Expressivity within second-order transitive-closure logic Jonni - PowerPoint PPT Presentation

Expressivity within second-order transitive-closure logic Expressivity within second-order transitive-closure logic Jonni Virtema Transitive closure FO(TC) & SO(TC) Jonni Virtema Examples Expressivity Hasselt University, Belgium MSO(TC) and jonni.virtema@gmail.com counting Order invariant MSO Joint work with Jan Van den Bussche and Flavio Ferrarotti To be presented in CSL 2018 Open questions Horizons of Logic, Computation and Definability Symposium in Honour of Lauri Hella’s 60th birthday July 6, 2018 1/ 22

Expressivity within Transitive closure second-order transitive-closure logic Jonni Virtema Transitive closure FO(TC) & SO(TC) The transitive closure TC ( R ) of a binary relation R ⊆ A × A is defined as follows Examples Expressivity TC ( R ) := { ( a , b ) ∈ A × A | ∃ n > 0 and e 0 , . . . , e n ∈ A MSO(TC) and counting such that a = e 0 , b = e n , and ( e i , e i +1 ) ∈ R for all i < n } . Order invariant MSO In our setting A is set of tuples ( a 1 , . . . a n ), where each a i is either an element or Open questions a relation over some domain D . 2/ 22

Expressivity within Transitive closure second-order transitive-closure logic Jonni Virtema Transitive closure FO(TC) & SO(TC) The transitive closure TC ( R ) of a binary relation R ⊆ A × A is defined as follows Examples Expressivity TC ( R ) := { ( a , b ) ∈ A × A | ∃ n > 0 and e 0 , . . . , e n ∈ A MSO(TC) and counting such that a = e 0 , b = e n , and ( e i , e i +1 ) ∈ R for all i < n } . Order invariant MSO In our setting A is set of tuples ( a 1 , . . . a n ), where each a i is either an element or Open questions a relation over some domain D . 2/ 22

Expressivity within Transitive closure second-order transitive-closure logic Jonni Virtema Example Transitive closure Let G = ( V , E ) be an undirected graph. Then ( a , b ) ∈ TC ( E ) if a and b are in FO(TC) & SO(TC) the same component of G , or equivalently, if there is a path from a to b in G . Examples Expressivity Example MSO(TC) and counting A graph G = ( V , E ) has a Hamiltonian cycle if the following holds: Order invariant MSO 1. There is a relation R such that Open questions Z ′ = Z ∪ { z ′ } , z ′ / ( Z , z , Z ′ , z ′ ) ∈ R ∈ Z and ( z , z ′ ) ∈ E . iff 2. The tuple ( { x } , x , V , y ) is in the transitive closure of R , for some x , y ∈ V such that ( y , x ) ∈ E . 3/ 22

Expressivity within Transitive closure second-order transitive-closure logic Jonni Virtema Example Transitive closure Let G = ( V , E ) be an undirected graph. Then ( a , b ) ∈ TC ( E ) if a and b are in FO(TC) & SO(TC) the same component of G , or equivalently, if there is a path from a to b in G . Examples Expressivity Example MSO(TC) and counting A graph G = ( V , E ) has a Hamiltonian cycle if the following holds: Order invariant MSO 1. There is a relation R such that Open questions Z ′ = Z ∪ { z ′ } , z ′ / ( Z , z , Z ′ , z ′ ) ∈ R ∈ Z and ( z , z ′ ) ∈ E . iff 2. The tuple ( { x } , x , V , y ) is in the transitive closure of R , for some x , y ∈ V such that ( y , x ) ∈ E . 3/ 22

Expressivity within Logics with transitive closure operator second-order transitive-closure logic First-order transitive closure logic FO ( TC ): Jonni Virtema y , � y ′ ) , ϕ ::= x = y | X ( x 1 , . . . , x k ) | ¬ ϕ | ( ϕ ∨ ϕ ) | ∃ x ϕ | [ TC � x ′ ϕ ]( � Transitive closure x ,� FO(TC) & SO(TC) y ′ are tuples of first-order variables of the same length. x , � y , and � x ′ , � where � Examples Expressivity Semantics for the TC operator: MSO(TC) and counting y , � y ) , s ( � a , � y ′ ) iff � y ′ ) � a ′ ) | A | A | = s [ TC � x ′ ϕ ]( � s ( � ∈ TC ( { ( � = s ( � a ′ ) ϕ } ) Order invariant x ,� a ,� x ′ �→ � x �→ � MSO Open questions Example The sentence ∀ x ∀ y [ TC z , z ′ E ( z , z ′ )]( x , y ) expresses connectivity of graphs ( V , E ). 4/ 22

Expressivity within Logics with transitive closure operator second-order transitive-closure logic First-order transitive closure logic FO ( TC ): Jonni Virtema y , � y ′ ) , ϕ ::= x = y | X ( x 1 , . . . , x k ) | ¬ ϕ | ( ϕ ∨ ϕ ) | ∃ x ϕ | [ TC � x ′ ϕ ]( � Transitive closure x ,� FO(TC) & SO(TC) y ′ are tuples of first-order variables of the same length. x , � y , and � x ′ , � where � Examples Expressivity Semantics for the TC operator: MSO(TC) and counting y , � y ) , s ( � a , � y ′ ) iff � y ′ ) � a ′ ) | A | A | = s [ TC � x ′ ϕ ]( � s ( � ∈ TC ( { ( � = s ( � a ′ ) ϕ } ) Order invariant x ,� a ,� x ′ �→ � x �→ � MSO Open questions Example The sentence ∀ x ∀ y [ TC z , z ′ E ( z , z ′ )]( x , y ) expresses connectivity of graphs ( V , E ). 4/ 22

Expressivity within Logics with transitive closure operator second-order transitive-closure logic Jonni Virtema Second-order transitive closure logic SO ( TC ): Transitive closure FO(TC) & SO(TC) X ′ ϕ ]( � Y , � Y ′ ) , ϕ ::= x = y | X ( x 1 , . . . , x k ) | ¬ ϕ | ( ϕ ∨ ϕ ) | ∃ x ϕ | ∃ Y ϕ | [ TC � X , � Examples Expressivity Y ′ are tuples of first-order and second-order variables of the where � X , � X ′ , � Y , and � MSO(TC) and counting same length and sort. Order invariant MSO Semantics for the TC operator: Open questions X ′ ϕ ]( � Y , � s ( � Y ) , s ( � ∈ TC ( { ( � A , � Y ′ ) iff � Y ′ ) � B ′ ) | A | A | = s [ TC � = s ( � A ′ ) ϕ } ) X , � X �→ � A , � X ′ �→ � MSO ( TC ) is the fragment of SO ( TC ) in which all second-order variables have arity 1. 5/ 22

Expressivity within Logics with transitive closure operator second-order transitive-closure logic Jonni Virtema Second-order transitive closure logic SO ( TC ): Transitive closure FO(TC) & SO(TC) X ′ ϕ ]( � Y , � Y ′ ) , ϕ ::= x = y | X ( x 1 , . . . , x k ) | ¬ ϕ | ( ϕ ∨ ϕ ) | ∃ x ϕ | ∃ Y ϕ | [ TC � X , � Examples Expressivity Y ′ are tuples of first-order and second-order variables of the where � X , � X ′ , � Y , and � MSO(TC) and counting same length and sort. Order invariant MSO Semantics for the TC operator: Open questions X ′ ϕ ]( � Y , � s ( � Y ) , s ( � ∈ TC ( { ( � A , � Y ′ ) iff � Y ′ ) � B ′ ) | A | A | = s [ TC � = s ( � A ′ ) ϕ } ) X , � X �→ � A , � X ′ �→ � MSO ( TC ) is the fragment of SO ( TC ) in which all second-order variables have arity 1. 5/ 22

Expressivity within The H¨ artig quantifier second-order transitive-closure logic Jonni Virtema A | = s H xy ( ϕ ( x ) , ψ ( y )) ⇔ the sets { a ∈ A | A | = s ( x �→ a ) ϕ ( x ) } and Transitive closure { b ∈ A | A | = s ( y �→ b ) ψ ( y ) } have the same cardinality FO(TC) & SO(TC) Examples Expressivity Example (The H¨ artig quantifier can be expressed in MSO(TC).) MSO(TC) and Let ψ decrement denote an FO -formula expressing that s ( X ′ ) = s ( X ) \ { a } and counting Order invariant s ( Y ′ ) = s ( Y ) \ { b } for some a and b . Define MSO Open questions �� � ∧ [ TC X , Y , X ′ , Y ′ ψ decrement ]( Z , Z ′ , X ∅ , X ∅ ) ψ ec := ∃ X ∅ � ∀ x ¬ X ∅ ( x ) . Now ψ ec holds under s if and only if the cardinalities of s ( Z ) and s ( Z ′ ) are the same. Therefore H xy ( ϕ ( x ) , ψ ( y )) is equivalent with the formula ∃ Z ∃ Z ′ � ∀ x ( ϕ ( x ) ↔ Z ( x )) ∧ ∀ y ( ψ ( y ) ↔ Z ′ ( y )) ∧ ψ ec � . 6/ 22

Expressivity within The H¨ artig quantifier second-order transitive-closure logic Jonni Virtema A | = s H xy ( ϕ ( x ) , ψ ( y )) ⇔ the sets { a ∈ A | A | = s ( x �→ a ) ϕ ( x ) } and Transitive closure { b ∈ A | A | = s ( y �→ b ) ψ ( y ) } have the same cardinality FO(TC) & SO(TC) Examples Expressivity Example (The H¨ artig quantifier can be expressed in MSO(TC).) MSO(TC) and Let ψ decrement denote an FO -formula expressing that s ( X ′ ) = s ( X ) \ { a } and counting Order invariant s ( Y ′ ) = s ( Y ) \ { b } for some a and b . Define MSO Open questions �� � ∧ [ TC X , Y , X ′ , Y ′ ψ decrement ]( Z , Z ′ , X ∅ , X ∅ ) ψ ec := ∃ X ∅ � ∀ x ¬ X ∅ ( x ) . Now ψ ec holds under s if and only if the cardinalities of s ( Z ) and s ( Z ′ ) are the same. Therefore H xy ( ϕ ( x ) , ψ ( y )) is equivalent with the formula ∃ Z ∃ Z ′ � ∀ x ( ϕ ( x ) ↔ Z ( x )) ∧ ∀ y ( ψ ( y ) ↔ Z ′ ( y )) ∧ ψ ec � . 6/ 22