Hanf normal form for first-order logic with unary counting - - PowerPoint PPT Presentation

Hanf normal form for first-order logic with unary counting quantifiers

Lucas Heimberg1, Dietrich Kuske2, Nicole Schweikardt1

1Humboldt-Universität zu Berlin, 2Technische 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)

• n 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 ϕEVEN := ∃0 mod 2y y=y

• 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)

r a A r-neighbourhood of a in A NA

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)

r a A r-neighbourhood of a in A NA

r (a) :=

all nodes with distance ≤ r from a r-sphere of a in A SA

r (a) :=

• A[NA

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)

r a2 a1 A r-neighbourhood of a in A NA

r (a) :=

all nodes with distance ≤ r from a r-sphere of a in A SA

r (a) :=

• A[NA

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 #τ(A) := |{u ∈ A : SA

r (u) ∼

= 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)

• r

#τ(A), #τ(B) ≥ 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 #τ(A) := |{u ∈ A : SA

r (u) ∼

= 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)

• r

#τ(A), #τ(B) ≥ 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 ∃≥ky sphτ(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 A | = ∃≥ky sphτ(y) ⇐ ⇒ #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 A | = ∃≥ky sphτ(y) ⇐ ⇒ #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 2d2O(|

|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 A | = ∃≥ky sphτ(y) ⇐ ⇒ #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 2d2O(|

|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 py 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, A | = ∃0 mod py ϕ(y) ⇐ ⇒ |{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, A | = ∃0 mod py ϕ(y) ⇐ ⇒ |{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 ∃≥ky sphτ(y) ∃r mod py sphτ(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, A | = ∃0 mod py ϕ(y) ⇐ ⇒ |{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 A | = ∃≥ky sphτ(y) ⇐ ⇒ #τ(A) ≥ k , A | = ∃r mod py sphτ(y) ⇐ ⇒ #τ(A) ≡ r 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, A | = ∃0 mod py ϕ(y) ⇐ ⇒ |{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 A | = ∃≥ky sphτ(y) ⇐ ⇒ #τ(A) ≥ k , A | = ∃r mod py sphτ(y) ⇐ ⇒ #τ(A) ≡ r mod p .

How can Hanf normal forms for FO(∃0 mod p) be computed?

• 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, A | = ∃0 mod py ϕ(y) ⇐ ⇒ |{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 A | = ∃≥ky sphτ(y) ⇐ ⇒ #τ(A) ≥ k , A | = ∃r mod py sphτ(y) ⇐ ⇒ #τ(A) ≡ r mod p . Theorem Let D denote the set of all modulo-counting quantifiers. There is an algorithm which, on input of

• a degree bound d ≥ 0 and
• an FO(D)-sentence ϕ,

computes a d-equivalent Hanf normal form in time 2d2O(|

|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

6

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

HNF for FO + unary counting quantifiers

Unary counting quantifier For every S ⊆ N, Sy ϕ(y)

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

7

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

HNF for FO + unary counting quantifiers

Unary counting quantifier For every S ⊆ N, A | = Sy ϕ(y) ⇐ ⇒ |{u ∈ A : A | = ϕ[u]}| ∈ S . For every set Q ⊆ P(N), FO(Q) := extension of FO by all quantifiers S ∈ Q.

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

7

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

HNF for FO + unary counting quantifiers

Unary counting quantifier For every S ⊆ N, A | = Sy ϕ(y) ⇐ ⇒ |{u ∈ A : A | = ϕ[u]}| ∈ S . For every set Q ⊆ P(N), FO(Q) := extension of FO by all quantifiers S ∈ Q. Hanf normal form for FO(Q) Boolean combination of sentences of the shape (S+k)y sphτ(y) where S ∈ Q and (S+k) := {n + k : n ∈ S}.

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

7

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

HNF for FO + unary counting quantifiers

Unary counting quantifier For every S ⊆ N, A | = Sy ϕ(y) ⇐ ⇒ |{u ∈ A : A | = ϕ[u]}| ∈ S . For every set Q ⊆ P(N), FO(Q) := extension of FO by all quantifiers S ∈ Q. Hanf normal form for FO(Q) Boolean combination of sentences of the shape A | = (S+k)y sphτ(y) ⇐ ⇒ |{u ∈ A : A | = ϕ[u]}| ∈ (S+k) where S ∈ Q and (S+k) := {n + k : n ∈ S}.

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

7

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

HNF for FO + unary counting quantifiers

Unary counting quantifier For every S ⊆ N, A | = Sy ϕ(y) ⇐ ⇒ |{u ∈ A : A | = ϕ[u]}| ∈ S . For every set Q ⊆ P(N), FO(Q) := extension of FO by all quantifiers S ∈ Q. Hanf normal form for FO(Q) Boolean combination of sentences of the shape A | = (S+k)y sphτ(y) ⇐ ⇒ |{u ∈ A : A | = ϕ[u]}| ∈ (S+k) where S ∈ Q and (S+k) := {n + k : n ∈ S}. Q ⊆ P(N) permits Hanf normal form iff for each d ≥ 0 and every FO(Q)-sentence, there is a d-equivalent FO(Q)-sentence in Hanf normal form Examples: ∅ and D permit Hanf normal form.

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

7

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

Results

Which sets Q ⊆ P(N) permit Hanf normal form?

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

8

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

Results

Theorem For every set Q ⊆ P(N) of unary counting quantifiers, Q permits Hanf normal form = ⇒ all S ∈ Q are “ultimately periodic”.

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

8

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

Results

Theorem For every set Q ⊆ P(N) of unary counting quantifiers, Q permits Hanf normal form = ⇒ all S ∈ Q are “ultimately periodic”.

How to compute such Hanf normal forms?

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

8

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

Results

Theorem For every set Q ⊆ P(N) of unary counting quantifiers, Q permits Hanf normal form ⇐ ⇒ all S ∈ Q are “ultimately periodic”. Theorem There is an algorithm which, on input of

• a degree bound d ≥ 0 and
• an FO(Q)-formula ϕ (where all S ∈ Q are “ultimately periodic”),

computes a d-equivalent Hanf normal form in time 2d2O(|

|ϕ| |)

∈ 3-exp(| |ϕ| |).

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

8

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

Results

Theorem For every set Q ⊆ P(N) of unary counting quantifiers, Q permits Hanf normal form ⇐ ⇒ all S ∈ Q are “ultimately periodic”. Theorem There is an algorithm which, on input of

• a degree bound d ≥ 0 and
• an FO(Q)-formula ϕ (where all S ∈ Q are “ultimately periodic”),

computes a d-equivalent Hanf normal form in time 2d2O(|

|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

8

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

Results

Theorem For every set Q ⊆ P(N) of unary counting quantifiers, Q permits Hanf normal form ⇐ ⇒ all S ∈ Q are “ultimately periodic”. Theorem There is an algorithm which, on input of

• a degree bound d ≥ 0 and
• an FO(Q)-formula ϕ (where all S ∈ Q are “ultimately periodic”),

computes a d-equivalent Hanf normal form in time 2d2O(|

|ϕ| |)

∈ 3-exp(| |ϕ| |). For d ≥ 3, this is worst-case optimal. An application: Generalisation of Seese’s model-checking algorithm for graphs of bounded degree If all S ∈ Q are “ultimately periodic”, then, for each d ≥ 0, model-checking for FO(Q) on graphs of degree ≤ d is fixed-parameter tractable.

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

8

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

Ultimately periodic quantifiers

S ⊆ N is ultimately periodic iff there is an offset n0 ≥ 0 and a period p ≥ 1 such that for all n ≥ n0, n ∈ S ⇐ ⇒ n + p ∈ S.

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

9

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

Ultimately periodic quantifiers

S ⊆ N is ultimately periodic iff there is an offset n0 ≥ 0 and a period p ≥ 1 such that for all n ≥ n0, n ∈ S ⇐ ⇒ n + p ∈ S. 1 1

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

9

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

Ultimately periodic quantifiers

S ⊆ N is ultimately periodic iff there is an offset n0 ≥ 0 and a period p ≥ 1 such that for all n ≥ n0, n ∈ S ⇐ ⇒ n + p ∈ S. 1 1 1 1

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

9

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

Ultimately periodic quantifiers

S ⊆ N is ultimately periodic iff there is an offset n0 ≥ 0 and a period p ≥ 1 such that for all n ≥ n0, n ∈ S ⇐ ⇒ n + p ∈ S. 1 1 1 1 1 1

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

9

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

Ultimately periodic quantifiers

S ⊆ N is ultimately periodic iff there is an offset n0 ≥ 0 and a period p ≥ 1 such that for all n ≥ n0, n ∈ S ⇐ ⇒ n + p ∈ S. 1 1 1 1 1 1 1 1 · · ·

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

9

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

Ultimately periodic quantifiers

S ⊆ N is ultimately periodic iff there is an offset n0 ≥ 0 and a period p ≥ 1 such that for all n ≥ n0, n ∈ S ⇐ ⇒ n + p ∈ S. 1 1 1 1 1 1 1 1 · · · Examples:

• ∃ and ∃0 mod p are ultimately periodic,
• {n2 : n ∈ N} and {n : n is prime} are not ultimately periodic.
• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

9

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

Ultimately periodic quantifiers

S ⊆ N is ultimately periodic iff there is an offset n0 ≥ 0 and a period p ≥ 1 such that for all n ≥ n0, n ∈ S ⇐ ⇒ n + p ∈ S. 1 1 1 1 1 1 1 1 · · · Examples:

• ∃ and ∃0 mod p are ultimately periodic,
• {n2 : n ∈ N} and {n : n is prime} are not ultimately periodic.

Only ultimately periodic quantifiers permit Hanf normal form Let σ := {P} and let Q ⊆ P(N) with S ∈ Q not ultimately periodic. There is no Hanf normal form δ ∈ FO(Q)[σ] such that for every σ-structures A, A | = δ ⇐ ⇒ |A| ∈ S. Consequence: There is no Hanf normal form for Sy y=y.

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

9

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

δ ∈ FO(Q)[σ]: Boolean combination of Hanf-sentences (R+ℓ)y P(y) and (R+ℓ)y ¬P(y) where R ∈ {R1, . . . , Rn} ⊆ Q and ℓ ∈ {0, . . . , k} for a k ≥ 0.

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

10

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

δ ∈ FO(Q)[σ]: Boolean combination of Hanf-sentences (R+ℓ)y P(y) and (R+ℓ)y ¬P(y) where R ∈ {R1, . . . , Rn} ⊆ Q and ℓ ∈ {0, . . . , k} for a k ≥ 0. For each Ri and a ≥ k: · · ·

a k

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

10

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

δ ∈ FO(Q)[σ]: Boolean combination of Hanf-sentences (R+ℓ)y P(y) and (R+ℓ)y ¬P(y) where R ∈ {R1, . . . , Rn} ⊆ Q and ℓ ∈ {0, . . . , k} for a k ≥ 0. For each Ri and a ≥ k: · · ·

a k

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

10

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

δ ∈ FO(Q)[σ]: Boolean combination of Hanf-sentences (R+ℓ)y P(y) and (R+ℓ)y ¬P(y) where R ∈ {R1, . . . , Rn} ⊆ Q and ℓ ∈ {0, . . . , k} for a k ≥ 0. For each Ri and a ≥ k: · · ·

a k

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

10

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

δ ∈ FO(Q)[σ]: Boolean combination of Hanf-sentences (R+ℓ)y P(y) and (R+ℓ)y ¬P(y) where R ∈ {R1, . . . , Rn} ⊆ Q and ℓ ∈ {0, . . . , k} for a k ≥ 0. For each Ri and a ≥ k: · · ·

a k

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

10

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

δ ∈ FO(Q)[σ]: Boolean combination of Hanf-sentences (R+ℓ)y P(y) and (R+ℓ)y ¬P(y) where R ∈ {R1, . . . , Rn} ⊆ Q and ℓ ∈ {0, . . . , k} for a k ≥ 0. For each Ri and a ≥ k: · · ·

a k

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

10

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

δ ∈ FO(Q)[σ]: Boolean combination of Hanf-sentences (R+ℓ)y P(y) and (R+ℓ)y ¬P(y) where R ∈ {R1, . . . , Rn} ⊆ Q and ℓ ∈ {0, . . . , k} for a k ≥ 0. For each Ri and a ≥ k: · · ·

a k

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

10

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

δ ∈ FO(Q)[σ]: Boolean combination of Hanf-sentences (R+ℓ)y P(y) and (R+ℓ)y ¬P(y) where R ∈ {R1, . . . , Rn} ⊆ Q and ℓ ∈ {0, . . . , k} for a k ≥ 0. For each Ri and a ≥ k: · · ·

a k

wa :=

R1(a−k, a] R2(a−k, a]

· · ·

Ri (a−k, a]

· · ·

Rn(a−k, a] ∈ {0, 1}k·n

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

10

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

δ ∈ FO(Q)[σ]: Boolean combination of Hanf-sentences (R+ℓ)y P(y) and (R+ℓ)y ¬P(y) where R ∈ {R1, . . . , Rn} ⊆ Q and ℓ ∈ {0, . . . , k} for a k ≥ 0. For each Ri and a ≥ k: · · ·

a k

wa :=

R1(a−k, a] R2(a−k, a]

· · ·

Ri (a−k, a]

· · ·

Rn(a−k, a] ∈ {0, 1}k·n

• ex. b > a with wb :=

R1(b−k, b] R2(b−k, b]

· · ·

Ri (b−k, b]

· · ·

Rn(b−k, b] ∈ {0, 1}k·n

=

A, B: σ-structures with |PA| = a, |PB| = b .

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

10

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

δ ∈ FO(Q)[σ]: Boolean combination of Hanf-sentences (R+ℓ)y P(y) and (R+ℓ)y ¬P(y) where R ∈ {R1, . . . , Rn} ⊆ Q and ℓ ∈ {0, . . . , k} for a k ≥ 0. For each Ri and a ≥ k: · · ·

a k

wa :=

R1(a−k, a] R2(a−k, a]

· · ·

Ri (a−k, a]

· · ·

Rn(a−k, a] ∈ {0, 1}k·n

• ex. b > a with wb :=

R1(b−k, b] R2(b−k, b]

· · ·

Ri (b−k, b]

· · ·

Rn(b−k, b] ∈ {0, 1}k·n

=

Recall: S is not ultimately periodic. Hence, there

• ex. c ≥ 0 such that

a + c ∈ S ⇐ ⇒ b + c ∈ S. A, B: σ-structures with |PA| = a, |PB| = b, |A \ PA| = |B \ PB| = c. A | = δ ⇐ ⇒ B | = δ and |A| ∈ S ⇐ ⇒ |B| ∈ S

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

10

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

Constructing Hanf normal forms for FO(D)

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

11

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

Constructing Hanf normal forms for FO(D)

Hanf normal form for formulas with free variables Boolean combination of (modulo-)counting-sentences and sphere-formulas sphτ(x) (where τ has |x| centres)

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

11

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

Constructing Hanf normal forms for FO(D)

Hanf normal form for formulas with free variables Boolean combination of (modulo-)counting-sentences and sphere-formulas (A, a) | = sphτ(x) ⇐ ⇒ SA

r (a) ∼

= τ . (where τ has |x| centres)

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

11

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

Constructing Hanf normal forms for FO(D)

Hanf normal form for formulas with free variables Boolean combination of (modulo-)counting-sentences and sphere-formulas (A, a) | = sphτ(x) ⇐ ⇒ SA

r (a) ∼

= τ . (where τ has |x| centres) Task: On input of d ≥ 0 and ϕ ∈ FO(D), compute a d-equivalent HNF ψ.

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

11

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

Constructing Hanf normal forms for FO(D)

Hanf normal form for formulas with free variables Boolean combination of (modulo-)counting-sentences and sphere-formulas (A, a) | = sphτ(x) ⇐ ⇒ SA

r (a) ∼

= τ . (where τ has |x| centres) Task: On input of d ≥ 0 and ϕ ∈ FO(D), compute a d-equivalent HNF ψ. Interesting case ϕ(x) = Qy ψ′(x, y) with ψ′ in HNF s.t. each sphere in ψ′ has radius ≤ r.

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

11

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

Constructing Hanf normal forms for FO(D)

Hanf normal form for formulas with free variables Boolean combination of (modulo-)counting-sentences and sphere-formulas (A, a) | = sphτ(x) ⇐ ⇒ SA

r (a) ∼

= τ . (where τ has |x| centres) Task: On input of d ≥ 0 and ϕ ∈ FO(D), compute a d-equivalent HNF ψ. Interesting case ϕ(x) = Qy ψ′(x, y) with ψ′ in HNF s.t. each sphere in ψ′ has radius ≤ r. ϕ(x) is d-equivalent to . . . Qy ψ′(x, y)

• HNF
• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

11

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

Constructing Hanf normal forms for FO(D)

Hanf normal form for formulas with free variables Boolean combination of (modulo-)counting-sentences and sphere-formulas (A, a) | = sphτ(x) ⇐ ⇒ SA

r (a) ∼

= τ . (where τ has |x| centres) Task: On input of d ≥ 0 and ϕ ∈ FO(D), compute a d-equivalent HNF ψ. Interesting case ϕ(x) = Qy ψ′(x, y) with ψ′ in HNF s.t. each sphere in ψ′ has radius ≤ r. ϕ(x) is d-equivalent to . . . Qy

• τ∈T
• sphτ(x, y) ∧

ψ′

τ

• HNF
• where

T := all (up to isomorphism) d-bounded r-spheres with |x, y| centres.

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

11

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

Constructing Hanf normal forms for FO(D)

Hanf normal form for formulas with free variables Boolean combination of (modulo-)counting-sentences and sphere-formulas (A, a) | = sphτ(x) ⇐ ⇒ SA

r (a) ∼

= τ . (where τ has |x| centres) Task: On input of d ≥ 0 and ϕ ∈ FO(D), compute a d-equivalent HNF ψ. Interesting case ϕ(x) = Qy ψ′(x, y) with ψ′ in HNF s.t. each sphere in ψ′ has radius ≤ r. ϕ(x) is d-equivalent to . . . a Boolean combination of formulas ψ′

τ

• HNF

∧ (S+k)y sphτ(x, y)

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

11

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

Constructing Hanf normal forms for FO(D)

Hanf normal form for formulas with free variables Boolean combination of (modulo-)counting-sentences and sphere-formulas (A, a) | = sphτ(x) ⇐ ⇒ SA

r (a) ∼

= τ . (where τ has |x| centres) Task: On input of d ≥ 0 and ϕ ∈ FO(D), compute a d-equivalent HNF ψ. Interesting case ϕ(x) = Qy ψ′(x, y) with ψ′ in HNF s.t. each sphere in ψ′ has radius ≤ r. ϕ(x) is d-equivalent to . . . a Boolean combination of formulas ψ′

τ ∧ ατ(x)

• HNF
• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

11

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

Summary

Theorem For every set Q ⊆ P(N) of unary counting quantifiers, Q permits Hanf normal form ⇐ ⇒ all S ∈ Q are ultimately periodic. Theorem There is an algorithm which, on input of

• a degree bound d ≥ 0 and
• an FO(Q)-formula ϕ (where all S ∈ Q are ultimately periodic),

computes a d-equivalent Hanf normal form in time 2d2O(|

|ϕ| |)

∈ 3-exp(| |ϕ| |). For d ≥ 3, this is worst-case optimal. An application: Generalisation of Seese’s model-checking algorithm for graphs of bounded degree If all S ∈ Q are ultimately periodic, then, for each d ≥ 0, model-checking for FO(Q) on graphs of degree ≤ d is fixed-parameter tractable.

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

12

Introduction Hanf normal form Results Ultimately periodic quantifiers Algorithm Summary

Thank you very much for your attention!

Theorem For every set Q ⊆ P(N) of unary counting quantifiers, Q permits Hanf normal form ⇐ ⇒ all S ∈ Q are ultimately periodic. Theorem There is an algorithm which, on input of

• a degree bound d ≥ 0 and
• an FO(Q)-formula ϕ (where all S ∈ Q are ultimately periodic),

computes a d-equivalent Hanf normal form in time 2d2O(|

|ϕ| |)

∈ 3-exp(| |ϕ| |). For d ≥ 3, this is worst-case optimal. An application: Generalisation of Seese’s model-checking algorithm for graphs of bounded degree If all S ∈ Q are ultimately periodic, then, for each d ≥ 0, model-checking for FO(Q) on graphs of degree ≤ d is fixed-parameter tractable.

• L. Heimberg, D. Kuske, N. Schweikardt: Hanf normal forms for unary counting quantifiers

12