On the Expressiveness and Complexity of ATL

Transcription

1 On the Expressiveness and Complexity of ATL François Laroussinie, Nicolas Markey, Ghassan Oreiby LSV, CNRS & ENS-Cachan Recherches en vérification automatique March 14, 2006

2 Overview of CTL CTL A Kripke structure Quantification over paths (E/ A) out of gate q 0 q 3 in gate out of gate request q 1 q 2 out of gate grant Example There exists a run such that the train is always out of gate. EG out of gate

3 Overview of CTL CTL A Kripke structure Quantification over paths (E/ A) out of gate q 0 q 3 in gate out of gate request q 1 q 2 out of gate grant Example There exists a run such that the train is always out of gate. EG out of gate

4 Overview of ATL ATL A multi-agent system. Quantification over strategies of agents out of gate q 0 q 3 in gate out of gate request train ctr q 1 ctr train q 2 out of gate grant Example Whenever the train is out of gate, the controller cannot force it to enter the gate. AG (out of gate = ctr F in gate)

5 Overview of ATL ATL A multi-agent system. Quantification over strategies of agents out of gate q 0 q 3 in gate out of gate request train ctr q 1 ctr train q 2 out of gate grant Example Whenever the train is out of gate, the controller cannot force it to enter the gate. AG (out of gate = ctr F in gate)

6 Outline of the talk 1 Introduction Overview of CTL and ATL 2 Definitions Multi-agent models Strategy and outcomes ATL (Alternating-time Temporal Logic) 3 Expressiveness Weak Until 4 Complexity Model checking ATL on CGSs Model checking ATL on ATSs 5 Conclusion

7 Outline of the talk 1 Introduction Overview of CTL and ATL 2 Definitions Multi-agent models Strategy and outcomes ATL (Alternating-time Temporal Logic) 3 Expressiveness Weak Until 4 Complexity Model checking ATL on CGSs Model checking ATL on ATSs 5 Conclusion

8 CGS definition Definition A CGS C is a 5-tuple (Loc, Lab, Agt, Chc, Edg) s.t: Loc: a finite set of locations; Lab: Loc 2 AP : a labeling function; Agt = {A 1,, A k }: a set of agents (or players); Chc: Loc Agt N 1 the choice function. Chc(l, A i ) = number of possible moves for A i from l. Edg: Loc N k Loc: the transition table.

9 Example of a CGS Example p p, r r, s s Start s r, p s, r p q 0 r s, s p, p r Player 1 Player 2 q 0 p r s p q 0 q 1 q 2 r q 2 q 0 q 1 s q 1 q 2 q 0 2 Win q 2 q 1 1 Win Figure: Paper, rock and scissors

10 Semantics of CGSs From a location l, each agent A i chooses some m Ai with m Ai < Chc(l, A i ). Edg(l, m A1,, m Ak ) gives the new location. Notations: Next(l) = {Edg(l, m Ai ) m Ai 1 i k} Next(l, A j, m) = { Edg(l,, m Aj 1, m, m Aj+1 ) }

11 CGS example Example p p, r r, s s Start s r, p s, r p q 0 r s, s p, p r Player 1 Player 2 q 0 p r s p q 0 q 1 q 2 r q 2 q 0 q 1 s q 1 q 2 q 0 2 Win q 2 q 1 1 Win Figure: Paper, rock and scissors

12 CGS example Example p p, r r, s s Start s r, p s, r p q 0 r s, s p, p r Player 1 Player 2 q 0 p r s p q 0 q 1 q 2 r q 2 q 0 q 1 s q 1 q 2 q 0 2 Win q 2 q 1 1 Win Figure: Paper, rock and scissors

13 CGS example Example p p, r r, s s Start s r, p s, r p q 0 r s, s p, p r Player 1 Player 2 q 0 p r s p q 0 q 1 q 2 r q 2 q 0 q 1 s q 1 q 2 q 0 2 Win q 2 q 1 1 Win Figure: Paper, rock and scissors

14 ATS definition Definition An ATS A is a 4-tuple (Loc, Lab, Agt, Chc) where: Loc, Lab and Agt are the same as in CGSs; a move is a set of locations: Chc: Loc Agt P(P(Loc)) with the following requirement: for any location l and for moves Q i Chc(l, A i ), Q i must be a singleton. i k The next location is precisely the location that belongs to all the choices of the agents. Next(l) and Next(l, A i, m) are defined in the obvious way.

15 ATS example Example out of gate q 0 q 3 in gate out of gate request train ctr q 1 ctr train q 2 out of gate grant δ(q 0, train) = {{q 0 }, {q 1 }}. δ(q 1, ctr) = {{q 0 }, {q 1 }, {q 2 }}. δ(q 2, train) = {{q 0 }, {q 3 }}. δ(q 3, ctr) = {{q 0 }, {q 3 }}. δ(q 0, ctr) = δ(q 1, train) = δ(q 2, ctr) = δ(q 3, train) = {Loc}. Figure: Train controller

16 Translation CGS ATS 3.1 A D 2.2, B C Naive approach Move Player 1 Player 2 1 {B, D } {A, B, D } 2 {C, D } {C, D } 3 {A, D } {C, D } Figure: Converting an CGS into an ATS Cost of the translation: polynomial CGS exponential ATS

17 Translation CGS ATS 3.1 A D 2.2, B C Correct approach Move Player 1 Player 2 1 {B 1.1, D 1.2, D 1.3 } {A 3.1, B 1.1, D 2.1 } 2 {C 2.2, C 2.3, D 2.1 } {C 2.2, D 1.2, D 3.2 } 3 {A 3.1, D 3.2, D 3.3 } {C 2.3, D 1.3, D 3.3 } Figure: Converting an CGS into an ATS Cost of the translation: polynomial CGS exponential ATS

18 Translation CGS ATS 3.1 A D 2.2, B C Correct approach Move Player 1 Player 2 1 {B 1.1, D 1.2, D 1.3 } {A 3.1, B 1.1, D 2.1 } 2 {C 2.2, C 2.3, D 2.1 } {C 2.2, D 1.2, D 3.2 } 3 {A 3.1, D 3.2, D 3.3 } {C 2.3, D 1.3, D 3.3 } Figure: Converting an CGS into an ATS Cost of the translation: polynomial CGS exponential ATS

19 Strategies and outcomes Definition A computation is an infinite sequence ρ = l 0 l 1 such that i, l i+1 Next(l i ). A strategy is a function f Ai s.t. f Ai (l 0,, l m ) = a possible move for A i from l m. The outcomes Out(l, f Ai ) are the set of computations from l that agree with the strategy f Ai of A i. Notations: given A Agt, we note: F A = {f Ai A i A} Out(l, F A )

20 Syntax of ATL Definition ([AHK97]) The syntax of ATL is defined by the following grammar: ATL ϕ s, ψ s ::= p ϕ s ϕ s ψ s A ϕ p ϕ p ::= X ϕ s G ϕ s ϕ s U ψ s. where p ranges over the set AP and A over the subsets of Agt. ATL subsumes CTL, since we have: Eϕ p Agt ϕ p, Aϕ p ϕ p.

21 Semantics Semantics l = A ϕ p iff F A Strat(A). ρ Out(l, F A ). ρ = ϕ p ρ = ϕ s U ψ s iff i.ρ[i] = ψ s and 0 j < i.ρ[j] = ϕ s We have A ϕ Agt A ϕ, but A ϕ Agt A ϕ. We denote A ϕ for A ϕ

22 Outline of the talk 1 Introduction Overview of CTL and ATL 2 Definitions Multi-agent models Strategy and outcomes ATL (Alternating-time Temporal Logic) 3 Expressiveness Weak Until 4 Complexity Model checking ATL on CGSs Model checking ATL on ATSs 5 Conclusion

23 Can ATL express weak until? Definition ϕ W ψ ϕ U ψ G ϕ CTL Eϕ W ψ EG ϕ Eϕ U ψ Aϕ W ψ E( ψ) U ( ϕ ψ) Question Can we express A a W b in ATL? A (G ϕ ϕ U ψ) is not an ATL formula, A ϕ W ψ A G ϕ A ϕ U ψ.

24 Can ATL express weak until? Definition ϕ W ψ ϕ U ψ G ϕ CTL Eϕ W ψ EG ϕ Eϕ U ψ Aϕ W ψ E( ψ) U ( ϕ ψ) Question Can we express A a W b in ATL? A (G ϕ ϕ U ψ) is not an ATL formula, A ϕ W ψ A G ϕ A ϕ U ψ.

25 Can ATL express weak until? Question Can we express A a W b in ATL? Theorem Answer: No Formula ϕ = A a W b cannot be expressed in ATL. Idea: We present two families of models that cannot be distinguished by ATL formulae of any given size. One model satisfies A a W b while the other does not.

26 Can ATL express weak until? Question Can we express A a W b in ATL? Theorem Answer: No Formula ϕ = A a W b cannot be expressed in ATL. Idea: We present two families of models that cannot be distinguished by ATL formulae of any given size. One model satisfies A a W b while the other does not.

27 Can ATL express weak until? Question Can we express A a W b in ATL? Theorem Answer: No Formula ϕ = A a W b cannot be expressed in ATL. Idea: We present two families of models that cannot be distinguished by ATL formulae of any given size. One model satisfies A a W b while the other does not.

28 Can ATL express weak until? s i , , , 4.2 a , 1.3, 2.1, 3.2, 3.3 s i a s a b i b a a ai ai 1 a a b 1 1.2, 1.3, 2.1, 3.2, 3.3 b s i 1.2, 1.3 a a 2.1, 3.2, 3.3 si a s1 1.2, , 3.2, 3.3 a, b s 0 Lemma i > 0, ψ ATL with ψ i we have: s i = ψ iff s i = ψ.

29 Outline of the talk 1 Introduction Overview of CTL and ATL 2 Definitions Multi-agent models Strategy and outcomes ATL (Alternating-time Temporal Logic) 3 Expressiveness Weak Until 4 Complexity Model checking ATL on CGSs Model checking ATL on ATSs 5 Conclusion

30 ATL model checking over CGSs Theorem ([AHK02]) Model checking ATL over CGSs is PTIME-complete. Pre (A, L) = {l m A Next(l, A, m A ) L} ϕ = A θ 1 U θ 2 L := [false]; T := [θ 2 ]; while T L do L := L T ; T := Pre(A, L) [θ 1 ] od; [ϕ] := T Overall complexity: O( Edg ϕ ), thus PTIME.

31 Implicit CGS Definition An implicit CGS is a CGS where: The transition function: in each l it s given ((ϕ 0, l 0 ),, (ϕ n, l n )) where l i Loc, ϕ i is a boolean combination of propositions A j = c. Edg (l, m A1,, m Ak ) = l j s.t. j = min (i ϕ i (l, m A1,, m Ak ) = ). It is required that the last formula, ϕ n, be. explicit CGS exponential polynomial ATS polynomial exponential polynomial exponential implicit CGS

32 Implicit CGS Definition An implicit CGS is a CGS where: The transition function: in each l it s given ((ϕ 0, l 0 ),, (ϕ n, l n )) where l i Loc, ϕ i is a boolean combination of propositions A j = c. Edg (l, m A1,, m Ak ) = l j s.t. j = min (i ϕ i (l, m A1,, m Ak ) = ). It is required that the last formula, ϕ n, be. explicit CGS exponential polynomial ATS polynomial exponential polynomial exponential implicit CGS

33 ATL model checking over implicit CGSs Theorem Model checking ATL over implicit CGSs in 3 -complete.

34 ATL model checking over implicit CGSs Theorem Model checking ATL over implicit CGSs in 3 -complete. NP Σ 2 = NP NP PTIME 2 = PTIME NP 3 = PTIME Σ 2 PSPACE co-np Π 2 = co-np NP co- Polynomial-time hierarchy PH

35 ATL model checking over implicit CGSs Theorem Model checking ATL over implicit CGSs in 3 -complete. Membership in 3. Σ 2 algorithm proposed in [JD05]: correctly handles positive formulas (i.e. of the form A ϕ). That algorithm is used as an oracle, called a polynomial number of times.

36 ATL model checking over implicit CGSs Theorem Model checking ATL over implicit CGSs in 3 -complete. Hardness in Σ 2. [JD05] EQSAT 2 : Input: a boolean formula ϕ over variables in X Y. Output: true iff X. Y. ϕ(x, Y )

37 ATL model checking over implicit CGSs Theorem Model checking ATL over implicit CGSs in 3 -complete. Hardness in Σ 2. [JD05] EQSAT 2 : Input: a boolean formula ϕ over variables in X Y. Output: true iff X. Y. ϕ(x, Y ) 1 player A i per variable in X, 1 player B j per variable in Y.

38 ATL model checking over implicit CGSs Theorem Model checking ATL over implicit CGSs in 3 -complete. Hardness in Σ 2. [JD05] EQSAT 2 : Input: a boolean formula ϕ over variables in X Y. Output: true iff X. Y. ϕ(x, Y ) 1 player A i per variable in X, 1 player B j per variable in Y. Lemma The instance of EQSAT 2 is positive iff q 0 ϕ(a,b) ϕ(a,b) q q 0 = A 1,, A n X q. q

39 ATL model checking over implicit CGSs Theorem Model checking ATL over implicit CGSs in 3 -complete. Hardness in Π 2. AQSAT 2 : Input: a boolean formula ϕ over variables in X Y. Output: true iff X. Y. ϕ(x, Y ) 1 player A i per variable in X, 1 player B j per variable in Y. Lemma The instance of AQSAT 2 is positive iff q 0 ϕ(a,b) ϕ(a,b) q q 0 = A 1,, A n X q. q

40 ATL model checking over implicit CGSs Theorem Model checking ATL over implicit CGSs in 3 -complete. Hardness in 3 (sketch). SNSAT 2 : Input: formulas ϕ i over variables in X i Y i {z 1,, z i 1 }. Output: the value of z m defined by: def z 1 = X 1. Y 1. ϕ 1 (X 1, Y 1 ) def z 2 = X 2. Y 2. ϕ 2 (z 1, X 2, Y 2 ) def z 3 = X 3. Y 3. ϕ 3 (z 1, z 2, X 3, Y 3 )... def = X m. Y m. ϕ m (z 1,, z m 1, X m, Y m ) z m

41 ATL model checking over implicit CGSs Theorem Model checking ATL over implicit CGSs in 3 -complete. Hardness in 3 (sketch). z m 1 z m 2 s s z m z m 1 z m 2 ϕ m(c m 1,A m,b m) ϕ m(c m 1,A m,b m) q q q q q q ψ m = AC ( s) U (q EX (s EX ψ m 1 )).

42 ATL model checking over ATSs Theorem ([AHK97]) Model checking ATL over ATSs is PTIME-complete. Proof. Similar to the case of CGSs. But Transitions of an ATS are not given explicitely. The algorithm is polynomial in the size of the underlying CGS (which might be exponential). Theorem ([JD05]) Model checking ATL over ATSs is PTIME-complete if the number of agents is fixed.

43 ATL model checking over ATSs Theorem ([AHK97]) Model checking ATL over ATSs is PTIME-complete. Proof. Similar to the case of CGSs. But Transitions of an ATS are not given explicitely. The algorithm is polynomial in the size of the underlying CGS (which might be exponential). Theorem ([JD05]) Model checking ATL over ATSs is PTIME-complete if the number of agents is fixed.

44 ATL model checking over ATSs Theorem ([AHK97]) Model checking ATL over ATSs is PTIME-complete. Proof. Similar to the case of CGSs. But Transitions of an ATS are not given explicitely. The algorithm is polynomial in the size of the underlying CGS (which might be exponential). Theorem ([JD05]) Model checking ATL over ATSs is PTIME-complete if the number of agents is fixed.

45 ATL model checking over ATSs Theorem Model checking ATL over ATSs is 2 -complete.

46 ATL model checking over ATSs Theorem Model checking ATL over ATSs is 2 -complete. NP-hardness: Reduction from 3-SAT.

47 ATL model checking over ATSs Theorem Model checking ATL over ATSs is 2 -complete. NP-hardness: Reduction from 3-SAT. C = p q r c 0 = p q r c 1 = p q r c 2 = p q r c 3 = p q r c 4 = p q r c 5 = p q r c 6 = p q r c 7 = p q r

48 ATL model checking over ATSs Theorem Model checking ATL over ATSs is 2 -complete. q 0 c 1 0 c 1 1 c 1 2 c 1 3 c 1 4 c 1 5 c 1 6 c 2 0 c 2 1 c 2 2 c 2 3 c 2 4 c 2 5 c 2 6 c n 0 c n 1 c n 2 c n 3 c n 4 c n 5 c n 6 c 7 1 c 7 2 c 7 n C 1 C 2 C n

49 ATL model checking over ATSs Theorem Model checking ATL over ATSs is 2 -complete. 1 player (P 1 to P k ) per atomic proposition: p { c i j c i j not made true by p } p { c i j c i j not made true by p } c 1 0 c 1 1 c 1 2 c 1 3 c 1 4 c 1 5 c 1 6 c 2 0 c 2 1 c 2 2 c 2 3 c 2 4 c 2 5 c 2 6 q 0 c n 0 c n 1 c n 2 c n 3 c n 4 c n 5 c n 6 c 7 1 c 7 2 c 7 n C 1 C 2 C n

50 ATL model checking over ATSs Theorem Model checking ATL over ATSs is 2 -complete. 1 player (P 1 to P k ) per atomic proposition: p { c i j c i j not made true by p } p { c i j c i j not made true by p } Once those players have chosen their moves, exactly one clause cj i per original clause C i belongs to the intersection of the chosen sets. E.g. p = q = r = p q r c 1 0 c 1 1 c 1 2 c 1 3 c 1 4 c 1 5 c 1 6 c 2 0 c 2 1 c 2 2 c 2 3 c 2 4 c 2 5 c 2 6 q 0 c n 0 c n 1 c n 2 c n 3 c n 4 c n 5 c n 6 c 7 1 c 7 2 c 7 n C 1 C 2 C n

51 ATL model checking over ATSs Theorem Model checking ATL over ATSs is 2 -complete. 1 player (P 1 to P k ) per atomic proposition: p { c i j c i j not made true by p } p { c i j c i j not made true by p } 1 extra player chooses one set among {c 1 0,, c1 7 } to {cn 0,, cn 7 } c 1 0 c 1 1 c 1 2 c 1 3 c 1 4 c 1 5 c 1 6 c 2 0 c 2 1 c 2 2 c 2 3 c 2 4 c 2 5 c 2 6 q 0 c n 0 c n 1 c n 2 c n 3 c n 4 c n 5 c n 6 c 7 1 c 7 2 c 7 n C 1 C 2 C n

52 ATL model checking over ATSs Theorem Model checking ATL over ATSs is 2 -complete. 1 player (P 1 to P k ) per atomic proposition: p { c i j c i j not made true by p } p { c i j c i j not made true by p } 1 extra player chooses one set among {c 1 0,, c1 7 } to {cn 0,, cn 7 } Lemma The 3-SAT instance is true iff q 0 = P 1,, P k X c 1 0 c 1 1 c 1 2 c 1 3 c 1 4 c 1 5 c 1 6 c 2 0 c 2 1 c 2 2 c 2 3 c 2 4 c 2 5 c 2 6 q 0 c n 0 c n 1 c n 2 c n 3 c n 4 c n 5 c n 6 c 7 1 c 7 2 c 7 n C 1 C 2 C n

53 Conclusion Expressiveness Complexity results ATL W > ATL CGS fixed CGS ATS ATL PTIME 3 2 ATL Future Work Fairness constraints Timed models