Complexity of propositional logics in team semantics Jonni Virtema Complexity of propositional logics in team semantics Movativation History Team Semantics Dependency atoms Jonni Virtema Complexity Results From 3SAT to University of Helsinki, Finland ADQBF jonni.virtema@gmail.com References Joint work with Miika Hannula 1 , Martin L¨ uck 2 , Juha Kontinen 3 , and Heribert Vollmer 2 Related to papers in GandALF 2016, MFCS 2015, Information and Computation 2016 1 The University of Auckland, 2 University of Hanover, 3 University of Helsinki, 8th of December, 2016 – Computational Logic Day 2016 1/ 14
Complexity of Core of Team Semantics propositional logics in team semantics Jonni Virtema Movativation ◮ In most studied logics formulae are evaluated in a single state of affairs. History Team Semantics E.g., Dependency atoms ◮ a first-order assignment in first-order logic, Complexity Results ◮ a propositional assignment in propositional logic, ◮ a possible world of a Kripke structure in modal logic. From 3SAT to ADQBF ◮ In team semantics sets of states of affairs are considered. References E.g., ◮ a set of first-order assignments in first-order logic, ◮ a set of propositional assignments in propositional logic, ◮ a set of possible worlds of a Kripke structure in modal logic. ◮ These sets of things are called teams. 2/ 14
Complexity of Core of Team Semantics propositional logics in team semantics Jonni Virtema Movativation ◮ In most studied logics formulae are evaluated in a single state of affairs. History Team Semantics E.g., Dependency atoms ◮ a first-order assignment in first-order logic, Complexity Results ◮ a propositional assignment in propositional logic, ◮ a possible world of a Kripke structure in modal logic. From 3SAT to ADQBF ◮ In team semantics sets of states of affairs are considered. References E.g., ◮ a set of first-order assignments in first-order logic, ◮ a set of propositional assignments in propositional logic, ◮ a set of possible worlds of a Kripke structure in modal logic. ◮ These sets of things are called teams. 2/ 14
Complexity of Core of Team Semantics propositional logics in team semantics Jonni Virtema Movativation ◮ In most studied logics formulae are evaluated in a single state of affairs. History Team Semantics E.g., Dependency atoms ◮ a first-order assignment in first-order logic, Complexity Results ◮ a propositional assignment in propositional logic, ◮ a possible world of a Kripke structure in modal logic. From 3SAT to ADQBF ◮ In team semantics sets of states of affairs are considered. References E.g., ◮ a set of first-order assignments in first-order logic, ◮ a set of propositional assignments in propositional logic, ◮ a set of possible worlds of a Kripke structure in modal logic. ◮ These sets of things are called teams. 2/ 14
Complexity of Team Semantics: Motivation and History propositional logics in team semantics Logical modelling of uncertainty, imperfect information, and different notions of Jonni Virtema dependence such as functional dependence and independence. Related to similar Movativation History concepts in statistics, database theory etc. Team Semantics Historical development: Dependency atoms Complexity Results ◮ Branching quantifiers by Henkin 1959. From 3SAT to ◮ Independence-friendly logic by Hintikka and Sandu 1989. ADQBF References ◮ Compositional semantics for independence-friendly logic by Hodges 1997. (Origin of team semantics.) ◮ Dependence logic by V¨ a¨ an¨ anen 2007. ◮ Modal dependence logic by V¨ a¨ an¨ anen 2008. ◮ Introduction of other dependency notions to team semantics such as inclusion, exclusion, and independence. Galliani, Gr¨ adel, V¨ a¨ an¨ anen. ◮ Generalized atoms by Kuusisto (derived from generalised quantifiers). 3/ 14
Complexity of Team Semantics: Motivation and History propositional logics in team semantics Logical modelling of uncertainty, imperfect information, and different notions of Jonni Virtema dependence such as functional dependence and independence. Related to similar Movativation History concepts in statistics, database theory etc. Team Semantics Historical development: Dependency atoms Complexity Results ◮ Branching quantifiers by Henkin 1959. From 3SAT to ◮ Independence-friendly logic by Hintikka and Sandu 1989. ADQBF References ◮ Compositional semantics for independence-friendly logic by Hodges 1997. (Origin of team semantics.) ◮ Dependence logic by V¨ a¨ an¨ anen 2007. ◮ Modal dependence logic by V¨ a¨ an¨ anen 2008. ◮ Introduction of other dependency notions to team semantics such as inclusion, exclusion, and independence. Galliani, Gr¨ adel, V¨ a¨ an¨ anen. ◮ Generalized atoms by Kuusisto (derived from generalised quantifiers). 3/ 14
Complexity of Propositional logic propositional logics in team semantics Jonni Virtema Syntax of propositional logic: Movativation History ϕ ::= p | ¬ p | ( ϕ ∧ ϕ ) | ( ϕ ∨ ϕ ) Team Semantics Dependency atoms Semantics via propositional assignments: Complexity Results ” name ” p q r From 3SAT to s | = ( q ∧ r ) ADQBF 0 1 1 s References Team semantics / semantics via sets of assignments: ” name ” p q r s 0 1 1 { s , t , u } | = q , { s , t } | = ( p ∨ r ) 1 1 0 t u 0 1 0 4/ 14
Complexity of Team semantics propositional logics in team semantics Jonni Virtema We want that for each formula ϕ of propositional logic and for each team X Movativation History X | = ϕ iff ∀ s ∈ X : s | = ϕ. Team Semantics Dependency atoms We define that Complexity Results From 3SAT to ADQBF X | = p iff ∀ s ∈ X : s ( p ) = 1 References X | = ¬ p iff ∀ s ∈ X : s ( p ) = 0 X | = ϕ ∧ ψ X | = ϕ and X | iff = ψ X | = ϕ ∨ ψ iff Y | = ϕ and Z | = ψ, for some Y , Z ⊆ X such that Y ∪ Z = X . 5/ 14
Complexity of Team semantics propositional logics in team semantics Jonni Virtema We want that for each formula ϕ of propositional logic and for each team X Movativation History X | = ϕ iff ∀ s ∈ X : s | = ϕ. Team Semantics Dependency atoms We define that Complexity Results From 3SAT to ADQBF X | = p iff ∀ s ∈ X : s ( p ) = 1 References X | = ¬ p iff ∀ s ∈ X : s ( p ) = 0 X | = ϕ ∧ ψ X | = ϕ and X | iff = ψ X | = ϕ ∨ ψ iff Y | = ϕ and Z | = ψ, for some Y , Z ⊆ X such that Y ∪ Z = X . 5/ 14
Complexity of Extensions of propositional logic propositional logics in team semantics Jonni Virtema We extend PL by adding atomic formulae that describe properties of teams. Movativation History Dependence atoms: dep ( p , q , r ) Team Semantics ”the truth value of r is functionally determined by the truth values of p and q ”. Dependency atoms Complexity Results p q r From 3SAT to 0 1 1 s ADQBF t 1 1 0 References 0 1 0 u { s , u } �| = dep ( p , r ) , { s , t } | = dep ( p , q ) , { s , t , u } | = dep ( q ) , { s , t , u } | = dep ( r ) ∨ dep ( r ) . 6/ 14
Complexity of Extensions of propositional logic propositional logics in team semantics Jonni Virtema Movativation History We extend PL by adding atomic formulae that describe properties of teams. Team Semantics Dependency atoms Inclusion atoms: ( p 1 , p 2 ) ⊆ ( q 1 , q 2 ) Complexity Results ”truth values that appear for p 1 , p 2 also appear as truth values for q 1 , q 2 ”. From 3SAT to ADQBF p q r References 0 1 1 s { s , t } �| = p ⊆ q , { s , t } | = q ⊆ r , { s , t , u } | = ( p , q ) ⊆ ( r , q ) t 1 1 0 u 0 1 0 6/ 14
Complexity of Extensions of propositional logic propositional logics in team semantics Jonni Virtema We extend PL by adding atomic formulae that describe properties of teams. Movativation History Syntax of propositional dependence logic PD: Team Semantics Dependency atoms ϕ ::= p | ¬ p | ( ϕ ∧ ϕ ) | ( ϕ ∨ ϕ ) | dep ( p 1 , . . . , p n , q ) Complexity Results From 3SAT to Syntax of propositional inclusion logic PLInc: ADQBF References ϕ ::= p | ¬ p | ( ϕ ∧ ϕ ) | ( ϕ ∨ ϕ ) | ( p 1 , . . . , p n ) ⊆ ( q 1 , . . . , q n ) Syntax of propositional team logic PTL: ϕ ::= p | ¬ p | ( ϕ ∧ ϕ ) | ( ϕ ∨ ϕ ) | ∼ ϕ, with the semantics X | = ∼ ϕ iff X �| = ϕ . 6/ 14
Complexity of Important decision problems propositional logics in team semantics Jonni Virtema Movativation History Model checking: Team Semantics Input: A team X and a formula ϕ . Dependency atoms Output: Does X | = ϕ hold? Complexity Results From 3SAT to Satisfiability: ADQBF Input: A formula ϕ . References Output: Does there exists a non-empty team X s.t. X | = ϕ ? Validity: Input: A formula ϕ . Output: Does X | = ϕ hold for every non-empty team X ? 7/ 14
Complexity of Complexity results propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Satisfiability Validity Model checking Dependency atoms Complexity Results NC 1 PL NP coNP From 3SAT to ADQBF PD NP NEXPTIME NP References PLInc EXPTIME coNP P PTL AEXPTIME(poly) AEXPTIME(poly) PSPACE 8/ 14
Complexity of Source of hardness: propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms A well-known NP-complete problem: Complexity Results 3SAT: From 3SAT to Input: A 3CNF-formula ϕ ADQBF (e.g.,( p 2 ∨ ¬ p 7 ) ∧ ( ¬ p 1 ∨ p 3 ∨ p 2 ) ∧ ( p 3 ∨ ¬ p 4 ∨ ¬ p 2 ) ∧ p 2 ). References Output: Does there exists an assignment s s.t. s | = ϕ ? 9/ 14
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