tatl: implementation of an atl tableau-based decision procedure

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