tatl: implementation of an atl tableau-based decision procedure
TRANSCRIPT
TATL: Implementation of anATL Tableau-Based Decision Procedure
Tableaux 2013
Amelie David
Laboratoire IBISCEvry - France
19 September 2013
Amelie David TATL: Tableaux for ATL 19 September 2013 1 / 10
Introduction
TATL : Tableaux for ATL
1st implementation of asatisfiability decision procedurefor ATL
using tableau-based decisionprocedure by V. Goranko andD. Shkatov (2009)
Amelie David TATL: Tableaux for ATL 19 September 2013 1 / 10
ATL
1997 - 2002: Alur, Henzinger et Kupferman
ATL: Alternating-time Temporal Logic
à Extends CTL with notion of agents and coalitions of agents
à Path quantifiers are labeled with agent coalition
CTL ATLE�ϕ III 〈〈a1, a2〉〉�ϕ
à Syntax :
F := p | ¬F | F∧F | F∨F | 〈〈A〉〉©F | 〈〈A〉〉�F | 〈〈A〉〉F UF | 〈〈A〉〉♦F
Amelie David TATL: Tableaux for ATL 19 September 2013 2 / 10
ATL
1997 - 2002: Alur, Henzinger et Kupferman
ATL: Alternating-time Temporal Logic
à Extends CTL with notion of agents and coalitions of agents
à Path quantifiers are labeled with agent coalition
CTL ATLE�ϕ III 〈〈a1, a2〉〉�ϕ
à Syntax :
F := p | ¬F | F∧F | F∨F | 〈〈A〉〉©F | 〈〈A〉〉�F | 〈〈A〉〉F UF | 〈〈A〉〉♦F
Amelie David TATL: Tableaux for ATL 19 September 2013 2 / 10
ATL
1997 - 2002: Alur, Henzinger et Kupferman
ATL: Alternating-time Temporal Logic
à Extends CTL with notion of agents and coalitions of agents
à Path quantifiers are labeled with agent coalition
CTL ATLE�ϕ III 〈〈a1, a2〉〉�ϕ
à Syntax :
F := p | ¬F | F∧F | F∨F | 〈〈A〉〉©F | 〈〈A〉〉�F | 〈〈A〉〉F UF | 〈〈A〉〉♦F
Amelie David TATL: Tableaux for ATL 19 September 2013 2 / 10
ATL
1997 - 2002: Alur, Henzinger et Kupferman
ATL: Alternating-time Temporal Logic
à Extends CTL with notion of agents and coalitions of agents
à Path quantifiers are labeled with agent coalition
CTL ATLE�ϕ III 〈〈a1, a2〉〉�ϕ
à Syntax :
F := p | ¬F | F∧F | F∨F | 〈〈A〉〉©F | 〈〈A〉〉�F | 〈〈A〉〉F UF | 〈〈A〉〉♦F
Amelie David TATL: Tableaux for ATL 19 September 2013 2 / 10
ATL - Example
System of 2agents
P1
P2
Concurrent Game Structure
s0
∅
s2{ca 2 1}
s1 {ca 1 2}
s3
{c 1 2}
0, 0
1, 0
1, 1
0, 1 0, 1
0, 0
1, 0
0, 0 1, 1
0, 0
1, 0
0, 1
1, 0
1, 1
0, 1 0, 1
0, 0
1, 0
0, 0 1, 1
0, 0
1, 0
0, 1
〈〈P1,P2〉〉♦c 1 2
Amelie David TATL: Tableaux for ATL 19 September 2013 3 / 10
ATL - Example
System of 2agents
P1
P2
Concurrent Game Structure
s0
∅
s2{ca 2 1}
s1 {ca 1 2}
s3
{c 1 2}
0, 0
1, 0
1, 1
0, 1 0, 1
0, 0
1, 0
0, 0 1, 1
0, 0
1, 0
0, 1
1, 0
1, 1
0, 1 0, 1
0, 0
1, 0
0, 0 1, 1
0, 0
1, 0
0, 1
〈〈P1,P2〉〉♦c 1 2
Amelie David TATL: Tableaux for ATL 19 September 2013 3 / 10
ATL - Example
System of 2agents
P1
P2
Concurrent Game Structure
s0
∅
s2{ca 2 1}
s1 {ca 1 2}
s3
{c 1 2}
0, 0
1, 0
1, 1
0, 1
0, 1
0, 0
1, 0
0, 0 1, 1
0, 0
1, 0
0, 1
1, 0
1, 1
0, 1 0, 1
0, 0
1, 0
0, 0 1, 1
0, 0
1, 0
0, 1
〈〈P1,P2〉〉♦c 1 2
Amelie David TATL: Tableaux for ATL 19 September 2013 3 / 10
ATL - Example
System of 2agents
P1
P2
Concurrent Game Structure
s0
∅
s2{ca 2 1}
s1 {ca 1 2}
s3
{c 1 2}
0, 0
1, 0
1, 1
0, 1
0, 1
0, 0
1, 0
0, 0 1, 1
0, 0
1, 0
0, 1
1, 0
1, 1
0, 1 0, 1
0, 0
1, 0
0, 0 1, 1
0, 0
1, 0
0, 1
〈〈P1,P2〉〉♦c 1 2
Amelie David TATL: Tableaux for ATL 19 September 2013 3 / 10
ATL - Example
System of 2agents
P1
P2
Concurrent Game Structure
s0
∅
s2{ca 2 1}
s1 {ca 1 2}
s3
{c 1 2}
0, 0
1, 0
1, 1
0, 1 0, 1
0, 0
1, 0
0, 0 1, 1
0, 0
1, 0
0, 1
1, 0
1, 1
0, 1 0, 1
0, 0
1, 0
0, 0 1, 1
0, 0
1, 0
0, 1
〈〈P1,P2〉〉♦c 1 2
Amelie David TATL: Tableaux for ATL 19 September 2013 3 / 10
ATL - Example
System of 2agents
P1
P2
Concurrent Game Structure
s0
∅
s2{ca 2 1}
s1 {ca 1 2}
s3
{c 1 2}
0, 0
1, 0
1, 1
0, 1 0, 1
0, 0
1, 0
0, 0 1, 1
0, 0
1, 0
0, 1
1, 0
1, 1
0, 1 0, 1
0, 0
1, 0
0, 0 1, 1
0, 0
1, 0
0, 1
〈〈P1,P2〉〉♦c 1 2
Amelie David TATL: Tableaux for ATL 19 September 2013 3 / 10
ATL - Example
System of 2agents
P1
P2
Concurrent Game Structure
s0
∅
s2{ca 2 1}
s1 {ca 1 2}
s3
{c 1 2}
0, 0
1, 0
1, 1
0, 1 0, 1
0, 0
1, 0
0, 0 1, 1
0, 0
1, 0
0, 1
1, 0
1, 1
0, 1 0, 1
0, 0
1, 0
0, 0 1, 1
0, 0
1, 0
0, 1
〈〈P1,P2〉〉♦c 1 2
Amelie David TATL: Tableaux for ATL 19 September 2013 3 / 10
Tableaux for ATL
Characteristics of Tableaux for ATL
à Tableaux for ATL are graphs
à Nodes of the graph are sets of formulas
à Procedure in 2 phases : construction then elimination of statesIII Application of several rules
à Tableaux for ATL are open iff at least one state with the initialformula remains at the end of the procedure
Amelie David TATL: Tableaux for ATL 19 September 2013 4 / 10
Tableaux for ATL - Example
θ = 〈〈1〉〉�¬q ∧ 〈〈2〉〉p Uq
θ
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q, q
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,q
〈〈1〉〉�¬q 〈〈1〉〉�¬q,〈〈2〉〉p Uq
> 〈〈2〉〉p Uq
〈〈1〉〉�¬q, ¬q,〈〈1〉〉 © 〈〈1〉〉�¬q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,
〈〈1〉〉© 〈〈1〉〉�¬q, q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,
q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
〈〈1〉〉�¬q,¬q,
〈〈2〉〉p Uq,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
>,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, q,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, p,〈〈2〉〉 © 〈〈2〉〉p Uq
0, 0 0, 1 1, 01, 1
0, 0 0, 1 1, 01, 1
0, 00, 1
1, 0
1, 1
0, 0
0, 0
0, 0
0, 0
0, 00, 0
0, 0
0, 0
0, 0
0, 1
0, 1
0, 1
1, 0
0, 0 1, 11, 1
0, 1
0, 01, 0 1, 1
1, 1
Unsatisfiable
Construction
à Static Ruleà Next Rule
Elimination
à Prestateà F ,¬Fà Eventualityà Choices
Amelie David TATL: Tableaux for ATL 19 September 2013 5 / 10
Tableaux for ATL - Example
θ = 〈〈1〉〉�¬q ∧ 〈〈2〉〉p Uq
θ
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q, q
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,q
〈〈1〉〉�¬q 〈〈1〉〉�¬q,〈〈2〉〉p Uq
> 〈〈2〉〉p Uq
〈〈1〉〉�¬q, ¬q,〈〈1〉〉 © 〈〈1〉〉�¬q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,
〈〈1〉〉© 〈〈1〉〉�¬q, q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,
q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
〈〈1〉〉�¬q,¬q,
〈〈2〉〉p Uq,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
>,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, q,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, p,〈〈2〉〉 © 〈〈2〉〉p Uq
0, 0 0, 1 1, 01, 1
0, 0 0, 1 1, 01, 1
0, 00, 1
1, 0
1, 1
0, 0
0, 0
0, 0
0, 0
0, 00, 0
0, 0
0, 0
0, 0
0, 1
0, 1
0, 1
1, 0
0, 0 1, 11, 1
0, 1
0, 01, 0 1, 1
1, 1
Unsatisfiable
Construction
à Static Ruleà Next Rule
Elimination
à Prestateà F ,¬Fà Eventualityà Choices
Amelie David TATL: Tableaux for ATL 19 September 2013 5 / 10
Tableaux for ATL - Example
θ = 〈〈1〉〉�¬q ∧ 〈〈2〉〉p Uq
θ
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q, q
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,q
〈〈1〉〉�¬q 〈〈1〉〉�¬q,〈〈2〉〉p Uq
> 〈〈2〉〉p Uq
〈〈1〉〉�¬q, ¬q,〈〈1〉〉 © 〈〈1〉〉�¬q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,
〈〈1〉〉© 〈〈1〉〉�¬q, q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,
q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
〈〈1〉〉�¬q,¬q,
〈〈2〉〉p Uq,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
>,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, q,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, p,〈〈2〉〉 © 〈〈2〉〉p Uq
0, 0 0, 1 1, 01, 1
0, 0 0, 1 1, 01, 1
0, 00, 1
1, 0
1, 1
0, 0
0, 0
0, 0
0, 0
0, 00, 0
0, 0
0, 0
0, 0
0, 1
0, 1
0, 1
1, 0
0, 0 1, 11, 1
0, 1
0, 01, 0 1, 1
1, 1
Unsatisfiable
Construction
à Static Ruleà Next Rule
Elimination
à Prestateà F ,¬Fà Eventualityà Choices
Amelie David TATL: Tableaux for ATL 19 September 2013 5 / 10
Tableaux for ATL - Example
θ = 〈〈1〉〉�¬q ∧ 〈〈2〉〉p Uq
θ
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q, q
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,q
〈〈1〉〉�¬q 〈〈1〉〉�¬q,〈〈2〉〉p Uq
> 〈〈2〉〉p Uq
〈〈1〉〉�¬q, ¬q,〈〈1〉〉 © 〈〈1〉〉�¬q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,
〈〈1〉〉© 〈〈1〉〉�¬q, q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,
q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
〈〈1〉〉�¬q,¬q,
〈〈2〉〉p Uq ,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
>,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, q,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, p,〈〈2〉〉 © 〈〈2〉〉p Uq
0, 0 0, 1 1, 01, 1
0, 0 0, 1 1, 01, 1
0, 00, 1
1, 0
1, 1
0, 0
0, 0
0, 0
0, 0
0, 00, 0
0, 0
0, 0
0, 0
0, 1
0, 1
0, 1
1, 0
0, 0 1, 11, 1
0, 1
0, 01, 0 1, 1
1, 1
Unsatisfiable
Construction
à Static Ruleà Next Rule
Elimination
à Prestateà F ,¬Fà Eventualityà Choices
Amelie David TATL: Tableaux for ATL 19 September 2013 5 / 10
Tableaux for ATL - Example
θ = 〈〈1〉〉�¬q ∧ 〈〈2〉〉p Uq
θ
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q, q
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,q
〈〈1〉〉�¬q 〈〈1〉〉�¬q,〈〈2〉〉p Uq
> 〈〈2〉〉p Uq
〈〈1〉〉�¬q, ¬q,〈〈1〉〉 © 〈〈1〉〉�¬q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,
〈〈1〉〉© 〈〈1〉〉�¬q, q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,
q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
〈〈1〉〉�¬q,¬q,
〈〈2〉〉p Uq ,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
>,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, q,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, p,〈〈2〉〉 © 〈〈2〉〉p Uq
0, 0 0, 1 1, 01, 1
0, 0 0, 1 1, 01, 1
0, 00, 1
1, 0
1, 1
0, 0
0, 0
0, 0
0, 0
0, 00, 0
0, 0
0, 0
0, 0
0, 1
0, 1
0, 1
1, 0
0, 0 1, 11, 1
0, 1
0, 01, 0 1, 1
1, 1
Unsatisfiable
Construction
à Static Ruleà Next Rule
Elimination
à Prestateà F ,¬Fà Eventualityà Choices
Amelie David TATL: Tableaux for ATL 19 September 2013 5 / 10
Tableaux for ATL - Example
θ = 〈〈1〉〉�¬q ∧ 〈〈2〉〉p Uq
θ
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q, q
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,q
〈〈1〉〉�¬q 〈〈1〉〉�¬q,〈〈2〉〉p Uq
> 〈〈2〉〉p Uq
〈〈1〉〉�¬q, ¬q,〈〈1〉〉 © 〈〈1〉〉�¬q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,
〈〈1〉〉© 〈〈1〉〉�¬q, q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,
q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
〈〈1〉〉�¬q,¬q,
〈〈2〉〉p Uq ,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
>,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, q,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, p,〈〈2〉〉 © 〈〈2〉〉p Uq
0, 0 0, 1 1, 01, 1
0, 0 0, 1 1, 01, 1
0, 00, 1
1, 0
1, 1
0, 0
0, 0
0, 0
0, 0
0, 00, 0
0, 0
0, 0
0, 0
0, 1
0, 1
0, 1
1, 0
0, 0 1, 11, 1
0, 1
0, 01, 0 1, 1
1, 1
Unsatisfiable
Construction
à Static Ruleà Next Rule
Elimination
à Prestateà F ,¬Fà Eventualityà Choices
Amelie David TATL: Tableaux for ATL 19 September 2013 5 / 10
Tableaux for ATL - Example
θ = 〈〈1〉〉�¬q ∧ 〈〈2〉〉p Uq
θ
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q, q
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,q
〈〈1〉〉�¬q 〈〈1〉〉�¬q,〈〈2〉〉p Uq
> 〈〈2〉〉p Uq
〈〈1〉〉�¬q, ¬q,〈〈1〉〉 © 〈〈1〉〉�¬q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,
〈〈1〉〉© 〈〈1〉〉�¬q, q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,
q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
〈〈1〉〉�¬q,¬q,
〈〈2〉〉p Uq ,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
>,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, q,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, p,〈〈2〉〉 © 〈〈2〉〉p Uq
0, 0 0, 1 1, 01, 1
0, 0 0, 1 1, 01, 1
0, 00, 1
1, 0
1, 1
0, 0
0, 0
0, 0
0, 0
0, 00, 0
0, 0
0, 0
0, 0
0, 1
0, 1
0, 1
1, 0
0, 0 1, 11, 1
0, 1
0, 01, 0 1, 1
1, 1
Unsatisfiable
Construction
à Static Ruleà Next Rule
Elimination
à Prestateà F ,¬Fà Eventualityà Choices
Amelie David TATL: Tableaux for ATL 19 September 2013 5 / 10
Tableaux for ATL - Example
θ = 〈〈1〉〉�¬q ∧ 〈〈2〉〉p Uq
θ
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q, q
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,q
〈〈1〉〉�¬q 〈〈1〉〉�¬q,〈〈2〉〉p Uq
> 〈〈2〉〉p Uq
〈〈1〉〉�¬q, ¬q,〈〈1〉〉 © 〈〈1〉〉�¬q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,
〈〈1〉〉© 〈〈1〉〉�¬q, q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,
q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
〈〈1〉〉�¬q,¬q,
〈〈2〉〉p Uq ,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
>,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, q,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, p,〈〈2〉〉 © 〈〈2〉〉p Uq
0, 0 0, 1 1, 01, 1
0, 0 0, 1 1, 01, 1
0, 00, 1
1, 0
1, 1
0, 0
0, 0
0, 0
0, 0
0, 00, 0
0, 0
0, 0
0, 0
0, 1
0, 1
0, 1
1, 0
0, 0 1, 11, 1
0, 1
0, 01, 0 1, 1
1, 1
Unsatisfiable
Construction
à Static Ruleà Next Rule
Elimination
à Prestateà F ,¬Fà Eventualityà Choices
Amelie David TATL: Tableaux for ATL 19 September 2013 5 / 10
Tableaux for ATL - Example
θ = 〈〈1〉〉�¬q ∧ 〈〈2〉〉p Uq
θ
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q, q
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,q
〈〈1〉〉�¬q 〈〈1〉〉�¬q,〈〈2〉〉p Uq
> 〈〈2〉〉p Uq
〈〈1〉〉�¬q, ¬q,〈〈1〉〉 © 〈〈1〉〉�¬q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,
〈〈1〉〉© 〈〈1〉〉�¬q, q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,
q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
〈〈1〉〉�¬q,¬q,
〈〈2〉〉p Uq ,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
>,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, q,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, p,〈〈2〉〉 © 〈〈2〉〉p Uq
0, 0 0, 1 1, 01, 1
0, 0 0, 1 1, 01, 1
0, 00, 1
1, 0
1, 1
0, 0
0, 0
0, 0
0, 0
0, 00, 0
0, 0
0, 0
0, 0
0, 1
0, 1
0, 1
1, 0
0, 0 1, 11, 1
0, 1
0, 01, 0 1, 1
1, 1
Unsatisfiable
Construction
à Static Ruleà Next Rule
Elimination
à Prestateà F ,¬Fà Eventualityà Choices
Amelie David TATL: Tableaux for ATL 19 September 2013 5 / 10
Tableaux for ATL - Example
θ = 〈〈1〉〉�¬q ∧ 〈〈2〉〉p Uq
θ
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q, q
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,q
〈〈1〉〉�¬q 〈〈1〉〉�¬q,〈〈2〉〉p Uq
> 〈〈2〉〉p Uq
〈〈1〉〉�¬q, ¬q,〈〈1〉〉 © 〈〈1〉〉�¬q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,
〈〈1〉〉© 〈〈1〉〉�¬q, q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,
q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
〈〈1〉〉�¬q,¬q,
〈〈2〉〉p Uq ,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
>,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, q,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, p,〈〈2〉〉 © 〈〈2〉〉p Uq
0, 0 0, 1 1, 01, 1
0, 0 0, 1 1, 01, 1
0, 00, 1
1, 0
1, 1
0, 0
0, 0
0, 0
0, 0
0, 00, 0
0, 0
0, 0
0, 0
0, 1
0, 1
0, 1
1, 0
0, 0 1, 11, 1
0, 1
0, 01, 0 1, 1
1, 1
Unsatisfiable
Construction
à Static Ruleà Next Rule
Elimination
à Prestateà F ,¬Fà Eventualityà Choices
Amelie David TATL: Tableaux for ATL 19 September 2013 5 / 10
Tableaux for ATL - Example
θ = 〈〈1〉〉�¬q ∧ 〈〈2〉〉p Uq
θ
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q, q
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,q
〈〈1〉〉�¬q 〈〈1〉〉�¬q,〈〈2〉〉p Uq
> 〈〈2〉〉p Uq
〈〈1〉〉�¬q, ¬q,〈〈1〉〉 © 〈〈1〉〉�¬q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,
〈〈1〉〉© 〈〈1〉〉�¬q, q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,
q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
〈〈1〉〉�¬q,¬q,
〈〈2〉〉p Uq ,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
>,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, q,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, p,〈〈2〉〉 © 〈〈2〉〉p Uq
0, 0 0, 1 1, 01, 1
0, 0 0, 1 1, 01, 1
0, 00, 1
1, 0
1, 1
0, 0
0, 0
0, 0
0, 0
0, 00, 0
0, 0
0, 0
0, 0
0, 1
0, 1
0, 1
1, 0
0, 0 1, 11, 1
0, 1
0, 01, 0 1, 1
1, 1
Unsatisfiable
Construction
à Static Ruleà Next Rule
Elimination
à Prestateà F ,¬Fà Eventualityà Choices
Amelie David TATL: Tableaux for ATL 19 September 2013 5 / 10
Tableaux for ATL - Example
θ = 〈〈1〉〉�¬q ∧ 〈〈2〉〉p Uq
θ
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q, q
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,q
〈〈1〉〉�¬q 〈〈1〉〉�¬q,〈〈2〉〉p Uq
> 〈〈2〉〉p Uq
〈〈1〉〉�¬q, ¬q,〈〈1〉〉 © 〈〈1〉〉�¬q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,
〈〈1〉〉© 〈〈1〉〉�¬q, q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,
q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
〈〈1〉〉�¬q,¬q,
〈〈2〉〉p Uq ,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
>,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, q,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, p,〈〈2〉〉 © 〈〈2〉〉p Uq
0, 0 0, 1 1, 01, 1
0, 0 0, 1 1, 01, 1
0, 00, 1
1, 0
1, 1
0, 0
0, 0
0, 0
0, 0
0, 00, 0
0, 0
0, 0
0, 0
0, 1
0, 1
0, 1
1, 0
0, 0 1, 11, 1
0, 1
0, 01, 0 1, 1
1, 1
Unsatisfiable
Construction
à Static Ruleà Next Rule
Elimination
à Prestateà F ,¬Fà Eventualityà Choices
Amelie David TATL: Tableaux for ATL 19 September 2013 5 / 10
Tableaux for ATL - Example
θ = 〈〈1〉〉�¬q ∧ 〈〈2〉〉p Uq
θ
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉 © 〈〈2〉〉p Uq
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q, q
θ, 〈〈1〉〉�¬q, 〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,q
〈〈1〉〉�¬q 〈〈1〉〉�¬q,〈〈2〉〉p Uq
> 〈〈2〉〉p Uq
〈〈1〉〉�¬q, ¬q,〈〈1〉〉 © 〈〈1〉〉�¬q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,
〈〈1〉〉© 〈〈1〉〉�¬q, q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q ,〈〈1〉〉 © 〈〈1〉〉�¬q,
q
〈〈1〉〉�¬q,〈〈2〉〉p Uq,¬q,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
〈〈1〉〉�¬q,¬q,
〈〈2〉〉p Uq ,〈〈1〉〉 © 〈〈1〉〉�¬q,p, 〈〈2〉〉© 〈〈2〉〉p Uq
>,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, q,〈〈1, 2〉〉 © >
〈〈2〉〉p Uq, p,〈〈2〉〉 © 〈〈2〉〉p Uq
0, 0 0, 1 1, 01, 1
0, 0 0, 1 1, 01, 1
0, 00, 1
1, 0
1, 1
0, 0
0, 0
0, 0
0, 0
0, 00, 0
0, 0
0, 0
0, 0
0, 1
0, 1
0, 1
1, 0
0, 0 1, 11, 1
0, 1
0, 01, 0 1, 1
1, 1
Unsatisfiable
Construction
à Static Ruleà Next Rule
Elimination
à Prestateà F ,¬Fà Eventualityà Choices
Amelie David TATL: Tableaux for ATL 19 September 2013 5 / 10
Implementation
à Tableaux implementation: Ocaml ('1200 lines)à HMI: PHP
Amelie David TATL: Tableaux for ATL 19 September 2013 6 / 10
Tests of the Implementation
à No other implementation
à No benchmark
Creation of 42 formulas
à goal: test that the application is working correctly.
Amelie David TATL: Tableaux for ATL 19 September 2013 7 / 10
Tests of the Implementation
à No other implementation
à No benchmark
Creation of 42 formulas
à goal: test that the application is working correctly.
Amelie David TATL: Tableaux for ATL 19 September 2013 7 / 10
Tests of the Implementation
à No other implementation
à No benchmark
Creation of 42 formulas
à goal: test that the application is working correctly.
Amelie David TATL: Tableaux for ATL 19 September 2013 7 / 10
Demonstration
à atila.ibisc.univ-evry/tableau ATL
Amelie David TATL: Tableaux for ATL 19 September 2013 8 / 10
Perspectives
à Create a benchmark for the application
à Improve the implementation to be able to extract models fromthe tableau
à Implemente of an on-the-fly version
à Extend the implementation to other extensions of ATL
Amelie David TATL: Tableaux for ATL 19 September 2013 9 / 10
Thank you for your attention
Any questions ?
atila.ibisc.univ-evry/tableau ATL
Amelie David TATL: Tableaux for ATL 19 September 2013 10 / 10