This document discusses temporal logics for multi-agent systems. It introduces Computation Tree Logic (CTL) and Alternating-Time Temporal Logic (ATL) for expressing properties of reactive and multi-agent systems. It then presents ATL with strategy contexts (ATLsc), which extends ATL with additional strategy quantifiers to express properties involving strategies. ATLsc is shown to be strictly more expressive than ATL and as expressive as Quantified CTL. The document discusses examples of properties that can be expressed in ATLsc, such as interactions between clients and servers accessing a shared resource and existence of Nash equilibria.

1. Temporal logics for multi-agent systems
Nicolas Markey
LSV – ENS Cachan
(based on joint works with Thomas Brihaye,
Arnaud Da Costa-Lopes, François Laroussinie)
« Formalisation des Activités Concurrentes »
Toulouse, 16 April 2014

2. Model checking and synthesis
system:
[http://www.embedded.com]
3
propriété
a!
b?
a?
b!
AG( : B.overfull
^ :B.dried_up)
model-checking
algorithm
yes/no

3. Model checking and synthesis
system:
[http://www.embedded.com]
3
propriété
a!
b! ? AG( : B.overfull
b?
a?
^ :B.dried_up)
synthesis
algorithm
a?
b!

4. Outline of the presentation
1 Introduction
2 Basics of CTL and ATL
expressing properties of reactive systems
efficient verification algorithms
3 Temporal logics for multi-agent systems
specifying properties of complex interacting systems
expressive power of ATLsc
translation into Quantified CTL (QCTL)
algorithms for ATLsc
4 Conclusions and future works

5. Outline of the presentation
1 Introduction
2 Basics of CTL and ATL
expressing properties of reactive systems
efficient verification algorithms
3 Temporal logics for multi-agent systems
specifying properties of complex interacting systems
expressive power of ATLsc
translation into Quantified CTL (QCTL)
algorithms for ATLsc
4 Conclusions and future works

6. Computation-Tree Logic (CTL)
atomic propositions: , , ...

7. Computation-Tree Logic (CTL)
atomic propositions: , , ...
boolean combinators: :', ' _ , ' ^ , ...

8. Computation-Tree Logic (CTL)
atomic propositions: , , ...
boolean combinators: :', ' _ , ' ^ , ...
temporal modalities:
X ' ' “next '”
' U ' ' “' until ”

9. Computation-Tree Logic (CTL)
atomic propositions: , , ...
boolean combinators: :', ' _ , ' ^ , ...
temporal modalities:
X ' ' “next '”
' U ' ' “' until ”
true U ' F ' ' “eventually '”
: F :' G ' ' ' ' ' ' “always '”

10. Computation-Tree Logic (CTL)
atomic propositions: , , ...
boolean combinators: :', ' _ , ' ^ , ...
temporal modalities:
X ' ' “next '”
' U ' ' “' until ”
true U ' F ' ' “eventually '”
: F :' G ' ' ' ' ' ' “always '”
path quantifiers:
E'
' A'
' ' ' ' ' '

11. Examples of CTL and CTL formulas
In CTL, each temporal modality is in the immediate scope of a
path quantifier.

12. Examples of CTL and CTL formulas
In CTL, each temporal modality is in the immediate scope of a
path quantifier.
E F is reachable

13. Examples of CTL and CTL formulas
In CTL, each temporal modality is in the immediate scope of a
path quantifier.
E F is reachable
3
3
3

14. Examples of CTL and CTL formulas
In CTL, each temporal modality is in the immediate scope of a
path quantifier.
EG( : ^ E F ) there is a path along which is always
reachable, but never reached

15. Examples of CTL and CTL formulas
In CTL, each temporal modality is in the immediate scope of a
path quantifier.
EG( : ^ |E F{z }
p
) there is a path along which is always
reachable, but never reached
p
p p

16. Examples of CTL and CTL formulas
In CTL, each temporal modality is in the immediate scope of a
path quantifier.
EG( : ^ |E F{z }
p
) there is a path along which is always
reachable, but never reached
3
p
3
p
p

17. Examples of CTL and CTL formulas
In CTL, each temporal modality is in the immediate scope of a
path quantifier.
Theorem ([CE81,QS82])
CTL model checking is PTIME-complete.
[CE81] Clarke, Emerson. Design and Synthesis of Synchronization Skeletons using
Branching-Time Temporal Logic. LOP’81.
[QS82] Queille, Sifakis. Specification and verification of concurrent systems in CESAR.
SOP’82.

18. Examples of CTL and CTL formulas
In CTL, each temporal modality is in the immediate scope of a
path quantifier.
Theorem ([CE81,QS82])
CTL model checking is PTIME-complete.
Theorem ([KVW94])
CTL model checking on product structures is
PSPACE-complete.
[CE81] Clarke, Emerson. Design and Synthesis of Synchronization Skeletons using
Branching-Time Temporal Logic. LOP’81.
[QS82] Queille, Sifakis. Specification and verification of concurrent systems in CESAR.
SOP’82.
[KVW94] Kupferman, Vardi, Wolper. An automata-theoretic approach to branching-time
model checking. CAV’94.

19. Examples of CTL and CTL formulas
In CTL, we have no restriction on modalities and quantifiers.

20. Examples of CTL and CTL formulas
In CTL, we have no restriction on modalities and quantifiers.
EG F there is a path visiting infinitely many times
3
3
3

21. Examples of CTL and CTL formulas
In CTL, we have no restriction on modalities and quantifiers.
A(G F ) G(: )) any path that visits infinitely many times,
never visits

22. Examples of CTL and CTL formulas
In CTL, we have no restriction on modalities and quantifiers.
A(G F ) G(: )) any path that visits infinitely many times,
never visits
3
3
3
3

23. Examples of CTL and CTL formulas
In CTL, we have no restriction on modalities and quantifiers.
Theorem ([EH86,KVW94])
CTL model checking is PSPACE-complete.
Theorem ([KVW94])
CTL model checking on product structures is PSPACE-complete.
[EH86] Emerson, Halpern. Sometimes and Not Never Revisited: On Branching versus
Linear Time Temporal Logic. J.ACM, 1986.
[KVW94] Kupferman, Vardi, Wolper. An automata-theoretic approach to branching-time
model checking. CAV’94.

24. Reasoning about open systems

25. Reasoning about open systems
Concurrent games
A concurrent game is made of
a transition system;
q0
q1
q2

26. Reasoning about open systems
Concurrent games
A concurrent game is made of
a transition system;
a set of agents (or players);
q0
q1
q2

27. Reasoning about open systems
Concurrent games
A concurrent game is made of
a transition system;
a set of agents (or players);
a table indicating the transition to be taken given the actions
of the players.
q0
q1
q2
player 1
q0 q2 q1
q1 q0 q2
q2 q1 q0
player 2

28. Reasoning about open systems
Concurrent games
A concurrent game is made of
a transition system;
a set of agents (or players);
a table indicating the transition to be taken given the actions
of the players.
Turn-based games
A turn-based game is a game
where only one agent plays at
a time.

29. Reasoning about open systems
Strategies
A strategy for a given player is a function telling what to play
depending on what has happened previously.

30. Reasoning about open systems
Strategies
A strategy for a given player is a function telling what to play
depending on what has happened previously.
Example
Strategy for player :
alternately go to and .

31. Reasoning about open systems
Strategies
A strategy for a given player is a function telling what to play
depending on what has happened previously.
Example
Strategy for player :
alternately go to and .
...
...
...
...

32. Temporal logics for games: ATL
ATL extends CTL with strategy quantifiers
hhAii ' expresses that A has a strategy to enforce '.
[AHK02] Alur, Henzinger, Kupferman. Alternating-time Temporal Logic. J. ACM, 2002.

33. Temporal logics for games: ATL
ATL extends CTL with strategy quantifiers
hhAii ' expresses that A has a strategy to enforce '.
hh ii F
[AHK02] Alur, Henzinger, Kupferman. Alternating-time Temporal Logic. J. ACM, 2002.

34. Temporal logics for games: ATL
ATL extends CTL with strategy quantifiers
hhAii ' expresses that A has a strategy to enforce '.
3 3
3 3
hh ii F
[AHK02] Alur, Henzinger, Kupferman. Alternating-time Temporal Logic. J. ACM, 2002.

35. Temporal logics for games: ATL
ATL extends CTL with strategy quantifiers
hhAii ' expresses that A has a strategy to enforce '.
hh ii F
hh ii F
[AHK02] Alur, Henzinger, Kupferman. Alternating-time Temporal Logic. J. ACM, 2002.

36. Temporal logics for games: ATL
ATL extends CTL with strategy quantifiers
hhAii ' expresses that A has a strategy to enforce '.
3
3
hh ii F
hh ii F
[AHK02] Alur, Henzinger, Kupferman. Alternating-time Temporal Logic. J. ACM, 2002.

37. Temporal logics for games: ATL
ATL extends CTL with strategy quantifiers
hhAii ' expresses that A has a strategy to enforce '.
hh ii F
hh ii F
hh ii G( hh ii F )
[AHK02] Alur, Henzinger, Kupferman. Alternating-time Temporal Logic. J. ACM, 2002.

38. Temporal logics for games: ATL
ATL extends CTL with strategy quantifiers
hhAii ' expresses that A has a strategy to enforce '.
p
p
hh ii F
hh ii F
hh ii G( hh ii F ) hh ii G p
p
[AHK02] Alur, Henzinger, Kupferman. Alternating-time Temporal Logic. J. ACM, 2002.

39. Temporal logics for games: ATL
ATL extends CTL with strategy quantifiers
hhAii ' expresses that A has a strategy to enforce '.
p
p
hh ii F
hh ii F
hh ii G( hh ii F ) hh ii G p
p
Theorem ([AHK02])
Model checking ATL is PTIME-complete.
Model checking ATL is 2-EXPTIME-complete.
[AHK02] Alur, Henzinger, Kupferman. Alternating-time Temporal Logic. J. ACM, 2002.

40. Outline of the presentation
1 Introduction
2 Basics of CTL and ATL
expressing properties of reactive systems
efficient verification algorithms
3 Temporal logics for multi-agent systems
specifying properties of complex interacting systems
expressive power of ATLsc
translation into Quantified CTL (QCTL)
algorithms for ATLsc
4 Conclusions and future works

41. ATL with strategy contexts [BDLM09]
hh ii G( hh ii F )
[BDLM09] Brihaye, Da Costa, Laroussinie, M. ATL with strategy contexts. LFCS, 2009.

42. ATL with strategy contexts [BDLM09]
hh ii G( hh ii F )
consider the following strategy
of Player : “always go to ”;
[BDLM09] Brihaye, Da Costa, Laroussinie, M. ATL with strategy contexts. LFCS, 2009.

43. ATL with strategy contexts [BDLM09]
hh ii G( hh ii F )
consider the following strategy
of Player : “always go to ”;
[BDLM09] Brihaye, Da Costa, Laroussinie, M. ATL with strategy contexts. LFCS, 2009.

44. ATL with strategy contexts [BDLM09]
hh ii G( hh ii F )
consider the following strategy
of Player : “always go to ”;
in the remaining tree, Player
can always enforce a visit to .
[BDLM09] Brihaye, Da Costa, Laroussinie, M. ATL with strategy contexts. LFCS, 2009.

45. ATL with strategy contexts
Definition
ATLsc has two new strategy quantifiers: hAi ' and h-A-i '.
hAi is similar to hhAii but assigns the corresponding strategy
to A for evaluating ';

46. ATL with strategy contexts
Definition
ATLsc has two new strategy quantifiers: hAi ' and h-A-i '.
hAi is similar to hhAii but assigns the corresponding strategy
to A for evaluating ';
h-A-i drops the assigned strategies for A.

47. ATL with strategy contexts
Definition
ATLsc has two new strategy quantifiers: hAi ' and h-A-i '.
hAi is similar to hhAii but assigns the corresponding strategy
to A for evaluating ';
h-A-i drops the assigned strategies for A.
[A] is dual to hAi :
[A] ' : hAi :'
[A] ' which states that any strategy for A has an outcome
along which ' holds.

48. What ATLsc can express
Client-server interactions for accessing a shared resource:
hServeri G
2
66664
^
c2Clients
hci F accessc
^
:
^
c6=c0
accessc ^ accessc0
3
77775

49. What ATLsc can express
Client-server interactions for accessing a shared resource:
hServeri G
2
66664
^
c2Clients
hci F accessc
^
:
^
c6=c0
accessc ^ accessc0
3
77775
Existence of Nash equilibria:
hA1; :::; Ani
^
i
( hAi i 'Ai ) 'Ai )

50. What ATLsc can express
Client-server interactions for accessing a shared resource:
hServeri G
2
66664
^
c2Clients
hci F accessc
^
:
^
c6=c0
accessc ^ accessc0
3
77775
Existence of Nash equilibria:
hA1; :::; Ani
^
i
( hAi i 'Ai ) 'Ai )
Existence of dominating strategy:
hAi [B] (:' ) [A] :')

51. More expressiveness results
Theorem
ATLsc is strictly more expressive than ATL,
The operator h-A-i does not add expressive power,
ATLsc is as expressive as ATL
sc .

52. More expressiveness results
Theorem
ATLsc is strictly more expressive than ATL,
The operator h-A-i does not add expressive power,
ATLsc is as expressive as ATL
sc .
Proof
hhAii ' h-Agt-i hAi ^'

53. More expressiveness results
Theorem
ATLsc is strictly more expressive than ATL,
The operator h-A-i does not add expressive power,
ATLsc is as expressive as ATL
sc .
Proof
h1i ( h2i X a ^ h2i X b) is only true in the second game.
But ATL cannot distinguish between these two games.
s
h1:1i;h2:2i h1:1i;h2:2i;h3:3i
h1:2i h2:1i h1:2i;h1:3i;h3:2i h2:1i;h2:3i;h3:1i
a b
s0
a b

54. More expressiveness results
Theorem
ATLsc is strictly more expressive than ATL,
The operator h-A-i does not add expressive power,
ATLsc is as expressive as ATL
sc .
Proof
Replace implicit quantification with explicit one:
h1i ' h1i [Agt n f1g] h;i b'
; we can always assume that the context is full.

55. More expressiveness results
Theorem
ATLsc is strictly more expressive than ATL,
The operator h-A-i does not add expressive power,
ATLsc is as expressive as ATL
sc .
Proof
Replace implicit quantification with explicit one:
h1i ' h1i [Agt n f1g] h;i b'
; we can always assume that the context is full.
h-A-i ' is then equivalent to [A] h;i ';
h;i can be inserted between two temporal modalities.

56. Outline of the presentation
1 Introduction
2 Basics of CTL and ATL
expressing properties of reactive systems
efficient verification algorithms
3 Temporal logics for multi-agent systems
specifying properties of complex interacting systems
expressive power of ATLsc
translation into Quantified CTL (QCTL)
algorithms for ATLsc
4 Conclusions and future works

57. Quantified CTL [ES84,Kup95,Fre01]
QCTL extends CTL with propositional quantifiers
9p: ' means that there exists a labelling of the model
with p under which ' holds.
[ES84] Emerson and Sistla. Deciding Full Branching Time Logic. Information Control, 1984.
[Kup95] Kupferman. Augmenting Branching Temporal Logics with Existential Quantification
over Atomic Propositions. CAV, 1995.
[Fre01] French. Decidability of Quantifed Propositional Branching Time Logics. AJCAI, 2001.

58. Quantified CTL [ES84,Kup95,Fre01]
QCTL extends CTL with propositional quantifiers
9p: ' means that there exists a labelling of the model
with p under which ' holds.
E F ^ 8p:
E F(p ^ ) ) AG( ) p)
[ES84] Emerson and Sistla. Deciding Full Branching Time Logic. Information Control, 1984.
[Kup95] Kupferman. Augmenting Branching Temporal Logics with Existential Quantification
over Atomic Propositions. CAV, 1995.
[Fre01] French. Decidability of Quantifed Propositional Branching Time Logics. AJCAI, 2001.

59. Quantified CTL [ES84,Kup95,Fre01]
QCTL extends CTL with propositional quantifiers
9p: ' means that there exists a labelling of the model
with p under which ' holds.
E F ^ 8p:
E F(p ^ ) ) AG( ) p)
uniq( )
[ES84] Emerson and Sistla. Deciding Full Branching Time Logic. Information Control, 1984.
[Kup95] Kupferman. Augmenting Branching Temporal Logics with Existential Quantification
over Atomic Propositions. CAV, 1995.
[Fre01] French. Decidability of Quantifed Propositional Branching Time Logics. AJCAI, 2001.

60. Quantified CTL [ES84,Kup95,Fre01]
QCTL extends CTL with propositional quantifiers
9p: ' means that there exists a labelling of the model
with p under which ' holds.
E F ^ 8p:
E F(p ^ ) ) AG( ) p)
uniq( )
; true if we label the Kripke structure;
; false if we label the computation tree;
[ES84] Emerson and Sistla. Deciding Full Branching Time Logic. Information Control, 1984.
[Kup95] Kupferman. Augmenting Branching Temporal Logics with Existential Quantification
over Atomic Propositions. CAV, 1995.
[Fre01] French. Decidability of Quantifed Propositional Branching Time Logics. AJCAI, 2001.

61. Semantics of QCTL
structure semantics:
j=s 9p:' ,
p
j= '

62. Semantics of QCTL
structure semantics:
j=s 9p:' ,
p
j= '
tree semantics:
j=t 9p:' , p
p p
p
j= '

63. Expressiveness of QCTL
QCTL can “count”:
EX1 ' EX ' ^ 8p: [EX(p ^ ') ) AX(' ) p)]
EX2 ' 9q: [EX1(' ^ q) ^ EX1(' ^ :q)]
[DLM12] Da Costa, Laroussinie, M. Quantified CTL: expressiveness and model checking.
CONCUR, 2012.

64. Expressiveness of QCTL
QCTL can “count”:
EX1 ' EX ' ^ 8p: [EX(p ^ ') ) AX(' ) p)]
EX2 ' 9q: [EX1(' ^ q) ^ EX1(' ^ :q)]
QCTL can express (least or greatest) fixpoints:
T:'(T) 9t: [AG(t () '(t))^
(8t:0(AG(t0 () '(t0)) ) AG(t ) t0)))]
[DLM12] Da Costa, Laroussinie, M. Quantified CTL: expressiveness and model checking.
CONCUR, 2012.

65. Expressiveness of QCTL
QCTL can “count”:
EX1 ' EX ' ^ 8p: [EX(p ^ ') ) AX(' ) p)]
EX2 ' 9q: [EX1(' ^ q) ^ EX1(' ^ :q)]
QCTL can express (least or greatest) fixpoints:
T:'(T) 9t: [AG(t () '(t))^
(8t:0(AG(t0 () '(t0)) ) AG(t ) t0)))]
Theorem
QCTL, QCTL and MSO are equally expressive (under both
semantics).
[DLM12] Da Costa, Laroussinie, M. Quantified CTL: expressiveness and model checking.
CONCUR, 2012.

66. QCTL with structure semantics
Theorem
Model checking QCTL for the structure semantics is
PSPACE-complete.
[DLM12] Da Costa, Laroussinie, M. Quantified CTL: expressiveness and model checking.
CONCUR, 2012.

67. QCTL with structure semantics
Theorem
Model checking QCTL for the structure semantics is
PSPACE-complete.
Proof
Membership:
Iteratively
(nondeterministically) pick a labelling,
check the subformula.
Hardness:
QBF is a special case (without even using temporal modalities).
[DLM12] Da Costa, Laroussinie, M. Quantified CTL: expressiveness and model checking.
CONCUR, 2012.

68. QCTL with structure semantics
Theorem
Model checking QCTL for the structure semantics is
PSPACE-complete.
Proof
Membership:
Iteratively
(nondeterministically) pick a labelling,
check the subformula.
Hardness:
QBF is a special case (without even using temporal modalities).
Theorem
QCTL satisfiability for the structure semantics is undecidable.
[DLM12] Da Costa, Laroussinie, M. Quantified CTL: expressiveness and model checking.
CONCUR, 2012.

69. QCTL with tree semantics
Theorem
Model checking QCTL with k quantifiers in the tree semantics
is k-EXPTIME-complete.
Satisfiability of QCTL with k quantifiers in the tree semantics
is (k+1)-EXPTIME-complete.
[DLM12] Da Costa, Laroussinie, M. Quantified CTL: expressiveness and model checking.
CONCUR, 2012.
[LM13a] Laroussinie, M. Quantified CTL: expressiveness and complexity. Submitted, 2013.

70. QCTL with tree semantics
Theorem
Model checking QCTL with k quantifiers in the tree semantics
is k-EXPTIME-complete.
Satisfiability of QCTL with k quantifiers in the tree semantics
is (k+1)-EXPTIME-complete.
Proof
Using (alternating) parity tree automata:

71. QCTL with tree semantics
Theorem
Model checking QCTL with k quantifiers in the tree semantics
is k-EXPTIME-complete.
Satisfiability of QCTL with k quantifiers in the tree semantics
is (k+1)-EXPTIME-complete.
Proof
Using (alternating) parity tree automata:

72. QCTL with tree semantics
Theorem
Model checking QCTL with k quantifiers in the tree semantics
is k-EXPTIME-complete.
Satisfiability of QCTL with k quantifiers in the tree semantics
is (k+1)-EXPTIME-complete.
Proof
Using (alternating) parity tree automata:
(q0; ) = (q0; q1) _ (q1; q0)
(q0; ) = (q1; q1)
(q0; ) = (q2; q2)
(q1; ? ) = (q1; q1)
(q2; ? ) = (q2; q2)

73. QCTL with tree semantics
Theorem
Model checking QCTL with k quantifiers in the tree semantics
is k-EXPTIME-complete.
Satisfiability of QCTL with k quantifiers in the tree semantics
is (k+1)-EXPTIME-complete.
Proof
Using (alternating) parity tree automata:
q0
(q0; ) = (q0; q1) _ (q1; q0)
(q0; ) = (q1; q1)
(q0; ) = (q2; q2)
(q1; ? ) = (q1; q1)
(q2; ? ) = (q2; q2)

74. QCTL with tree semantics
Theorem
Model checking QCTL with k quantifiers in the tree semantics
is k-EXPTIME-complete.
Satisfiability of QCTL with k quantifiers in the tree semantics
is (k+1)-EXPTIME-complete.
Proof
Using (alternating) parity tree automata:
q0
q0 q1
(q0; ) = (q0; q1) _ (q1; q0)
(q0; ) = (q1; q1)
(q0; ) = (q2; q2)
(q1; ? ) = (q1; q1)
(q2; ? ) = (q2; q2)

75. QCTL with tree semantics
Theorem
Model checking QCTL with k quantifiers in the tree semantics
is k-EXPTIME-complete.
Satisfiability of QCTL with k quantifiers in the tree semantics
is (k+1)-EXPTIME-complete.
Proof
Using (alternating) parity tree automata:
q0
q0 q1
q1 q0
(q0; ) = (q0; q1) _ (q1; q0)
(q0; ) = (q1; q1)
(q0; ) = (q2; q2)
(q1; ? ) = (q1; q1)
(q2; ? ) = (q2; q2)

76. QCTL with tree semantics
Theorem
Model checking QCTL with k quantifiers in the tree semantics
is k-EXPTIME-complete.
Satisfiability of QCTL with k quantifiers in the tree semantics
is (k+1)-EXPTIME-complete.
Proof
Using (alternating) parity tree automata:
q0
q0 q1
q1 q0 q1 q1
(q0; ) = (q0; q1) _ (q1; q0)
(q0; ) = (q1; q1)
(q0; ) = (q2; q2)
(q1; ? ) = (q1; q1)
(q2; ? ) = (q2; q2)

77. QCTL with tree semantics
Theorem
Model checking QCTL with k quantifiers in the tree semantics
is k-EXPTIME-complete.
Satisfiability of QCTL with k quantifiers in the tree semantics
is (k+1)-EXPTIME-complete.
Proof
Using (alternating) parity tree automata:
q0
q0 q1
q1 q0 q1 q1
q1 q1 q1 q1
(q0; ) = (q0; q1) _ (q1; q0)
(q0; ) = (q1; q1)
(q0; ) = (q2; q2)
(q1; ? ) = (q1; q1)
(q2; ? ) = (q2; q2)

78. QCTL with tree semantics
Theorem
Model checking QCTL with k quantifiers in the tree semantics
is k-EXPTIME-complete.
Satisfiability of QCTL with k quantifiers in the tree semantics
is (k+1)-EXPTIME-complete.
Proof
Using (alternating) parity tree automata:
q0
q0 q1
q1 q0 q1 q1
q1 q1 q1 q1 q1 q1
(q0; ) = (q0; q1) _ (q1; q0)
(q0; ) = (q1; q1)
(q0; ) = (q2; q2)
(q1; ? ) = (q1; q1)
(q2; ? ) = (q2; q2)

79. QCTL with tree semantics
Theorem
Model checking QCTL with k quantifiers in the tree semantics
is k-EXPTIME-complete.
Satisfiability of QCTL with k quantifiers in the tree semantics
is (k+1)-EXPTIME-complete.
Proof
Using (alternating) parity tree automata:
q0
q0 q1
q1 q0 q1 q1
q1 q1 q1 q1 q1 q1 q1 q1
(q0; ) = (q0; q1) _ (q1; q0)
(q0; ) = (q1; q1)
(q0; ) = (q2; q2)
(q1; ? ) = (q1; q1)
(q2; ? ) = (q2; q2)

80. QCTL with tree semantics
Theorem
Model checking QCTL with k quantifiers in the tree semantics
is k-EXPTIME-complete.
Satisfiability of QCTL with k quantifiers in the tree semantics
is (k+1)-EXPTIME-complete.
Proof
Using (alternating) parity tree automata:
q0
q0 q1
q1 q0 q1 q1
q1 q1 q1 q1 q1 q1 q1 q1
(q0; ) = (q0; q1) _ (q1; q0)
(q0; ) = (q1; q1)
(q0; ) = (q2; q2)
(q1; ? ) = (q1; q1)
(q2; ? ) = (q2; q2)
This automaton corresponds to E U

81. QCTL with tree semantics
Theorem
Model checking QCTL with k quantifiers in the tree semantics
is k-EXPTIME-complete.
Satisfiability of QCTL with k quantifiers in the tree semantics
is (k+1)-EXPTIME-complete.
Proof
polynomial-size automata for CTL;
quantification is handled by projection, which first requires
removing alternation (exponential blowup);
an automaton equivalent to a QCTL formula can be built
inductively;
emptiness of an alternating parity tree automaton can be
decided in exponential time.

82. Translating ATLsc into QCTL
player A has moves mA
1 , ..., mA
n ;
from the transition table, we can compute the
set Next( ; A;mA
i ) of states that can be
reached from when player A plays mA
i .
[DLM12] Da Costa, Laroussinie, M. Quantified CTL: expressiveness and model checking.
CONCUR, 2012.

83. Translating ATLsc into QCTL
player A has moves mA
1 , ..., mA
n ;
from the transition table, we can compute the
set Next( ; A;mA
i ) of states that can be
reached from when player A plays mA
i .
hAi ' can be encoded as follows:
9mA
1 : 9mA
2 : : : 9mA
n :
this corresponds to a strategy: AG(mA
i ,
V
:mA
j );
the outcomes all satisfy ':
G(q ^ mA
A
i ) X Next(q; A;mA
i )) ) '
.
[DLM12] Da Costa, Laroussinie, M. Quantified CTL: expressiveness and model checking.
CONCUR, 2012.

84. Translating ATLsc into QCTL
player A has moves mA
1 , ..., mA
n ;
from the transition table, we can compute the
set Next( ; A;mA
i ) of states that can be
reached from when player A plays mA
i .
Corollary
ATLsc model checking is decidable, with non-elementary complexity
(TOWER-complete).
Corollary
ATL0
sc (quantification restricted to memoryless strategies) model
checking is PSPACE-complete.
[DLM12] Da Costa, Laroussinie, M. Quantified CTL: expressiveness and model checking.
CONCUR, 2012.

85. What about satisfiability?
Theorem
QCTL satisfiability is decidable (for the tree semantics).

86. What about satisfiability?
Theorem
QCTL satisfiability is decidable (for the tree semantics).
But
Theorem ([TW12])
ATLsc satisfiability is undecidable.
[TW12] Troquard, Walther. On Satisfiability in ATL with Strategy Contexts. JELIA, 2012.

87. What about satisfiability?
Theorem
QCTL satisfiability is decidable (for the tree semantics).
But
Theorem ([TW12])
ATLsc satisfiability is undecidable.
Why?
The translation from ATLsc to QCTL assumes
that the game structure is given!
[TW12] Troquard, Walther. On Satisfiability in ATL with Strategy Contexts. JELIA, 2012.

88. Satisfiability for turn-based games
Theorem (LM13b)
When restricted to turn-based games, ATLsc satisfiability is
decidable.
[LM13b] Laroussinie, M. Satisfiability of ATL with strategy contexts. Gandalf, 2013.

89. Satisfiability for turn-based games
Theorem (LM13b)
When restricted to turn-based games, ATLsc satisfiability is
decidable.
player has moves , and .
a strategy can be encoded by marking some of
the nodes of the tree with proposition movA.
hAi ' can be encoded as follows:
9movA:
it corresponds to a strategy: AG(turnA ) EX1 movA);
the outcomes all satisfy ': A
G(turnA ^ X movA) ) '
.
[LM13b] Laroussinie, M. Satisfiability of ATL with strategy contexts. Gandalf, 2013.

90. What about Strategy Logic? [CHP07,MMV10]
Strategy logic
Explicit quantification over strategies + strategy assignement
Example
hAi ' 91:assign(1; A):'
Strategy logic can also be translated into QCTL.
Theorem
Strategy-logic model-checking is decidable.
Strategy-logic satisfiability is decidable when restricted to
turn-based games.
[CHP07] Chatterjee, Henzinger, Piterman. Strategy Logic. CONCUR, 2007.
[MMV10] Mogavero, Murano, Vardi. Reasoning about strategies. FSTTCS, 2010.

91. Conclusions and future works
Conclusions
QCTL is a powerful extension of CTL;
it is equivalent to MSO over finite graphs and regular trees;
it is a nice tool to understand temporal logics for games (ATL
with strategy contexts, Strategy Logic, ...);

92. Conclusions and future works
Conclusions
QCTL is a powerful extension of CTL;
it is equivalent to MSO over finite graphs and regular trees;
it is a nice tool to understand temporal logics for games (ATL
with strategy contexts, Strategy Logic, ...);
Future directions
Defining interesting (expressive yet tractable) fragments of
those logics;
Obtaining practicable algorithms.
Considering randomised strategies.