[PPT] - Expressive Power of Monadic Second-Order Logic on Finite Structures PowerPoint Presentation

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Expressive Power of Monadic Second-Order Logic on Finite Structures

Michael Elberfeld

RWTH Aachen University

AutoMathA 2015 Leipzig, 9 May, 2015 (Comments: Slides of a 50-minutes invited talk)

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Formulas Define Properties of Graphs

Graph G is a Logical Structure

V(G) is the universe, we say vertices
E(G) ⊆ V(G)×V(G) is a binary relation, we say edges

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Formulas Define Properties of Graphs

Graph G is a Logical Structure

V(G) is the universe, we say vertices
E(G) ⊆ V(G)×V(G) is a binary relation, we say edges

First-Order Logic (FO-Logic)

ϕ CLIQUE := ∀u,v ∈ V(G)

u = v ↔ (u,v) ∈ E(G)
defines cliques, e.g.

| = ϕ CLIQUE

SLIDE 4

2 of 23

Formulas Define Properties of Graphs

Graph G is a Logical Structure

V(G) is the universe, we say vertices
E(G) ⊆ V(G)×V(G) is a binary relation, we say edges

First-Order Logic (FO-Logic)

ϕ CLIQUE := ∀u,v ∈ V(G)

u = v ↔ (u,v) ∈ E(G)
defines cliques, e.g.

| = ϕ CLIQUE

Monadic Second-Order Logic (MSO-Logic)

ϕ EVEN-PATH := ϕ PATH ∧∃R,B ⊆ V(G)

ϕ ALTERNATING-COLORING(R,B)
defines paths with even universe, e.g.

| = ϕ EVEN-PATH

SLIDE 5

2 of 23

Formulas Define Properties of Graphs

Graph G is a Logical Structure

V(G) is the universe, we say vertices
E(G) ⊆ V(G)×V(G) is a binary relation, we say edges

First-Order Logic (FO-Logic)

ϕ CLIQUE := ∀u,v ∈ V(G)

u = v ↔ (u,v) ∈ E(G)
defines cliques, e.g.

| = ϕ CLIQUE

Monadic Second-Order Logic (MSO-Logic)

ϕ EVEN-PATH := ϕ PATH ∧∃R,B ⊆ V(G)

ϕ ALTERNATING-COLORING(R,B)
defines paths with even universe, e.g.

| = ϕ EVEN-PATH

SLIDE 6

2 of 23

Formulas Define Properties of Graphs

Graph G is a Logical Structure

V(G) is the universe, we say vertices
E(G) ⊆ V(G)×V(G) is a binary relation, we say edges

First-Order Logic (FO-Logic)

ϕ CLIQUE := ∀u,v ∈ V(G)

u = v ↔ (u,v) ∈ E(G)
defines cliques, e.g.

| = ϕ CLIQUE

Monadic Second-Order Logic (MSO-Logic)

ϕ EVEN-PATH := ϕ PATH ∧∃R,B ⊆ V(G)

ϕ ALTERNATING-COLORING(R,B)
defines paths with even universe, e.g.

| = ϕ EVEN-PATH

Guarded Second-Order Logic (GSO-Logic)

ϕ EVEN-CLIQUE := ϕ CLIQUE ∧∃F ⊆ E(G)

ϕ EVEN-PATH(F)
defines cliques with even universe, e.g.

| = ϕ EVEN-CLIQUE

SLIDE 7

2 of 23

Formulas Define Properties of Graphs

Graph G is a Logical Structure

V(G) is the universe, we say vertices
E(G) ⊆ V(G)×V(G) is a binary relation, we say edges

First-Order Logic (FO-Logic)

ϕ CLIQUE := ∀u,v ∈ V(G)

u = v ↔ (u,v) ∈ E(G)
defines cliques, e.g.

| = ϕ CLIQUE

Monadic Second-Order Logic (MSO-Logic)

ϕ EVEN-PATH := ϕ PATH ∧∃R,B ⊆ V(G)

ϕ ALTERNATING-COLORING(R,B)
defines paths with even universe, e.g.

| = ϕ EVEN-PATH

Guarded Second-Order Logic (GSO-Logic)

ϕ EVEN-CLIQUE := ϕ CLIQUE ∧∃F ⊆ E(G)

ϕ EVEN-PATH(F)
defines cliques with even universe, e.g.

| = ϕ EVEN-CLIQUE

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Expressive Powers of Logics Differ

Expressive Power

FO := {property P : t.e. FO-formula ϕ s.t. f.e. G we have G | = ϕ iff G ∈ P} MSO and GSO are defined in the same way

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Expressive Powers of Logics Differ

Expressive Power

FO := {property P : t.e. FO-formula ϕ s.t. f.e. G we have G | = ϕ iff G ∈ P} MSO and GSO are defined in the same way

Increasing Expressive Power

FO MSO GSO

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Expressive Powers of Logics Differ

Expressive Power

FO := {property P : t.e. FO-formula ϕ s.t. f.e. G we have G | = ϕ iff G ∈ P} MSO and GSO are defined in the same way

Increasing Expressive Power

FO MSO GSO

Proof.

FO ⊆ MSO and EVEN-PATH /

∈ FO, but ∈ MSO

MSO ⊆ GSO and EVEN-CLIQUE /

∈ MSO, but ∈ GSO

[Ebbinghaus and Flum, 1999, Libkin, 2004]

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Expressive Power Depends on Classes of Graphs

Expressive Power on Class of Graphs C

FO on C := {property P : t.e. FO-formula ϕ s.t. f.e. G ∈ C we have G | = ϕ iff G ∈ P} MSO on C and GSO on C are defined in the same way

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Expressive Power Depends on Classes of Graphs

Expressive Power on Class of Graphs C

FO on C := {property P : t.e. FO-formula ϕ s.t. f.e. G ∈ C we have G | = ϕ iff G ∈ P} MSO on C and GSO on C are defined in the same way

Expressive Power on Paths

FO MSO = GSO on class C of all

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Expressive Power Depends on Classes of Graphs

Expressive Power on Class of Graphs C

FO on C := {property P : t.e. FO-formula ϕ s.t. f.e. G ∈ C we have G | = ϕ iff G ∈ P} MSO on C and GSO on C are defined in the same way

Expressive Power on Paths

FO MSO = GSO on class C of all

Expressive Power on Cliques

FO = MSO GSO on class C of all (or )

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Interplay between Logics and Graphs is Ubiquitous

Talk’s Topic

What is the influence of graph classes on expressivity?

Where do logics coincide?
Where do logics differ?

Applications

Meta Theorems in Algorithm Design
Automata Theory (on Graphs)

[Courcelle and Engelfriet, 2012]

Descriptive Complexity Theory
Database Query Optimization

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Interplay between Logics and Graphs is Ubiquitous

Talk’s Topic

What is the influence of graph classes on expressivity?

Where do logics coincide?
Where do logics differ?

Applications

Meta Theorems in Algorithm Design
Automata Theory (on Graphs)

[Courcelle and Engelfriet, 2012]

Descriptive Complexity Theory
Database Query Optimization

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Interplay between Logics and Graphs is Ubiquitous

Talk’s Topic

What is the influence of graph classes on expressivity?

Where do logics coincide?
Where do logics differ?

Applications

Meta Theorems in Algorithm Design
Automata Theory (on Graphs)

[Courcelle and Engelfriet, 2012]

Descriptive Complexity Theory
Database Query Optimization

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Interplay between Logics and Graphs is Ubiquitous

Talk’s Topic

What is the influence of graph classes on expressivity?

Where do logics coincide?
Where do logics differ?

Applications

Meta Theorems in Algorithm Design
Automata Theory (on Graphs)

[Courcelle and Engelfriet, 2012]

Descriptive Complexity Theory
Database Query Optimization

SLIDE 18

5 of 23

Interplay between Logics and Graphs is Ubiquitous

Talk’s Topic

What is the influence of graph classes on expressivity?

Where do logics coincide?
Where do logics differ?

Applications

Meta Theorems in Algorithm Design
Automata Theory (on Graphs)

[Courcelle and Engelfriet, 2012]

Descriptive Complexity Theory
Database Query Optimization

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5 of 23

Interplay between Logics and Graphs is Ubiquitous

Talk’s Topic

What is the influence of graph classes on expressivity?

Where do logics coincide?
Where do logics differ?

Applications

Meta Theorems in Algorithm Design
Automata Theory (on Graphs)

[Courcelle and Engelfriet, 2012]

Descriptive Complexity Theory
Database Query Optimization

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Interplay between Logics and Graphs is Ubiquitous

Talk’s Topic

What is the influence of graph classes on expressivity?

Where do logics coincide?
Where do logics differ?

Applications

Meta Theorems in Algorithm Design
Automata Theory (on Graphs)

[Courcelle and Engelfriet, 2012]

Descriptive Complexity Theory
Database Query Optimization

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Talk’s Content

1 Collapsing MSO- to FO-Logic on Graph Classes 2 Separating MSO- from FO-Logic on Graph Classes 3 Collapsing GSO- to MSO-Logic on Graph Classes 4 Collapsing Logics via Definable Decompositions

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Talk’s Content

1 Collapsing MSO- to FO-Logic on Graph Classes 2 Separating MSO- from FO-Logic on Graph Classes 3 Collapsing GSO- to MSO-Logic on Graph Classes 4 Collapsing Logics via Definable Decompositions

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FO = MSO on Graphs Containing only Short Paths

Tree Depth of a Graph

[Neˇ setˇ ril and Ossona de Mendez, 2006]

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FO = MSO on Graphs Containing only Short Paths

Tree Depth of a Graph

Single vertex tree-depth = 1

[Neˇ setˇ ril and Ossona de Mendez, 2006]

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FO = MSO on Graphs Containing only Short Paths

Tree Depth of a Graph

Single vertex tree-depth = 1 Disconnected tree-depth( . . . ) = max

component

tree-depth( )

[Neˇ setˇ ril and Ossona de Mendez, 2006]

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FO = MSO on Graphs Containing only Short Paths

Tree Depth of a Graph

Single vertex tree-depth = 1 Disconnected tree-depth( . . . ) = max

component

tree-depth( ) Connected tree-depth

. . .
=

minvertex tree-depth( . . . )+1

[Neˇ setˇ ril and Ossona de Mendez, 2006]

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FO = MSO on Graphs Containing only Short Paths

Tree Depth of a Graph

Single vertex tree-depth = 1 Disconnected tree-depth( . . . ) = max

component

tree-depth( ) Connected tree-depth

. . .
=

minvertex tree-depth( . . . )+1

[Neˇ setˇ ril and Ossona de Mendez, 2006]

Fact

C’s graphs have bounded tree depth if, and only if, C’s graphs contain only paths of bounded length

[Neˇ setˇ ril and Ossona de Mendez, 2008]

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FO = MSO on Graphs Containing only Short Paths

Theorem

FO = MSO = GSO on graph classes C of bounded tree depth

[Elberfeld, Grohe, and Tantau, 2012]

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FO = MSO on Graphs Containing only Short Paths

Theorem

FO = MSO = GSO on graph classes C of bounded tree depth

[Elberfeld, Grohe, and Tantau, 2012]

Proof via Recursive Type Composition.

1 FO-define tree depth’s recursive vertex deletion 2 FO-evaluate GSO-formula by recursively

define set of satisfying GSO-formulas (type), and
compose by counting types up to a threshold

. . .

SLIDE 30

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FO = MSO on Graphs Containing only Short Paths

Theorem

FO = MSO = GSO on graph classes C of bounded tree depth

[Elberfeld, Grohe, and Tantau, 2012]

Proof via Recursive Type Composition.

1 FO-define tree depth’s recursive vertex deletion 2 FO-evaluate GSO-formula by recursively

define set of satisfying GSO-formulas (type), and
compose by counting types up to a threshold

. . .

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FO = MSO on Graphs Containing only Short Paths

Theorem

FO = MSO = GSO on graph classes C of bounded tree depth

[Elberfeld, Grohe, and Tantau, 2012]

Proof via Recursive Type Composition.

1 FO-define tree depth’s recursive vertex deletion 2 FO-evaluate GSO-formula by recursively

define set of satisfying GSO-formulas (type), and
compose by counting types up to a threshold

. . . | = ϕ1, | = ϕ5, . . . | = ϕ1, | = ϕ2, . . . | = ϕ2, | = ϕ6, . . .

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FO = MSO on Graphs Containing only Short Paths

Theorem

FO = MSO = GSO on graph classes C of bounded tree depth

[Elberfeld, Grohe, and Tantau, 2012]

Proof via Recursive Type Composition.

1 FO-define tree depth’s recursive vertex deletion 2 FO-evaluate GSO-formula by recursively

define set of satisfying GSO-formulas (type), and
compose by counting types up to a threshold

. . . | = ϕ1, | = ϕ5, . . . | = ϕ1, | = ϕ2, . . . | = ϕ2, | = ϕ6, . . . | = ϕ1, | = ϕ8,...

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FO = MSO on Artificial Graphs with Long Paths

Lemma

FO = MSO = GSO on a graph class C of unbounded tree depth

[Dawar and Hella, 1995]

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FO = MSO on Artificial Graphs with Long Paths

Lemma

FO = MSO = GSO on a graph class C of unbounded tree depth

[Dawar and Hella, 1995]

Proof Diagonalizes along Formulas.

Enumeration of GSO-formulas ϕ1, ϕ2, ϕ3, . . .
Infinite class C := C1 ∩C2 ∩... of paths
{G ∈ C : G |

= ϕi} or {G ∈ C : G | = ϕi} is finite for each ϕi

Hardwire graphs into equivalent FO-formula ψi

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Talk’s Content

1 Collapsing MSO- to FO-Logic on Graph Classes 2 Separating MSO- from FO-Logic on Graph Classes 3 Collapsing GSO- to MSO-Logic on Graph Classes 4 Collapsing Logics via Definable Decompositions

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Characterization Based on Subgraph Closure

Closure Under Taking Subgraphs

If ∈ C, then , , , ∈ C

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Characterization Based on Subgraph Closure

Closure Under Taking Subgraphs

If ∈ C, then , , , ∈ C

Lemma

Let C be closed under subgraphs. If C has unbounded tree depth, then FO MSO on C.

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Characterization Based on Subgraph Closure

Closure Under Taking Subgraphs

If ∈ C, then , , , ∈ C

Lemma

Let C be closed under subgraphs. If C has unbounded tree depth, then FO MSO on C.

Proof.

C’s graphs contain paths of unbounded length
C contains all

SLIDE 39

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Characterization Based on Subgraph Closure

Closure Under Taking Subgraphs

If ∈ C, then , , , ∈ C

Lemma

Let C be closed under subgraphs. If C has unbounded tree depth, then FO MSO on C.

Proof.

C’s graphs contain paths of unbounded length
C contains all

SLIDE 40

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Characterization Based on Subgraph Closure

Closure Under Taking Subgraphs

If ∈ C, then , , , ∈ C

Lemma

Let C be closed under subgraphs. If C has unbounded tree depth, then FO MSO on C.

Proof.

C’s graphs contain paths of unbounded length
C contains all

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Characterization Based on Subgraph Closure

Closure Under Taking Subgraphs

If ∈ C, then , , , ∈ C

Lemma

Let C be closed under subgraphs. If C has unbounded tree depth, then FO MSO on C.

Corollary

Let C be closed under subgraphs. Then

C has bounded tree depth,
FO = MSO on C, and
FO = GSO on C

are equivalent

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Characterization Based on Induced Subgraphs

Closure Under Taking Induced Subgraphs

If ∈ C, then , , ∈ C

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Characterization Based on Induced Subgraphs

Closure Under Taking Induced Subgraphs

If ∈ C, then , , ∈ C

Lemma

Let C be closed under induced subgraphs. If C has unbounded tree depth, then FO GSO on C

[Elberfeld, Grohe, and Tantau, 2012]

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Characterization Based on Induced Subgraphs

Closure Under Taking Induced Subgraphs

If ∈ C, then , , ∈ C

Lemma

Let C be closed under induced subgraphs. If C has unbounded tree depth, then FO GSO on C

[Elberfeld, Grohe, and Tantau, 2012]

Proof via Ramsey’s Theorem.

C’s graphs contain paths of unbounded length
C contains all

, or , or via Ramsey’s Th.

SLIDE 45

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Characterization Based on Induced Subgraphs

Closure Under Taking Induced Subgraphs

If ∈ C, then , , ∈ C

Lemma

Let C be closed under induced subgraphs. If C has unbounded tree depth, then FO GSO on C

[Elberfeld, Grohe, and Tantau, 2012]

Proof via Ramsey’s Theorem.

C’s graphs contain paths of unbounded length
C contains all

, or , or via Ramsey’s Th.

SLIDE 46

12 of 23

Characterization Based on Induced Subgraphs

Closure Under Taking Induced Subgraphs

If ∈ C, then , , ∈ C

Lemma

Let C be closed under induced subgraphs. If C has unbounded tree depth, then FO GSO on C

[Elberfeld, Grohe, and Tantau, 2012]

Proof via Ramsey’s Theorem.

C’s graphs contain paths of unbounded length
C contains all

, or , or via Ramsey’s Th.

SLIDE 47

12 of 23

Characterization Based on Induced Subgraphs

Closure Under Taking Induced Subgraphs

If ∈ C, then , , ∈ C

Lemma

Let C be closed under induced subgraphs. If C has unbounded tree depth, then FO GSO on C

[Elberfeld, Grohe, and Tantau, 2012]

Proof via Ramsey’s Theorem.

C’s graphs contain paths of unbounded length
C contains all

, or , or via Ramsey’s Th.

SLIDE 48

12 of 23

Characterization Based on Induced Subgraphs

Closure Under Taking Induced Subgraphs

If ∈ C, then , , ∈ C

Lemma

Let C be closed under induced subgraphs. If C has unbounded tree depth, then FO GSO on C

[Elberfeld, Grohe, and Tantau, 2012]

Proof via Ramsey’s Theorem.

C’s graphs contain paths of unbounded length
C contains all

, or , or via Ramsey’s Th.

SLIDE 49

12 of 23

Characterization Based on Induced Subgraphs

Closure Under Taking Induced Subgraphs

If ∈ C, then , , ∈ C

Lemma

Let C be closed under induced subgraphs. If C has unbounded tree depth, then FO GSO on C

[Elberfeld, Grohe, and Tantau, 2012]

Proof via Ramsey’s Theorem.

C’s graphs contain paths of unbounded length
C contains all

, or , or via Ramsey’s Th.

SLIDE 50

12 of 23

Characterization Based on Induced Subgraphs

Closure Under Taking Induced Subgraphs

If ∈ C, then , , ∈ C

Lemma

Let C be closed under induced subgraphs. If C has unbounded tree depth, then FO GSO on C

[Elberfeld, Grohe, and Tantau, 2012]

Proof via Ramsey’s Theorem.

C’s graphs contain paths of unbounded length
C contains all

, or , or via Ramsey’s Th.

SLIDE 51

12 of 23

Characterization Based on Induced Subgraphs

Closure Under Taking Induced Subgraphs

If ∈ C, then , , ∈ C

Lemma

Let C be closed under induced subgraphs. If C has unbounded tree depth, then FO GSO on C

[Elberfeld, Grohe, and Tantau, 2012]

Proof via Ramsey’s Theorem.

C’s graphs contain paths of unbounded length
C contains all

, or , or via Ramsey’s Th.

SLIDE 52

12 of 23

Characterization Based on Induced Subgraphs

Closure Under Taking Induced Subgraphs

If ∈ C, then , , ∈ C

Lemma

Let C be closed under induced subgraphs. If C has unbounded tree depth, then FO GSO on C

[Elberfeld, Grohe, and Tantau, 2012]

Corollary

Let C be a graph class closed under induced subgraphs. Then

C’s graphs have bounded tree depth, and
FO = GSO on C

are equivalent

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Characterization Based on Induced Subgraphs

Closure Under Taking Induced Subgraphs

If ∈ C, then , , ∈ C

Fact

FO = MSO on every C of bounded shrub depth.

[Gajarsk´ y and Hlinen´ y, 2012]

SLIDE 54

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Characterization Based on Induced Subgraphs

Closure Under Taking Induced Subgraphs

If ∈ C, then , , ∈ C

Fact

FO = MSO on every C of bounded shrub depth.

[Gajarsk´ y and Hlinen´ y, 2012]

Conjecture

Let C be closed under induced subgraphs. Then

C has bounded shrub depth, and
FO = MSO on C

are equivalent

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Talk’s Content

1 Collapsing MSO- to FO-Logic on Graph Classes 2 Separating MSO- from FO-Logic on Graph Classes 3 Collapsing GSO- to MSO-Logic on Graph Classes 4 Collapsing Logics via Definable Decompositions

SLIDE 56

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MSO = GSO on Degenerate Graphs

Degenerate Graphs

G is d-degenerate if all subgraphs have a vertex of degree ≤ d

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MSO = GSO on Degenerate Graphs

Degenerate Graphs

G is d-degenerate if all subgraphs have a vertex of degree ≤ d

Classes that are Degenerate

C of all

via d := 1

C with bounded tree depth via d := tree-depth−1
C of planar graphs (also, excluding a minor)

SLIDE 58

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MSO = GSO on Degenerate Graphs

Degenerate Graphs

G is d-degenerate if all subgraphs have a vertex of degree ≤ d

Theorem

MSO = GSO on every C of d-degenerate graphs

[Courcelle, 2003]

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MSO = GSO on Degenerate Graphs

Degenerate Graphs

G is d-degenerate if all subgraphs have a vertex of degree ≤ d

Theorem

MSO = GSO on every C of d-degenerate graphs

[Courcelle, 2003]

Proof Based on Degree-Bounded Orientations.

Orient with out-degree ≤ d via MSO-definable colorings

[Neˇ setˇ ril et al., 1997]

Totally order successors via MSO-definable colorings
Turn edge set quantification into d vertex sets

SLIDE 60

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MSO = GSO on Degenerate Graphs

Degenerate Graphs

G is d-degenerate if all subgraphs have a vertex of degree ≤ d

Theorem

MSO = GSO on every C of d-degenerate graphs

[Courcelle, 2003]

Proof Based on Degree-Bounded Orientations.

Orient with out-degree ≤ d via MSO-definable colorings

[Neˇ setˇ ril et al., 1997]

Totally order successors via MSO-definable colorings
Turn edge set quantification into d vertex sets

SLIDE 61

14 of 23

MSO = GSO on Degenerate Graphs

Degenerate Graphs

G is d-degenerate if all subgraphs have a vertex of degree ≤ d

Theorem

MSO = GSO on every C of d-degenerate graphs

[Courcelle, 2003]

Proof Based on Degree-Bounded Orientations.

Orient with out-degree ≤ d via MSO-definable colorings

[Neˇ setˇ ril et al., 1997]

Totally order successors via MSO-definable colorings
Turn edge set quantification into d vertex sets

SLIDE 62

14 of 23

MSO = GSO on Degenerate Graphs

Degenerate Graphs

G is d-degenerate if all subgraphs have a vertex of degree ≤ d

Theorem

MSO = GSO on every C of d-degenerate graphs

[Courcelle, 2003]

Proof Based on Degree-Bounded Orientations.

Orient with out-degree ≤ d via MSO-definable colorings

[Neˇ setˇ ril et al., 1997]

Totally order successors via MSO-definable colorings
Turn edge set quantification into d vertex sets

SLIDE 63

14 of 23

MSO = GSO on Degenerate Graphs

Degenerate Graphs

G is d-degenerate if all subgraphs have a vertex of degree ≤ d

Theorem

MSO = GSO on every C of d-degenerate graphs

[Courcelle, 2003]

Proof Based on Degree-Bounded Orientations.

Orient with out-degree ≤ d via MSO-definable colorings

[Neˇ setˇ ril et al., 1997]

Totally order successors via MSO-definable colorings
Turn edge set quantification into d vertex sets

SLIDE 64

14 of 23

MSO = GSO on Degenerate Graphs

Degenerate Graphs

G is d-degenerate if all subgraphs have a vertex of degree ≤ d

Theorem

MSO = GSO on every C of d-degenerate graphs

[Courcelle, 2003]

Open Question

Let C be closed under taking ???. Then

C has ???, and
MSO = GSO on C

are equivalent

SLIDE 65

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Talk’s Content

1 Collapsing MSO- to FO-Logic on Graph Classes 2 Separating MSO- from FO-Logic on Graph Classes 3 Collapsing GSO- to MSO-Logic on Graph Classes 4 Collapsing Logics via Definable Decompositions

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Definable Decomposition Conjecture is Still Open

Graphs with Bounded Tree Width

graph G

1 11 a b c d 111 a b c d 12 a b c d 121 a b c d 122 a b c d

width-4 D = (T,B)

1 1,11,a 1,11,a,b,d 1,11,b,d,c 11 11,111,a 11,111,a,b,d 11,111,b,d,c 1,12,a 1,12,a,b,d 1,12,b,d,c 12 12,121,a 12,121,a,b,d 12,121,b,d,c 12,122,a 12,122,a,b,d 12,122,b,d,c

Decomposition D satisfies cover and connectedness conditions

SLIDE 67

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Definable Decomposition Conjecture is Still Open

Graphs with Bounded Tree Width

graph G

1 11 a b c d 111 a b c d 12 a b c d 121 a b c d 122 a b c d

width-4 D = (T,B)

1 1,11,a 1,11,a,b,d 1,11,b,d,c 11 11,111,a 11,111,a,b,d 11,111,b,d,c 1,12,a 1,12,a,b,d 1,12,b,d,c 12 12,121,a 12,121,a,b,d 12,121,b,d,c 12,122,a 12,122,a,b,d 12,122,b,d,c

Decomposition D satisfies cover and connectedness conditions

Definable Decomposition Conjecture (Simplified)

There are MSO-formulas with parameters ϕw

TREE-NODE(u)

ϕw

TREE-EDGE(u,v)

ϕw

BAG(u,x)

defining tree decompositions of tree-width-w graphs

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Defining Decompositions Applies to Build-In Arithmetic

GSO with Build-In Modulo Counting

mod-GSO is GSO with relations |X| ≡ 0 mod m

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Defining Decompositions Applies to Build-In Arithmetic

GSO with Build-In Modulo Counting

mod-GSO is GSO with relations |X| ≡ 0 mod m

Conjecture’s Application to Language Theory

Recognizability = mod-GSO on C of bounded tree width

[Courcelle, 1990]

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Defining Decompositions Applies to Build-In Arithmetic

GSO with Build-In Modulo Counting

mod-GSO is GSO with relations |X| ≡ 0 mod m

Conjecture’s Application to Language Theory

Recognizability = mod-GSO on C of bounded tree width

[Courcelle, 1990]

GSO with Build-In Invariant Ordering

<-inv-GSO is GSO with invariantly used x < y

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Defining Decompositions Applies to Build-In Arithmetic

GSO with Build-In Modulo Counting

mod-GSO is GSO with relations |X| ≡ 0 mod m

Conjecture’s Application to Language Theory

Recognizability = mod-GSO on C of bounded tree width

[Courcelle, 1990]

GSO with Build-In Invariant Ordering

<-inv-GSO is GSO with invariantly used x < y

Conjecture’s Application to Build-In Arithmetic

mod-GSO= <-inv-GSO on C of bounded tree width

[Benedikt and Segoufin, 2009]

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Decomposition Conjecture Holds for Tree Depth

Theorem

There are FO-formulas with parameters ϕd

TREE-NODE(u)

ϕd

TREE-EDGE(u,v)

ϕd

BAG(u,x)

defining tree decompositions of tree-depth-d graphs

[Eickmeyer et al., 2014]

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Decomposition Conjecture Holds for Tree Depth

Theorem

There are FO-formulas with parameters ϕd

TREE-NODE(u)

ϕd

TREE-EDGE(u,v)

ϕd

BAG(u,x)

defining tree decompositions of tree-depth-d graphs

[Eickmeyer et al., 2014]

Proof.

There are ≤ f(d) root vertices start tree depth recursion

[Bouland et al., 2012, Dvoˇ r´ ak et al., 2012]

Put them into the root bag, use one to represent the bag
Recursion also handles back-edges to root vertices

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Decomposition Conjecture Holds for Tree Depth

Theorem

There are FO-formulas with parameters ϕd

TREE-NODE(u)

ϕd

TREE-EDGE(u,v)

ϕd

BAG(u,x)

defining tree decompositions of tree-depth-d graphs

[Eickmeyer et al., 2014]

Proof.

There are ≤ f(d) root vertices start tree depth recursion

[Bouland et al., 2012, Dvoˇ r´ ak et al., 2012]

Put them into the root bag, use one to represent the bag
Recursion also handles back-edges to root vertices

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Decomposition Conjecture Holds for Tree Depth

Theorem

There are FO-formulas with parameters ϕd

TREE-NODE(u)

ϕd

TREE-EDGE(u,v)

ϕd

BAG(u,x)

defining tree decompositions of tree-depth-d graphs

[Eickmeyer et al., 2014]

Proof.

There are ≤ f(d) root vertices start tree depth recursion

[Bouland et al., 2012, Dvoˇ r´ ak et al., 2012]

Put them into the root bag, use one to represent the bag
Recursion also handles back-edges to root vertices

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Decomposition Conjecture Holds for Tree Depth

Theorem

There are FO-formulas with parameters ϕd

TREE-NODE(u)

ϕd

TREE-EDGE(u,v)

ϕd

BAG(u,x)

defining tree decompositions of tree-depth-d graphs

[Eickmeyer et al., 2014]

Proof.

There are ≤ f(d) root vertices start tree depth recursion

[Bouland et al., 2012, Dvoˇ r´ ak et al., 2012]

Put them into the root bag, use one to represent the bag
Recursion also handles back-edges to root vertices

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Graph Structure Influences Expressivity of Logics

Summary

FO = MSO = GSO on C is linked to C’s tree depth
MSO = GSO on C is linked to C’s degeneracy
Arithmetic predicates are linked to defining decompositions

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Graph Structure Influences Expressivity of Logics

Summary

FO = MSO = GSO on C is linked to C’s tree depth
MSO = GSO on C is linked to C’s degeneracy
Arithmetic predicates are linked to defining decompositions

SLIDE 79

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Graph Structure Influences Expressivity of Logics

Summary

FO = MSO = GSO on C is linked to C’s tree depth
MSO = GSO on C is linked to C’s degeneracy
Arithmetic predicates are linked to defining decompositions

SLIDE 80

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Graph Structure Influences Expressivity of Logics

Summary

FO = MSO = GSO on C is linked to C’s tree depth
MSO = GSO on C is linked to C’s degeneracy
Arithmetic predicates are linked to defining decompositions

SLIDE 81

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Graph Structure Influences Expressivity of Logics

Summary

FO = MSO = GSO on C is linked to C’s tree depth
MSO = GSO on C is linked to C’s degeneracy
Arithmetic predicates are linked to defining decompositions

References

M. Elberfeld, M. Grohe, and T. Tantau (2012).

Where first-order and monadic second-order logic coincide. In Proceedings of LICS 2012, pages 265–274. Eickmeyer, K., Elberfeld, M., and Harwath, F. (2014). Expressivity and succinctness of order-invariant logics on depth-bounded structures. In Proceedings of MFCS 2014, pages 256–266.

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References I

Benedikt, M. and Segoufin, L. (2009). Regular tree languages definable in FO and in FOmod. ACM Trans. Comput. Logic, 11:4:1–4:32. Bouland, A., Dawar, A., and Kopczynski, E. (2012). On tractable parameterizations of graph isomorphism. In Proc. IPEC 2012, pages 218–230. Courcelle, B. (1990). The monadic second-order logic of graphs. i. recognizable sets of finite graphs. Information and Computation, 85(1):12–75. Courcelle, B. (2003). The monadic second-order logic of graphs xiv: uniformly sparse graphs and edge set quantifications. Theoretical Computer Science, 299(1–3):1–36.

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References II

Courcelle, B. and Engelfriet, J. (2012). Graph structure and monadic second-order logic, a language theoretic approach. Cambridge University Press. to be published. Dawar, A. and Hella, L. (1995). The expressive power of finitely many generalized quantifiers. Information and Computation, 123(2):172–184. Dvoˇ r´ ak, Z., Giannopoulou, A. C., and Thilikos, D. M. (2012). Forbidden graphs for tree-depth.

Eur. J. Comb., 33(5):969–979.

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References III

Ebbinghaus, H.-D. and Flum, J. (1999). Finite model theory. Springer. Eickmeyer, K., Elberfeld, M., and Harwath, F. (2014). Expressivity and succinctness of order-invariant logics on depth-bounded structures. In Mathematical Foundations of Computer Science 2014 - 39th International Symposium, MFCS 2014, Budapest, Hungary, August 25-29, 2014. Proceedings, Part I, pages 256–266.

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References IV

Elberfeld, M., Grohe, M., and Tantau, T. (2012). Where first-order and monadic second-order logic coincide. In Proceedings of the 27th Annual IEEE/ACM Symposium

n Logic in Computer Science (LICS 2012), LICS ’12,

pages 265–274. IEEE Computer Society. Gajarsk´ y, J. and Hlinen´ y, P . (2012). Faster deciding MSO properties of trees of fixed height, and some consequences. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2012, December 15-17, 2012, Hyderabad, India, pages 112–123.

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References V

Libkin, L. (2004). Elements Of Finite Model Theory. Springer. Neˇ setˇ ril, J. and Ossona de Mendez, P . (2006). Tree-depth, subgraph coloring and homomorphism bounds. European Journal of Combinatorics, 27(6):1022–1041. Neˇ setˇ ril, J. and Ossona de Mendez, P . (2008). Grad and classes with bounded expansion I. Decompositions. European Journal of Combinatorics, 29(3):760–776. Neˇ setˇ ril, J., Sopena, E., and Vignal, L. (1997). T-preserving homomorphisms of oriented graphs.

Comment. Math. Univ. Carolin, 38(1):125–136.