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Complexity Optimal Decision Procedure for PDL with Parallel Composition Joseph Boudou IRIT, Toulouse University, France IJCAR 2016, Coimbra Propositional Dynamic Logic (PDL) Syntax , ::= a | ( ; ) | ( ) | | ?

  1. Complexity Optimal Decision Procedure for PDL with Parallel Composition Joseph Boudou IRIT, Toulouse University, France IJCAR 2016, Coimbra

  2. Propositional Dynamic Logic (PDL) Syntax α,β ::= a | ( α ; β ) | ( α ∪ β ) | α ∗ | ϕ ? (programs) ϕ,ψ ::= p | ⊥ | ϕ → ψ | � α � ϕ (formulas) Semantics q , r a b a � a � p p p , q b b p , q Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 2 / 13

  3. Propositional Dynamic Logic (PDL) Syntax α,β ::= a | ( α ; β ) | ( α ∪ β ) | α ∗ | ϕ ? (programs) ϕ,ψ ::= p | ⊥ | ϕ → ψ | � α � ϕ (formulas) Semantics q , r a b a [ a ] q p p , q b b p , q Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 2 / 13

  4. Propositional Dynamic Logic (PDL) Syntax α,β ::= a | ( α ; β ) | ( α ∪ β ) | α ∗ | ϕ ? (programs) ϕ,ψ ::= p | ⊥ | ϕ → ψ | � α � ϕ (formulas) Semantics q , r a b a � a ; b � r p p , q b b p , q Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 2 / 13

  5. Propositional Dynamic Logic (PDL) Syntax α,β ::= a | ( α ; β ) | ( α ∪ β ) | α ∗ | ϕ ? (programs) ϕ,ψ ::= p | ⊥ | ϕ → ψ | � α � ϕ (formulas) Semantics q , r a b a [ a ∪ b ] q p p , q b b p , q Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 2 / 13

  6. Propositional Dynamic Logic (PDL) Syntax α,β ::= a | ( α ; β ) | ( α ∪ β ) | α ∗ | ϕ ? (programs) ϕ,ψ ::= p | ⊥ | ϕ → ψ | � α � ϕ (formulas) Semantics q , r a b a � p ? �⊤ p p , q b b p , q Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 2 / 13

  7. Propositional Dynamic Logic (PDL) Syntax α,β ::= a | ( α ; β ) | ( α ∪ β ) | α ∗ | ϕ ? (programs) ϕ,ψ ::= p | ⊥ | ϕ → ψ | � α � ϕ (formulas) Semantics q , r a b a � b ∗ � r p p , q b b p , q Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 2 / 13

  8. Propositional Dynamic Logic (PDL) Syntax α,β ::= a | ( α ; β ) | ( α ∪ β ) | α ∗ | ϕ ? (programs) ϕ,ψ ::= p | ⊥ | ϕ → ψ | � α � ϕ (formulas) Semantics q , r a b a � ( p ? ; b ) ∗ � r p p , q b b p , q Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 2 / 13

  9. Propositional Dynamic Logic (PDL) Properties ◮ Satisfiability problem is EXPTIME-complete. ◮ Tree-like model property. Fischer-Ladner closure q , r a b � a � p a p , q p p b b p , q Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 3 / 13

  10. Propositional Dynamic Logic (PDL) Properties ◮ Satisfiability problem is EXPTIME-complete. ◮ Tree-like model property. Fischer-Ladner closure q , r a b [ a ] q a p , q p q b b p , q Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 3 / 13

  11. Propositional Dynamic Logic (PDL) Properties ◮ Satisfiability problem is EXPTIME-complete. ◮ Tree-like model property. Fischer-Ladner closure q , r a b � a ; b � r a p , q p � a �� b � r � b � r b b p , q Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 3 / 13

  12. Propositional Dynamic Logic (PDL) Properties ◮ Satisfiability problem is EXPTIME-complete. ◮ Tree-like model property. Fischer-Ladner closure q , r a b [ a ∪ b ] q a p , q p [ a ] q [ b ] q q b b p , q Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 3 / 13

  13. Propositional Dynamic Logic (PDL) Properties ◮ Satisfiability problem is EXPTIME-complete. ◮ Tree-like model property. Fischer-Ladner closure q , r a b � p ? �⊤ a p , q p p b b p , q Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 3 / 13

  14. Propositional Dynamic Logic (PDL) Properties ◮ Satisfiability problem is EXPTIME-complete. ◮ Tree-like model property. Fischer-Ladner closure q , r a b � b ∗ � r a p , q p � b �� b ∗ � r � b ∗ � r r b b p , q Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 3 / 13

  15. Propositional Dynamic Logic (PDL) Properties ◮ Satisfiability problem is EXPTIME-complete. ◮ Tree-like model property. Fischer-Ladner closure q , r � ( p ? ; b ) ∗ � r a b � p ? ; b �� ( p ? ; b ) ∗ � r r a � p ? �� b �� ( p ? ; b ) ∗ � r p , q p � b �� ( p ? ; b ) ∗ � r p � ( p ? ; b ) ∗ � r b b p , q Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 3 / 13

  16. Interleaving PDL Syntax α,β ::= a | ( α ; β ) | ( α ∪ β ) | α ∗ | ϕ ? | ( α | β ) (programs) ϕ,ψ ::= p | ⊥ | ϕ → ψ | � α � ϕ (formulas) Semantics L ( α | β ) � L ( α ) ✁ L ( β ) For instance: � a | b � ϕ ↔ � ( a ; b ) ∪ ( b ; a ) � ϕ Complexity The satisfiability problem is 2EXPTIME-complete. Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 4 / 13

  17. PDL with intersection Syntax α,β ::= a | ( α ; β ) | ( α ∪ β ) | α ∗ | ϕ ? | ( α ∩ β ) (programs) ϕ,ψ ::= p | ⊥ | ϕ → ψ | � α � ϕ (formulas) Semantics α α ∩ β w x We have: � α ∩ β � ϕ → � α � ϕ ∧ � β � ϕ β Complexity The satisfiability problem is 2EXPTIME-complete. Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 5 / 13

  18. Concurrency and cooperation Sometimes only parallel programs are executable. a b ≤ 1 ≤ 1 � a � b �⊤ ∧ [ a ] ⊥ ∧ [ b ] ⊥ Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 6 / 13

  19. PDL with separating Parallel composition (PPDL) Syntax α,β ::= a | ( α ; β ) | ( α ∪ β ) | α ∗ | ϕ ? | ( α � β ) (programs) ϕ,ψ ::= p | ⊥ | ϕ → ψ | � α � ϕ (formulas) Semantics w 1 w 3 α A model is a tuple M = ( W , R , ⊳ , V ) where: α � β y x ◮ ( W , R , V ) is a PDL model, ◮ ⊳ is a ternary relation over W . β w 2 w 4 Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 7 / 13

  20. PDL with deterministic separating Parallel composition (PPDL det ) Definition A model is ⊳ -deterministic i ff there is at most one way to merge any pair of states: if w 1 ⊳ ( x , y ) and w 2 ⊳ ( x , y ) then w 1 = w 2 Rationale ◮ There is a partial binary operator • such that w ⊳ ( x , y ) ⇔ w = x • y . ◮ Usual constraint in modal logics with a binary modality (arrow logics, ambient logics, separation logic...). Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 8 / 13

  21. The Petri net example a b a b Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 9 / 13

  22. Adaptation of the Fischer-Ladner closure [ α ] ... α [ α � β ] ϕ β [ β ] ... Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 10 / 13

  23. Adaptation of the Fischer-Ladner closure [ α ] L 1 α [ α � β ] ϕ β [ β ] R 1 Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 10 / 13

  24. Adaptation of the Fischer-Ladner closure [ α ] L 1 L 1 α [ α � β ] ϕ β [ β ] R 1 R 1 Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 10 / 13

  25. Adaptation of the Fischer-Ladner closure [ α ] L 1 L 1 α ϕ [ α � β ] ϕ β [ β ] R 1 R 1 Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 10 / 13

  26. The neat model property α � β γ � δ Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 11 / 13

  27. Elimination of Hintikka sets procedure Hintikka sets Maximal consistent subsets of the Fischer-Ladner closure. Plugs Triples ( H 1 , H 2 , H 3 ) of Hintikka sets s.t. L i ∈ H 2 and R i ∈ H 3 for some i ∈ { 1 , 2 } . Sockets Sets of zero, one or two plugs. States of the pseudo-models Pairs ( H , S ) where H is a Hintikka set and S is a socket. ( H 1 , S 1 ) ⊳ (( H 2 , S 2 ) , ( H 3 , S 3 )) i ff ( H 1 , H 2 , H 3 ) ∈ S 2 = S 3 Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 12 / 13

  28. Conclusion Theoretical result The addition of deterministic separating parallel compositions to PDL does not increase the complexity of the satisfiability problem. Future works ◮ Design an optimal and implementable decision procedure. ◮ Add commutativity and associativity. Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 13 / 13

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