Real-time Synthesis is Hard!
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1 Real-time Synthesis is Highlights 2016 Benjamin Monmege (LIF, Aix-Marseille Université) Thomas Brihaye, Morgane Estiévenart, Hsi-Ming Ho (UMONS) Gilles Geeraerts (ULB) Nathalie Sznajder (UPMC, LIP6) 7 September 2016 Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 1 / 10
2 Reactive synthesis Σ = Σ C Σ E controllable actions owned by controller C: {MoveUp, MoveDown, OpenDoor, Opened,...} uncontrollable actions owned by environment E: {0F-Up, 0F-Down,..., -1F, 0F,... Open, Close,...} Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 2 / 10
3 Reactive synthesis Σ = Σ C Σ E controllable actions owned by controller C: {MoveUp, MoveDown, OpenDoor, Opened,...} uncontrollable actions owned by environment E: {0F-Up, 0F-Down,..., -1F, 0F,... Open, Close,...} + state (at which floor, opening,... ) Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 2 / 10
4 Reactive synthesis Σ = Σ C Σ E controllable actions owned by controller C: {MoveUp, MoveDown, OpenDoor, Opened,...} uncontrollable actions owned by environment E: {0F-Up, 0F-Down,..., -1F, 0F,... Open, Close,...} + state (at which floor, opening,... ) + timing restrictions (latency,... ) Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 2 / 10
5 Reactive synthesis Σ = Σ C Σ E controllable actions owned by controller C: {MoveUp, MoveDown, OpenDoor, Opened,...} uncontrollable actions owned by environment E: {0F-Up, 0F-Down,..., -1F, 0F,... Open, Close,...} + state (at which floor, opening,... ) + timing restrictions (latency,... ) = Plant P: a DTA over Σ Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 2 / 10
6 Reactive synthesis Σ = Σ C Σ E controllable actions owned by controller C: {MoveUp, MoveDown, OpenDoor, Opened,...} uncontrollable actions owned by environment E: {0F-Up, 0F-Down,..., -1F, 0F,... Open, Close,...} + state (at which floor, opening,... ) + timing restrictions (latency,... ) = Plant P: a DTA over Σ Σ E Σ E 0 TO 1 Arrive x = 5 1F MoveUp x := 0 Σ E Σ E 0F MoveDown x := 0 0 TO -1 Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 2 / 10
7 Reactive synthesis A run of P can be seen as a play of the timed game between C and E. Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 3 / 10
8 Reactive synthesis A run of P can be seen as a play of the timed game between C and E. In each round, each player proposes a pair (delay, action) enabled in P: Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 3 / 10
9 Reactive synthesis A run of P can be seen as a play of the timed game between C and E. In each round, each player proposes a pair (delay, action) enabled in P: CloseDoor x := 0 CLOSING Closed x = 2 Open Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 3 / 10
10 Reactive synthesis A run of P can be seen as a play of the timed game between C and E. In each round, each player proposes a pair (delay, action) enabled in P: CloseDoor x := 0 CLOSING Closed x = 2 Open ( C, Closed) ( E, Open) Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 3 / 10
11 Reactive synthesis A run of P can be seen as a play of the timed game between C and E. In each round, each player proposes a pair (delay, action) enabled in P: CloseDoor x := 0 CLOSING Closed x = 2 Open ( C, Closed) ( E, Open) Only action(s) with the shortest delay min( C, E ) may be played. Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 3 / 10
12 Reactive synthesis The lift responds to any call in less than 10 t.u. Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 4 / 10
13 Reactive synthesis The lift responds to any call in less than 10 t.u. 1F-Up x := 0-1F-Up x := 0 MoveUp x < 10 MoveDown x < 10 Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 4 / 10
14 Reactive synthesis The lift responds to any call in less than 10 t.u. 1F-Up x := 0-1F-Up x := 0 MoveUp x < 10 MoveDown x < 10 Specification L: a set of timed words over Σ Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 4 / 10
15 Reactive synthesis The lift responds to any call in less than 10 t.u. 1F-Up x := 0-1F-Up x := 0 MoveUp x < 10 MoveDown x < 10 Specification L: a set of timed words over Σ Reactive synthesis problem (RS) Given plant P and specification L, find a strategy of Controller such that no matter what Environment does, every play satisfies the specification. Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 4 / 10
16 Metric Temporal Logic (MTL) ϕ ::= a ϕ ϕ ϕ ϕ U I ϕ with a Σ, I [0, ) with bounds in Q {+ } Models: finite (or infinite) timed words σ = (a 1, t 1 )(a 2, t 2 ) Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 5 / 10
17 Metric Temporal Logic (MTL) ϕ ::= a ϕ ϕ ϕ ϕ U I ϕ with a Σ, I [0, ) with bounds in Q {+ } Models: finite (or infinite) timed words σ = (a 1, t 1 )(a 2, t 2 ) Theorem: [Doyen, Geeraerts, Raskin, and Reichert, 2009] Reactive synthesis problem is undecidable for ECL (hence, MTL) specifications, even without plant. Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 5 / 10
18 Reactive synthesis for MTL Reactive synthesis Undec. [Doyen, Geeraerts, Raskin, and Reichert, 2009] Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 6 / 10
19 Reactive synthesis for MTL Reactive synthesis Undec. [Doyen, Geeraerts, Raskin, and Reichert, 2009] Controller = timed automaton Implementable reactive synthesis Undec. [Bouyer, Bozzelli, and Chevalier, 2006] Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 6 / 10
20 Reactive synthesis for MTL Reactive synthesis Undec. [Doyen, Geeraerts, Raskin, and Reichert, 2009] Controller = timed automaton Implementable reactive synthesis Undec. [Bouyer, Bozzelli, and Chevalier, 2006] Bounding resources of controller Clocks- and precision-bounded reactive synthesis Dec. & Non-elem. over finite words [Bouyer, Bozzelli, and Chevalier, 2006] Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 6 / 10
21 Reactive synthesis for MTL Reactive synthesis Undec. [Doyen, Geeraerts, Raskin, and Reichert, 2009] Controller = timed automaton Implementable reactive synthesis Undec. [Bouyer, Bozzelli, and Chevalier, 2006] Clocks-bounded reactive synthesis Precision-bounded reactive synthesis Bounding resources of controller Clocks- and precision-bounded reactive synthesis Dec. & Non-elem. over finite words [Bouyer, Bozzelli, and Chevalier, 2006] Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 6 / 10
22 Reactive synthesis for MTL Reactive synthesis Undec. [Doyen, Geeraerts, Raskin, and Reichert, 2009] Controller = timed automaton Implementable reactive synthesis Undec. [Bouyer, Bozzelli, and Chevalier, 2006] Clocks-bounded reactive synthesis Undec. Precision-bounded reactive synthesis Undec. Bounding resources of controller Clocks- and precision-bounded reactive synthesis Dec. & Non-elem. over finite words [Bouyer, Bozzelli, and Chevalier, 2006] Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 6 / 10
23 Regaining hope? Less expressive specifications Undecidability proofs heavily use punctuality of MTL: request =1 grant Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 7 / 10
24 Regaining hope? Less expressive specifications Undecidability proofs heavily use punctuality of MTL: request =1 grant request [1,2] grant request 3 grant Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 7 / 10
25 Regaining hope? Less expressive specifications Undecidability proofs heavily use punctuality of MTL: request =1 grant request [1,2] grant request 3 grant MITL = non-punctual fragment of MTL: ϕ ::= a ϕ ϕ ϕ ϕ U I ϕ with a Σ, I [0, ) is a non-singular with bounds in Q {+ } Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 7 / 10
26 Contribution: reactive synthesis for MITL Reactive synthesis Implementable reactive synthesis Controller = timed automaton Clocks-bounded reactive synthesis Precision-bounded reactive synthesis Bounding resources of controller Clocks- and precision-bounded reactive synthesis Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 8 / 10
27 Contribution: reactive synthesis for MITL Reactive synthesis Undec. (even without plant) Implementable reactive synthesis Controller = timed automaton Clocks-bounded reactive synthesis Precision-bounded reactive synthesis Bounding resources of controller Clocks- and precision-bounded reactive synthesis Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 8 / 10
28 Contribution: reactive synthesis for MITL Reactive synthesis Undec. (even without plant) Controller = timed automaton Clocks-bounded reactive synthesis Implementable reactive synthesis Precision-bounded reactive synthesis Undec. Bounding resources of controller Clocks- and precision-bounded reactive synthesis Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 8 / 10
29 Contribution: reactive synthesis for MITL Reactive synthesis Undec. (even without plant) Controller = timed automaton Clocks-bounded reactive synthesis Undec. Implementable reactive synthesis Precision-bounded reactive synthesis Undec. Undec. Bounding resources of controller Clocks- and precision-bounded reactive synthesis Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 8 / 10
30 Contribution: reactive synthesis for MITL Reactive synthesis Undec. (even without plant) Controller = timed automaton Clocks-bounded reactive synthesis Undec. Implementable reactive synthesis Precision-bounded reactive synthesis Undec. Undec. Bounding resources of controller Clocks- and precision-bounded reactive synthesis Dec. & 3-EXPTIME over finite words [D Souza and Madhusudan, 2002] Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 8 / 10
31 Contribution: reactive synthesis for MITL Reactive synthesis Undec. (even without plant) Controller = timed automaton Clocks-bounded reactive synthesis Undec. Implementable reactive synthesis Precision-bounded reactive synthesis Undec. Undec. Bounding resources of controller Clocks- and precision-bounded reactive synthesis Dec. & 3-EXPTIME over finite words [D Souza and Madhusudan, 2002] On-the-fly algo: interval sem. [Brihaye, Estiévenart, and Geeraerts, 2013] Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 8 / 10
32 Conclusion For almost all reactive synthesis problems, MITL is as hard as MTL... Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 9 / 10
33 Conclusion For almost all reactive synthesis problems, MITL is as hard as MTL except for resources-bounded problem over finite words: Non-elementary for MTL; 3-EXPTIME for MITL; on-the-fly algorithm Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 9 / 10
34 Conclusion For almost all reactive synthesis problems, MITL is as hard as MTL except for resources-bounded problem over finite words: Non-elementary for MTL; 3-EXPTIME for MITL; on-the-fly algorithm Other fragments?? Hopeless! implementable RS clock-bounded RS precision-bounded RS Safety-MTL coflat-mtl Open-MITL Closed-MITL Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 9 / 10
35 Conclusion For almost all reactive synthesis problems, MITL is as hard as MTL except for resources-bounded problem over finite words: Non-elementary for MTL; 3-EXPTIME for MITL; on-the-fly algorithm Other fragments?? Hopeless! Safety-MTL coflat-mtl Open-MITL Closed-MITL implementable RS undec. undec. undec. undec. clock-bounded RS undec. undec. undec. undec. precision-bounded RS undec. undec. undec. undec. Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 9 / 10
36 Conclusion For almost all reactive synthesis problems, MITL is as hard as MTL except for resources-bounded problem over finite words: Non-elementary for MTL; 3-EXPTIME for MITL; on-the-fly algorithm Other fragments?? Hopeless! Safety-MTL coflat-mtl Open-MITL Closed-MITL implementable RS undec. undec. undec. undec. clock-bounded RS undec. undec. undec. undec. precision-bounded RS undec. undec. undec. undec. Possible future directions: Decidable fragments for BPrecRS/BClockRS Heuristics for speed-up for the on-the-fly algorithm Experiments of the on-the-fly algorithm over the fragments Robustness of controllers Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 9 / 10
37 Conclusion For almost all reactive synthesis problems, MITL is as hard as MTL except for resources-bounded problem over finite words: Non-elementary for MTL; 3-EXPTIME for MITL; on-the-fly algorithm Other fragments?? Hopeless! Safety-MTL coflat-mtl Open-MITL Closed-MITL implementable RS undec. undec. undec. undec. clock-bounded RS undec. undec. undec. undec. precision-bounded RS undec. undec. undec. undec. Possible future directions: Decidable fragments for BPrecRS/BClockRS Heuristics for speed-up for the on-the-fly algorithm Experiments of the on-the-fly algorithm over the fragments Robustness of controllers Thank you for your attention! Questions? Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 9 / 10
38 References Patricia Bouyer, Laura Bozzelli, and Fabrice Chevalier. Controller synthesis for MTL specifications. In Proceedings of the 17th International Conference on Concurrency Theory (CONCUR 06), volume 4137 of Lecture Notes in Computer Science, pages Springer, Patricia Bouyer, Nicolas Markey, Joël Ouaknine, and James Worrell. The cost of punctuality. In Proceedings of the 22nd Annual IEEE Symposium on Logic in Computer Science (LICS 07), pages IEEE Computer Society, Thomas Brihaye, Morgane Estiévenart, and Gilles Geeraerts. On MITL and alternating timed automata. In Proceedings of the 11th international conference on Formal Modeling and Analysis of Timed Systems (FORMATS 13), volume 8053 of Lecture Notes in Computer Science, pages Springer, Thomas Brihaye, Morgane Estiévenart, Gilles Geeraerts, Hsi-Ming Ho, Benjamin Monmege, and Nathalie Sznajder. Real-time synthesis is hard! In Martin Fränzle and Nicolas Markey, editors, Proceedings of the 14th International Conference on Formal Modeling and Analysis of Timed Systems (FORMATS 16), volume 9884 of Lecture Notes in Computer Science, pages Springer, August doi: / Laurent Doyen, Gilles Geeraerts, Jean-François Raskin, and Julien Reichert. Realizability of real-time logics. In Proceedings of the 7th International Conference on Formal Modeling and Analysis of Timed Systems (FORMATS 09), volume 5813 of Lecture Notes in Computer Science, pages Springer, Deepak D Souza and P. Madhusudan. Timed control synthesis for external specifications. In Proceedings of the 19th Annual conference on Theoretical Aspects of Computer Science (STACS 02), volume 2285 of Lecture Notes in Computer Science, pages Springer, Joël Ouaknine and James Worrell. Safety metric temporal logic is fully decidable. In Proceedings of the 12th International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS 06), volume 3920 of Lecture Notes in Computer Science, pages Springer, Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 10 / 10
39 Implementable reactive synthesis (IRS) C = deterministic symbolic transition system T set of locations; if finite T is a DTA finite set of clocks X finite set of possible clock constraints precision (m, K): g ::= g g x < α/m x α/m x = α/m x α/m x > α/m with x X, m N and 0 α K. Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 11 / 10
40 Implementable reactive synthesis (IRS) C = deterministic symbolic transition system T set of locations; if finite T is a DTA finite set of clocks X finite set of possible clock constraints precision (m, K): g ::= g g x < α/m x α/m x = α/m x α/m x > α/m with x X, m N and 0 α K. Implementable reactive synthesis problem (IRS) Given P and L, find such a T that no matter what E does, every play satisfies the specification. Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 11 / 10
41 Recovering decidability... Clock constraints in T : g ::= g g x < α/m x α/m x = α/m x α/m x > α/m with x X, m N and 0 α K. Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 12 / 10
42 Recovering decidability... Clock constraints in T : g ::= g g x < α/m x α/m x = α/m x α/m x > α/m with x X, m N and 0 α K. Fix X and (m, K) = the alphabet of T is given! Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 12 / 10
43 Recovering decidability... Clock constraints in T : g ::= g g x < α/m x α/m x = α/m x α/m x > α/m with x X, m N and 0 α K. Fix X and (m, K) = the alphabet of T is given! Bounded-resources reactive synthesis problem (BResRS) Given P, L, and a set of clocks X and precision (m, K), find such a resourcebounded T that no matter what E does, every play satisfies the specification. Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 12 / 10
44 Other fragments of MTL IRS/BPrecRS/BClockRS for MITL are undecidable. What about other fragments of MTL? Theorem: [Ouaknine and Worrell, 2006] Satisfiability and model-checking for Safety-MTL are decidable. Theorem: [Bouyer, Markey, Ouaknine, and Worrell, 2007] Model-checking for coflat-mtl is decidable. However with some more efforts, we can rewrite the formula in these fragments. IRS/BPrecRS/BClockRS over infinite timed words: Safety-MTL coflat-mtl desired undec. undec. undesired undec. undec. Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 13 / 10
45 On the decidable side Theorem: [D Souza and Madhusudan, 2002] BResRS is decidable for undesired specifications given in TA. Theorem: [Bouyer, Bozzelli, and Chevalier, 2006] BResRS is decidable for MTL specifications with non-primitive recursive complexity (over finite timed words). We propose an on-the-fly algorithm that solves BResRS for MITL specifications with 3-EXPTIME complexity (over finite words). Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 14 / 10
46 From MTL to OCATA ϕ = (a [1,2] b) b a a y := 0 b y [1, 2] Execution on the timed word (a, 0.5)(a, 0.6)(a, 1.2)(b, 2.3): y = y = y = Benjamin Monmege (LIF, Aix-Marseille Université) Real-time Synthesis is Hard! 15 / 10
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