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 | hαi ϕ
(formulas)
Semantics
q, r
a
hai p
b
a
p
b
p, q
b
p, q
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 | hαi ϕ
(formulas)
Semantics
q, r
a
[a] q
b
a
p
b
p, q
b
p, q
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 | hαi ϕ
(formulas)
Semantics
q, r
b
a
ha ; b i r
a
p
b
p, q
b
p, q
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 | hαi ϕ
(formulas)
Semantics
q, r
a
[a ∪ b ] q
b
a
p
p, q
b
b
p, q
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 | hαi ϕ
(formulas)
Semantics
q, r
a
hp?i >
b
a
p
b
p, q
b
p, q
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 | hαi ϕ
(formulas)
Semantics
q, r
b
a
hb ∗ i r
a
p
b
p, q
b
p, q
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 | hαi ϕ
(formulas)
Semantics
q, r
b
a
h(p? ; b )∗ i r
a
p
b
p, q
b
p, q
Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition
2 / 13
Propositional Dynamic Logic (PDL)
Properties
I
Satisfiability problem is EXPTIME-complete.
I
Tree-like model property.
Fischer-Ladner closure
q, r
a
hai p
p
b
a
p
b
p, q
b
p, q
Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition
3 / 13
Propositional Dynamic Logic (PDL)
Properties
I
Satisfiability problem is EXPTIME-complete.
I
Tree-like model property.
Fischer-Ladner closure
q, r
a
[a] q
q
b
a
p
b
p, q
b
p, q
Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition
3 / 13
Propositional Dynamic Logic (PDL)
Properties
I
Satisfiability problem is EXPTIME-complete.
I
Tree-like model property.
Fischer-Ladner closure
q, r
b
a
ha ; b i r
hai hb i r hb i r
a
p
b
p, q
b
p, q
Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition
3 / 13
Propositional Dynamic Logic (PDL)
Properties
I
Satisfiability problem is EXPTIME-complete.
I
Tree-like model property.
Fischer-Ladner closure
q, r
a
[a ∪ b ] q
[a] q [b ] q q
b
a
p
p, q
b
b
p, q
Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition
3 / 13
Propositional Dynamic Logic (PDL)
Properties
I
Satisfiability problem is EXPTIME-complete.
I
Tree-like model property.
Fischer-Ladner closure
q, r
a
hp?i >
p
b
a
p
b
p, q
b
p, q
Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition
3 / 13
Propositional Dynamic Logic (PDL)
Properties
I
Satisfiability problem is EXPTIME-complete.
I
Tree-like model property.
Fischer-Ladner closure
q, r
b
a
r
hb ∗ i r
hb i hb ∗ i r
∗
hb i r
a
p
b
p, q
b
p, q
Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition
3 / 13
Propositional Dynamic Logic (PDL)
Properties
I
Satisfiability problem is EXPTIME-complete.
I
Tree-like model property.
Fischer-Ladner closure
h(p? ; b )∗ i r
r hp? ; b i h(p? ; b )∗ i r
hp?i hb i h(p? ; b )∗ i r
p hb i h(p? ; b )∗ i r
h(p? ; b )∗ i r
q, r
b
a
a
p
b
p, q
b
p, q
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)
ϕ, ψ ::= p | ⊥ | ϕ → ψ | hαi ϕ
(programs)
(formulas)
Semantics
L (α | β) L(α)
L(β)
For instance: ha | b i ϕ ↔ h(a ; b ) ∪ (b ; a)i 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 | hαi ϕ
(formulas)
Semantics
α
We have: α ∩ β ϕ → hαi ϕ ∧ β ϕ
w
α∩β
x
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.
a
≤1
b
≤1
ha k b i > ∧ [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 k β)
(programs)
ϕ, ψ ::= p | ⊥ | ϕ → ψ | hαi ϕ
(formulas)
Semantics
w1
A model is a tuple M = (W , R , C, V )
where:
I
(W , R , V ) is a PDL model,
I
C is a ternary relation over W .
α
w3
αkβ
x
w2
Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition
β
y
w4
7 / 13
PDL with deterministic separating Parallel composition (PPDLdet )
Definition
A model is C-deterministic iff there is at most one way to merge any pair
of states:
if w1 C (x, y) and w2 C (x, y) then w1 = w2
Rationale
I
There is a partial binary operator • such that w C (x, y) ⇔ w = x • y.
I
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
Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition
b
9 / 13
Adaptation of the Fischer-Ladner closure
[α] . . .
α
[α k β] ϕ
[β] . . .
β
Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition
10 / 13
Adaptation of the Fischer-Ladner closure
[α] L1
α
[α k β] ϕ
[β] R1
β
Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition
10 / 13
Adaptation of the Fischer-Ladner closure
[α] L1
α
L1
β
R1
[α k β] ϕ
[β] R1
Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition
10 / 13
Adaptation of the Fischer-Ladner closure
[α] L1
α
L1
ϕ
[α k β] ϕ
[β] R1
β
R1
Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition
10 / 13
The neat model property
αkβ
γ kδ
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 ) C ((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
I
Design an optimal and implementable decision procedure.
I
Add commutativity and associativity.
Joseph Boudou: Complexity Optimal Decision Procedure for PDL with Parallel Composition
13 / 13