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

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 � = 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 � = 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 � = 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 � = 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 � = 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 [Neˇ setˇ ril 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 [Neˇ setˇ ril 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 . . . Disconnected tree - depth ( ) = max tree - depth ( ) component [Neˇ setˇ ril 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 . . . Disconnected tree - depth ( ) = max tree - depth ( ) component � . . . � Connected tree - depth = . . . min vertex tree - depth ( )+ 1 [Neˇ setˇ ril 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 . . . Disconnected tree - depth ( ) = max tree - depth ( ) component � . . . � Connected tree - depth = . . . min vertex 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] 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