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

Expressivity within second-order transitive-closure logic Jonni Virtema Expressivity within second-order transitive-closure logic Background Transitive closure FO(TC) & SO(TC) Examples Jonni Virtema Expressivity Hasselt University, Belgium MSO(TC) and counting jonni.virtema@gmail.com Order invariant MSO Joint work with Jan Van den Bussche and Flavio Ferrarotti Open questions CSL 2018 – September 5th 2018 1/ 24

Expressivity within Descriptive Complexity second-order transitive-closure logic Jonni Virtema Background Transitive closure ◮ Offers a machine independent description of complexity classes: FO(TC) & SO(TC) ◮ Time/Space used by a machine to decide a problem Examples ⇒ richness of the logical language needed to describe the problem. Expressivity MSO(TC) and ◮ Complexity classes can/could be then separated by separating logics. counting ◮ Many characterisations are known: Order invariant MSO ◮ Fagin’s Theorem 1973: Existential second-order logic characterises NP. Open questions 2/ 24

Expressivity within Descriptive Complexity second-order transitive-closure logic Jonni Virtema ◮ Offers a machine independent description of complexity classes: Background ◮ Time/Space used by a machine to decide a problem Transitive closure ⇒ richness of the logical language needed to describe the problem. FO(TC) & SO(TC) Examples ◮ Complexity classes can/could be then separated by separating logics. Expressivity ◮ Many characterisations are known: MSO(TC) and ◮ Fagin’s Theorem 1973: Existential second-order logic characterises NP. counting Order invariant MSO ”A graph is three colourable” = Open questions � ∃ R ∃ B ∃ G ”each node is labeled by exactly one colour” � ∧ ”adjacent nodes are always coloured with distinct colours” 2/ 24

Expressivity within Descriptive Complexity second-order transitive-closure logic Jonni Virtema ◮ Offers a machine independent description of complexity classes: Background ◮ Time/Space used by a machine to decide a problem Transitive closure ⇒ richness of the logical language needed to describe the problem. FO(TC) & SO(TC) ◮ Complexity classes can/could be then separated by separating logics. Examples Expressivity ◮ Many characterisations are known: MSO(TC) and ◮ Fagin’s Theorem 1973: Existential second-order logic characterises NP. counting ◮ Least fixed point logic LFP characterises P on ordered structures. Order invariant MSO ◮ First-order transitive closure logic characterises NLOGSPACE on ordered Open questions structures. ◮ Second-order logic characterises the polynomial time hierarchy. ◮ Second-order transitive closure logic characterises PSPACE. ◮ ... 2/ 24

Expressivity within Second-order transitive closure logic SO(TC) second-order transitive-closure logic Jonni Virtema Background ◮ Expressive declarative language – can express exactly all PSPACE properties. Transitive closure ◮ Can express step-wise defined properties in a natural and elegant manner. FO(TC) & SO(TC) Examples ◮ Recursive properties of graphs: Determine whether a graph G can be built Expressivity starting from some graph pattern G p by some recursive procedure. MSO(TC) and ◮ Already the monadic fragment MSO ( TC ) can express many interesting counting properties: Order invariant MSO ◮ On strings it characterises nondeterministic linear space. Open questions ◮ Can express NP-complete problems (e.g., QBF). ◮ Can express counting. 3/ 24

Expressivity within Transitive closure second-order transitive-closure logic Jonni Virtema Background 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 | there exists a finite directed R -path from a to b } . MSO(TC) and counting In our setting A is set of tuples ( a 1 , . . . a n ), where each a i is either an element or Order invariant MSO a relation over some domain D . Open questions 4/ 24

Expressivity within Transitive closure second-order transitive-closure logic Jonni Virtema Background 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 | there exists a finite directed R -path from a to b } . MSO(TC) and counting In our setting A is set of tuples ( a 1 , . . . a n ), where each a i is either an element or Order invariant MSO a relation over some domain D . Open questions 4/ 24

Expressivity within Transitive closure second-order transitive-closure logic Example Jonni Virtema Background Let G = ( V , E ) be an undirected graph. Then ( a , b ) ∈ TC ( E ) if a and b are in Transitive closure the same component of G , or equivalently, if there is a path from a to b in G . FO(TC) & SO(TC) Examples Example Expressivity A graph G = ( V , E ) has a Hamiltonian cycle (cycle that visits every node MSO(TC) and counting exactly once) 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 . 5/ 24

Expressivity within Transitive closure second-order transitive-closure logic Example Jonni Virtema Background Let G = ( V , E ) be an undirected graph. Then ( a , b ) ∈ TC ( E ) if a and b are in Transitive closure the same component of G , or equivalently, if there is a path from a to b in G . FO(TC) & SO(TC) Examples Example Expressivity A graph G = ( V , E ) has a Hamiltonian cycle (cycle that visits every node MSO(TC) and counting exactly once) 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 . 5/ 24

Expressivity within Definable relations second-order transitive-closure logic Jonni Virtema Background Transitive closure Let � x and � y be k -tuples of first-order variables, ϕ ( � x , � y ) an FO -formula, and A a FO(TC) & SO(TC) Examples model. Expressivity ◮ ϕ ( � x , � y ) defines a 2 k -ary relation on A . MSO(TC) and ◮ We consider this 2 k -ary relation as a binary relation over k -tuples. counting Order invariant ◮ We denote by BIN � � ϕ ( � x , � y ) this binary relation. MSO Open questions 6/ 24

Expressivity within Logics with transitive closure operator second-order transitive-closure logic First-order transitive closure logic FO ( TC ): Jonni Virtema y , � ϕ ::= x = y | X ( x 1 , . . . , x k ) | ¬ ϕ | ( ϕ ∨ ϕ ) | ∃ x ϕ | [ TC � x ′ ϕ ]( � y ′ ) , Background x ,� Transitive closure y ′ are tuples of first-order variables of the same length. x , � y , and � where � x ′ , � FO(TC) & SO(TC) Examples Semantics for the TC operator: Expressivity MSO(TC) and � �� y , � y ) , s ( � x , � � � � A | = s [ TC � x ′ ϕ ]( � y ′ ) iff s ( � y ′ ) ∈ TC ϕ ( � x ′ ) BIN counting x ,� Order invariant MSO Open questions Example The sentence ∀ x ∀ y x = y ∨ [ TC z , z ′ E ( z , z ′ )]( x , y ) expresses connectivity of graphs ( V , E ). 7/ 24

Expressivity within Logics with transitive closure operator second-order transitive-closure logic First-order transitive closure logic FO ( TC ): Jonni Virtema y , � ϕ ::= x = y | X ( x 1 , . . . , x k ) | ¬ ϕ | ( ϕ ∨ ϕ ) | ∃ x ϕ | [ TC � x ′ ϕ ]( � y ′ ) , Background x ,� Transitive closure y ′ are tuples of first-order variables of the same length. x , � y , and � where � x ′ , � FO(TC) & SO(TC) Examples Semantics for the TC operator: Expressivity MSO(TC) and � �� y , � y ) , s ( � x , � � � � A | = s [ TC � x ′ ϕ ]( � y ′ ) iff s ( � y ′ ) ∈ TC ϕ ( � x ′ ) BIN counting x ,� Order invariant MSO Open questions Example The sentence ∀ x ∀ y x = y ∨ [ TC z , z ′ E ( z , z ′ )]( x , y ) expresses connectivity of graphs ( V , E ). 7/ 24

Expressivity within Logics with transitive closure operator second-order transitive-closure logic Jonni Virtema Second-order transitive closure logic SO ( TC ): Background X ′ ϕ ]( � Y , � Transitive closure ϕ ::= x = y | X ( x 1 , . . . , x k ) | ¬ ϕ | ( ϕ ∨ ϕ ) | ∃ x ϕ | ∃ Y ϕ | [ TC � Y ′ ) , X , � FO(TC) & SO(TC) Examples where � X , � X ′ , � Y , and � Y ′ are tuples of first-order and second-order variables of the Expressivity same length and sort. MSO(TC) and counting Semantics for the TC operator: Order invariant MSO � �� X ′ ϕ ]( � Y , � s ( � Y ) , s ( � ϕ ( � X , � A | Y ′ ) iff � Y ′ ) � ∈ TC � X ′ ) Open questions = s [ TC � BIN X , � MSO ( TC ) is the fragment of SO ( TC ) in which all second-order variables have arity 1. 8/ 24