Strategy Synthesis for Multi-dimensional Quantitative Objectives Krishnendu Chatterjee 1 Mickael Randour 2 cois Raskin 3 Jean-Fran¸ 1 IST Austria 2 UMONS 3 ULB 04.09.2012 CONCUR 2012: 23rd International Conference on Concurrency Theory
MEPGs & MMPPGs Mem. bounds Synthesis Randomization Conclusion Aim of this work system environment functional properties description description (e.g., no deadlock) quantitative model as model as requirements winning a game (e.g., mean response objectives time, fuel consumption) synthesis is there a winning strategy ? no yes empower system capabilities strategy = or weaken controller specification requirements Strat. Synth. for Multi Quant. Obj. Chatterjee, Randour, Raskin 1 / 29
MEPGs & MMPPGs Mem. bounds Synthesis Randomization Conclusion Aim of this work system environment functional properties description description (e.g., no deadlock) quantitative model as model as requirements winning a game (e.g., mean response objectives time, fuel consumption) synthesis � restriction to finite-memory strategies. is there a winning strategy ? no yes empower system capabilities strategy = or weaken controller specification requirements Strat. Synth. for Multi Quant. Obj. Chatterjee, Randour, Raskin 1 / 29
MEPGs & MMPPGs Mem. bounds Synthesis Randomization Conclusion Aim of this work Study games with � multi-dimensional quantitative objectives (energy and mean-payoff) � and a parity objective. � First study of such a conjunction. Address questions that revolve around strategies : � bounds on memory, � synthesis algorithm, ? � randomness ∼ memory. Strat. Synth. for Multi Quant. Obj. Chatterjee, Randour, Raskin 2 / 29
MEPGs & MMPPGs Mem. bounds Synthesis Randomization Conclusion Results Overview Memory bounds MEPGs MMPPGs optimal finite-memory optimal optimal exp. exp. infinite [CDHR10] Strategy synthesis (finite memory) MEPGs MMPPGs EXPTIME EXPTIME Randomness as a substitute for finite memory MEGs EPGs MMP(P)Gs MPPGs √ √ × × one-player √ two-player × × × Strat. Synth. for Multi Quant. Obj. Chatterjee, Randour, Raskin 3 / 29
MEPGs & MMPPGs Mem. bounds Synthesis Randomization Conclusion 1 Multi energy and mean-payoff parity games 2 Memory bounds 3 Strategy synthesis 4 Randomization as a substitute to finite-memory 5 Conclusion Strat. Synth. for Multi Quant. Obj. Chatterjee, Randour, Raskin 4 / 29
MEPGs & MMPPGs Mem. bounds Synthesis Randomization Conclusion 1 Multi energy and mean-payoff parity games 2 Memory bounds 3 Strategy synthesis 4 Randomization as a substitute to finite-memory 5 Conclusion Strat. Synth. for Multi Quant. Obj. Chatterjee, Randour, Raskin 5 / 29
MEPGs & MMPPGs Mem. bounds Synthesis Randomization Conclusion Turn-based games s 0 G = ( S 1 , S 2 , s init , E ) s 1 s 2 S = S 1 ∪ S 2 , S 1 ∩ S 2 = ∅ , E ⊆ S × S P 1 states = P 2 states = s 3 Plays, prefixes, pure strategies. s 4 s 5 Strat. Synth. for Multi Quant. Obj. Chatterjee, Randour, Raskin 6 / 29
MEPGs & MMPPGs Mem. bounds Synthesis Randomization Conclusion Integer k -dim. payoff function s 0 G = ( S 1 , S 2 , s init , E , w ) (2 , 1) (1 , − 2) w : E → Z k , model changes in s 1 s 2 (0 , 0) (1 , 0) quantities Energy level (0 , − 2) ( − 3 , 3) EL( ρ ) = v 0 + � i = n − 1 w ( s i , s i +1 ) i =0 s 3 Mean-payoff MP( π ) = lim inf n →∞ 1 n EL( π ( n )) (0 , 1) (1 , − 1) s 4 s 5 Strat. Synth. for Multi Quant. Obj. Chatterjee, Randour, Raskin 6 / 29
MEPGs & MMPPGs Mem. bounds Synthesis Randomization Conclusion Energy and mean-payoff problems Unknown initial credit s 0 ∃ ? v 0 ∈ N k , λ 1 ∈ Λ 1 s.t. (2 , 1) (1 , − 2) s 1 s 2 (0 , 0) (1 , 0) (0 , − 2) ( − 3 , 3) Mean-payoff threshold s 3 Given v ∈ Q k , ∃ ? λ 1 ∈ Λ 1 s.t. (0 , 1) (1 , − 1) s 4 s 5 Strat. Synth. for Multi Quant. Obj. Chatterjee, Randour, Raskin 6 / 29
MEPGs & MMPPGs Mem. bounds Synthesis Randomization Conclusion Parity problem s 0 p = 1 � � G p = S 1 , S 2 , s init , E , w , p (2 , 1) (1 , − 2) p : S → N s 1 s 2 (0 , 0) (1 , 0) Par( π ) = min { p ( s ) | s ∈ Inf( π ) } p = 1 p = 3 Even parity (0 , − 2) ( − 3 , 3) ∃ ? λ 1 ∈ Λ 1 s.t. the parity is even s 3 p = 2 � canonical way to express ω -regular objectives (0 , 1) (1 , − 1) s 4 s 5 p = 0 p = 2 Strat. Synth. for Multi Quant. Obj. Chatterjee, Randour, Raskin 6 / 29
MEPGs & MMPPGs Mem. bounds Synthesis Randomization Conclusion Known results Memory ( P 1 ) Decision problem 1-dim memoryless NP ∩ coNP [CdAHS03, BFL + 08] k -dim Energy finite coNP-c [CDHR10] 1-dim + parity exponential NP ∩ coNP [CD10] 1-dim memoryless NP ∩ coNP [EM79, LL69] k -dim Mean-payoff infinite coNP-c (fin.) [CDHR10] 1-dim + parity infinite NP ∩ coNP [CHJ05, BMOU11] Strat. Synth. for Multi Quant. Obj. Chatterjee, Randour, Raskin 7 / 29
MEPGs & MMPPGs Mem. bounds Synthesis Randomization Conclusion Infinite memory? Example for MMPGs, even with only one player! [CDHR10] (2 , 0) (0 , 2) (0 , 0) s 0 s 1 (0 , 0) � To obtain MP( π ) = (1 , 1) (with lim sup, (2 , 2) !), P 1 has to visit s 0 and s 1 for longer and longer intervals before jumping from one to the other. � Any finite-memory strategy alternating between these edges induces an ultimately periodic play s.t. MP( π ) = ( x , y ), x + y < 2. Strat. Synth. for Multi Quant. Obj. Chatterjee, Randour, Raskin 8 / 29
MEPGs & MMPPGs Mem. bounds Synthesis Randomization Conclusion Restriction to finite memory Infinite memory: � needed for MMPGs & MPPGs, � practical implementation is unrealistic. Strat. Synth. for Multi Quant. Obj. Chatterjee, Randour, Raskin 9 / 29
MEPGs & MMPPGs Mem. bounds Synthesis Randomization Conclusion Restriction to finite memory Infinite memory: � needed for MMPGs & MPPGs, � practical implementation is unrealistic. Finite memory: � preserves game determinacy, � provides equivalence between energy and mean-payoff settings, � the way to go for strategy synthesis. Strat. Synth. for Multi Quant. Obj. Chatterjee, Randour, Raskin 9 / 29
MEPGs & MMPPGs Mem. bounds Synthesis Randomization Conclusion 1 Multi energy and mean-payoff parity games 2 Memory bounds 3 Strategy synthesis 4 Randomization as a substitute to finite-memory 5 Conclusion Strat. Synth. for Multi Quant. Obj. Chatterjee, Randour, Raskin 10 / 29
MEPGs & MMPPGs Mem. bounds Synthesis Randomization Conclusion Obtained results MEPGs MMPPGs optimal finite-memory optimal optimal exp. exp. infinite [CDHR10] By [CDHR10], we only have to consider MEPGs. Recall that the unknown initial credit decision problem for MEGs (without parity) is coNP-complete. Strat. Synth. for Multi Quant. Obj. Chatterjee, Randour, Raskin 11 / 29
MEPGs & MMPPGs Mem. bounds Synthesis Randomization Conclusion Upper memory bound: even-parity SCTs s 0 A winning strategy λ 1 for initial 2 credit v 0 = (2 , 0) is � λ 1 ( ∗ s 1 s 3 ) = s 4 , ( − 1 , 1) (0 , 2) � λ 1 ( ∗ s 2 s 3 ) = s 5 , s 1 s 2 � λ 1 ( ∗ s 5 s 3 ) = s 5 . (0 , − 1) 3 1 (0 , 1) (0 , 0) s 3 2 (2 , 0) (1 , − 1) ( − 2 , 1) ( − 2 , 1) s 4 s 5 3 0 Strat. Synth. for Multi Quant. Obj. Chatterjee, Randour, Raskin 12 / 29
MEPGs & MMPPGs Mem. bounds Synthesis Randomization Conclusion Upper memory bound: even-parity SCTs s 0 A winning strategy λ 1 for initial 2 credit v 0 = (2 , 0) is � λ 1 ( ∗ s 1 s 3 ) = s 4 , ( − 1 , 1) (0 , 2) � λ 1 ( ∗ s 2 s 3 ) = s 5 , s 1 s 2 � λ 1 ( ∗ s 5 s 3 ) = s 5 . (0 , − 1) 3 1 Lemma: To win, P 1 must be able (0 , 1) (0 , 0) to enforce positive cycles of even parity. s 3 � Self-covering paths on VASS 2 (2 , 0) [Rac78, RY86]. (1 , − 1) ( − 2 , 1) ( − 2 , 1) � Self-covering trees (SCTs) on reachability games over VASS s 4 s 5 [BJK10]. 3 0 Strat. Synth. for Multi Quant. Obj. Chatterjee, Randour, Raskin 12 / 29
MEPGs & MMPPGs Mem. bounds Synthesis Randomization Conclusion Upper memory bound: even-parity SCTs s 0 � s 0 , (0 , 0) � 2 ( − 1 , 1) (0 , 2) � s 1 , ( − 1 , 1) � � s 2 , (0 , 2) � s 1 s 2 (0 , − 1) 3 1 � s 3 , ( − 1 , 2) � � s 3 , (0 , 2) � (0 , 1) (0 , 0) s 3 � s 4 , (0 , 1) � � s 5 , ( − 2 , 3) � 2 (2 , 0) (1 , − 1) ( − 2 , 1) ( − 2 , 1) � s 0 , (0 , 0) � � s 3 , (0 , 3) � s 4 s 5 3 0 Pebble moves ⇒ strategy. Strat. Synth. for Multi Quant. Obj. Chatterjee, Randour, Raskin 12 / 29
MEPGs & MMPPGs Mem. bounds Synthesis Randomization Conclusion Upper memory bound: even-parity SCTs T = ( Q , R ) is an epSCT for s 0 , � s 0 , (0 , 0) � Θ : Q �→ S × Z k is a labeling function. Root labeled � s 0 , (0 , . . . , 0) � . � s 1 , ( − 1 , 1) � � s 2 , (0 , 2) � Non-leaf nodes have � unique child if P 1 , � s 3 , ( − 1 , 2) � � s 3 , (0 , 2) � � all possible children if P 2 . Leafs have even-descendance � s 4 , (0 , 1) � � s 5 , ( − 2 , 3) � energy ancestors : ancestors with lower label and minimal priority even on the downward � s 0 , (0 , 0) � � s 3 , (0 , 3) � path. Pebble moves ⇒ strategy. Strat. Synth. for Multi Quant. Obj. Chatterjee, Randour, Raskin 12 / 29
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