![Page 1: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/1.jpg)
PreliminariesModel checking and satisfiability
Some translations
On modal µ-calculus in S5 and applications
Giovanna D’AgostinoDIMI, Universita di Udine, Italy
Giacomo Lenzi (speaker)DipMat, Universita di Salerno, Italy
CILC 2011Pescara, Italy
August 31-September 2
D’Agostino-Lenzi On modal µ-calculus
![Page 2: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/2.jpg)
PreliminariesModel checking and satisfiability
Some translations
Introduction
The modal µ-calculus is modal logic plus extremal fixpoints.
The modal µ-calculus is important in system verification, becauseit lies at the heart of the Model Checking technique.
Model theoretic and algorithmic properties of µ-calculus areinteresting, both on arbitrary graphs and on subclasses of graphs.
In this talk we will consider two important subclasses of graphs, S5and K 4.
D’Agostino-Lenzi On modal µ-calculus
![Page 3: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/3.jpg)
PreliminariesModel checking and satisfiability
Some translations
Introduction
The modal µ-calculus is modal logic plus extremal fixpoints.
The modal µ-calculus is important in system verification, becauseit lies at the heart of the Model Checking technique.
Model theoretic and algorithmic properties of µ-calculus areinteresting, both on arbitrary graphs and on subclasses of graphs.
In this talk we will consider two important subclasses of graphs, S5and K 4.
D’Agostino-Lenzi On modal µ-calculus
![Page 4: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/4.jpg)
PreliminariesModel checking and satisfiability
Some translations
Introduction
The modal µ-calculus is modal logic plus extremal fixpoints.
The modal µ-calculus is important in system verification, becauseit lies at the heart of the Model Checking technique.
Model theoretic and algorithmic properties of µ-calculus areinteresting, both on arbitrary graphs and on subclasses of graphs.
In this talk we will consider two important subclasses of graphs, S5and K 4.
D’Agostino-Lenzi On modal µ-calculus
![Page 5: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/5.jpg)
PreliminariesModel checking and satisfiability
Some translations
Introduction
The modal µ-calculus is modal logic plus extremal fixpoints.
The modal µ-calculus is important in system verification, becauseit lies at the heart of the Model Checking technique.
Model theoretic and algorithmic properties of µ-calculus areinteresting, both on arbitrary graphs and on subclasses of graphs.
In this talk we will consider two important subclasses of graphs, S5and K 4.
D’Agostino-Lenzi On modal µ-calculus
![Page 6: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/6.jpg)
PreliminariesModel checking and satisfiability
Some translations
Introduction
The modal µ-calculus is modal logic plus extremal fixpoints.
The modal µ-calculus is important in system verification, becauseit lies at the heart of the Model Checking technique.
Model theoretic and algorithmic properties of µ-calculus areinteresting, both on arbitrary graphs and on subclasses of graphs.
In this talk we will consider two important subclasses of graphs, S5and K 4.
D’Agostino-Lenzi On modal µ-calculus
![Page 7: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/7.jpg)
PreliminariesModel checking and satisfiability
Some translations
Contents
1 Preliminaries
2 Model checking and satisfiability
3 Some translations
D’Agostino-Lenzi On modal µ-calculus
![Page 8: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/8.jpg)
PreliminariesModel checking and satisfiability
Some translations
Contents
1 Preliminaries
2 Model checking and satisfiability
3 Some translations
D’Agostino-Lenzi On modal µ-calculus
![Page 9: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/9.jpg)
PreliminariesModel checking and satisfiability
Some translations
Contents
1 Preliminaries
2 Model checking and satisfiability
3 Some translations
D’Agostino-Lenzi On modal µ-calculus
![Page 10: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/10.jpg)
PreliminariesModel checking and satisfiability
Some translations
Contents
1 Preliminaries
2 Model checking and satisfiability
3 Some translations
D’Agostino-Lenzi On modal µ-calculus
![Page 11: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/11.jpg)
PreliminariesModel checking and satisfiability
Some translations
Scalar µ-calculus
Variables, atoms and negated atoms are formulas;
∧,∨ are the boolean operators;
[ ], 〈 〉 are the modal operators;
µX .φ(X ) is the least fixpoint;
νX .φ(X ) is the greatest fixpoint.
D’Agostino-Lenzi On modal µ-calculus
![Page 12: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/12.jpg)
PreliminariesModel checking and satisfiability
Some translations
Scalar µ-calculus
Variables, atoms and negated atoms are formulas;
∧,∨ are the boolean operators;
[ ], 〈 〉 are the modal operators;
µX .φ(X ) is the least fixpoint;
νX .φ(X ) is the greatest fixpoint.
D’Agostino-Lenzi On modal µ-calculus
![Page 13: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/13.jpg)
PreliminariesModel checking and satisfiability
Some translations
Scalar µ-calculus
Variables, atoms and negated atoms are formulas;
∧,∨ are the boolean operators;
[ ], 〈 〉 are the modal operators;
µX .φ(X ) is the least fixpoint;
νX .φ(X ) is the greatest fixpoint.
D’Agostino-Lenzi On modal µ-calculus
![Page 14: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/14.jpg)
PreliminariesModel checking and satisfiability
Some translations
Scalar µ-calculus
Variables, atoms and negated atoms are formulas;
∧,∨ are the boolean operators;
[ ], 〈 〉 are the modal operators;
µX .φ(X ) is the least fixpoint;
νX .φ(X ) is the greatest fixpoint.
D’Agostino-Lenzi On modal µ-calculus
![Page 15: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/15.jpg)
PreliminariesModel checking and satisfiability
Some translations
Scalar µ-calculus
Variables, atoms and negated atoms are formulas;
∧,∨ are the boolean operators;
[ ], 〈 〉 are the modal operators;
µX .φ(X ) is the least fixpoint;
νX .φ(X ) is the greatest fixpoint.
D’Agostino-Lenzi On modal µ-calculus
![Page 16: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/16.jpg)
PreliminariesModel checking and satisfiability
Some translations
Scalar µ-calculus
Variables, atoms and negated atoms are formulas;
∧,∨ are the boolean operators;
[ ], 〈 〉 are the modal operators;
µX .φ(X ) is the least fixpoint;
νX .φ(X ) is the greatest fixpoint.
D’Agostino-Lenzi On modal µ-calculus
![Page 17: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/17.jpg)
PreliminariesModel checking and satisfiability
Some translations
Vectorial µ-calculus
We consider vectorial µ-terms of the form
T :
x1 =θ1 f1(x1, . . . , xn, y1, . . . , ym)
. . .xn =θn fn(x1, . . . , xn, y1, . . . , ym)
where fi are modal formulas and θi is µ or ν.
A vectorial µ-term T is by definition equivalent to a n-tuple offormulas (Sol1(T ), . . . ,Soln(T )).
D’Agostino-Lenzi On modal µ-calculus
![Page 18: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/18.jpg)
PreliminariesModel checking and satisfiability
Some translations
Vectorial µ-calculus
We consider vectorial µ-terms of the form
T :
x1 =θ1 f1(x1, . . . , xn, y1, . . . , ym)
. . .xn =θn fn(x1, . . . , xn, y1, . . . , ym)
where fi are modal formulas and θi is µ or ν.
A vectorial µ-term T is by definition equivalent to a n-tuple offormulas (Sol1(T ), . . . ,Soln(T )).
D’Agostino-Lenzi On modal µ-calculus
![Page 19: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/19.jpg)
PreliminariesModel checking and satisfiability
Some translations
Vectorial µ-calculus
We consider vectorial µ-terms of the form
T :
x1 =θ1 f1(x1, . . . , xn, y1, . . . , ym)
. . .xn =θn fn(x1, . . . , xn, y1, . . . , ym)
where fi are modal formulas and θi is µ or ν.
A vectorial µ-term T is by definition equivalent to a n-tuple offormulas (Sol1(T ), . . . ,Soln(T )).
D’Agostino-Lenzi On modal µ-calculus
![Page 20: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/20.jpg)
PreliminariesModel checking and satisfiability
Some translations
Vectorial µ-calculus
We consider vectorial µ-terms of the form
T :
x1 =θ1 f1(x1, . . . , xn, y1, . . . , ym)
. . .xn =θn fn(x1, . . . , xn, y1, . . . , ym)
where fi are modal formulas and
θi is µ or ν.
A vectorial µ-term T is by definition equivalent to a n-tuple offormulas (Sol1(T ), . . . ,Soln(T )).
D’Agostino-Lenzi On modal µ-calculus
![Page 21: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/21.jpg)
PreliminariesModel checking and satisfiability
Some translations
Vectorial µ-calculus
We consider vectorial µ-terms of the form
T :
x1 =θ1 f1(x1, . . . , xn, y1, . . . , ym)
. . .xn =θn fn(x1, . . . , xn, y1, . . . , ym)
where fi are modal formulas and θi is µ or ν.
A vectorial µ-term T is by definition equivalent to a n-tuple offormulas (Sol1(T ), . . . ,Soln(T )).
D’Agostino-Lenzi On modal µ-calculus
![Page 22: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/22.jpg)
PreliminariesModel checking and satisfiability
Some translations
Vectorial µ-calculus
We consider vectorial µ-terms of the form
T :
x1 =θ1 f1(x1, . . . , xn, y1, . . . , ym)
. . .xn =θn fn(x1, . . . , xn, y1, . . . , ym)
where fi are modal formulas and θi is µ or ν.
A vectorial µ-term T is by definition equivalent to a n-tuple offormulas (Sol1(T ), . . . ,Soln(T )).
D’Agostino-Lenzi On modal µ-calculus
![Page 23: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/23.jpg)
PreliminariesModel checking and satisfiability
Some translations
Fixpoint hierarchy
In scalar and vectorial µ-calculus we let:
Π0 = Σ0 = formulas without fixpoints;
Πn+1 = closure of Πn ∪ Σn under composition and greatestfixpoints;
Σn+1 = closure of Πn ∪ Σn under composition and least fixpoints;
∆n = Σn ∩ Πn.
D’Agostino-Lenzi On modal µ-calculus
![Page 24: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/24.jpg)
PreliminariesModel checking and satisfiability
Some translations
Fixpoint hierarchy
In scalar and vectorial µ-calculus we let:
Π0 = Σ0 = formulas without fixpoints;
Πn+1 = closure of Πn ∪ Σn under composition and greatestfixpoints;
Σn+1 = closure of Πn ∪ Σn under composition and least fixpoints;
∆n = Σn ∩ Πn.
D’Agostino-Lenzi On modal µ-calculus
![Page 25: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/25.jpg)
PreliminariesModel checking and satisfiability
Some translations
Fixpoint hierarchy
In scalar and vectorial µ-calculus we let:
Π0 = Σ0 = formulas without fixpoints;
Πn+1 = closure of Πn ∪ Σn under composition and greatestfixpoints;
Σn+1 = closure of Πn ∪ Σn under composition and least fixpoints;
∆n = Σn ∩ Πn.
D’Agostino-Lenzi On modal µ-calculus
![Page 26: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/26.jpg)
PreliminariesModel checking and satisfiability
Some translations
Fixpoint hierarchy
In scalar and vectorial µ-calculus we let:
Π0 = Σ0 = formulas without fixpoints;
Πn+1 = closure of Πn ∪ Σn under composition and greatestfixpoints;
Σn+1 = closure of Πn ∪ Σn under composition and least fixpoints;
∆n = Σn ∩ Πn.
D’Agostino-Lenzi On modal µ-calculus
![Page 27: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/27.jpg)
PreliminariesModel checking and satisfiability
Some translations
Fixpoint hierarchy
In scalar and vectorial µ-calculus we let:
Π0 = Σ0 = formulas without fixpoints;
Πn+1 = closure of Πn ∪ Σn under composition and greatestfixpoints;
Σn+1 = closure of Πn ∪ Σn under composition and least fixpoints;
∆n = Σn ∩ Πn.
D’Agostino-Lenzi On modal µ-calculus
![Page 28: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/28.jpg)
PreliminariesModel checking and satisfiability
Some translations
Fixpoint hierarchy
In scalar and vectorial µ-calculus we let:
Π0 = Σ0 = formulas without fixpoints;
Πn+1 = closure of Πn ∪ Σn under composition and greatestfixpoints;
Σn+1 = closure of Πn ∪ Σn under composition and least fixpoints;
∆n = Σn ∩ Πn.
D’Agostino-Lenzi On modal µ-calculus
![Page 29: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/29.jpg)
PreliminariesModel checking and satisfiability
Some translations
Kripke semantics
We need valued graphs (G , val), where
G = (V ,R) is a graph;
val is a function from atoms and variables to P(V ).
For every formula φ, ||φ||(G , val) is a subset of V , defined byinduction on φ.
D’Agostino-Lenzi On modal µ-calculus
![Page 30: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/30.jpg)
PreliminariesModel checking and satisfiability
Some translations
Kripke semantics
We need valued graphs (G , val), where
G = (V ,R) is a graph;
val is a function from atoms and variables to P(V ).
For every formula φ, ||φ||(G , val) is a subset of V , defined byinduction on φ.
D’Agostino-Lenzi On modal µ-calculus
![Page 31: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/31.jpg)
PreliminariesModel checking and satisfiability
Some translations
Kripke semantics
We need valued graphs (G , val), where
G = (V ,R) is a graph;
val is a function from atoms and variables to P(V ).
For every formula φ, ||φ||(G , val) is a subset of V , defined byinduction on φ.
D’Agostino-Lenzi On modal µ-calculus
![Page 32: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/32.jpg)
PreliminariesModel checking and satisfiability
Some translations
Kripke semantics
We need valued graphs (G , val), where
G = (V ,R) is a graph;
val is a function from atoms and variables to P(V ).
For every formula φ, ||φ||(G , val) is a subset of V , defined byinduction on φ.
D’Agostino-Lenzi On modal µ-calculus
![Page 33: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/33.jpg)
PreliminariesModel checking and satisfiability
Some translations
Kripke semantics
We need valued graphs (G , val), where
G = (V ,R) is a graph;
val is a function from atoms and variables to P(V ).
For every formula φ, ||φ||(G , val) is a subset of V , defined byinduction on φ.
D’Agostino-Lenzi On modal µ-calculus
![Page 34: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/34.jpg)
PreliminariesModel checking and satisfiability
Some translations
Bisimulation
B is a bisimulation between (G , val) and (G ′, val ′) ifB ⊆ V (G )× V (G ′) and whenever xBx ′:
x ∈ val(A) if and only if x ′ ∈ val ′(A);
if xRy , then there is y ′ such that x ′R ′y ′ and yBy ′;
if x ′R ′y ′, then there is y such that xRy and yBy ′.
D’Agostino-Lenzi On modal µ-calculus
![Page 35: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/35.jpg)
PreliminariesModel checking and satisfiability
Some translations
Bisimulation
B is a bisimulation between (G , val) and (G ′, val ′) ifB ⊆ V (G )× V (G ′)
and whenever xBx ′:
x ∈ val(A) if and only if x ′ ∈ val ′(A);
if xRy , then there is y ′ such that x ′R ′y ′ and yBy ′;
if x ′R ′y ′, then there is y such that xRy and yBy ′.
D’Agostino-Lenzi On modal µ-calculus
![Page 36: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/36.jpg)
PreliminariesModel checking and satisfiability
Some translations
Bisimulation
B is a bisimulation between (G , val) and (G ′, val ′) ifB ⊆ V (G )× V (G ′) and whenever xBx ′:
x ∈ val(A) if and only if x ′ ∈ val ′(A);
if xRy , then there is y ′ such that x ′R ′y ′ and yBy ′;
if x ′R ′y ′, then there is y such that xRy and yBy ′.
D’Agostino-Lenzi On modal µ-calculus
![Page 37: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/37.jpg)
PreliminariesModel checking and satisfiability
Some translations
Bisimulation
B is a bisimulation between (G , val) and (G ′, val ′) ifB ⊆ V (G )× V (G ′) and whenever xBx ′:
x ∈ val(A) if and only if x ′ ∈ val ′(A);
if xRy , then there is y ′ such that x ′R ′y ′ and yBy ′;
if x ′R ′y ′, then there is y such that xRy and yBy ′.
D’Agostino-Lenzi On modal µ-calculus
![Page 38: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/38.jpg)
PreliminariesModel checking and satisfiability
Some translations
Bisimulation
B is a bisimulation between (G , val) and (G ′, val ′) ifB ⊆ V (G )× V (G ′) and whenever xBx ′:
x ∈ val(A) if and only if x ′ ∈ val ′(A);
if xRy , then there is y ′ such that x ′R ′y ′ and yBy ′;
if x ′R ′y ′, then there is y such that xRy and yBy ′.
D’Agostino-Lenzi On modal µ-calculus
![Page 39: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/39.jpg)
PreliminariesModel checking and satisfiability
Some translations
Bisimulation
B is a bisimulation between (G , val) and (G ′, val ′) ifB ⊆ V (G )× V (G ′) and whenever xBx ′:
x ∈ val(A) if and only if x ′ ∈ val ′(A);
if xRy , then there is y ′ such that x ′R ′y ′ and yBy ′;
if x ′R ′y ′, then there is y such that xRy and yBy ′.
D’Agostino-Lenzi On modal µ-calculus
![Page 40: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/40.jpg)
PreliminariesModel checking and satisfiability
Some translations
K4 and S5
K 4 is the class of all transitive graphs.
S5 is the class of all equivalence relations, i.e. of reflexive,symmetric, transitive graphs.
K 4 is important in many contexts (e.g. temporal reasoning),whereas S5 is often used as an epistemic logic.
D’Agostino-Lenzi On modal µ-calculus
![Page 41: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/41.jpg)
PreliminariesModel checking and satisfiability
Some translations
K4 and S5
K 4 is the class of all transitive graphs.
S5 is the class of all equivalence relations, i.e. of reflexive,symmetric, transitive graphs.
K 4 is important in many contexts (e.g. temporal reasoning),whereas S5 is often used as an epistemic logic.
D’Agostino-Lenzi On modal µ-calculus
![Page 42: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/42.jpg)
PreliminariesModel checking and satisfiability
Some translations
K4 and S5
K 4 is the class of all transitive graphs.
S5 is the class of all equivalence relations, i.e. of reflexive,symmetric, transitive graphs.
K 4 is important in many contexts (e.g. temporal reasoning),whereas S5 is often used as an epistemic logic.
D’Agostino-Lenzi On modal µ-calculus
![Page 43: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/43.jpg)
PreliminariesModel checking and satisfiability
Some translations
K4 and S5
K 4 is the class of all transitive graphs.
S5 is the class of all equivalence relations, i.e. of reflexive,symmetric, transitive graphs.
K 4 is important in many contexts (e.g. temporal reasoning),whereas S5 is often used as an epistemic logic.
D’Agostino-Lenzi On modal µ-calculus
![Page 44: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/44.jpg)
PreliminariesModel checking and satisfiability
Some translations
Parity games
Two players c and d (after Arnold);
G = (Vc ,Vd ,E , v0,Ω : Vc ∪ Vd → 1, . . . , n);
players move along edges;
if either player has no move, the other wins;
otherwise, d wins if the largest value of Ω occurring infinitely oftenis even.
D’Agostino-Lenzi On modal µ-calculus
![Page 45: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/45.jpg)
PreliminariesModel checking and satisfiability
Some translations
Parity games
Two players c and d (after Arnold);
G = (Vc ,Vd ,E , v0,Ω : Vc ∪ Vd → 1, . . . , n);
players move along edges;
if either player has no move, the other wins;
otherwise, d wins if the largest value of Ω occurring infinitely oftenis even.
D’Agostino-Lenzi On modal µ-calculus
![Page 46: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/46.jpg)
PreliminariesModel checking and satisfiability
Some translations
Parity games
Two players c and d (after Arnold);
G = (Vc ,Vd ,E , v0,Ω : Vc ∪ Vd → 1, . . . , n);
players move along edges;
if either player has no move, the other wins;
otherwise, d wins if the largest value of Ω occurring infinitely oftenis even.
D’Agostino-Lenzi On modal µ-calculus
![Page 47: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/47.jpg)
PreliminariesModel checking and satisfiability
Some translations
Parity games
Two players c and d (after Arnold);
G = (Vc ,Vd ,E , v0,Ω : Vc ∪ Vd → 1, . . . , n);
players move along edges;
if either player has no move, the other wins;
otherwise, d wins if the largest value of Ω occurring infinitely oftenis even.
D’Agostino-Lenzi On modal µ-calculus
![Page 48: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/48.jpg)
PreliminariesModel checking and satisfiability
Some translations
Parity games
Two players c and d (after Arnold);
G = (Vc ,Vd ,E , v0,Ω : Vc ∪ Vd → 1, . . . , n);
players move along edges;
if either player has no move, the other wins;
otherwise, d wins if the largest value of Ω occurring infinitely oftenis even.
D’Agostino-Lenzi On modal µ-calculus
![Page 49: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/49.jpg)
PreliminariesModel checking and satisfiability
Some translations
Parity games
Two players c and d (after Arnold);
G = (Vc ,Vd ,E , v0,Ω : Vc ∪ Vd → 1, . . . , n);
players move along edges;
if either player has no move, the other wins;
otherwise, d wins if the largest value of Ω occurring infinitely oftenis even.
D’Agostino-Lenzi On modal µ-calculus
![Page 50: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/50.jpg)
PreliminariesModel checking and satisfiability
Some translations
Contents
1 Preliminaries
2 Model checking and satisfiability
3 Some translations
D’Agostino-Lenzi On modal µ-calculus
![Page 51: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/51.jpg)
PreliminariesModel checking and satisfiability
Some translations
A reduction theorem
Theorem
Given a model (G , val) and a vectorial µ-term T , there is a model(G ′, val ′) and a vectorial µ-term T ′, such that:
T ′ is existential (i.e. box free);
G ′ is of class S5;
(G ′,Val ′) and T ′ are built in time polynomial in the size of(G , val) plus the size of T ;
(G , val) verifies sol1(T ) if and only if (G ′, val ′) verifiessol1(T ′).
D’Agostino-Lenzi On modal µ-calculus
![Page 52: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/52.jpg)
PreliminariesModel checking and satisfiability
Some translations
A reduction theorem
Theorem
Given a model (G , val) and a vectorial µ-term T ,
there is a model(G ′, val ′) and a vectorial µ-term T ′, such that:
T ′ is existential (i.e. box free);
G ′ is of class S5;
(G ′,Val ′) and T ′ are built in time polynomial in the size of(G , val) plus the size of T ;
(G , val) verifies sol1(T ) if and only if (G ′, val ′) verifiessol1(T ′).
D’Agostino-Lenzi On modal µ-calculus
![Page 53: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/53.jpg)
PreliminariesModel checking and satisfiability
Some translations
A reduction theorem
Theorem
Given a model (G , val) and a vectorial µ-term T , there is a model(G ′, val ′) and a vectorial µ-term T ′, such that:
T ′ is existential (i.e. box free);
G ′ is of class S5;
(G ′,Val ′) and T ′ are built in time polynomial in the size of(G , val) plus the size of T ;
(G , val) verifies sol1(T ) if and only if (G ′, val ′) verifiessol1(T ′).
D’Agostino-Lenzi On modal µ-calculus
![Page 54: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/54.jpg)
PreliminariesModel checking and satisfiability
Some translations
A reduction theorem
Theorem
Given a model (G , val) and a vectorial µ-term T , there is a model(G ′, val ′) and a vectorial µ-term T ′, such that:
T ′ is existential (i.e. box free);
G ′ is of class S5;
(G ′,Val ′) and T ′ are built in time polynomial in the size of(G , val) plus the size of T ;
(G , val) verifies sol1(T ) if and only if (G ′, val ′) verifiessol1(T ′).
D’Agostino-Lenzi On modal µ-calculus
![Page 55: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/55.jpg)
PreliminariesModel checking and satisfiability
Some translations
A reduction theorem
Theorem
Given a model (G , val) and a vectorial µ-term T , there is a model(G ′, val ′) and a vectorial µ-term T ′, such that:
T ′ is existential (i.e. box free);
G ′ is of class S5;
(G ′,Val ′) and T ′ are built in time polynomial in the size of(G , val) plus the size of T ;
(G , val) verifies sol1(T ) if and only if (G ′, val ′) verifiessol1(T ′).
D’Agostino-Lenzi On modal µ-calculus
![Page 56: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/56.jpg)
PreliminariesModel checking and satisfiability
Some translations
A reduction theorem
Theorem
Given a model (G , val) and a vectorial µ-term T , there is a model(G ′, val ′) and a vectorial µ-term T ′, such that:
T ′ is existential (i.e. box free);
G ′ is of class S5;
(G ′,Val ′) and T ′ are built in time polynomial in the size of(G , val) plus the size of T ;
(G , val) verifies sol1(T ) if and only if (G ′, val ′) verifiessol1(T ′).
D’Agostino-Lenzi On modal µ-calculus
![Page 57: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/57.jpg)
PreliminariesModel checking and satisfiability
Some translations
A reduction theorem
Theorem
Given a model (G , val) and a vectorial µ-term T , there is a model(G ′, val ′) and a vectorial µ-term T ′, such that:
T ′ is existential (i.e. box free);
G ′ is of class S5;
(G ′,Val ′) and T ′ are built in time polynomial in the size of(G , val) plus the size of T ;
(G , val) verifies sol1(T ) if and only if (G ′, val ′) verifiessol1(T ′).
D’Agostino-Lenzi On modal µ-calculus
![Page 58: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/58.jpg)
PreliminariesModel checking and satisfiability
Some translations
Corollaries
Corollary
If there is a polytime translation from box-free vectorial µ-calculusin S5 to vectorial modal logic in S5, then the µ-calculus modelchecking problem is in P.
Corollary
If there is a polytime translation from box-free vectorial µ-calculusin K 4 to vectorial Π2 in K 4, then the µ-calculus model checkingproblem is in P.
D’Agostino-Lenzi On modal µ-calculus
![Page 59: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/59.jpg)
PreliminariesModel checking and satisfiability
Some translations
Corollaries
Corollary
If there is a polytime translation from box-free vectorial µ-calculusin S5 to vectorial modal logic in S5,
then the µ-calculus modelchecking problem is in P.
Corollary
If there is a polytime translation from box-free vectorial µ-calculusin K 4 to vectorial Π2 in K 4, then the µ-calculus model checkingproblem is in P.
D’Agostino-Lenzi On modal µ-calculus
![Page 60: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/60.jpg)
PreliminariesModel checking and satisfiability
Some translations
Corollaries
Corollary
If there is a polytime translation from box-free vectorial µ-calculusin S5 to vectorial modal logic in S5, then the µ-calculus modelchecking problem is in P.
Corollary
If there is a polytime translation from box-free vectorial µ-calculusin K 4 to vectorial Π2 in K 4, then the µ-calculus model checkingproblem is in P.
D’Agostino-Lenzi On modal µ-calculus
![Page 61: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/61.jpg)
PreliminariesModel checking and satisfiability
Some translations
Corollaries
Corollary
If there is a polytime translation from box-free vectorial µ-calculusin S5 to vectorial modal logic in S5, then the µ-calculus modelchecking problem is in P.
Corollary
If there is a polytime translation from box-free vectorial µ-calculusin K 4 to vectorial Π2 in K 4,
then the µ-calculus model checkingproblem is in P.
D’Agostino-Lenzi On modal µ-calculus
![Page 62: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/62.jpg)
PreliminariesModel checking and satisfiability
Some translations
Corollaries
Corollary
If there is a polytime translation from box-free vectorial µ-calculusin S5 to vectorial modal logic in S5, then the µ-calculus modelchecking problem is in P.
Corollary
If there is a polytime translation from box-free vectorial µ-calculusin K 4 to vectorial Π2 in K 4, then the µ-calculus model checkingproblem is in P.
D’Agostino-Lenzi On modal µ-calculus
![Page 63: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/63.jpg)
PreliminariesModel checking and satisfiability
Some translations
Satisfiability in S5
Lemma
If a µ-calculus formula φ has a model in S5, then it has one withsize linear in φ.
Our proof uses parity games.
Theorem
The satisfiability problem for the µ-calculus in S5 is NP-complete.
In fact, NP-hardness is because the µ-calculus includespropositional logic; an NP algorithm is given by guessing a modelof a formula and then running an NP model checking algorithm.
D’Agostino-Lenzi On modal µ-calculus
![Page 64: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/64.jpg)
PreliminariesModel checking and satisfiability
Some translations
Satisfiability in S5
Lemma
If a µ-calculus formula φ has a model in S5, then it has one withsize linear in φ.
Our proof uses parity games.
Theorem
The satisfiability problem for the µ-calculus in S5 is NP-complete.
In fact, NP-hardness is because the µ-calculus includespropositional logic; an NP algorithm is given by guessing a modelof a formula and then running an NP model checking algorithm.
D’Agostino-Lenzi On modal µ-calculus
![Page 65: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/65.jpg)
PreliminariesModel checking and satisfiability
Some translations
Satisfiability in S5
Lemma
If a µ-calculus formula φ has a model in S5, then it has one withsize linear in φ.
Our proof uses parity games.
Theorem
The satisfiability problem for the µ-calculus in S5 is NP-complete.
In fact, NP-hardness is because the µ-calculus includespropositional logic; an NP algorithm is given by guessing a modelof a formula and then running an NP model checking algorithm.
D’Agostino-Lenzi On modal µ-calculus
![Page 66: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/66.jpg)
PreliminariesModel checking and satisfiability
Some translations
Satisfiability in S5
Lemma
If a µ-calculus formula φ has a model in S5, then it has one withsize linear in φ.
Our proof uses parity games.
Theorem
The satisfiability problem for the µ-calculus in S5 is NP-complete.
In fact, NP-hardness is because the µ-calculus includespropositional logic; an NP algorithm is given by guessing a modelof a formula and then running an NP model checking algorithm.
D’Agostino-Lenzi On modal µ-calculus
![Page 67: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/67.jpg)
PreliminariesModel checking and satisfiability
Some translations
Satisfiability in S5
Lemma
If a µ-calculus formula φ has a model in S5, then it has one withsize linear in φ.
Our proof uses parity games.
Theorem
The satisfiability problem for the µ-calculus in S5 is NP-complete.
In fact, NP-hardness is because the µ-calculus includespropositional logic;
an NP algorithm is given by guessing a modelof a formula and then running an NP model checking algorithm.
D’Agostino-Lenzi On modal µ-calculus
![Page 68: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/68.jpg)
PreliminariesModel checking and satisfiability
Some translations
Satisfiability in S5
Lemma
If a µ-calculus formula φ has a model in S5, then it has one withsize linear in φ.
Our proof uses parity games.
Theorem
The satisfiability problem for the µ-calculus in S5 is NP-complete.
In fact, NP-hardness is because the µ-calculus includespropositional logic; an NP algorithm is given by guessing a modelof a formula and then running an NP model checking algorithm.
D’Agostino-Lenzi On modal µ-calculus
![Page 69: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/69.jpg)
PreliminariesModel checking and satisfiability
Some translations
On existential µ-calculus
Lemma
The satisfiability problem for existential µ-calculus is polynomialtime equivalent to the same problem on S5.
In fact, if φ has a model M, then it is true on the S5 closure of M.From the previous theorem it follows:
Corollary
The satisfiability problem for existential µ-calculus in S5 isNP-complete.
Corollary
The satisfiability problem for existential µ-calculus is NP-complete.
D’Agostino-Lenzi On modal µ-calculus
![Page 70: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/70.jpg)
PreliminariesModel checking and satisfiability
Some translations
On existential µ-calculus
Lemma
The satisfiability problem for existential µ-calculus is polynomialtime equivalent to the same problem on S5.
In fact, if φ has a model M, then it is true on the S5 closure of M.From the previous theorem it follows:
Corollary
The satisfiability problem for existential µ-calculus in S5 isNP-complete.
Corollary
The satisfiability problem for existential µ-calculus is NP-complete.
D’Agostino-Lenzi On modal µ-calculus
![Page 71: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/71.jpg)
PreliminariesModel checking and satisfiability
Some translations
On existential µ-calculus
Lemma
The satisfiability problem for existential µ-calculus is polynomialtime equivalent to the same problem on S5.
In fact, if φ has a model M, then it is true on the S5 closure of M.
From the previous theorem it follows:
Corollary
The satisfiability problem for existential µ-calculus in S5 isNP-complete.
Corollary
The satisfiability problem for existential µ-calculus is NP-complete.
D’Agostino-Lenzi On modal µ-calculus
![Page 72: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/72.jpg)
PreliminariesModel checking and satisfiability
Some translations
On existential µ-calculus
Lemma
The satisfiability problem for existential µ-calculus is polynomialtime equivalent to the same problem on S5.
In fact, if φ has a model M, then it is true on the S5 closure of M.From the previous theorem it follows:
Corollary
The satisfiability problem for existential µ-calculus in S5 isNP-complete.
Corollary
The satisfiability problem for existential µ-calculus is NP-complete.
D’Agostino-Lenzi On modal µ-calculus
![Page 73: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/73.jpg)
PreliminariesModel checking and satisfiability
Some translations
On existential µ-calculus
Lemma
The satisfiability problem for existential µ-calculus is polynomialtime equivalent to the same problem on S5.
In fact, if φ has a model M, then it is true on the S5 closure of M.From the previous theorem it follows:
Corollary
The satisfiability problem for existential µ-calculus in S5 isNP-complete.
Corollary
The satisfiability problem for existential µ-calculus is NP-complete.
D’Agostino-Lenzi On modal µ-calculus
![Page 74: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/74.jpg)
PreliminariesModel checking and satisfiability
Some translations
On existential µ-calculus
Lemma
The satisfiability problem for existential µ-calculus is polynomialtime equivalent to the same problem on S5.
In fact, if φ has a model M, then it is true on the S5 closure of M.From the previous theorem it follows:
Corollary
The satisfiability problem for existential µ-calculus in S5 isNP-complete.
Corollary
The satisfiability problem for existential µ-calculus is NP-complete.
D’Agostino-Lenzi On modal µ-calculus
![Page 75: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/75.jpg)
PreliminariesModel checking and satisfiability
Some translations
Contents
1 Preliminaries
2 Model checking and satisfiability
3 Some translations
D’Agostino-Lenzi On modal µ-calculus
![Page 76: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/76.jpg)
PreliminariesModel checking and satisfiability
Some translations
The Alberucci-Facchini rank
A µ-calculus sentence φ is well named if it is guarded and, for anyvariable X , no two distinct occurrences of fixpoint operators in φbind X , and the atom X occurs only once in φ.
Define now an ordinal rank on formulas:
rank(A) = rank(¬A) = 1;
rank(〈 〉φ) = rank([ ]φ) = rank(φ) + 1;
rank(φ ∧ ψ) = rank(φ ∨ ψ) = maxrank(φ), rank(ψ)+ 1;
rank(µX .φ(X )) = rank(νX .φ(X )) =suprank(φn(X )) + 1; n ∈ N.
D’Agostino-Lenzi On modal µ-calculus
![Page 77: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/77.jpg)
PreliminariesModel checking and satisfiability
Some translations
The Alberucci-Facchini rank
A µ-calculus sentence φ is well named if it is guarded and, for anyvariable X , no two distinct occurrences of fixpoint operators in φbind X , and the atom X occurs only once in φ.
Define now an ordinal rank on formulas:
rank(A) = rank(¬A) = 1;
rank(〈 〉φ) = rank([ ]φ) = rank(φ) + 1;
rank(φ ∧ ψ) = rank(φ ∨ ψ) = maxrank(φ), rank(ψ)+ 1;
rank(µX .φ(X )) = rank(νX .φ(X )) =suprank(φn(X )) + 1; n ∈ N.
D’Agostino-Lenzi On modal µ-calculus
![Page 78: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/78.jpg)
PreliminariesModel checking and satisfiability
Some translations
The Alberucci-Facchini rank
A µ-calculus sentence φ is well named if it is guarded and, for anyvariable X , no two distinct occurrences of fixpoint operators in φbind X , and the atom X occurs only once in φ.
Define now an ordinal rank on formulas:
rank(A) = rank(¬A) = 1;
rank(〈 〉φ) = rank([ ]φ) = rank(φ) + 1;
rank(φ ∧ ψ) = rank(φ ∨ ψ) = maxrank(φ), rank(ψ)+ 1;
rank(µX .φ(X )) = rank(νX .φ(X )) =suprank(φn(X )) + 1; n ∈ N.
D’Agostino-Lenzi On modal µ-calculus
![Page 79: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/79.jpg)
PreliminariesModel checking and satisfiability
Some translations
The Alberucci-Facchini rank
A µ-calculus sentence φ is well named if it is guarded and, for anyvariable X , no two distinct occurrences of fixpoint operators in φbind X , and the atom X occurs only once in φ.
Define now an ordinal rank on formulas:
rank(A) = rank(¬A) = 1;
rank(〈 〉φ) = rank([ ]φ) = rank(φ) + 1;
rank(φ ∧ ψ) = rank(φ ∨ ψ) = maxrank(φ), rank(ψ)+ 1;
rank(µX .φ(X )) = rank(νX .φ(X )) =suprank(φn(X )) + 1; n ∈ N.
D’Agostino-Lenzi On modal µ-calculus
![Page 80: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/80.jpg)
PreliminariesModel checking and satisfiability
Some translations
The Alberucci-Facchini rank
A µ-calculus sentence φ is well named if it is guarded and, for anyvariable X , no two distinct occurrences of fixpoint operators in φbind X , and the atom X occurs only once in φ.
Define now an ordinal rank on formulas:
rank(A) = rank(¬A) = 1;
rank(〈 〉φ) = rank([ ]φ) = rank(φ) + 1;
rank(φ ∧ ψ) = rank(φ ∨ ψ) = maxrank(φ), rank(ψ)+ 1;
rank(µX .φ(X )) = rank(νX .φ(X )) =suprank(φn(X )) + 1; n ∈ N.
D’Agostino-Lenzi On modal µ-calculus
![Page 81: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/81.jpg)
PreliminariesModel checking and satisfiability
Some translations
The Alberucci-Facchini translation
It is a function t given by induction on the rank:
t(A) = A, t(¬A) = ¬A;
t(true) = true, t(false) = false;
t(〈 〉φ) = 〈 〉t(φ);
t([ ]φ) = [ ]t(φ);
t(φ ∧ ψ) = t(φ) ∧ t(ψ);
t(φ ∨ ψ) = t(φ) ∨ t(ψ);
t(µX .φ(X )) = t((φ(φ(false))∗);
t(νX .φ(X )) = t((φ(φ(true))∗),
where (φ(φ(false))∗, (φ(φ(true))∗ denote the well named formulasobtained from φ(φ(false)), φ(φ(true)) by renaming repeated boundvariables.
D’Agostino-Lenzi On modal µ-calculus
![Page 82: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/82.jpg)
PreliminariesModel checking and satisfiability
Some translations
The Alberucci-Facchini translation
It is a function t given by induction on the rank:
t(A) = A, t(¬A) = ¬A;
t(true) = true, t(false) = false;
t(〈 〉φ) = 〈 〉t(φ);
t([ ]φ) = [ ]t(φ);
t(φ ∧ ψ) = t(φ) ∧ t(ψ);
t(φ ∨ ψ) = t(φ) ∨ t(ψ);
t(µX .φ(X )) = t((φ(φ(false))∗);
t(νX .φ(X )) = t((φ(φ(true))∗),
where (φ(φ(false))∗, (φ(φ(true))∗ denote the well named formulasobtained from φ(φ(false)), φ(φ(true)) by renaming repeated boundvariables.
D’Agostino-Lenzi On modal µ-calculus
![Page 83: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/83.jpg)
PreliminariesModel checking and satisfiability
Some translations
The Alberucci-Facchini translation
It is a function t given by induction on the rank:
t(A) = A, t(¬A) = ¬A;
t(true) = true, t(false) = false;
t(〈 〉φ) = 〈 〉t(φ);
t([ ]φ) = [ ]t(φ);
t(φ ∧ ψ) = t(φ) ∧ t(ψ);
t(φ ∨ ψ) = t(φ) ∨ t(ψ);
t(µX .φ(X )) = t((φ(φ(false))∗);
t(νX .φ(X )) = t((φ(φ(true))∗),
where (φ(φ(false))∗, (φ(φ(true))∗ denote the well named formulasobtained from φ(φ(false)), φ(φ(true)) by renaming repeated boundvariables.
D’Agostino-Lenzi On modal µ-calculus
![Page 84: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/84.jpg)
PreliminariesModel checking and satisfiability
Some translations
The Alberucci-Facchini translation
It is a function t given by induction on the rank:
t(A) = A, t(¬A) = ¬A;
t(true) = true, t(false) = false;
t(〈 〉φ) = 〈 〉t(φ);
t([ ]φ) = [ ]t(φ);
t(φ ∧ ψ) = t(φ) ∧ t(ψ);
t(φ ∨ ψ) = t(φ) ∨ t(ψ);
t(µX .φ(X )) = t((φ(φ(false))∗);
t(νX .φ(X )) = t((φ(φ(true))∗),
where (φ(φ(false))∗, (φ(φ(true))∗ denote the well named formulasobtained from φ(φ(false)), φ(φ(true)) by renaming repeated boundvariables.
D’Agostino-Lenzi On modal µ-calculus
![Page 85: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/85.jpg)
PreliminariesModel checking and satisfiability
Some translations
The Alberucci-Facchini translation
It is a function t given by induction on the rank:
t(A) = A, t(¬A) = ¬A;
t(true) = true, t(false) = false;
t(〈 〉φ) = 〈 〉t(φ);
t([ ]φ) = [ ]t(φ);
t(φ ∧ ψ) = t(φ) ∧ t(ψ);
t(φ ∨ ψ) = t(φ) ∨ t(ψ);
t(µX .φ(X )) = t((φ(φ(false))∗);
t(νX .φ(X )) = t((φ(φ(true))∗),
where (φ(φ(false))∗, (φ(φ(true))∗ denote the well named formulasobtained from φ(φ(false)), φ(φ(true)) by renaming repeated boundvariables.
D’Agostino-Lenzi On modal µ-calculus
![Page 86: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/86.jpg)
PreliminariesModel checking and satisfiability
Some translations
The Alberucci-Facchini translation
It is a function t given by induction on the rank:
t(A) = A, t(¬A) = ¬A;
t(true) = true, t(false) = false;
t(〈 〉φ) = 〈 〉t(φ);
t([ ]φ) = [ ]t(φ);
t(φ ∧ ψ) = t(φ) ∧ t(ψ);
t(φ ∨ ψ) = t(φ) ∨ t(ψ);
t(µX .φ(X )) = t((φ(φ(false))∗);
t(νX .φ(X )) = t((φ(φ(true))∗),
where (φ(φ(false))∗, (φ(φ(true))∗ denote the well named formulasobtained from φ(φ(false)), φ(φ(true)) by renaming repeated boundvariables.
D’Agostino-Lenzi On modal µ-calculus
![Page 87: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/87.jpg)
PreliminariesModel checking and satisfiability
Some translations
The complexity of the translation
Lemma
If φ is a well named formula, then the length of t(φ) is at most 2|φ|.
The exponential bound is tight.
D’Agostino-Lenzi On modal µ-calculus
![Page 88: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/88.jpg)
PreliminariesModel checking and satisfiability
Some translations
The complexity of the translation
Lemma
If φ is a well named formula, then the length of t(φ) is at most 2|φ|.
The exponential bound is tight.
D’Agostino-Lenzi On modal µ-calculus
![Page 89: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/89.jpg)
PreliminariesModel checking and satisfiability
Some translations
The complexity of the translation
Lemma
If φ is a well named formula, then the length of t(φ) is at most 2|φ|.
The exponential bound is tight.
D’Agostino-Lenzi On modal µ-calculus
![Page 90: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/90.jpg)
PreliminariesModel checking and satisfiability
Some translations
An alternative translation
Let φ be a µ-calculus formula containing a set At of atoms. Then,up to bisimulation, there are finitely many S5 models colored withAt, more precisely exponentially many of them. We note that inS5, characteristic formulas of models are modal.
So, φ is equivalent to the finite disjunction of the characteristicformulas of the bisimulation classes of the models of φ. Note thatthis alternative translation is also (at most) exponential.
D’Agostino-Lenzi On modal µ-calculus
![Page 91: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/91.jpg)
PreliminariesModel checking and satisfiability
Some translations
An alternative translation
Let φ be a µ-calculus formula containing a set At of atoms.
Then,up to bisimulation, there are finitely many S5 models colored withAt, more precisely exponentially many of them. We note that inS5, characteristic formulas of models are modal.
So, φ is equivalent to the finite disjunction of the characteristicformulas of the bisimulation classes of the models of φ. Note thatthis alternative translation is also (at most) exponential.
D’Agostino-Lenzi On modal µ-calculus
![Page 92: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/92.jpg)
PreliminariesModel checking and satisfiability
Some translations
An alternative translation
Let φ be a µ-calculus formula containing a set At of atoms. Then,up to bisimulation, there are finitely many S5 models colored withAt, more precisely exponentially many of them.
We note that inS5, characteristic formulas of models are modal.
So, φ is equivalent to the finite disjunction of the characteristicformulas of the bisimulation classes of the models of φ. Note thatthis alternative translation is also (at most) exponential.
D’Agostino-Lenzi On modal µ-calculus
![Page 93: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/93.jpg)
PreliminariesModel checking and satisfiability
Some translations
An alternative translation
Let φ be a µ-calculus formula containing a set At of atoms. Then,up to bisimulation, there are finitely many S5 models colored withAt, more precisely exponentially many of them. We note that inS5, characteristic formulas of models are modal.
So, φ is equivalent to the finite disjunction of the characteristicformulas of the bisimulation classes of the models of φ. Note thatthis alternative translation is also (at most) exponential.
D’Agostino-Lenzi On modal µ-calculus
![Page 94: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/94.jpg)
PreliminariesModel checking and satisfiability
Some translations
An alternative translation
Let φ be a µ-calculus formula containing a set At of atoms. Then,up to bisimulation, there are finitely many S5 models colored withAt, more precisely exponentially many of them. We note that inS5, characteristic formulas of models are modal.
So, φ is equivalent to the finite disjunction of the characteristicformulas of the bisimulation classes of the models of φ.
Note thatthis alternative translation is also (at most) exponential.
D’Agostino-Lenzi On modal µ-calculus
![Page 95: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/95.jpg)
PreliminariesModel checking and satisfiability
Some translations
An alternative translation
Let φ be a µ-calculus formula containing a set At of atoms. Then,up to bisimulation, there are finitely many S5 models colored withAt, more precisely exponentially many of them. We note that inS5, characteristic formulas of models are modal.
So, φ is equivalent to the finite disjunction of the characteristicformulas of the bisimulation classes of the models of φ. Note thatthis alternative translation is also (at most) exponential.
D’Agostino-Lenzi On modal µ-calculus
![Page 96: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/96.jpg)
PreliminariesModel checking and satisfiability
Some translations
A corollary
Corollary
If the reduction theorem specializes to scalar µ-calculus and thereis a polytime translation from the µ-calculus to modal logic in S5,then the µ-calculus model checking is in P.
D’Agostino-Lenzi On modal µ-calculus
![Page 97: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/97.jpg)
PreliminariesModel checking and satisfiability
Some translations
A corollary
Corollary
If the reduction theorem specializes to scalar µ-calculus
and thereis a polytime translation from the µ-calculus to modal logic in S5,then the µ-calculus model checking is in P.
D’Agostino-Lenzi On modal µ-calculus
![Page 98: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/98.jpg)
PreliminariesModel checking and satisfiability
Some translations
A corollary
Corollary
If the reduction theorem specializes to scalar µ-calculus and thereis a polytime translation from the µ-calculus to modal logic in S5,
then the µ-calculus model checking is in P.
D’Agostino-Lenzi On modal µ-calculus
![Page 99: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/99.jpg)
PreliminariesModel checking and satisfiability
Some translations
A corollary
Corollary
If the reduction theorem specializes to scalar µ-calculus and thereis a polytime translation from the µ-calculus to modal logic in S5,then the µ-calculus model checking is in P.
D’Agostino-Lenzi On modal µ-calculus
![Page 100: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/100.jpg)
PreliminariesModel checking and satisfiability
Some translations
Conclusion
Simplicity of S5 gives good satisfiability bounds, but no goodmodel checking bounds.
Are there “natural” translations from vectorial to scalar terms inS5?
Complexity of satisfiability of µ in K 4 can be obtained by reducingto arbitrary graphs; what about better bounds?
D’Agostino-Lenzi On modal µ-calculus
![Page 101: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/101.jpg)
PreliminariesModel checking and satisfiability
Some translations
Conclusion
Simplicity of S5 gives good satisfiability bounds, but no goodmodel checking bounds.
Are there “natural” translations from vectorial to scalar terms inS5?
Complexity of satisfiability of µ in K 4 can be obtained by reducingto arbitrary graphs; what about better bounds?
D’Agostino-Lenzi On modal µ-calculus
![Page 102: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/102.jpg)
PreliminariesModel checking and satisfiability
Some translations
Conclusion
Simplicity of S5 gives good satisfiability bounds, but no goodmodel checking bounds.
Are there “natural” translations from vectorial to scalar terms inS5?
Complexity of satisfiability of µ in K 4 can be obtained by reducingto arbitrary graphs; what about better bounds?
D’Agostino-Lenzi On modal µ-calculus
![Page 103: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/103.jpg)
PreliminariesModel checking and satisfiability
Some translations
Conclusion
Simplicity of S5 gives good satisfiability bounds, but no goodmodel checking bounds.
Are there “natural” translations from vectorial to scalar terms inS5?
Complexity of satisfiability of µ in K 4 can be obtained by reducingto arbitrary graphs; what about better bounds?
D’Agostino-Lenzi On modal µ-calculus
![Page 104: On modal -calculus in S5 and applications · Giovanna D’Agostino DIMI, Universit a di Udine, Italy giovanna.dagostino@dimi.uniud.it Giacomo Lenzi (speaker) DipMat, Universit a di](https://reader034.vdocument.in/reader034/viewer/2022052005/601965acc0e8093dc615e8f7/html5/thumbnails/104.jpg)
PreliminariesModel checking and satisfiability
Some translations
Thank you!
D’Agostino-Lenzi On modal µ-calculus