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)
1 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
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 6= v ↔ (u, v ) ∈ E(G)
defines cliques, e.g.
|= ϕ CLIQUE
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 6= 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
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 6= 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
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 6= 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
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 6= 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
2 of 23
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
3 of 23
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
3 of 23
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]
3 of 23
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
4 of 23
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
4 of 23
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
)
4 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
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
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
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
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
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
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
5 of 23
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
6 of 23
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
7 of 23
FO = MSO on Graphs Containing only Short Paths
Tree Depth of a Graph
[Nešetřil and Ossona de Mendez, 2006]
8 of 23
FO = MSO on Graphs Containing only Short Paths
Tree Depth of a Graph
Single vertex tree-depth
=1
[Nešetřil and Ossona de Mendez, 2006]
8 of 23
FO = MSO on Graphs Containing only Short Paths
Tree Depth of a Graph
Single vertex tree-depth
Disconnected tree-depth(
max
component
=1
...
)=
tree-depth( )
[Nešetřil and Ossona de Mendez, 2006]
8 of 23
FO = MSO on Graphs Containing only Short Paths
Tree Depth of a Graph
Single vertex tree-depth
Disconnected tree-depth(
max
component
=1
...
)=
tree-depth( )
...
Connected tree-depth
minvertex tree-depth(
=
...
) +1
[Nešetřil and Ossona de Mendez, 2006]
8 of 23
FO = MSO on Graphs Containing only Short Paths
Tree Depth of a Graph
Single vertex tree-depth
Disconnected tree-depth(
max
component
=1
...
)=
tree-depth( )
...
Connected tree-depth
minvertex tree-depth(
=
...
) +1
[Nešetřil 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
[Nešetřil and Ossona de Mendez, 2008]
8 of 23
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]
8 of 23
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
2
FO-define tree depth’s recursive vertex deletion
FO-evaluate GSO-formula by recursively
• define set of satisfying GSO-formulas (type), and
• compose by counting types up to a threshold
...
8 of 23
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
2
FO-define tree depth’s recursive vertex deletion
FO-evaluate GSO-formula by recursively
• define set of satisfying GSO-formulas (type), and
• compose by counting types up to a threshold
...
8 of 23
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
2
FO-define tree depth’s recursive vertex deletion
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 ,
...
8 of 23
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
2
FO-define tree depth’s recursive vertex deletion
FO-evaluate GSO-formula by recursively
• define set of satisfying GSO-formulas (type), and
• compose by counting types up to a threshold
|= ϕ1 , |= ϕ8 , . . .
...
|= ϕ1 ,
|= ϕ5 ,
...
|= ϕ1 ,
|= ϕ2 ,
...
|= ϕ2 ,
|= ϕ6 ,
...
8 of 23
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]
9 of 23
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 6|= ϕi } is finite for each ϕi
• Hardwire graphs into equivalent FO-formula ψi
9 of 23
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
10 of 23
Characterization Based on Subgraph Closure
Closure Under Taking Subgraphs
If
∈ C, then ,
,
,
∈C
11 of 23
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.
11 of 23
Characterization Based on Subgraph Closure
Closure Under Taking Subgraphs
∈ C, then ,
If
,
,
∈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
11 of 23
Characterization Based on Subgraph Closure
Closure Under Taking Subgraphs
∈ C, then ,
If
,
,
∈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
11 of 23
Characterization Based on Subgraph Closure
Closure Under Taking Subgraphs
∈ C, then ,
If
,
,
∈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
11 of 23
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
11 of 23
Characterization Based on Induced Subgraphs
Closure Under Taking Induced Subgraphs
If
∈ C, then ,
,
∈C
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]
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.
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.
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.
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.
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.
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.
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.
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.
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
12 of 23
Characterization Based on Induced Subgraphs
Closure Under Taking Induced Subgraphs
If
∈ C, then ,
,
∈C
Fact
FO = MSO on every C of bounded shrub depth.
[Gajarský and Hlinený, 2012]
12 of 23
Characterization Based on Induced Subgraphs
Closure Under Taking Induced Subgraphs
∈ C, then ,
If
,
∈C
Fact
FO = MSO on every C of bounded shrub depth.
[Gajarský and Hlinený, 2012]
Conjecture
Let C be closed under induced subgraphs. Then
• C has bounded shrub depth, and
• FO = MSO on C
are equivalent
12 of 23
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
13 of 23
MSO = GSO on Degenerate Graphs
Degenerate Graphs
G is d-degenerate if all subgraphs have a vertex of degree ≤ d
14 of 23
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)
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]
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
[Nešetřil et al., 1997]
• Totally order successors via MSO-definable colorings
• Turn edge set quantification into d vertex sets
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
[Nešetřil et al., 1997]
• Totally order successors via MSO-definable colorings
• Turn edge set quantification into d vertex sets
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
[Nešetřil et al., 1997]
• Totally order successors via MSO-definable colorings
• Turn edge set quantification into d vertex sets
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
[Nešetřil et al., 1997]
• Totally order successors via MSO-definable colorings
• Turn edge set quantification into d vertex sets
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
[Nešetřil et al., 1997]
• Totally order successors via MSO-definable colorings
• Turn edge set quantification into d vertex sets
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
14 of 23
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
15 of 23
Definable Decomposition Conjecture is Still Open
Graphs with Bounded Tree Width
width-4 D = (T , B)
graph G
1
b
a
1
b
c
d
1,11,a
c
1,11,a,b,d
a
d
1,11,b,d,c
11
b
a
12
b
c
d
111
a
121
b
c
d
a
c
122
1,12,a,b,d
1,12,b,d,c
11
12
11,111,a
12,121,a 12,122,a
11,111,a,b,d
d
1,12,a
11,111,b,d,c
12,121,a,b,d
12,121,b,d,c
12,122,a,b,d
12,122,b,d,c
Decomposition D satisfies cover and connectedness conditions
16 of 23
Definable Decomposition Conjecture is Still Open
Graphs with Bounded Tree Width
width-4 D = (T , B)
graph G
1
b
a
1
b
c
d
1,11,a
c
1,11,a,b,d
a
d
a
12
b
c
d
111
a
1,12,a,b,d
1,11,b,d,c
11
b
1,12,a
b
c
d
121
a
c
12
11,111,a
12,121,a 12,122,a
11,111,a,b,d
d
122
1,12,b,d,c
11
11,111,b,d,c
12,121,a,b,d
12,121,b,d,c
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
w
ϕTREE
- NODE (u)
w
ϕTREE
- EDGE (u, v )
w (u, x)
ϕBAG
defining tree decompositions of tree-width-w graphs
16 of 23
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
d
ϕTREE
- NODE (u)
d
ϕTREE
- EDGE (u, v )
d (u, x)
ϕBAG
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
d
ϕTREE
- NODE (u)
d
ϕTREE
- EDGE (u, v )
d (u, x)
ϕBAG
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, Dvořák 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
d
ϕTREE
- NODE (u)
d
ϕTREE
- EDGE (u, v )
d (u, x)
ϕBAG
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, Dvořák 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
d
ϕTREE
- NODE (u)
d
ϕTREE
- EDGE (u, v )
d (u, x)
ϕBAG
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, Dvořák 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
d
ϕTREE
- NODE (u)
d
ϕTREE
- EDGE (u, v )
d (u, x)
ϕBAG
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, Dvořák 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
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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
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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
ACM Trans. Comput. Logic, 11:4:1–4:32.
FO mod .
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.
Dvořák, 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
on Logic in Computer Science (LICS 2012), LICS ’12,
pages 265–274. IEEE Computer Society.
Gajarský, J. and Hlinený, 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.
Nešetřil, J. and Ossona de Mendez, P. (2006).
Tree-depth, subgraph coloring and homomorphism bounds.
European Journal of Combinatorics, 27(6):1022–1041.
Nešetřil, J. and Ossona de Mendez, P. (2008).
Grad and classes with bounded expansion I.
Decompositions.
European Journal of Combinatorics, 29(3):760–776.
Nešetřil, J., Sopena, E., and Vignal, L. (1997).
T-preserving homomorphisms of oriented graphs.
Comment. Math. Univ. Carolin, 38(1):125–136.
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