The first order definability of finite graphs Oleg Verbitsky Humboldt Universit¨ at IAPMM and Berlin, Germany Lviv, Ukraine Bertinoro, October 2009 Based on joint work with Oleg Pikhurko and Joel Spencer, with important contributions by Tom Bohman, Alan Frieze, Martin Grohe, Jeong Han Kim, Tomasz � Luczak, Clifford Smyth, and Helmut Veith.
Outline • Basic concepts (the logical depth, width, and length of a graph) • Main tools (the Ehrenfeucht game and the Weisfeiler-Lehman algorithm) • Upper bounds – trees and graphs with bounded treewidth – planar graphs – general case (Ramsey?) • Random graph – applications to the 0-1 law • Succinctly definable graphs – definitions with no quantifier alternation (Ramsey!) 1
Language of the first order theory of graphs • variables ( x , y , y 1 , etc), ranging through the vertex set of a graph; • the relations = (equality) and ∼ (vertex adjacency); • the quantifiers ∀ (universality) and ∃ (existence); • the Boolean connectives ∧ (and), ∨ (or), and ¬ (negation). Example. The following first order formula ∆ n ( x, y ) says that vertices x any y lie at distance no more than n : def ∆ 1 ( x, y ) = x ∼ y ∨ x = y n − 2 “ ” def ^ ∆ n ( x, y ) = ∃ z 1 . . . ∃ z n − 1 ∆ 1 ( x, z 1 ) ∧ ∆ 1 ( z i , z i +1 ) ∧ ∆ 1 ( z n − 1 , y ) i =1 2
Basics A sentence Φ distinguishes a graph G from another graph H if Φ is true on G but false on H . Example. 1. The sentence ∀ x ∀ y ∆ 1 ( x, y ) distinguishes a complete graph K n from any other graph H that is not complete. 2. The sentence ∀ x ∀ y ∆ n − 1 ( x, y ) distinguishes P n , a path on n vertices, from any longer path P m , m > n . 3
Basics A sentence Φ defines a graph G (up to isomorphism) if Φ distinguishes G from every non-isomorphic graph H . Example. P n is defined by ∀ x ∀ y ∆ n − 1 ( x, y ) ∧ ¬∀ x ∀ y ∆ n − 2 ( x, y ) to say that the diameter = n − 1 “^ ” ^ ∧ ∀ x ¬∃ y 1 ∃ y 2 ∃ y 3 i =1 , 2 , 3 x ∼ y i ∧ i � = j y i � = y j to say that the maximum degree ≤ 2 “^ ” ∧ ∃ x ¬∃ y 1 ∃ y 2 i =1 , 2 x ∼ y i ∧ y 1 � = y 2 to say that the minimum degree ≤ 1 (thereby distinguishing from cycles C 2 n − 2 and C 2 n − 1 ) 4
Basics Succinctness measures of a formula Φ : the length L (Φ) , the quantifier depth D (Φ) , and the width W (Φ) Definition. W (Φ) is the number of variables used in Φ (different occurrences of the same variable are not counted!) Example. W (∆ n ) = n + 1 . However, rewriting it as def ∆ ′ = ∃ z (∆ 1 ( x, z ) ∧ ∆ ′ n ( x, y ) n − 1 ( z, y )) , where def ∆ ′ = ∃ x (∆ 1 ( z, x ) ∧ ∆ ′ n − 1 ( z, y ) n − 2 ( x, y )) and so on, we get W (∆ ′ n ) = 3 . 5
Basics Definition. D (Φ) , the quantifier depth of Φ , is the maximum number of nested quantifiers in Φ . Example. D (∆ n ) = n − 1 . However, rewriting it as “ ” def ∆ ′′ ∆ ′′ ⌊ n/ 2 ⌋ ( x, z ) ∧ ∆ ′′ n ( x, y ) = ∃ z ⌈ n/ 2 ⌉ ( z, y ) , we get D (∆ ′′ n ) = log n + O (1) . 6
Main Definition Definition (the logical length, depth, and width of a graph). L ( G ) (resp. D ( G ) , W ( G ) ) is the minimum L (Φ) (resp. D (Φ) , W (Φ) ) over all Φ defining G . Example. W ( P n ) ≤ 4 , D ( P n ) ≤ log n + O (1) . 7
Basics Definition. Let G �∼ = H . Then D ( G, H ) (resp. W ( G, H ) ) is the minimum D (Φ) (resp. W (Φ) ) over all Φ distinguishing G from H . Proposition. 1. D ( G ) = max H D ( G, H ) 2. W ( G ) = max H W ( G, H ) 8
Variations of a logic Fragments of first order logic • Bounded number of quantifier alternations (later). • Bounded number of variables. D k ( G ) denotes the logical depth of G in the k -variable logic An extension of first order logic: Counting quantifiers ∃ m x Ψ( x ) means that there are at least m vertices x having property Ψ . D # ( G ) and W # ( G ) will denote the logical depth and width of a graph G in the counting logic. D k # ( G ) denotes the variant of D k ( G ) for the k -variable counting logic. 9
Ehrenfeucht’s game Immerman-Poizat: G and H are distinguishable with k variables and quantifier depth r iff Spoiler wins the Ehrenfeucht game with k pebbles in r moves. Rules of the Game Players: Spoiler and Duplicator Resources: k pebbles, each in duplicate A round: Spoiler puts a pebble on a vertex in G or H Duplicator puts the other copy on a vertex in the other graph G H Duplicator’s objective: after each round the pebbling should determine a partial isomorphism between G and H 10
k -dimensional Weisfeiler-Lehman algorithm 1-dim WL = color refinement procedure 1 1 1 1 1 1 1 1 1 1 1 1 Initial coloring 11
1-dim WL = color refinement procedure 1,11 1,111 1,11 1,111 1,111 1,1111 1,1111 1,111 1,1 1,1 1,1 1,1 Refine coloring: for each vertex New Color = Old Color + Old Colors of all neighbors 12
1-dim WL = color refinement procedure 2 3 2 3 3 4 4 3 1 1 1 1 Simplify color names 13
1-dim WL = color refinement procedure 2,33 3,234 2,34 3,234 3,234 4,1133 4,1233 3,134 1,4 1,4 1,4 1,3 Refine coloring again. The multisets of colors differ, hence the graphs are non-isomorphic. 14
k -dim WL k -dim WL = the same idea, but now we color V k instead of V . The initial coloring of ( v 1 , . . . , v k ) is the isomorphism type of the subgraph induced on v 1 , . . . , v k . Theorem (Cai, F¨ urer, and Immerman) The r -round k -dim WL works correctly on any pair ( G, H ) if k = W # ( G ) − 1 and r = D k +1 # ( G ) − 1 . On the other hand, it is wrong for some ( G, H ) if k < W # ( G ) − 1 , whatever r . Theorem. Let k ≥ 2 be a constant. 1. Let C be a class of graphs G with D k # ( G ) = O (log n ) . Then Graph Isomorphism for C is solvable in TC 1 . 2. Let C be a class of graphs G with D k ( G ) = O (log n ) . Then Graph Isomorphism for C is solvable in AC 1 . 15
Example: Trees Theorem. D 3 # ( T ) ≤ 3 log n + 2 for every tree T on n vertices. Proof (a separator strategy): Let T ′ �∼ = T (and assume T ′ is a tree too). We need to show that Spoiler wins the 3-pebble game on T and T ′ in 3 log n + 2 moves. Step 1. Spoiler pebbles a separator v in T (every component of T − v has ≤ n/ 2 vertices). Step 2. Spoiler ensures pebbling u ∈ Γ( v ) and u ′ ∈ Γ( v ′ ) so that the corresponding components are non-isomorphic rooted trees. u´ u v v´ Spoiler forces further play on these components and T´ applies the same strategy. T 16
Proof - continuation A complication: the strategy is now applied to a graph with one vertex pebbled and we may need more than 3 pebbles. Step 3. If T − v and T ′ − v ′ differ only u´ 0 by the components u 0 with pebbled verti- v ces u 0 and u ′ 0 , then v´ Spoiler pebbles a v 1 in the v - u 0 -path such that T − v 1 T ′ − v ′ T´ and differ T 1 by components with no pebble. (The case that u´ u 1 1 d ( v ′ , u ′ d ( v, u 0 ) � = 0 ) u´ u 0 v 0 v´ v´ v or d ( v, v 1 ) � = d ( v ′ , v ′ 1 ) 1 1 is even easier for Spoiler.) 17
Isomorphism of trees (a revision of the history) GI for trees is solvable in • in LOG SPACE Lindell 92 • in AC 1 Miller-Reif 91 • in AC 1 if ∆ = O (log n ) Ruzzo 81 • in LIN TIME by 1-WL ( W # ( T ) = 2 ) Aho-Hopcroft-Ullman 74 Miller and Reif [SIAM J. Comput. 91]: “No polylogarithmic parallel algorithm was previously known for isomorphism of unbounded-degree trees.” However, the 3 log n -round 2 -WL solves TREE ISO in TC 1 ⊆ NC 2 and is known since 68 ! 18
Estimates for particular classes Theorem. If a graph G on n vertices has treewidth k , then D 4 k +4 ( G ) < 2( k + 1) log n + 8 k + 9 . # Consequently, isomorphism of graphs whose treewidth does not exceed k is recognizable by the (4 k + 3) -dim WL in TC 1 ⊆ NC 2 . Theorem. For a 3-connected planar graph G on n vertices we have D 15 ( G ) < 11 log 2 n + 45 . Consequently, the isomorphism problem for 3-connected planar graphs is solvable by the 14-dim WL in AC 1 . 19
General bounds Consider G �∼ = H , both with n vertices. It is easy to find example where W ( G, H ) ≥ n + 1 . 2 Moreover: Theorem (Cai, F¨ urer, and Immerman) There are pairs of graphs such that W # ( G, H ) = Ω( n ) . On the other hand: How to show that D ( G, H ) < n for all such pairs? 20
An initial approach with the Ramsey theorem D ( G, H ) < n − 1 4 log 2 n. Indeed, as D ( G, H ) = D ( ¯ G, ¯ H ) we can assume that G contains an independent set A with | A | > 1 2 log 2 n . ✉ ✉ ✉ ✉ ✉ ✉ A B ✬ ✩ ✬ ✩ • Spoiler selects all V ( G ) \ A . • Duplicator selects V ( H ) \ B for some B , | B | = | A | . G H ✫ ✪ ✫ ✪ Identify V ( G ) \ A and V ( H ) \ B according to the partial isomorphism established. Suppose B is independent too for else Spoiler wins in 2 moves. Given X ⊆ V ( G ) \ A , let m G ( X ) = | { v ∈ A : Γ( v ) = X } | and m H ( X ) = | { v ∈ B : Γ( v ) = X } | . Since G �∼ = H , there is X with m G ( X ) � = m H ( X ) . Spoiler can now win in min { m G ( X ) , m H ( X ) } +1 moves. Since P X m G ( X ) = P X m H ( X ) , there are at least two such sets X 1 and X 2 and for one of them min { m G ( X i ) , m H ( X i ) } < | A | / 2 . 21
The final bound Theorem. D ( G, H ) ≤ n +3 for all non-isomorphic G and H on n 2 vertices. 22
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