for
ThI Institute
Theoretical Computer Science
Complexity of Model
Checking for Modal
Dependence Logic
Johannes Ebbing
joint work with Peter Lohmann
Institut für Theoretische Informatik
Leibniz Universität Hannover
January 23, 2012
SOFSEM 2012
ThI
Institute for
Theoretical Computer Science
Overview
1 Introduction
2 Preliminaries
Kripke Structure
Modal Dependence Logic
Model Checking
3 Complexity Results on Model Checking for MDL
Unbounded arity fragments
Bounded arity fragments
4 Conclusion and open Questions
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 2
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Institute for
Theoretical Computer Science
Motivation
What is the motivation for Modal Dependence Logic?
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 3
ThI
Institute for
Theoretical Computer Science
Motivation
What is the motivation for Modal Dependence Logic?
Dependence atom extends ordinary modal logic
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 3
ThI
Institute for
Theoretical Computer Science
Motivation
What is the motivation for Modal Dependence Logic?
Dependence atom extends ordinary modal logic
CTL is used for verifying systems e.g. integrated circuits
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 3
ThI
Institute for
Theoretical Computer Science
Motivation
What is the motivation for Modal Dependence Logic?
Dependence atom extends ordinary modal logic
CTL is used for verifying systems e.g. integrated circuits
MDL seen as first approach of DCTL (= CTL + dependence atom)
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 3
ThI
Institute for
Theoretical Computer Science
Motivation
What is the motivation for Modal Dependence Logic?
Dependence atom extends ordinary modal logic
CTL is used for verifying systems e.g. integrated circuits
MDL seen as first approach of DCTL (= CTL + dependence atom)
MDL is based on teams, not on single assignments / worlds
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 3
ThI
Institute for
Theoretical Computer Science
Introduction
ML
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 4
ThI
Institute for
Theoretical Computer Science
Introduction
ML
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 4
ThI
Institute for
Theoretical Computer Science
Introduction
CTL
ML
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 4
ThI
Institute for
Theoretical Computer Science
Introduction
CTL
ML
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 4
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Institute for
Theoretical Computer Science
Introduction
CTL
MDL
extended by =(·)
ML
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 4
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Institute for
Theoretical Computer Science
Introduction
CTL
MDL
extended by =(·)
ML
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 4
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Theoretical Computer Science
Introduction
extended by =(·)
DCTL
CTL
MDL
extended by =(·)
ML
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 4
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Theoretical Computer Science
Introduction
extended by =(·)
DCTL
CTL
MDL
extended by =(·)
ML
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 4
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Institute for
Theoretical Computer Science
Kripke structure
Definition
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 5
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Institute for
Theoretical Computer Science
Kripke structure
Definition
A Kripke structure (or frame) over a set of atomic propositions AP (or
AP-Kripke structure) is a tuple W = (S, R, π), where
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 5
ThI
Institute for
Theoretical Computer Science
Kripke structure
Definition
A Kripke structure (or frame) over a set of atomic propositions AP (or
AP-Kripke structure) is a tuple W = (S, R, π), where
S 6= ∅
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 5
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Institute for
Theoretical Computer Science
Kripke structure
Definition
A Kripke structure (or frame) over a set of atomic propositions AP (or
AP-Kripke structure) is a tuple W = (S, R, π), where
S 6= ∅
R ⊆S ×S
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 5
ThI
Institute for
Theoretical Computer Science
Kripke structure
Definition
A Kripke structure (or frame) over a set of atomic propositions AP (or
AP-Kripke structure) is a tuple W = (S, R, π), where
S 6= ∅
R ⊆S ×S
π : S → P(AP)
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 5
ThI
Institute for
Theoretical Computer Science
Kripke structure
Definition
A Kripke structure (or frame) over a set of atomic propositions AP (or
AP-Kripke structure) is a tuple W = (S, R, π), where
S 6= ∅
R ⊆S ×S
π : S → P(AP)
If s ∈ S , s is called world. We call S the set of worlds. R is called the
transition relation between the worlds, and π is called the labeling function.
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 5
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Institute for
Theoretical Computer Science
Modal Dependence Logic
Syntax of Modal Logic
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 6
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Modal Dependence Logic
Syntax of Modal Logic
Let AP be a set of atomic propositions and p1 , . . . , pn , q ∈ AP. The syntax
of modal logic formulae is defined by the grammar:
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 6
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Institute for
Theoretical Computer Science
Modal Dependence Logic
Syntax of Modal Logic
Let AP be a set of atomic propositions and p1 , . . . , pn , q ∈ AP. The syntax
of modal logic formulae is defined by the grammar:
ϕ ::= q | ¬q | ϕ ∨ ϕ | ϕ ∧ ϕ | ϕ | ♦ ϕ
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 6
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Institute for
Theoretical Computer Science
Modal Dependence Logic
Syntax of Modal Dependence Logic
Let AP be a set of atomic propositions and p1 , . . . , pn , q ∈ AP. The syntax
of modal logic formulae is defined by the grammar:
ϕ ::= q | ¬q | ϕ ∨ ϕ | ϕ ∧ ϕ | ϕ | ♦ ϕ | =(p1 , . . . , pn ; q) |
¬ =(p1 , . . . , pn ; q)
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 6
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Modal Logic
Semantics of Modal Logic
Let W = (S, R, π) an AP-Kripke structure, T ⊆ S a set of initial states, and
ϕ an ML formula and p ∈ AP. Inductive over the structure of ϕ we define
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 7
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Modal Logic
Semantics of Modal Logic
Let W = (S, R, π) an AP-Kripke structure, T ⊆ S a set of initial states, and
ϕ an ML formula and p ∈ AP. Inductive over the structure of ϕ we define
W , T |= p
Introduction
Preliminaries
iff
p ∈ π(s) for all s ∈ T
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 7
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Institute for
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Modal Logic
Semantics of Modal Logic
Let W = (S, R, π) an AP-Kripke structure, T ⊆ S a set of initial states, and
ϕ an ML formula and p ∈ AP. Inductive over the structure of ϕ we define
W , T |= p
W , T |= ¬p
Introduction
Preliminaries
iff
iff
p ∈ π(s) for all s ∈ T
p∈
/ π(s) for all s ∈ T
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 7
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Institute for
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Modal Logic
Semantics of Modal Logic
Let W = (S, R, π) an AP-Kripke structure, T ⊆ S a set of initial states, and
ϕ an ML formula and p ∈ AP. Inductive over the structure of ϕ we define
W , T |= p
W , T |= ¬p
W , T |= ϕ ∨ ψ
Introduction
Preliminaries
iff
iff
iff
p ∈ π(s) for all s ∈ T
p∈
/ π(s) for all s ∈ T
there are sets T1 , T2 ⊆ T with
T1 ∪ T2 = T ,
W , T1 |= ϕ and W , T2 |= ψ
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 7
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Institute for
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Modal Logic
Semantics of Modal Logic
Let W = (S, R, π) an AP-Kripke structure, T ⊆ S a set of initial states, and
ϕ an ML formula and p ∈ AP. Inductive over the structure of ϕ we define
W , T |= p
W , T |= ¬p
W , T |= ϕ ∨ ψ
iff
iff
iff
W , T |= ϕ ∧ ψ
iff
Introduction
Preliminaries
p ∈ π(s) for all s ∈ T
p∈
/ π(s) for all s ∈ T
there are sets T1 , T2 ⊆ T with
T1 ∪ T2 = T ,
W , T1 |= ϕ and W , T2 |= ψ
W , T |= ϕ and W , T |= ψ
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 7
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Institute for
Theoretical Computer Science
Modal Logic
Semantics of Modal Logic
Let W = (S, R, π) an AP-Kripke structure, T ⊆ S a set of initial states, and
ϕ an ML formula and p ∈ AP. Inductive over the structure of ϕ we define
W , T |= p
W , T |= ¬p
W , T |= ϕ ∨ ψ
iff
iff
iff
W , T |= ϕ ∧ ψ
W , T |= ϕ
iff
iff
Introduction
Preliminaries
p ∈ π(s) for all s ∈ T
p∈
/ π(s) for all s ∈ T
there are sets T1 , T2 ⊆ T with
T1 ∪ T2 = T ,
W , T1 |= ϕ and W , T2 |= ψ
W , T |= ϕ and W , T |= ψ
W , {s 0 | ∃s ∈ T s.t.(s, s 0 ) ∈ R} |= ϕ
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 7
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Institute for
Theoretical Computer Science
Modal Logic
Semantics of Modal Logic
Let W = (S, R, π) an AP-Kripke structure, T ⊆ S a set of initial states, and
ϕ an ML formula and p ∈ AP. Inductive over the structure of ϕ we define
W , T |= p
W , T |= ¬p
W , T |= ϕ ∨ ψ
iff
iff
iff
W , T |= ϕ ∧ ψ
W , T |= ϕ
W , T |= ♦ ϕ
iff
iff
iff
Introduction
Preliminaries
p ∈ π(s) for all s ∈ T
p∈
/ π(s) for all s ∈ T
there are sets T1 , T2 ⊆ T with
T1 ∪ T2 = T ,
W , T1 |= ϕ and W , T2 |= ψ
W , T |= ϕ and W , T |= ψ
W , {s 0 | ∃s ∈ T s.t.(s, s 0 ) ∈ R} |= ϕ
there is a set T 0 ⊆ T s. t. W , T 0 |= ϕ
and for all a ∈ T there is an a0 ∈ T 0
with (a, a0 ) ∈ R.
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 7
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Institute for
Theoretical Computer Science
Modal Logic
Semantics of Modal Logic
Let W = (S, R, π) an AP-Kripke structure, T ⊆ S a set of initial states, and
ϕ an ML formula and p ∈ AP. Inductive over the structure of ϕ we define
W , T |= p
W , T |= ¬p
W , T |= ϕ ∨ ψ
iff
iff
iff
W , T |= ϕ ∧ ψ
W , T |= ϕ
W , T |= ♦ ϕ
iff
iff
iff
Introduction
Preliminaries
p ∈ π(s) for all s ∈ T
p∈
/ π(s) for all s ∈ T
there are sets T1 , T2 ⊆ T with
T1 ∪ T2 = T ,
W , T1 |= ϕ and W , T2 |= ψ
W , T |= ϕ and W , T |= ψ
W , {s 0 | ∃s ∈ T s.t.(s, s 0 ) ∈ R} |= ϕ
there is a set T 0 ⊆ T s. t. W , T 0 |= ϕ
and for all a ∈ T there is an a0 ∈ T 0
with (a, a0 ) ∈ R.
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 7
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Institute for
Theoretical Computer Science
Modal Dependence Logic
Semantics of the Dependence Atom
Let W = (S, R, π) an AP-Kripke structure, T ⊆ S the evaluation team. Let
p1 , . . . , pn , q ∈ AP. Then =(p1 , . . . , pn ; q) is called a dependence atom. The
truth of the dependence atom on W , T is defined by
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 8
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Modal Dependence Logic
Semantics of the Dependence Atom
Let W = (S, R, π) an AP-Kripke structure, T ⊆ S the evaluation team. Let
p1 , . . . , pn , q ∈ AP. Then =(p1 , . . . , pn ; q) is called a dependence atom. The
truth of the dependence atom on W , T is defined by
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 8
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Institute for
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Modal Dependence Logic
Semantics of the Dependence Atom
Let W = (S, R, π) an AP-Kripke structure, T ⊆ S the evaluation team. Let
p1 , . . . , pn , q ∈ AP. Then =(p1 , . . . , pn ; q) is called a dependence atom. The
truth of the dependence atom on W , T is defined by
W , T |= =(p1 , . . . , pn ; q)
iff for all s1 , s2 ∈ T it holds that
π(s1 ) ∩ {p1 , . . . , pn } =
6 π(s2 ) ∩ {p1 , . . . , pn }
or π(s1 ) ∩ {q} = π(s2 ) ∩ {q}
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 8
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Institute for
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Modal Dependence Logic
Semantics of the Dependence Atom
Let W = (S, R, π) an AP-Kripke structure, T ⊆ S the evaluation team. Let
p1 , . . . , pn , q ∈ AP. Then =(p1 , . . . , pn ; q) is called a dependence atom. The
truth of the dependence atom on W , T is defined by
W , T |= =(p1 , . . . , pn ; q)
iff for all s1 , s2 ∈ T it holds that
π(s1 ) ∩ {p1 , . . . , pn } =
6 π(s2 ) ∩ {p1 , . . . , pn }
or π(s1 ) ∩ {q} = π(s2 ) ∩ {q}
W , T |= ¬ =(p1 , . . . , pn ; q) iff T = ∅
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 8
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Modal Dependence Logic
Example: Let ϕ := ♦ =(p1 , . . . , p4 ; q) and T := {c1 , c2 , c3 }.
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 9
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Modal Dependence Logic
Example: Let ϕ := ♦ =(p1 , . . . , p4 ; q) and T := {c1 , c2 , c3 }.
s3
s̄3
s4
p3
p4 , q
p3 , q
s̄4
p4
c1
c2
c3
s̄2
Introduction
Preliminaries
s̄1
s1
s2
p1
p1 , q
p2 , q
Complexity Results on Model Checking for MDL
Conclusion and open Questions
p2
Page 9
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Modal Dependence Logic
Example: Let ϕ := ♦ =(p1 , . . . , p4 ; q) and T := {c1 , c2 , c3 }.
s3
s̄3
s4
p3
p4 , q
p3 , q
s̄4
p4
c1
c2
c3
s̄2
Introduction
Preliminaries
s̄1
s1
s2
p1
p1 , q
p2 , q
Complexity Results on Model Checking for MDL
Conclusion and open Questions
p2
Page 9
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Model Checking
Model Checking is a formal method to verify a given formula.
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 10
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Model Checking
Model Checking is a formal method to verify a given formula.
The Model Checking problem
Let M ⊆ {, ♦, ∧, ∨, ¬, =}.
Given: A Kripke structure W = (S, R, π), a formula ϕ ∈ MDL(M) and an
evaluation team T ⊆ S .
Problem: Check, whether W , T |= ϕ.
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 10
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Institute for
Theoretical Computer Science
Model Checking
Model Checking is a formal method to verify a given formula.
The Model Checking problem
Let M ⊆ {, ♦, ∧, ∨, ¬, =}.
Given: A Kripke structure W = (S, R, π), a formula ϕ ∈ MDL(M) and an
evaluation team T ⊆ S .
Problem: Check, whether W , T |= ϕ.
MDL-MC(M) :=
Introduction
Preliminaries
W = (S, R, π) an AP-Kripke structure,
hW , T , ϕi .
T ⊆ S , ϕ ∈ MDL(M) and W , T |= ϕ
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 10
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Complexity Results on MDL-MC
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 11
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Complexity Results on MDL-MC
Theorem
Model checking for modal dependence logic is NP-complete.
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 11
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Complexity Results on MDL-MC
Theorem
Model checking for modal dependence logic is NP-complete.
We have to show:
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 11
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Complexity Results on MDL-MC
Theorem
Model checking for modal dependence logic is NP-complete.
We have to show:
MDL-MC is in NP
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 11
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Complexity Results on MDL-MC
Theorem
Model checking for modal dependence logic is NP-complete.
We have to show:
MDL-MC is in NP
MDL-MC is NP-hard
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 11
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The MDL Model Checking problem is in NP
Algorithm
bool check(W = (S, R, π), ϕ, T )
case ϕ
when ϕ = p
foreach si ∈ T
if not p ∈ π(si ) then
return false;
return true;
when ϕ = ¬p
foreach si ∈ T
if p ∈ π(si ) then
return false;
return true;
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 12
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The MDL Model Checking problem is in NP
Algorithm
when ϕ = ψ ∨ θ
guess two sets A,B ⊆ T ;
if not A ∪ B = T then
return false;
return (check (W , ψ, A) and check (W , θ, B));
when ϕ = ψ ∧ θ
return (check (W , ψ, T ) and check (W , θ, T ));
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 13
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The MDL Model Checking problem is in NP
Algorithm
when ϕ = ♦ ψ
guess set of states T 0 ;
foreach s ∈ T
if there is no s 0 ∈ T 0 with (s, s 0 ) ∈ R then
return false;
return check (W , ψ, T 0 ) ;
when ϕ = ψ
T 0 := ∅;
foreach s 0 ∈ S
foreach s ∈ T
if (s, s 0 ) ∈ R then
T 0 ← T 0 ∪ {s 0 }
return check (W , ψ, T 0 ) ;
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 14
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The MDL Model Checking problem is in NP
Algorithm
when ϕ = =(p1 , . . . , pn ; q)
foreach (s, s 0 ) ∈ T × T
if π(s) ∩ {p1 , . . . , pn } = π(s 0 ) ∩ {p1 , . . . , pn } then
if not π(s) ∩ {q} = π(s 0 ) ∩ {q}
return false;
return true;
when ϕ = ¬ =(p1 , . . . , pn ; q)
return T = ∅
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 15
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The MDL Model Checking problem is NP-hard
Proposition
MDL-MC is NP-hard.
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 16
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The MDL Model Checking problem is NP-hard
Proposition
MDL-MC is NP-hard.
We reduce 3SAT onto MDL-MC, i.e. for every 3SAT formula φ, we find an
instance of MDL-MC which is at least as hard as φ.
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 16
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Complexity Results
∨
+
∗
+
−
+
Introduction
Preliminaries
∧
+
∗
∗
∗
−
Operators
♦ =(·)
∗ ∗
+
∗ +
+
+ ∗
+
∗ −
∗
− −
+
Complexity Results on Model Checking for MDL
Complexity
NP-complete
NP-complete
NP-complete
in P
in NP
Conclusion and open Questions
Page 17
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Institute for
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Complexity Results with bounded =(·)
∨
+
+
∗
+
−
−
∗
Operators
∧ ♦ =(·)
+ ∗ ∗
+
∗ ∗ +
+
+ ∗ +
+
∗ + ∗
+
− ∗ ∗
∗
∗ ∗ −
∗
− − −
∗
Complexity
NP-complete
NP-complete
NP-complete
NP-complete
in P
in P
in P
We limit the arity of the dependence atom, i.e. all dependence atoms are of
the form =(p1 , . . . , pj ; q) where j < k with a fixed k ∈ N.
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 18
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Complexity Results with bounded =(·)
∨
+
+
∗
+
−
−
∗
Operators
∧ ♦ =(·)
+ ∗ ∗
+
∗ ∗ +
+
+ ∗ +
+
∗ + ∗
+
− ∗ ∗
∗
∗ ∗ −
∗
− − −
∗
Complexity
NP-complete
NP-complete
NP-complete
NP-complete
in P
in P
in P
We limit the arity of the dependence atom, i.e. all dependence atoms are of
the form =(p1 , . . . , pj ; q) where j < k with a fixed k ∈ N.
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 18
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Open Questions
MDL-MC({¬, ∨, =(·)}) NP-complete?
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 19
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Open Questions
MDL-MC({¬, ∨, =(·)}) NP-complete?
Classification for DCTL-MC?
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 19
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Open Questions
MDL-MC({¬, ∨, =(·)}) NP-complete?
Classification for DCTL-MC?
Open case for MDL-MC(♦, ∧, =(·)) with bounded dependence atom,
is solved for k ≥ 0.
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
Page 19
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Open Questions
MDL-MC({¬, ∨, =(·)}) NP-complete?
Classification for DCTL-MC?
Open case for MDL-MC(♦, ∧, =(·)) with bounded dependence atom,
is solved for k ≥ 0.
Thank you!
Introduction
Preliminaries
Complexity Results on Model Checking for MDL
Conclusion and open Questions
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