freer, and Rehana Patel. What is a random structure? I ll start with two examples. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

3 Example 1: The random graph L = {E}. Build a graph with domain ω, probabilistically. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

4 Example 1: The random graph L = {E}. Build a graph with domain ω, probabilistically. For each pair i j, flip a coin: Heads iej Tails iej Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

5 Example 1: The random graph L = {E}. Build a graph with domain ω, probabilistically. For each pair i j, flip a coin: Heads iej Tails iej Symmetry: The probability of seeing a given induced subgraph on {i 1,..., i n } doesn t depend on the choice of {i 1,..., i n }. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

6 Example 1: The random graph L = {E}. Build a graph with domain ω, probabilistically. For each pair i j, flip a coin: Heads iej Tails iej Symmetry: The probability of seeing a given induced subgraph on {i 1,..., i n } doesn t depend on the choice of {i 1,..., i n }. Independence: If {i 1,..., i n } and {j 1,..., j m } are disjoint, the induced subgraphs on these sets are probabalistically independent. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

7 Example 1: The random graph L = {E}. Build a graph with domain ω, probabilistically. For each pair i j, flip a coin: Heads iej Tails iej Symmetry: The probability of seeing a given induced subgraph on {i 1,..., i n } doesn t depend on the choice of {i 1,..., i n }. Independence: If {i 1,..., i n } and {j 1,..., j m } are disjoint, the induced subgraphs on these sets are probabalistically independent. Concentration: The resulting graph is isomorphic to the random graph G R with probability 1. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

8 Example 2: The kaleidoscope random graph L = {E k k ω}. For each pair i j, flip a coin for each E k. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

9 Example 2: The kaleidoscope random graph L = {E k k ω}. For each pair i j, flip a coin for each E k. The result looks like countably many graphs, all stacked on the same domain, or a complete graph with edges labeled by P(ω). Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

10 Example 2: The kaleidoscope random graph L = {E k k ω}. For each pair i j, flip a coin for each E k. The result looks like countably many graphs, all stacked on the same domain, or a complete graph with edges labeled by P(ω). Symmetry: The probability of seeing a given induced subgraph on {i 1,..., i n } doesn t depend on the choice of {i 1,..., i n }. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

11 Example 2: The kaleidoscope random graph L = {E k k ω}. For each pair i j, flip a coin for each E k. The result looks like countably many graphs, all stacked on the same domain, or a complete graph with edges labeled by P(ω). Symmetry: The probability of seeing a given induced subgraph on {i 1,..., i n } doesn t depend on the choice of {i 1,..., i n }. Independence: If {i 1,..., i n } and {j 1,..., j m } are disjoint, the induced subgraphs on these sets are probabalistically independent. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

12 Example 2: The kaleidoscope random graph L = {E k k ω}. For each pair i j, flip a coin for each E k. The result looks like countably many graphs, all stacked on the same domain, or a complete graph with edges labeled by P(ω). Symmetry: The probability of seeing a given induced subgraph on {i 1,..., i n } doesn t depend on the choice of {i 1,..., i n }. Independence: If {i 1,..., i n } and {j 1,..., j m } are disjoint, the induced subgraphs on these sets are probabalistically independent. Anti-concentration: Every countable L-structure (indeed, every edge label from P(ω)) occurs with probability 0. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

13 Example 2: The kaleidoscope random graph L = {E k k ω}. For each pair i j, flip a coin for each E k. The result looks like countably many graphs, all stacked on the same domain, or a complete graph with edges labeled by P(ω). Symmetry: The probability of seeing a given induced subgraph on {i 1,..., i n } doesn t depend on the choice of {i 1,..., i n }. Independence: If {i 1,..., i n } and {j 1,..., j m } are disjoint, the induced subgraphs on these sets are probabalistically independent. Anti-concentration: Every countable L-structure (indeed, every edge label from P(ω)) occurs with probability 0. However, there is a reasonable complete first-order theory T such that we get a model of T with probability 1. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

14 The setting The space: Fix a countable relational language L. Str L = the space of L-structures with domain ω = R L 2 (ωar(r) ) Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

15 The setting The space: Fix a countable relational language L. Str L = the space of L-structures with domain ω = R L 2 (ωar(r) ) The logic action: S acts on Str L by permutations of ω. Orb(M) = {N σ S, σ(m) = N} = {N N = M} Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

16 The setting The space: Fix a countable relational language L. Str L = the space of L-structures with domain ω = R L 2 (ωar(r) ) The logic action: S acts on Str L by permutations of ω. Orb(M) = {N σ S, σ(m) = N} = {N N = M} Given a formula of ϕ in L ω1,ω and a tuple n from ω, [ϕ(n)] = {M Str L M = ϕ(n)} is a Borel set. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

17 Random structures Definition A random structure is a Borel probability measure on Str L which is invariant and ergodic for the logic action. Invariant: µ(x) = µ(σ[x]). Ergodic: µ(x σ[x]) = 0 for all σ = µ(x) = 0 or 1. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

18 Random structures Definition A random structure is a Borel probability measure on Str L which is invariant and ergodic for the logic action. Invariant: µ(x) = µ(σ[x]). Ergodic: µ(x σ[x]) = 0 for all σ = µ(x) = 0 or 1. A measure µ on Str L is uniquely determined by its restriction to the clopen sets quantifier-free formulas. Invariance Symmetry: µ([ϕ(n)]) = µ([ϕ(σ(n))]) Ergodicity Independence: µ([ϕ(n) ψ(m)]) = µ([ϕ(n)])µ([ψ(m)]) when n and m are disjoint. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

19 A dichotomy If ϕ is a sentence of L ω1,ω, then [ϕ] is invariant, so µ([ϕ]) = 0 or 1. Write µ = ϕ if µ([ϕ]) = 1, and Th Lω1,ω(µ) = {ϕ L ω1,ω µ = ϕ}. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

20 A dichotomy If ϕ is a sentence of L ω1,ω, then [ϕ] is invariant, so µ([ϕ]) = 0 or 1. Write µ = ϕ if µ([ϕ]) = 1, and Th Lω1,ω(µ) = {ϕ L ω1,ω µ = ϕ}. Theorem (Scott) Let M be a countable L-structure. Then there is a sentence θ M in L ω1,ω, the Scott sentence of M, such that for any countable L-structure N, N = θ M iff N = M, i.e. Orb(M) = [θ M ]. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

21 A dichotomy If ϕ is a sentence of L ω1,ω, then [ϕ] is invariant, so µ([ϕ]) = 0 or 1. Write µ = ϕ if µ([ϕ]) = 1, and Th Lω1,ω(µ) = {ϕ L ω1,ω µ = ϕ}. Theorem (Scott) Let M be a countable L-structure. Then there is a sentence θ M in L ω1,ω, the Scott sentence of M, such that for any countable L-structure N, N = θ M iff N = M, i.e. Orb(M) = [θ M ]. Let µ be a random structure. Either For some countable M, µ = θ M. We say µ concentrates on M. For all countable M, µ = θ M. We say µ is properly ergodic. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

22 Trivial dcl Definition Let M be a countable structure. M has trivial definable closure (dcl) if a M, b M \ a, σ Aut(M/a) such that σ(b) b. Let T be a complete theory in a countable fragment of L ω1,ω. T has trivial dcl if for all ψ it does not contain x! y ( n i=1 y x i ψ(x, y)). Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

23 Trivial dcl Definition Let M be a countable structure. M has trivial definable closure (dcl) if a M, b M \ a, σ Aut(M/a) such that σ(b) b. Let T be a complete theory in a countable fragment of L ω1,ω. T has trivial dcl if for all ψ it does not contain x! y ( n i=1 y x i ψ(x, y)). Trivial dcl is the only restriction imposed by randomness. Theorem (Ackerman-Freer-Patel) Let M be a countable structure. There exists a random structure µ concentrating on M iff M has trivial dcl. Let ϕ be a sentence of L ω1,ω, and let F be the countable fragment generated by ϕ. There exists a random structure µ = ϕ iff there is a complete consistent F -theory T, containing ϕ, with trivial dcl. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

24 Properly ergodic models What about properly ergodic models? In the kaleidoscope random graph, the measure is spread across a perfect tree of quantifier-free types. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

25 Properly ergodic models What about properly ergodic models? In the kaleidoscope random graph, the measure is spread across a perfect tree of quantifier-free types. All instances of proper ergodicity can be explained in this way - but the perfect tree of types may be hiding in a higher fragment of L ω1,ω. Theorem (Ackerman-Freer-K.-Patel) Let ϕ be a sentence of L ω1,ω. The following are equivalent: 1 There exists a properly ergodic random structure µ = ϕ. 2 There is a countable fragment F of L ω1,ω and a complete consistent F -theory T, containing ϕ, with trivial dcl, such that S F (T ) = 2 ℵ 0. S F (T ) = {p(x) p is a complete F -type realized in some model of T } Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

26 The Morley/Scott analysis Idea: Tie the concentration/anti-concentration dichotomy to properties of type spaces, via a Morley/Scott analysis. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

27 The Morley/Scott analysis Idea: Tie the concentration/anti-concentration dichotomy to properties of type spaces, via a Morley/Scott analysis. Given a random structure µ, build a ω 1 -length sequence of fragments. F 0 = first-order F λ = F α, λ a limit α<λ { } F α+1 = F α ϕ p S Fα (µ), ϕ p where S F (µ) = {p(x) p is a complete F -type such that µ(p) > 0}. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

28 The Morley/Scott analysis p S Fα (µ) splits later if there exists β > α such that for all q S Fβ (µ) with q p, µ(q) < µ(p). Proposition For every µ, there exists λ < ω 1 such that no type in S Fλ (µ) splits later. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

29 The Morley/Scott analysis p S Fα (µ) splits later if there exists β > α such that for all q S Fβ (µ) with q p, µ(q) < µ(p). Proposition For every µ, there exists λ < ω 1 such that no type in S Fλ (µ) splits later. n, p SF n (µ) µ(p) = 1 µ concentrates on some countable M. λ (F λ -types determine F λ+1 -types, µ gives measure 1 to a Scott sentence) n, p SF n (µ) µ(p) < 1 µ is properly ergodic. λ (There is a positive measure cloud.) Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

30 The Morley/Scott analysis p S Fα (µ) splits later if there exists β > α such that for all q S Fβ (µ) with q p, µ(q) < µ(p). Proposition For every µ, there exists λ < ω 1 such that no type in S Fλ (µ) splits later. n, p SF n (µ) µ(p) = 1 µ concentrates on some countable M. λ (F λ -types determine F λ+1 -types, µ gives measure 1 to a Scott sentence) n, p SF n (µ) µ(p) < 1 µ is properly ergodic. λ (There is a positive measure cloud.) Theorem (K.) If µ is properly ergodic, then Th Lω1,ω(µ) has no models of any cardinality. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

31 The AFP construction Theorem (Ackerman-Freer-K.-Patel) Let ϕ be a sentence of L ω1,ω. The following are equivalent: 1 There exists a properly ergodic random structure µ = ϕ. 2 There is a countable fragment F of L ω1,ω and a complete consistent F -theory T, containing ϕ, with trivial dcl, such that S F (T ) = 2 ℵ 0. (1) = (2): Clouds must have size 2 ℵ 0. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

32 The AFP construction Theorem (Ackerman-Freer-K.-Patel) Let ϕ be a sentence of L ω1,ω. The following are equivalent: 1 There exists a properly ergodic random structure µ = ϕ. 2 There is a countable fragment F of L ω1,ω and a complete consistent F -theory T, containing ϕ, with trivial dcl, such that S F (T ) = 2 ℵ 0. (1) = (2): Clouds must have size 2 ℵ 0. (2) = (1): A variant of the AFP construction. Build a Borel L-structure M, such that I.I.D. sampling of a countable substructure from M gives a model of T with probability 1. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

33 The AFP construction Theorem (Ackerman-Freer-K.-Patel) Let ϕ be a sentence of L ω1,ω. The following are equivalent: 1 There exists a properly ergodic random structure µ = ϕ. 2 There is a countable fragment F of L ω1,ω and a complete consistent F -theory T, containing ϕ, with trivial dcl, such that S F (T ) = 2 ℵ 0. (1) = (2): Clouds must have size 2 ℵ 0. (2) = (1): A variant of the AFP construction. Build a Borel L-structure M, such that I.I.D. sampling of a countable substructure from M gives a model of T with probability 1. Along the way, carefully split the measure across a perfect tree of n-types to create a cloud, ensuring proper ergodicity. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

34 An analogue of Vaught s conjecture Observation If a sentence ϕ has only countably many countable models {M i } i ω, then it has no properly ergodic model: 1 = µ([ϕ]) = µ( i ω [θ M i ]) = 0. Corollary (K.) If ϕ has a properly ergodic model, then it has 2 ℵ 0 -many countable models. This corollary also has an easy descriptive set theory proof. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

35 An analogue of Vaught s conjecture Observation If a sentence ϕ has only countably many countable models {M i } i ω, then it has no properly ergodic model: 1 = µ([ϕ]) = µ( i ω [θ M i ]) = 0. Corollary (K.) If ϕ has a properly ergodic model, then it has 2 ℵ 0 -many countable models. This corollary also has an easy descriptive set theory proof. Recall: Theorem If µ is properly ergodic, then Th Lω1,ω(µ) has no models of any cardinality. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

36 An analogue of Vaught s conjecture Observation If a sentence ϕ has only countably many countable models {M i } i ω, then it has no properly ergodic model: 1 = µ([ϕ]) = µ( i ω [θ M i ]) = 0. Corollary (K.) If ϕ has a properly ergodic model, then it has 2 ℵ 0 -many countable models. This corollary also has an easy descriptive set theory proof. Theorem (update) If µ is properly ergodic, then Th Lω1,ω(µ) has no models of any cardinality. But if F is a countable fragment, then Th F (µ) = {ϕ F µ = ϕ} has 2 ℵ 0 -many countable models. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

37 Why ergodic measures? Philosophy: An ergodic invariant measure plays the role of a single random structure, as opposed to a weighted average over a class of random structures. Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14

38 Why ergodic measures? Philosophy: An ergodic invariant measure plays the role of a single random structure, as opposed to a weighted average over a class of random structures. 1 Logic: Invariant measures do not have a zero-one law for sentences, in general, so no complete theory. 2 Ergodic theory: The ergodic measures are the extreme points in the space of invariant measures. 3 Probability: Ergodic measures arise naturally via I.I.D sampling from probabalistic structures. 4 Combinatorics: An ergodic measure on Str L is one way of representing the limit of a convergent sequence of finite structures. Other equivalent limit objects include graphons (Lovász and Szegedy in the case of graphs), hypergraph limits (Kallenberg, Austin, etc.), and flag algebra homomorphisms (Razborov). Alex Kruckman (UC Berkeley) Properly ergodic random structures June 11, / 14