Hanf normal form for first-order logic with unary counting quantifiers Lucas Heimberg 1 , Dietrich Kuske 2 , Nicole Schweikardt 1 1 Humboldt-Universität zu Berlin, 2 Technische Universität Ilmenau Highlights’16, Brussels
Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary In this talk Hanf normal form characterises the locality of first-order logic (FO) on classes of structures of bounded degree. • We generalise the notion of Hanf normal form to FO + sets Q of unary counting quantifiers Example: D consists of all modulo-counting quantifiers ∃ 0 mod 2 y y = y ϕ EVEN := • We provide a characterisation of the sets Q that permit generalised Hanf normal forms • We show how to compute generalised Hanf normal forms effectively and in worst-case optimal time L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers 2
Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary Neighbourhoods and spheres In this talk: All structures are finite, undirected and loop-free graphs. (but all results hold for arbitrary finite relational signatures) L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers 3
Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary Neighbourhoods and spheres In this talk: All structures are finite, undirected and loop-free graphs. (but all results hold for arbitrary finite relational signatures) A a L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers 3
Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary Neighbourhoods and spheres In this talk: All structures are finite, undirected and loop-free graphs. (but all results hold for arbitrary finite relational signatures) A r a r -neighbourhood of a in A N A r ( a ) := all nodes with distance ≤ r from a L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers 3
Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary Neighbourhoods and spheres In this talk: All structures are finite, undirected and loop-free graphs. (but all results hold for arbitrary finite relational signatures) A r a r -neighbourhood of a in A N A r ( a ) := all nodes with distance ≤ r from a r -sphere of a in A � � S A A [ N A r ( a ) := r ( a )] , a L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers 3
Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary Neighbourhoods and spheres In this talk: All structures are finite, undirected and loop-free graphs. (but all results hold for arbitrary finite relational signatures) A r a 1 a 2 r -neighbourhood of a in A N A r ( a ) := all nodes with distance ≤ r from a r -sphere of a in A � � S A A [ N A r ( a ) := r ( a )] , a L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers 3
Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary Hanf’s Theorem For each graph A and every r -sphere τ , let r ( u ) ∼ |{ u ∈ A : S A # τ ( A ) := = τ }| . Theorem (Hanf 1965; Fagin, Stockmeyer, Vardi 1995) For every degree bound d ≥ 0 and each quantifier rank q ≥ 0 , there is • a radius r ≥ 0 and • a threshold t ≥ 0 , such that for all graphs A , B with degree ≤ d, the following holds: If for every r-sphere τ with degree ≤ d, # τ ( A ) # τ ( B ) # τ ( A ) , # τ ( B ) ≥ = or t , then, for every FO-sentence ϕ with quantifier rank ≤ q, A | = ϕ ⇐ ⇒ B | = ϕ . L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers 4
Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary Hanf’s Theorem For each graph A and every r -sphere τ , let r ( u ) ∼ |{ u ∈ A : S A # τ ( A ) := = τ }| . Theorem (Hanf 1965; Fagin, Stockmeyer, Vardi 1995) For every degree bound d ≥ 0 and each quantifier rank q ≥ 0 , there is • a radius r ≥ 0 and • a threshold t ≥ 0 , such that for all graphs A , B with degree ≤ d, the following holds: If for every r-sphere τ with degree ≤ d, # τ ( A ) # τ ( B ) # τ ( A ) , # τ ( B ) ≥ = or t , then, for every FO-sentence ϕ with quantifier rank ≤ q, A | = ϕ ⇐ ⇒ B | = ϕ . Applications • Inexpressibility results • Algorithmic meta-theorems L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers 4
Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary Hanf normal form (HNF) L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers 5
Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary Hanf normal form (HNF) Consequence of Hanf’s Theorem For each d ≥ 0, every FO-sentence ϕ is d -equivalent to a Hanf normal form ψ , i.e., a Boolean combination of counting-sentences ∃ ≥ k y sph τ ( y ) Note: ϕ is d -equivalent to ψ iff A | = ϕ ⇐ ⇒ A | = ψ for all graphs A with degree ≤ d . L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers 5
Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary Hanf normal form (HNF) Consequence of Hanf’s Theorem For each d ≥ 0, every FO-sentence ϕ is d -equivalent to a Hanf normal form ψ , i.e., a Boolean combination of counting-sentences = ∃ ≥ k y sph τ ( y ) A | ⇐ ⇒ # τ ( A ) ≥ k Note: ϕ is d -equivalent to ψ iff A | = ϕ ⇐ ⇒ A | = ψ for all graphs A with degree ≤ d . L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers 5
Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary Hanf normal form (HNF) Consequence of Hanf’s Theorem For each d ≥ 0, every FO-sentence ϕ is d -equivalent to a Hanf normal form ψ , i.e., a Boolean combination of counting-sentences = ∃ ≥ k y sph τ ( y ) A | ⇐ ⇒ # τ ( A ) ≥ k Note: ϕ is d -equivalent to ψ iff A | = ϕ ⇐ ⇒ A | = ψ for all graphs A with degree ≤ d . Theorem (Bollig, Kuske 2012) There is an algorithm which, on input of • a degree bound d ≥ 0 and • an FO-sentence ϕ , computes a d -equivalent Hanf normal form in time 2 d 2 O ( | | ϕ | | ) ∈ 3-exp ( | | ϕ | | ) L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers 5
Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary Hanf normal form (HNF) Consequence of Hanf’s Theorem For each d ≥ 0, every FO-sentence ϕ is d -equivalent to a Hanf normal form ψ , i.e., a Boolean combination of counting-sentences = ∃ ≥ k y sph τ ( y ) A | ⇐ ⇒ # τ ( A ) ≥ k Note: ϕ is d -equivalent to ψ iff A | = ϕ ⇐ ⇒ A | = ψ for all graphs A with degree ≤ d . Theorem (Bollig, Kuske 2012) There is an algorithm which, on input of • a degree bound d ≥ 0 and • an FO-sentence ϕ , computes a d -equivalent Hanf normal form in time 2 d 2 O ( | | ϕ | | ) ∈ 3-exp ( | | ϕ | | ) For d ≥ 3, this is worst-case optimal. L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers 5
Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary HNF for FO + modulo-counting quantifiers Modulo-counting quantifier For a period p ≥ 2, ∃ 0 mod p y ϕ ( y ) L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers 6
Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary HNF for FO + modulo-counting quantifiers Modulo-counting quantifier For a period p ≥ 2, = ∃ 0 mod p y ϕ ( y ) A | ⇐ ⇒ |{ u ∈ A : A | = ϕ [ u ] }| ≡ 0 mod p . L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers 6
Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary HNF for FO + modulo-counting quantifiers Modulo-counting quantifier For a period p ≥ 2, = ∃ 0 mod p y ϕ ( y ) A | ⇐ ⇒ |{ u ∈ A : A | = ϕ [ u ] }| ≡ 0 mod p . Consequence of Nurmonen’s Theorem (Nurmonen 2000) For each d ≥ 0, each FO ( ∃ 0 mod p ) -sentence is d -equivalent to a HNF, i.e., a Boolean combination of counting-sentences and modulo-counting-sentences ∃ ≥ k y sph τ ( y ) ∃ r mod p y sph τ ( y ) L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers 6
Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary HNF for FO + modulo-counting quantifiers Modulo-counting quantifier For a period p ≥ 2, = ∃ 0 mod p y ϕ ( y ) A | ⇐ ⇒ |{ u ∈ A : A | = ϕ [ u ] }| ≡ 0 mod p . Consequence of Nurmonen’s Theorem (Nurmonen 2000) For each d ≥ 0, each FO ( ∃ 0 mod p ) -sentence is d -equivalent to a HNF, i.e., a Boolean combination of counting-sentences and modulo-counting-sentences = ∃ ≥ k y sph τ ( y ) A | ⇐ ⇒ # τ ( A ) ≥ k , = ∃ r mod p y sph τ ( y ) A | ⇐ ⇒ # τ ( A ) ≡ r mod p . L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers 6
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