First order logic on Galton-Watson trees

Transcription

1 First order logic on Galton-Watson trees Moumanti Podder Georgia Institute of Technology Joint work with Joel Spencer January 9, 2018 Mathematics Seminar, Indian Institute of Science, Bangalore 1 / 20

2 The Galton-Watson branching process Probability distribution χ on N {0}. Start with a single vertex, called root (denoted R). Let it have X 0 children where X 0 χ. If R has m children v 1,..., v m, then let v i have X i children, where X 1,..., X m independent and each follows χ. Continue like this. Call the random tree generated T χ. Assuming χ has finite expectation µ, if µ > 1 there is a positive probability that T χ survives. For µ 1, T χ dies almost surely. 2 / 20

3 Poisson offspring distribution We consider χ to be Poisson(λ), λ > 0, for this talk. If X Poisson(λ) then P[X = k] = e λ λk k!, k N {0}. When λ > 1, positive probability that T λ survives. If λ 1, then T λ dies out almost surely. Poisson thinning property makes computations easier. 3 / 20

4 Broad picture of our research Mathematical logic provides various classes of properties on underlying abstract structures, such as graphs and more specifically, trees. Example first order properties describe local, finite structures inside a rooted tree, where the root is considered a special vertex. Example second order properties describe more global structures inside a rooted tree. 4 / 20

5 First order (FO) properties Let s start with a few examples: The root R has at least 5 children. There exists a vertex inside the tree with exactly one child and exactly one grandchild. These describe finite-radius neighbourhoods of certain vertices of the tree. Hence first order. Remark In our analysis, the root is identified as a special vertex. 5 / 20

6 A more formal view of FO First order language on rooted trees comprises finite sentences consisting of: Special vertex root R, other vertices denoted by x, y, z... etc.; Relations: equality (x = y) and parent-child (π(y) = x, which denotes that x is the parent of y); Usual Boolean connectives,,, =, etc.; Quantifications: existential ( ) and universal ( ), allowed only over vertices. 6 / 20

7 Explaining with examples Consider A = {R has at least 2 children}. We write A = { x 1 x 2 [{π(x 1 ) = R} {π(x 2 ) = R}] }. Consider B = { a vertex with exactly 1 child}. We write { B = x y [ {π(y) = x} { z [π(z) = x = z = y]} ]}. Definition The quantifier depth of an FO property is the minimum number of nested quantifiers required to express the property. 7 / 20

8 Typically, simplest FO properties look like Fix a finite tree T 0. Define property A[T 0 ] = { a copy of T 0 inside the tree}. Fix a finite tree T 0, of depth d. Define property B[T 0 ] = {radius-d neighbourhood of R = T 0 }. 8 / 20

9 A = node with exactly one child and one grandchild Finite state space Σ = {,, }. : A holds; : root has one child, A holds; : all else. Node colour determined by count of children of each colour. ( 1,, ) (0, 1, 0) (0, 0, 1) x = Pr[ ], y = Pr[ ], z = Pr[ ]. Equations using Poisson thinning: x = 1 e xλ + yλe λ, y = zλe λ. Solution x = f A (λ) unique, nice function of λ. 9 / 20

10 1 f A (λ), as a function of λ 10 / 20

11 An example of a non-fo property B = the tree survives. This is a global property, and cannot be expressed as an FO property. : B holds; : B does not hold. ( 1, ). (0, ). x = Pr[ ], x = 1 e xλ. Solution x = f B (λ) not unique when λ > 1. Remark This is an example of existential monadic second order property. 11 / 20

12 Probability of immortality, as a function of λ 12 / 20

13 Let s play a game Ehrenfeucht games serve as bridge between mathematical logical and structural descriptions of logical properties. Different versions of the game needed for different classes of properties. The most standard version is needed for FO properties. This version also called pebble-move Ehrenfeucht game. The number of rounds k in the game is pre-fixed and denotes the maximum quantifier depth up to which we are considering FO properties. 13 / 20

14 Ehrenfeucht game for FO The game is played on two given trees T 1 (root R 1 ) and T 2 (root R 2 ), for k rounds, between players Spoiler and Duplicator. Each round consists of a move by Spoiler, followed by a move by Duplicator. Spoiler selects a vertex from either of T 1 and T 2 ; Duplicator then selects a vertex from the other tree. x i selected from T 1 and y i from T 2 in round i. Set x 0 = R 1 and y 0 = R 2 as roots are special vertices. Duplicator wins if for all 0 i, j k: xi = x j = y i = y j ; xi is the parent of x j iff y i is the parent of y j. 14 / 20

15 Connection of Ehrenfeucht games with FO properties Write T 1 k T 2 if Duplicator wins the game on T 1, T 2. k an equivalence relation on the space of rooted trees. Theorem (Well-known) Let A be any FO property of quantifier depth at most k. If T 1 k T 2, then either T 1 and T 2 both satisfy A, or T 1 and T 2 both satisfy A. Let Σ k set of all equivalence classes under k. By theorem above, enough for us to look at any representative tree from each equivalence class σ Σ k. Crucially, Σ k is finite. 15 / 20

16 Probabilities of FO properties Let P λ the measure induced by Galton-Watson branching process. We are interested in P λ [A] for any given FO property A of quantifier depth k (k arbitrary but fixed). Set Σ k (A) to be the set of all σ Σ k such that, if tree T is in σ, then A holds for T. Then P λ [A] = P λ [σ]. σ Σ k (A) Hence enough to analyze P λ [σ] for all σ Σ k. 16 / 20

17 Distributional map and fixed points Let D be the set of all probability distributions on Σ k. We define a natural distributional map Ψ : D D such that p(λ) = ( P λ (σ) : σ Σ k ) is a fixed point of Ψ. Theorem (P., Spencer) Ψ is a contraction. p(λ) is the unique fixed point of Ψ. p(λ) is analytic in λ. 17 / 20

18 The probabilities conditioned on survival are nice functions of λ and p λ (the survival probability of T λ ) compositions of polynomials and iterated exponentiations. 18 / 20 Our results on FO probabilities conditioned on survival Makes sense only when λ > 1. Recall A[T 0 ] = { copy of T 0 inside the entire tree}, for arbitrary but fixed finite tree T 0. Theorem (P., Spencer) Conditioned on T λ s survival, for every T 0, A[T 0 ] holds P λ -almost surely. Barring a bad set of infinite trees of P λ -measure 0, the FO properties of quantifier depth k that hold for a typical infinite tree are completely captured by the neighbourhood of the root of radius 3 k+2.

19 Conclusion and further work Previous results give us a complete description of probabilities of FO properties on rooted T λ. We have further explored existential monadic second order properties (EMSO s) on rooted Galton-Watson (GW) trees (and even more generally, random trees with certain characteristics). One element of our work has been to incorporate probability into purely logical questions. One class of problems studied extensively by logicians is expressibility, where they prove if a given property A is expressible in a given logical language L or not. Once we establish that A is not expressible in L, we next ask is A almost surely expressible in L? Here, the underlying probability measure is induced by the random structure (in our case, the rooted GW tree) of which A is a property. 19 / 20

20 Thank you! 20 / 20