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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

α,β ::= a | (α ; β) | (α ∪ β) | α∗ | ϕ? (programs) ϕ,ψ ::= p | ⊥ | ϕ → ψ | αϕ (formulas)

Semantics

ap

p q,r p,q p,q a b b b a

Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 2 / 13

Propositional Dynamic Logic (PDL)

Syntax

α,β ::= a | (α ; β) | (α ∪ β) | α∗ | ϕ? (programs) ϕ,ψ ::= p | ⊥ | ϕ → ψ | αϕ (formulas)

Semantics

[a]q

p q,r p,q p,q b b b a a

Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 2 / 13

Propositional Dynamic Logic (PDL)

Syntax

α,β ::= a | (α ; β) | (α ∪ β) | α∗ | ϕ? (programs) ϕ,ψ ::= p | ⊥ | ϕ → ψ | αϕ (formulas)

Semantics

a ; br

p q,r p,q p,q a b b a b

Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 2 / 13

Propositional Dynamic Logic (PDL)

Syntax

α,β ::= a | (α ; β) | (α ∪ β) | α∗ | ϕ? (programs) ϕ,ψ ::= p | ⊥ | ϕ → ψ | αϕ (formulas)

Semantics

[a ∪ b]q

p q,r p,q p,q b b a a b

Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 2 / 13

Propositional Dynamic Logic (PDL)

Syntax

α,β ::= a | (α ; β) | (α ∪ β) | α∗ | ϕ? (programs) ϕ,ψ ::= p | ⊥ | ϕ → ψ | αϕ (formulas)

Semantics

p?⊤

p q,r p,q p,q a a b b b

Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 2 / 13

Propositional Dynamic Logic (PDL)

Syntax

α,β ::= a | (α ; β) | (α ∪ β) | α∗ | ϕ? (programs) ϕ,ψ ::= p | ⊥ | ϕ → ψ | αϕ (formulas)

Semantics

b ∗r

p q,r p,q p,q a a b b b

Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 2 / 13

Propositional Dynamic Logic (PDL)

Syntax

α,β ::= a | (α ; β) | (α ∪ β) | α∗ | ϕ? (programs) ϕ,ψ ::= p | ⊥ | ϕ → ψ | αϕ (formulas)

Semantics

(p? ; b)∗r

p q,r p,q p,q a a b b b

Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 2 / 13

Propositional Dynamic Logic (PDL)

Properties

◮ Satisfiability problem is EXPTIME-complete. ◮ Tree-like model property.

Fischer-Ladner closure

ap p

p q,r p,q p,q a b b b a

Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 3 / 13

Propositional Dynamic Logic (PDL)

Properties

◮ Satisfiability problem is EXPTIME-complete. ◮ Tree-like model property.

Fischer-Ladner closure

[a]q q

p q,r p,q p,q b b b a a

Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 3 / 13

Propositional Dynamic Logic (PDL)

Properties

◮ Satisfiability problem is EXPTIME-complete. ◮ Tree-like model property.

Fischer-Ladner closure

a ; br abr br

p q,r p,q p,q a b b a b

Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 3 / 13

Propositional Dynamic Logic (PDL)

Properties

◮ Satisfiability problem is EXPTIME-complete. ◮ Tree-like model property.

Fischer-Ladner closure

[a ∪ b]q [a]q [b]q q

p q,r p,q p,q b b a a b

Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 3 / 13

Propositional Dynamic Logic (PDL)

Properties

◮ Satisfiability problem is EXPTIME-complete. ◮ Tree-like model property.

Fischer-Ladner closure

p?⊤ p

p q,r p,q p,q a a b b b

Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 3 / 13

Propositional Dynamic Logic (PDL)

Properties

◮ Satisfiability problem is EXPTIME-complete. ◮ Tree-like model property.

Fischer-Ladner closure

b ∗r r bb ∗r b ∗r

p q,r p,q p,q a a b b b

Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 3 / 13

Propositional Dynamic Logic (PDL)

Properties

◮ Satisfiability problem is EXPTIME-complete. ◮ Tree-like model property.

Fischer-Ladner closure

(p? ; b)∗r r p? ; b(p? ; b)∗r p?b(p? ; b)∗r p b(p? ; b)∗r (p? ; b)∗r

p q,r p,q p,q a a b b b

Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 3 / 13

Interleaving PDL

Syntax

α,β ::= a | (α ; β) | (α ∪ β) | α∗ | ϕ? | (α | β) (programs) ϕ,ψ ::= p | ⊥ | ϕ → ψ | αϕ (formulas)

Semantics

L(α | β) L(α) ✁ L(β) For instance: a | bϕ ↔ (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

PDL with intersection

Syntax

α,β ::= a | (α ; β) | (α ∪ β) | α∗ | ϕ? | (α ∩ β) (programs) ϕ,ψ ::= p | ⊥ | ϕ → ψ | αϕ (formulas)

Semantics

We have: α ∩ βϕ → αϕ ∧ βϕ w x

α ∩ β α β

Complexity

The satisfiability problem is 2EXPTIME-complete.

Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 5 / 13

Concurrency and cooperation

Sometimes only parallel programs are executable.

≤ 1 a ≤ 1 b a b⊤ ∧ [a]⊥ ∧ [b]⊥

Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 6 / 13

PDL with separating Parallel composition (PPDL)

Syntax

α,β ::= a | (α ; β) | (α ∪ β) | α∗ | ϕ? | (α β) (programs) ϕ,ψ ::= p | ⊥ | ϕ → ψ | αϕ (formulas)

Semantics

A model is a tuple M = (W,R,⊳,V) where:

◮ (W,R,V) is a PDL model, ◮ ⊳ is a ternary relation over W.

w2 x w1 w3 y w4

α β α β

Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 7 / 13

PDL with deterministic separating Parallel composition (PPDLdet)

Definition

A model is ⊳-deterministic iff there is at most one way to merge any pair

• f states:

if w1 ⊳ (x,y) and w2 ⊳ (x,y) then w1 = w2

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

The Petri net example a b a b

Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 9 / 13

Adaptation of the Fischer-Ladner closure [β] ... [α β]ϕ [α] ... α β

Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 10 / 13

Adaptation of the Fischer-Ladner closure [β]R1 [α β]ϕ [α]L1 α β

Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 10 / 13

Adaptation of the Fischer-Ladner closure [β]R1 [α β]ϕ [α]L1 L1 R1 α β

Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 10 / 13

Adaptation of the Fischer-Ladner closure [β]R1 [α β]ϕ [α]L1 L1 ϕ R1 α β

Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 10 / 13

The neat model property α β

γ δ

Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 11 / 13

Elimination of Hintikka sets procedure

Hintikka sets

Maximal consistent subsets of the Fischer-Ladner closure.

Plugs

Triples (H1,H2,H3) of Hintikka sets s.t. Li ∈ H2 and Ri ∈ H3 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. (H1,S1) ⊳ ((H2,S2),(H3,S3)) iff (H1,H2,H3) ∈ S2 = S3

Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition 12 / 13

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