There are no first-order sentences with quantifier depth 4 and an - PowerPoint PPT Presentation

There are no first-order sentences with quantifier depth 4 and an infinite spectrum Yury Yarovikov Moscow Institute of Physics and Technology June 9, 2019 Yury Yarovikov (Moscow Institute of Physics and Technology) There are no first-order sentences with quantifier depth 4 and an infinite spectrum June 9, 2019 1 / 40

Outline 1 First Order Spectra 2 Required theorems and constructions 3 How to play the Ehrenfeucht game 4 Proof sketch for 1 2 5 How to win for Spoiler 6 Future research Yury Yarovikov (Moscow Institute of Physics and Technology) There are no first-order sentences with quantifier depth 4 and an infinite spectrum June 9, 2019 2 / 40

Outline 1 First Order Spectra Yury Yarovikov (Moscow Institute of Physics and Technology) There are no first-order sentences with quantifier depth 4 and an infinite spectrum June 9, 2019 3 / 40

Erd˝ os–R´ enyi random graph model Definition Erd˝ os–R´ enyi random graph model is a probabilistic space G ( N, p ) = (Ω N , F N , P N,p ) , where N ∈ N , 0 � p � 1 , Ω N = {G = ( V N , E ) } − set of all graphs with V N = { 1 , 2 , . . . , N } , F N = 2 Ω N , P N,p ( G ) = p e ( G ) (1 − p ) C 2 N − e ( G ) . Yury Yarovikov (Moscow Institute of Physics and Technology) There are no first-order sentences with quantifier depth 4 and an infinite spectrum June 9, 2019 4 / 40

First-order graph properties Definition First-order graph property is a property defined by the first order formula with the folowing symbols: — predicate symbols: = , ∼ ; — logical symbols: → , ∧ , ∨ , ¬ , ...; — variables: x , y , ...; — quantifiers: ∃ , ∀ . Definition Quantifier depth of a first-order property Q is a minimal quantifier depth of a first-order formula that expresses Q . Yury Yarovikov (Moscow Institute of Physics and Technology) There are no first-order sentences with quantifier depth 4 and an infinite spectrum June 9, 2019 5 / 40

Zero-one law Definition For p = p ( N ) a zero-one law holds, if for each first-order property L P N,p ( L ) tends to either 0 or 1 . Consider probabilities p ( N ) = N − α . Theorem (J. Spencer, S. Shelah, 1988) Let p ( N ) = N − α . — Let α be a positive irrational. Then for p ( N ) a Zero-one law holds. � � l +1 , 1 + 1 1 — Let α be a positive rational. If α > 2 or α ∈ 1 + for some l ∈ N , l then for p ( N ) a Zero-one law holds. In all other cases a Zero-one law for p ( N ) does not hold. Yury Yarovikov (Moscow Institute of Physics and Technology) There are no first-order sentences with quantifier depth 4 and an infinite spectrum June 9, 2019 6 / 40

Zero-one k-law Definition For p = p ( N ) a zero-one k -law holds, if for each first-order property L with quantifier depth � k P N,p ( L ) tends to either 0 or 1 . Theorem (M. Zhukovskii, 2012) � � Let p ( N ) = N − α . If α ∈ 1 0 , , then for p ( N ) a zero-one k -law holds. k − 2 1 If α = k − 2 , then for p ( N ) a zero-one k -law does not hold. Yury Yarovikov (Moscow Institute of Physics and Technology) There are no first-order sentences with quantifier depth 4 and an infinite spectrum June 9, 2019 7 / 40

Fisrt-order spectra Definition We call a k -spectrum a set of all α ∈ (0 , 1) s.t. for p ( N ) = N − α a zero-one k -law does not hold. Consider a FO property Q . Definition Spectrum of Q is a set of all α ∈ (0 , 1) s.t. P N,p ( Q ) does not tend to either 0 or 1 for p ( N ) = N − α . A k -spectrum is clearly a union of all spectra of FO formulas Q with quantifier depth � k . Yury Yarovikov (Moscow Institute of Physics and Technology) There are no first-order sentences with quantifier depth 4 and an infinite spectrum June 9, 2019 8 / 40

Infinity of spectrum Theorem (J. Spencer, 1990) There exists a FO property of depth 14 with an infinite spectrum. Theorem (M. Zhukovskii, 2016) There exists a FO property of depth 5 with an infinite spectrum. The 5 -spectrum is finite. Theorem (M. Zhukovskii, A. Matushkin, 2017) The only possible limiting points of 4 -spectrum are 1 / 2 and 3 / 5 . Yury Yarovikov (Moscow Institute of Physics and Technology) There are no first-order sentences with quantifier depth 4 and an infinite spectrum June 9, 2019 9 / 40

Main result Theorem Points 1 2 and 3 5 cannot be limiting in 4 -spectrum. Therefore, 4 -spectrum is finite. Yury Yarovikov (Moscow Institute of Physics and Technology) There are no first-order sentences with quantifier depth 4 and an infinite spectrum June 9, 2019 10 / 40

Outline 2 Required theorems and constructions Yury Yarovikov (Moscow Institute of Physics and Technology) There are no first-order sentences with quantifier depth 4 and an infinite spectrum June 9, 2019 11 / 40

Ehrenfeucht game We now give a criterion of validity of the zero-one k -law. Theorem (A. Ehrenfeucht, 1960) Random graph G ( N, p ) obeys the zero-one k -law if and only if �� ( A, B ) : Duplicator has a winning strategy lim N,M →∞ P N,M,p �� in game EHR( A, B, k ) = 1 , где P N,M,p is a product of measures P N,p и P M,p . Yury Yarovikov (Moscow Institute of Physics and Technology) There are no first-order sentences with quantifier depth 4 and an infinite spectrum June 9, 2019 12 / 40

Threshold probabilities of properties “contain a copy of the subgraph” Let ρ ( G ) = e ( G ) v ( G ); ρ max ( G ) = max H ⊆ G ρ ( H ) . Theorem (Ruci´ nski A., Vince A, 1985) Let p 0 ( N ) = N − 1 /ρ max ( G ) . If p = o ( p 0 ) then G ( N, p ) a.a.s. does not contain a copy of G . If p 0 = o ( p ) then, on the contraty, G ( N, p ) a.a.s. does not contain a copy of G . Yury Yarovikov (Moscow Institute of Physics and Technology) There are no first-order sentences with quantifier depth 4 and an infinite spectrum June 9, 2019 13 / 40

Extensions Definition Let ( G, H ) and ( ˜ G, ˜ H ) , G ⊂ H , ˜ G ⊂ ˜ H be two pairs of graphs. Let V ( G ) = { x 1 , . . . , x m } , V ( H ) = { x 1 , . . . , x l } , V ( ˜ x m } , V ( ˜ G ) = { ˜ x 1 , . . . , ˜ H ) = { ˜ x 1 , . . . , ˜ x l } . x j ) ∈ E ( ˜ G ) \ E ( ˜ H ) , then ˜ If ( x i , x j ) ∈ E ( G ) \ E ( H ) ⇒ (˜ x i , ˜ G is called a ( G, H ) -extension of ˜ H . Yury Yarovikov (Moscow Institute of Physics and Technology) There are no first-order sentences with quantifier depth 4 and an infinite spectrum June 9, 2019 14 / 40

Extensions Fix α > 0 . Let v ( G, H ) = v ( G ) − v ( H ) , e ( G, H ) = e ( G ) − e ( H ) , f α ( G, H ) = v ( G, H ) − αe ( G, H ) . Yury Yarovikov (Moscow Institute of Physics and Technology) There are no first-order sentences with quantifier depth 4 and an infinite spectrum June 9, 2019 15 / 40

α -safe extensions f α ( G, H ) = v ( G, H ) − αe ( G, H ) . Pair ( G, H ) is called α -safe, if ∀ S ( H ⊂ S ⊆ G → f α ( S, H ) > 0) . 1 2 -safe If the pair ( G, H ) is α -safe then a.a.s. there is (there are many) a ( G, H ) -extension of each subgraph ˜ H in the random graph G ( N, p ) . Yury Yarovikov (Moscow Institute of Physics and Technology) There are no first-order sentences with quantifier depth 4 and an infinite spectrum June 9, 2019 16 / 40

α -rigid extensions f α ( G, H ) = v ( G, H ) − αe ( G, H ) . Pair ( G, H ) is called α -rigid, if ∀ S ( H ⊆ S ⊂ G → f α ( G, S ) < 0) . 1 3 -rigid Yury Yarovikov (Moscow Institute of Physics and Technology) There are no first-order sentences with quantifier depth 4 and an infinite spectrum June 9, 2019 17 / 40

α -neutral extensions f α ( G, H ) = v ( G, H ) − αe ( G, H ) . Pair ( G, H ) is called α -neutral, if ∀ S ( H ⊂ S ⊂ G → f α ( S, H ) > 0) и f α ( G, H ) = 0 3 5 -neutral Yury Yarovikov (Moscow Institute of Physics and Technology) There are no first-order sentences with quantifier depth 4 and an infinite spectrum June 9, 2019 18 / 40

Maximal extensions Question: Can there be any α -neutral or α -rigid extensions in a random graph? Answer: Yes, there can be. But we can choose new vertices so that there are none. α ∈ ( 1 2 , 1 2 + ε ) We can choose a ( G, H ) -extension such that there are no outer vertices of degree � 2 Yury Yarovikov (Moscow Institute of Physics and Technology) There are no first-order sentences with quantifier depth 4 and an infinite spectrum June 9, 2019 19 / 40

Outline 3 How to play the Ehrenfeucht game Yury Yarovikov (Moscow Institute of Physics and Technology) There are no first-order sentences with quantifier depth 4 and an infinite spectrum June 9, 2019 20 / 40

How to play the Ehrenfeucht game α ∈ ( 1 2 , 1 2 + ε ) Yury Yarovikov (Moscow Institute of Physics and Technology) There are no first-order sentences with quantifier depth 4 and an infinite spectrum June 9, 2019 21 / 40

How to play the Ehrenfeucht game α ∈ ( 1 2 , 1 2 + ε ) Yury Yarovikov (Moscow Institute of Physics and Technology) There are no first-order sentences with quantifier depth 4 and an infinite spectrum June 9, 2019 22 / 40

How to play the Ehrenfeucht game α ∈ ( 1 2 , 1 2 + ε ) Yury Yarovikov (Moscow Institute of Physics and Technology) There are no first-order sentences with quantifier depth 4 and an infinite spectrum June 9, 2019 23 / 40

How to play the Ehrenfeucht game α ∈ ( 1 2 , 1 2 + ε ) Yury Yarovikov (Moscow Institute of Physics and Technology) There are no first-order sentences with quantifier depth 4 and an infinite spectrum June 9, 2019 24 / 40