[PPT] - Infinite State Model-Checking of Propositional Dynamic Logics PowerPoint Presentation

SLIDE 1

Infinite State Model-Checking of Propositional Dynamic Logics

Stefan G¨

ller and Markus Lohrey

Universit¨ at Stuttgart August 25, 2006

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 2

Pushdown systems

A pushdown system is a tuple S = (P, Γ, ∆), where P is a finite set of control states, Γ is a finite stack alphabet, ∆ is a set of rewriting rules, where either pγ ֌ p′

r

pγ ֌ p′γ′γ.

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 3

Pushdown systems

A pushdown system is a tuple S = (P, Γ, ∆), where P is a finite set of control states, Γ is a finite stack alphabet, ∆ is a set of rewriting rules, where either pγ ֌ p′

r

pγ ֌ p′γ′γ. The pushdown graph G(S) has as nodes: PΓ∗ edges: pw → p′w′ if there is a rewriting rule in ∆ that can be applied to the prefixes accordingly.

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 4

Model-checking pushdown systems

INPUT: A pushdown system S, a configuration c, and a logical formula ϕ. QUESTION: (G(S), c) | = ϕ?

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 5

Model-checking pushdown systems

INPUT: A pushdown system S, a configuration c, and a logical formula ϕ. QUESTION: (G(S), c) | = ϕ? Related results: MSO: decidable (non-elementary) [Muller/Schupp 96] µ-calculus: EXP-complete [Walukiewicz 96, Kupfermann/Vardi 00] CTL: EXP-complete [Walukiewicz 00] EF: PSPACE-complete [Esparza et al. 97, Walukiewicz 00]

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 6

The logic PDL∩: Syntax

Fix some countable set A of atomic programs.

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 7

The logic PDL∩: Syntax

Fix some countable set A of atomic programs. Formulas ϕ and programs π of PDL∩ are given by the following grammar, where a ∈ A: ϕ ::= true | ¬ϕ | ϕ1 ∨ ϕ2 | πϕ π ::= a | π1 ∪ π2 | π1 ∩ π2 | π1 ◦ π2 | π∗ | ϕ?

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 8

The logic PDL∩: Syntax

Fix some countable set A of atomic programs. Formulas ϕ and programs π of PDL∩ are given by the following grammar, where a ∈ A: ϕ ::= true | ¬ϕ | ϕ1 ∨ ϕ2 | πϕ π ::= a | π1 ∪ π2 | π1 ∩ π2 | π1 ◦ π2 | π∗ | ϕ? Abbreviation: [π]ϕ = ¬π¬ϕ

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 9

The logic PDL∩: Semantics

A Kripke structure is a tuple K = (X, {→a| a ∈ A}), where X is a set of states, and →a⊆ X × X is a binary relation for each a ∈ A.

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 10

The logic PDL∩: Semantics

A Kripke structure is a tuple K = (X, {→a| a ∈ A}), where X is a set of states, and →a⊆ X × X is a binary relation for each a ∈ A. Define [ [π] ]K ⊆ X × X and [ [ϕ] ]K ⊆ X inductively: [ [a] ]K =→a [ [true] ]K = X [ [ϕ?] ]K = {(x, x) | x ∈ [ [ϕ] ]K} [ [¬ϕ] ]K = X \ [ [ϕ] ]K [ [π∗] ]K = [ [π] ]∗

K

[ [ϕ1 ∨ ϕ2] ]K = [ [ϕ1] ]K ∪ [ [ϕ2] ]K [ [π1 op π2] ]K = [ [π1] ]K op [ [π2] ]K where op ∈ {∪, ∩, ◦} [ [πϕ] ]K = {x | ∃y : (x, y) ∈ [ [π] ]K ∧ y ∈ [ [ϕ] ]K}

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 11

PDL∩: An example

The formula (a ◦ b∗ ◦ a) ∩ true? true enforces a cycle that begins with an a-labeled edge, followed by an arbitrary sequence of b-labeled edges, and ends with an a-labeled edge.

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 12

PDL∩: A non-trivial example

Let K = (X, {→a| a ∈ Σ}) be a deterministic Kripke structure.

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 13

PDL∩: A non-trivial example

Let K = (X, {→a| a ∈ Σ}) be a deterministic Kripke structure. We call a state x ∈ X a recovery state if, wherever we can get from x, we can always move back to x.

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 14

PDL∩: A non-trivial example

Let K = (X, {→a| a ∈ Σ}) be a deterministic Kripke structure. We call a state x ∈ X a recovery state if, wherever we can get from x, we can always move back to x. A node x ∈ X is a recovery state if and only if (K, x) | = [Σ∗]

a∈Σ
atrue ⇒ true? ∩ a ◦ Σ∗true
.

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 15

PDL∩: A non-trivial example

Let K = (X, {→a| a ∈ Σ}) be a deterministic Kripke structure. We call a state x ∈ X a recovery state if, wherever we can get from x, we can always move back to x. A node x ∈ X is a recovery state if and only if (K, x) | = [Σ∗]

a∈Σ
atrue ⇒ true? ∩ a ◦ Σ∗true
.

The recovery state property cannot be expressed in the modal µ-calculus.

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 16

The logic PDL∩: Properties and difficulties

PDL∩ does not have the tree model property, e.g. a ∩ true?true enforces

a

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 17

The logic PDL∩: Properties and difficulties

PDL∩ does not have the tree model property, e.g. a ∩ true?true enforces

a

is therefore not bisimulation invariant.

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 18

The logic PDL∩: Properties and difficulties

PDL∩ does not have the tree model property, e.g. a ∩ true?true enforces

a

is therefore not bisimulation invariant. does not have the finite model property.

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 19

The logic PDL∩: Properties and difficulties

PDL∩ does not have the tree model property, e.g. a ∩ true?true enforces

a

is therefore not bisimulation invariant. does not have the finite model property. satisfiability is 2EXP-complete [Danecki 84, Lange/Lutz 2005].

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 20

Complexity results of the model-checking problem

Basic process algebras Pushdown systems Pref.-recogn. systems

data P-complete EXP- complete EF expression PSPACE-complete PDL\? combined EXP- complete data P-complete PDL expression EXP-complete combined data PSPACE-hard in EXP EXP-complete PDL∩ expression PDL∩\? combined 2EXP-complete

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 21

PDL∩ over pushdown systems is in 2EXP (i)

A two-way alternating parity ω-tree automaton (TWAPTA) T is an automaton, that

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 22

PDL∩ over pushdown systems is in 2EXP (i)

A two-way alternating parity ω-tree automaton (TWAPTA) T is an automaton, that runs on infinite trees,

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 23

PDL∩ over pushdown systems is in 2EXP (i)

A two-way alternating parity ω-tree automaton (TWAPTA) T is an automaton, that runs on infinite trees, may use alternation,

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 24

PDL∩ over pushdown systems is in 2EXP (i)

A two-way alternating parity ω-tree automaton (TWAPTA) T is an automaton, that runs on infinite trees, may use alternation, can either move to some child, move to the parent node, or stay in the same node, and

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 25

PDL∩ over pushdown systems is in 2EXP (i)

A two-way alternating parity ω-tree automaton (TWAPTA) T is an automaton, that runs on infinite trees, may use alternation, can either move to some child, move to the parent node, or stay in the same node, and uses a parity acceptance condition.

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 26

PDL∩ over pushdown systems is in 2EXP (ii)

INPUT: A pushdown system S = (P, Γ, ∆), some configuration c, and a PDL∩ formula ϕ.

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 27

PDL∩ over pushdown systems is in 2EXP (ii)

INPUT: A pushdown system S = (P, Γ, ∆), some configuration c, and a PDL∩ formula ϕ. OUR GOAL: Find a TWAPTA T such that (G(S), c) | = ϕ if and only if T accepts the complete P ∪ Γ-tree ε

p1 p2 · · · p|P| γ1 γ2 · · · γ|Γ|

· · · · · · · · · · · · . . . . . . . . . . . . . . .

p1γ1 p2γ1 · · · p|P|γ1 γ1γ1 · · · γ|Γ|γ1

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 28

PDL∩ over pushdown systems is in 2EXP (iii)

Let Υ = P ∪ Γ.

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 29

PDL∩ over pushdown systems is in 2EXP (iii)

Let Υ = P ∪ Γ. We define [ [T ] ] = {w ∈ Υ∗ | T has a successful run starting from node w}

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 30

PDL∩ over pushdown systems is in 2EXP (iii)

Let Υ = P ∪ Γ. We define [ [T ] ] = {w ∈ Υ∗ | T has a successful run starting from node w} Translating formulas: For each formula ψ find a TWAPTA T (ψ) such that: [ [ψ] ]G(S) = [ [T (ψ)] ]

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 31

PDL∩ over pushdown systems is in 2EXP (iii)

Let Υ = P ∪ Γ. We define [ [T ] ] = {w ∈ Υ∗ | T has a successful run starting from node w} Translating formulas: For each formula ψ find a TWAPTA T (ψ) such that: [ [ψ] ]G(S) = [ [T (ψ)] ] How can we handle programs (i.e. binary relations)?

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 32

PDL∩ over pushdown systems is in 2EXP (iv)

T [s] : The same TWAPTA as T but with initial state s.

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 33

PDL∩ over pushdown systems is in 2EXP (iv)

T [s] : The same TWAPTA as T but with initial state s. An NFA A over a TWAPTA T (with state set S) is an ordinary NFA with one initial state and one final state over the alphabet Υ ∪ Υ ∪ {T [s]? | s ∈ S}

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 34

PDL∩ over pushdown systems is in 2EXP (iv)

T [s] : The same TWAPTA as T but with initial state s. An NFA A over a TWAPTA T (with state set S) is an ordinary NFA with one initial state and one final state over the alphabet Υ ∪ Υ ∪ {T [s]? | s ∈ S} Define [ [a] ] = {(w, aw) | w ∈ Υ∗} (a ∈ Υ) [ [a] ] = {(aw, w) | w ∈ Υ∗} (a ∈ Υ) [ [T [s]?] ] = {(w, w) | w ∈ [ [T [s]] ]} [ [A] ] = {[ [σ1] ] ◦ · · · ◦ [ [σn] ] | σ1 · · · σn ∈ L(A)}.

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 35

PDL∩ over pushdown systems is in 2EXP (v)

Translating programs: For each subprogram π, find an automaton A(π) over some TWAPTA T (π), such that: [ [A(π)] ] = [ [π] ]G(S)

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 36

PDL∩ over pushdown systems is in 2EXP (v)

Translating programs: For each subprogram π, find an automaton A(π) over some TWAPTA T (π), such that: [ [A(π)] ] = [ [π] ]G(S) if π = π1 ∪ π2, π = π1 ◦ π2, or π = π∗

1, then A(π) and T (π)

are built by applying standard automata constructions.

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 37

PDL∩ over pushdown systems is in 2EXP (v)

Translating programs: For each subprogram π, find an automaton A(π) over some TWAPTA T (π), such that: [ [A(π)] ] = [ [π] ]G(S) if π = π1 ∪ π2, π = π1 ◦ π2, or π = π∗

1, then A(π) and T (π)

are built by applying standard automata constructions. if π = π1 ∩ π2. GIVEN: A(πi) over T (πi) (i ∈ {1, 2}).

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 38

PDL∩ over pushdown systems is in 2EXP (v)

Translating programs: For each subprogram π, find an automaton A(π) over some TWAPTA T (π), such that: [ [A(π)] ] = [ [π] ]G(S) if π = π1 ∪ π2, π = π1 ◦ π2, or π = π∗

1, then A(π) and T (π)

are built by applying standard automata constructions. if π = π1 ∩ π2. GIVEN: A(πi) over T (πi) (i ∈ {1, 2}). GOAL: Find A(π) over some T (π) such that [ [A(π)] ] = [ [A(π1)] ] ∩ [ [A(π2)] ].

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 39

PDL∩ over pushdown systems is in 2EXP (v)

Translating programs: For each subprogram π, find an automaton A(π) over some TWAPTA T (π), such that: [ [A(π)] ] = [ [π] ]G(S) if π = π1 ∪ π2, π = π1 ◦ π2, or π = π∗

1, then A(π) and T (π)

are built by applying standard automata constructions. if π = π1 ∩ π2. GIVEN: A(πi) over T (πi) (i ∈ {1, 2}). GOAL: Find A(π) over some T (π) such that [ [A(π)] ] = [ [A(π1)] ] ∩ [ [A(π2)] ]. Problem: The runs of A(π1) and A(π2) may diverge.

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 40

PDL∩ over pushdown systems is in 2EXP (vi)

A(π1) A(π2)

x
y
inf(x, y)
w
z

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 41

PDL∩ over pushdown systems is in 2EXP (vi)

A(π1) A(π2)

x
y
inf(x, y)
w
z

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 42

PDL∩ over pushdown systems is in 2EXP (vi)

A(π1) A(π2)

x
y
inf(x, y)
w
z

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 43

PDL∩ over pushdown systems is in 2EXP (vi)

A(π1) A(π2)

x
y
inf(x, y)
w
z

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 44

PDL∩ over pushdown systems is in 2EXP (vi)

A(π1) A(π2)

x
y
inf(x, y)
w
z

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 45

PDL∩ over pushdown systems is in 2EXP (vi)

A(π1) A(π2)

x
y
inf(x, y)
w
z

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 46

PDL∩ over pushdown systems is in 2EXP (vi)

A(π1) A(π2)

x
y
inf(x, y)
w
z

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 47

PDL∩ over pushdown systems is in 2EXP (vi)

A(π1) A(π2)

x
y
inf(x, y)
w
z

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 48

PDL∩ over pushdown systems is in 2EXP (vi)

A(π1) A(π2)

x
y
inf(x, y)
w
z

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 49

PDL∩ over pushdown systems is in 2EXP (vi)

A(π1) A(π2)

x
y
inf(x, y)
w
z

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 50

PDL∩ over pushdown systems is in 2EXP (vi)

A(π1) A(π2)

x
y
inf(x, y)
w
z

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 51

PDL∩ over pushdown systems is in 2EXP (vi)

A(π1) A(π2)

x
y
inf(x, y)
w
z

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 52

PDL∩ over pushdown systems is in 2EXP (vi)

A(π1) A(π2)

x
y
inf(x, y)
w
z

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 53

PDL∩ over pushdown systems is in 2EXP (vi)

A(π1) A(π2)

x
y
inf(x, y)
w
z

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 54

PDL∩ over pushdown systems is in 2EXP (vi)

A(π1) A(π2)

x
y
inf(x, y)
w
z

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 55

PDL∩ over pushdown systems is in 2EXP (vi)

A(π1) A(π2)

x
y
inf(x, y)
w
z

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 56

PDL∩ over pushdown systems is in 2EXP (vi)

A(π1) A(π2)

x
y
inf(x, y)
w
z

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 57

PDL∩ over pushdown systems is in 2EXP (vi)

A(π1) A(π2)

x
y
inf(x, y)
w
z

qr

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 58

PDL∩ over pushdown systems is in 2EXP (vi)

A(π1) A(π2)

x
y
inf(x, y)
w
z

qr (w, w) ∈ [ [A(π2)[q, r]] ]

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 59

PDL∩ over pushdown systems is in 2EXP (vi)

A(π1) A(π2)

x
y
inf(x, y)
w
z

qr (w, w) ∈ [ [A(π2)[q, r]] ] q′ r′

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 60

PDL∩ over pushdown systems is in 2EXP (vi)

A(π1) A(π2)

x
y
inf(x, y)
w
z

qr (w, w) ∈ [ [A(π2)[q, r]] ] q′ r′ (z, z) ∈ [ [A(π1)[q′, r′]] ]

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 61

PDL∩ over pushdown systems is in 2EXP (vi)

A(π1) A(π2)

x
y
inf(x, y)
w
z

qr (w, w) ∈ [ [A(π2)[q, r]] ] q′ r′ (z, z) ∈ [ [A(π1)[q′, r′]] ]

Solution: For all pairs of states (q, r) of A(πi) find a TWAPTA T (q, r) such that: [ [T (q, r)] ] = {w ∈ Υ∗ | (w, w) ∈ [ [A(πi)[q, r]] ]}

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 62

PDL∩ over pushdown systems is in 2EXP (vi)

A(π1) A(π2)

x
y
inf(x, y)
w
z

qr (w, w) ∈ [ [A(π2)[q, r]] ] q′ r′ (z, z) ∈ [ [A(π1)[q′, r′]] ]

Solution: For all pairs of states (q, r) of A(πi) find a TWAPTA T (q, r) such that: [ [T (q, r)] ] = {w ∈ Υ∗ | (w, w) ∈ [ [A(πi)[q, r]] ]} For all pairs (q, r) of A(πi) add to A(πi) the transition q

T (q,r)?

− − − − − → r B(πi)

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 63

PDL∩ over pushdown systems is in 2EXP (vi)

B(π1) B(π2)

x
y
inf(x, y)
w
z

Solution: For all pairs of states (q, r) of A(πi) find a TWAPTA T (q, r) such that: [ [T (q, r)] ] = {w ∈ Υ∗ | (w, w) ∈ [ [A(πi)[q, r]] ]} For all pairs (q, r) of A(πi) add to A(πi) the transition q

T (q,r)?

− − − − − → r B(πi)

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 64

PDL∩ over pushdown systems is in 2EXP (vii)

A(π1 ∩ π2) is then the product automaton of B(π1) and B(π2), where

Υ- and Υ-transitions are done synchronously, and T [s]-transitions are done asynchronously.

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 65

PDL∩ over pushdown systems is in 2EXP (vii)

A(π1 ∩ π2) is then the product automaton of B(π1) and B(π2), where

Υ- and Υ-transitions are done synchronously, and T [s]-transitions are done asynchronously.

|A(π1 ∩ π2)| ∈ O

|A(π1)|2 + |A(π2)|2

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 66

PDL∩ over pushdown systems is in 2EXP (vii)

A(π1 ∩ π2) is then the product automaton of B(π1) and B(π2), where

Υ- and Υ-transitions are done synchronously, and T [s]-transitions are done asynchronously.

|A(π1 ∩ π2)| ∈ O

|A(π1)|2 + |A(π2)|2

|T (ϕ)| ∈ O

|S||ϕ|2

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 67

PDL∩ over pushdown systems is in 2EXP (vii)

A(π1 ∩ π2) is then the product automaton of B(π1) and B(π2), where

Υ- and Υ-transitions are done synchronously, and T [s]-transitions are done asynchronously.

|A(π1 ∩ π2)| ∈ O

|A(π1)|2 + |A(π2)|2

|T (ϕ)| ∈ O

|S||ϕ|2

Acceptance problem for TWAPTAs is EXP-complete [Vardi]

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 68

PDL∩ over pushdown systems is in 2EXP (vii)

A(π1 ∩ π2) is then the product automaton of B(π1) and B(π2), where

Υ- and Υ-transitions are done synchronously, and T [s]-transitions are done asynchronously.

|A(π1 ∩ π2)| ∈ O

|A(π1)|2 + |A(π2)|2

|T (ϕ)| ∈ O

|S||ϕ|2

Acceptance problem for TWAPTAs is EXP-complete [Vardi] ⇒ PDL∩ over pushdown systems is in 2EXP.

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 69

Future work

Close the gap between PSPACE and EXP for the data complexity of PDL∩ over BPAs.

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 70

Future work

Close the gap between PSPACE and EXP for the data complexity of PDL∩ over BPAs. Extend PDL∩ by allowing fixed point operators.

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics

SLIDE 71

Future work

Close the gap between PSPACE and EXP for the data complexity of PDL∩ over BPAs. Extend PDL∩ by allowing fixed point operators. Find the connection between infinite state model-checking and satisfiability for PDL∩.

Stefan G¨

ller and Markus Lohrey Universit¨

at Stuttgart Infinite State Model-Checking of Propositional Dynamic Logics