Timed Automata with Observers under Energy Constraints

Transcription

1 Timed Automata with Observers under Energy Constraints Patricia Bouyer-Decitre Uli Fahrenberg Kim G. Larsen Nicolas Markey LSV, CNRS & ENS Cachan, France Aalborg Universitet, Danmark /9

2 Introduction The standard timed automaton model [AD9,AD94] Example done, y 5 safe problem, x:= alarm repair, x 5 y:= 5 x 6 delayed, y:= repairing repair y x 56 y:= failsafe safe 3 safe problem alarm 5.6 alarm delayed failsafe x y [AD9] Alur, Dill. Automata for modeling real-time systems (ICALP 9). [AD94] Alur, Dill. A theory of timed automata (Theoretical Computer Science). /9

3 Modelling resources in timed systems Introduction System resources might be relevant and even crucial information 3/9

4 Modelling resources in timed systems Introduction System resources might be relevant and even crucial information energy consumption, price to pay, memory usage, benefits, bandwidth, temperature,... 3/9

5 Modelling resources in timed systems Introduction System resources might be relevant and even crucial information energy consumption, price to pay, memory usage, benefits, bandwidth, temperature,... timed automata are not powerful enough! 3/9

6 Modelling resources in timed systems Introduction System resources might be relevant and even crucial information energy consumption, price to pay, memory usage, benefits, bandwidth, temperature,... timed automata are not powerful enough! A possible solution: use hybrid automata 3/9

7 Modelling resources in timed systems Introduction System resources might be relevant and even crucial information energy consumption, price to pay, memory usage, benefits, bandwidth, temperature,... timed automata are not powerful enough! A possible solution: use hybrid automata The thermostat example Off Ṫ =.5T (T 8) T 9 T On Ṫ =.5.5T (T ) 3/9

8 Modelling resources in timed systems Introduction System resources might be relevant and even crucial information energy consumption, price to pay, memory usage, benefits, bandwidth, temperature,... timed automata are not powerful enough! A possible solution: use hybrid automata The thermostat example Off Ṫ =.5T (T 8) T 9 T On Ṫ =.5.5T (T ) time 3/9

9 Modelling resources in timed systems Introduction System resources might be relevant and even crucial information energy consumption, price to pay, memory usage, benefits, bandwidth, temperature,... timed automata are not powerful enough! A possible solution: use hybrid automata Theorem [HKPV95] The reachability problem is undecidable in hybrid automata. [HKPV95] Henzinger, Kopke, Puri, Varaiya. What s decidable wbout hybrid automata? (SToC 95). 3/9

10 Modelling resources in timed systems Introduction System resources might be relevant and even crucial information energy consumption, price to pay, memory usage, benefits, bandwidth, temperature,... timed automata are not powerful enough! A possible solution: use hybrid automata Theorem [HKPV95] The reachability problem is undecidable in hybrid automata. An alternative: weighted/priced timed automata [ALP,BFH+] hybrid variables are observer variables (they do not constrain a priori the system) [ALP] Alur, La Torre, Pappas. Optimal paths in weighted timed automata (HSCC ). [BFH+] Behrmann, Fehnker, Hune, Larsen, Pettersson, Romijn, Vaandrager. Minimum-cost reachability in priced timed automata (HSCC ). 3/9

11 A simple example of priced timed automata (PTA) Introduction [ALP,BFH+] Example (with a linear observer) 3 l x:= +6 l x= 6 l Run (l, ) delay( 6 ) (l, 6 ) (l, 6 ) delay( ) (l, 3 ) (l, 3 )... [ALP] Alur, La Torre, Pappas. Optimal paths in weighted timed automata (HSCC ). [BFH+] Behrmann, Fehnker, Hune, Larsen, Pettersson, Romijn, Vaandrager. Minimum-cost reachability in priced timed automata (HSCC ). 4/9

12 A simple example of priced timed automata (PTA) Introduction [ALP,BFH+] Example (with a linear observer) 4 3 l x:= +6 l x= 6 l 3 Run (l, ) delay( 6 ) (l, 6 ) (l, 6 ) delay( ) (l, 3 ) (l, 3 )... [ALP] Alur, La Torre, Pappas. Optimal paths in weighted timed automata (HSCC ). [BFH+] Behrmann, Fehnker, Hune, Larsen, Pettersson, Romijn, Vaandrager. Minimum-cost reachability in priced timed automata (HSCC ). 4/9

13 Beyond linear observers... Introduction Bank A: rate: % Bank B: rate: 5% Bank C: rate: 6% 5/9

14 Beyond linear observers... Introduction Example We also consider PTA with an exponential observer: 3 l +6 l 6 l x:= x= 5/9

15 Beyond linear observers... Introduction Example We also consider PTA with an exponential observer: 3 l +6 l 6 l x:= x= Rate 3 in location l means cost = 3 cost time cost = cost e 3 t 5/9

16 Beyond linear observers... Introduction Example We also consider PTA with an exponential observer: 3 l x:= +6 l x= 6 l 4 3 Rate 3 in location l means cost = 3 cost time cost = cost e 3 t 5/9

17 Relevant questions Introduction Various optimization questions (optimal reachability, optimal mean-cost or discounted infinite schedules, etc) an abundant literature since (for the linear observers only) 6/9

18 Relevant questions Introduction Various optimization questions (optimal reachability, optimal mean-cost or discounted infinite schedules, etc) an abundant literature since (for the linear observers only) Scheduling under energy constraints (resource management): are there scheduling policies/strategies when energy is constrained? [BFLMS8] Bouyer, Fahrenberg, Larsen, Markey, Srba. Infinite runs in weighted timed automata with energy constraints (FORMATS 8). 6/9

19 Introduction Relevant questions Various optimization questions (optimal reachability, optimal mean-cost or discounted infinite schedules, etc) an abundant literature since (for the linear observers only) Scheduling under energy constraints (resource management): are there scheduling policies/strategies when energy is constrained? problem statement and preliminary results in [BFLMS8] (for the linear observers only) exist. problem univ. problem games exist. problem univ. problem games exist. problem univ. problem games L P P UP coup P-hard L P P? L??? L+W P P NP conp P-hard L+W P P? L+W??? L+U PSPACE NP-hard P EXPTIME-c. L+U?? undecidable L+U?? undecidable [BFLMS8] Bouyer, Fahrenberg, Larsen, Markey, Srba. Infinite runs in weighted timed automata with energy constraints (FORMATS 8). 6/9

20 Introduction Relevant questions Various optimization questions (optimal reachability, optimal mean-cost or discounted infinite schedules, etc) an abundant literature since (for the linear observers only) Scheduling under energy constraints (resource management): are there scheduling policies/strategies when energy is constrained? problem statement and preliminary results in [BFLMS8] (for the linear observers only) exist. problem univ. problem games exist. problem univ. problem games exist. problem univ. problem games L P P UP coup P-hard L P P? L??? L+W P P NP conp P-hard L+W P P? L+W??? L+U PSPACE NP-hard P EXPTIME-c. L+U?? undecidable L+U?? undecidable go further in the understanding (+ exponential observers): this work [BFLMS8] Bouyer, Fahrenberg, Larsen, Markey, Srba. Infinite runs in weighted timed automata with energy constraints (FORMATS 8). 6/9

21 Scheduling under energy constraints Introduction Example 3 l x:= +6 l x= 6 l 7/9

22 Scheduling under energy constraints Introduction Example 4 3 l x:= +6 l x= 6 l 3 lost! energy is 7/9

23 Scheduling under energy constraints Introduction Example 4 3 l x:= +6 l x= 6 l 3 energy is 7/9

24 Scheduling under energy constraints Introduction Example 4 3 l x:= +6 l x= 6 l 3 energy lies within [,3] 7/9

25 Scheduling under energy constraints Introduction Example 4 3 l x:= +6 l x= 6 l 3 energy lies within [,3] In this work, we focus on: the energy constraint energy is ; single-clock PTA; a linear or exponential observer variable. 7/9

26 Use the corner-point abstraction? Introduction Idea: delay in the most profitable location the corner-point abstraction 8/9

27 Use the corner-point abstraction? Introduction Idea: delay in the most profitable location the corner-point abstraction Example x:= x= 8/9

28 Use the corner-point abstraction? Introduction Idea: delay in the most profitable location the corner-point abstraction Example x:= x= x= e= x=.3 e=. x=.3 e=. x=.8 e=3. x=.8 e=3. x= e=.9 8/9

29 Use the corner-point abstraction? Introduction Idea: delay in the most profitable location the corner-point abstraction Example x:= x= x= e= x=.3 e=. x=.3 e=. x=.8 e=3. x=.8 e=3. x= e=.9 3 x= e= x=ε e x=ε e x= ε e 7 x= ε e 7 6 x= e 7 8/9

30 Use the corner-point abstraction? Introduction Idea: delay in the most profitable location the corner-point abstraction Example x:= x= {}, 3 (,), (,), {}, {}, (,), +6 (,), {}, {}, 6 (,), (,), {}, 8/9

31 Use the corner-point abstraction? Introduction Idea: delay in the most profitable location the corner-point abstraction Theorem [BFLMS8] The corner-point abstraction is sound and complete for single-clock PTA with a linear observer and with no discrete costs. [BFLMS8] Bouyer, Fahrenberg, Larsen, Markey, Srba. Infinite runs in weighted timed automata with energy constraints (FORMATS 8). 8/9

32 Use the corner-point abstraction? Introduction Idea: delay in the most profitable location the corner-point abstraction Remark The corner-point abstraction is not correct with discrete costs x=,x:= 8/9

33 Use the corner-point abstraction? Introduction Idea: delay in the most profitable location the corner-point abstraction Remark The corner-point abstraction is not correct with discrete costs x=,x:= + {}, + (,), (,), {}, {}, (,), +4 (,), {}, 8/9

34 Use the corner-point abstraction? Introduction Idea: delay in the most profitable location the corner-point abstraction Remark The corner-point abstraction is not correct with discrete costs x=,x:= lost! 8/9

35 Use the corner-point abstraction? Introduction Idea: delay in the most profitable location the corner-point abstraction Remark The corner-point abstraction is not correct with discrete costs x=,x:= lost! /9

36 Outline of the presentation Introduction. Introduction. Solving the problem (and even more) along a unit path Linear observer Exponential observer 3. Solving the general problem 4. Conclusion 9/9

37 Outline of the presentation Linear observer. Introduction. Solving the problem (and even more) along a unit path Linear observer Exponential observer 3. Solving the general problem 4. Conclusion /9

38 Annotated unit path Linear observer should take t.u. x= + + x= /9

39 Annotated unit path Linear observer should take t.u. x= + + x= starting with initial credit, it is not possible to reach the last location; /9

40 Annotated unit path Linear observer should take t.u. x= + + x= starting with initial credit, it is not possible to reach the last location; starting with credit, we can exit with credit 5; /9

41 Annotated unit path Linear observer should take t.u. x= + + x= starting with initial credit, it is not possible to reach the last location; starting with credit, we can exit with credit 5; starting with credit 3, it is possible to exit with final credit 3. /9

42 Annotated unit path Linear observer should take t.u. x= + + x= starting with initial credit, it is not possible to reach the last location; starting with credit, we can exit with credit 5; starting with credit 3, it is possible to exit with final credit 3. we will compute the energy function initial credit maximal final credit /9

43 Linear observer Simplifying annotated unit paths l r l p p r b b l 3 r3 If (for some reason) no time should elapse in l, then this path is equivalent (regarding the final cost) to p +p r r 3 max(b,b p ) /9

44 Simplifying annotated unit paths Linear observer l r l p p r b b If (for some reason) no time should elapse in l, then this path is equivalent (regarding the final cost) to p +p r r 3 max(b,b p ) l 3 r3 Example x= + + x= is equivalent to x= + + x= 8 5 /9

45 Simplifying annotated unit paths Linear observer l r l p p r b b If (for some reason) no time should elapse in l, then this path is equivalent (regarding the final cost) to p +p r r 3 max(b,b p ) l 3 r3 Example x= + + x= is equivalent to x= + + x= 8 5 we can select locations with increasing rates (if some rate is positive...) /9

46 Normal form for unit paths Linear observer An annotated unit path x= p p p p n p n x= r r r n b b b b n b n is in normal form if one of the following cases holds: n = (trivial normal form); the rates r i are positive and increasing, and b i + p i < b i for all i n (positive normal form); the rates r i are negative and decreasing, and b i + p i > b i for all i n (negative normal form). 3/9

47 Normal form for unit paths Linear observer Example x= x= is not in normal form. 4/9

48 Normal form for unit paths Linear observer Example x= x= is not in normal form. x= x= is in (positive) normal form. 4/9

49 Normal form for unit paths Linear observer Example x= x= is not in normal form. x= x= is in (positive) normal form. Lemma Any annotated unit path can be transformed into an equivalent (w.r.t. maximal final cost) normal form path. 4/9

50 Computing optimal delays Linear observer Example x= x= /9

51 Computing optimal delays Linear observer Example x= x= t opt : 3 compute optimal delays t opt in l to l n ; 5/9

52 Computing optimal delays Linear observer Example x= x= t opt : 3 t : compute optimal delays t opt in l to l n ; compute optimal possible delays t in l to l n ; 5/9

53 Computing optimal delays Linear observer Example x= x= t opt : 3 t : compute optimal delays t opt in l to l n ; compute optimal possible delays t in l to l n ; 5/9

54 Computing optimal delays Linear observer Example x= x= t opt : 3 t : compute optimal delays t opt in l to l n ; compute optimal possible delays t in l to l n ; 5/9

55 Computing optimal delays Linear observer Example x= x= t opt : 3 t : compute optimal delays t opt in l to l n ; compute optimal possible delays t in l to l n ; 5/9

56 Computing optimal delays Linear observer Example x= x= t opt : 3 t : minimal initial credit required:.5, yields final credit 8. compute optimal delays t opt in l to l n ; compute optimal possible delays t in l to l n ; 5/9

57 Computing optimal delays Linear observer Example x= x= t opt : 3 t : compute optimal delays t opt in l to l n ; compute optimal possible delays t in l to l n ; compute other points on the energy function curve. 5/9

58 Computing optimal delays Linear observer Example x= x= t opt : 3 t : initial credit.5 + δ compute optimal delays t opt in l to l n ; compute optimal possible delays t in l to l n ; compute other points on the energy function curve. 5/9

59 Computing optimal delays Linear observer Example x= x= t opt : 3 t : initial credit.5 + δ δ 3 compute optimal delays t opt in l to l n ; compute optimal possible delays t in l to l n ; compute other points on the energy function curve. 5/9

60 Computing optimal delays Linear observer Example x= x= t opt : 3 t : initial credit.5 + δ δ 3 compute optimal delays t opt in l to l n ; compute optimal possible delays t in l to l n ; compute other points on the energy function curve. 5/9

61 Computing optimal delays Linear observer Example x= x= t opt : 3 t : initial credit.5 + δ δ 3 δ 3 compute optimal delays t opt in l to l n ; compute optimal possible delays t in l to l n ; compute other points on the energy function curve. 5/9

62 Computing optimal delays Linear observer Example x= x= t opt : 3 t : initial credit.5 + δ δ 3 δ 3 final credit δ compute optimal delays t opt in l to l n ; compute optimal possible delays t in l to l n ; compute other points on the energy function curve. 5/9

63 Computing optimal delays Linear observer Example x= x= t opt : 3 t : initial credit final credit compute optimal delays t opt in l to l n ; compute optimal possible delays t in l to l n ; compute other points on the energy function curve. 5/9

64 Computing optimal delays Linear observer Example x= x= t opt : 3 t : initial credit + δ δ 6 + δ 6 final credit δ compute optimal delays t opt in l to l n ; compute optimal possible delays t in l to l n ; compute other points on the energy function curve. 5/9

65 Computing optimal delays Linear observer Example x= x= t opt : 3 t : initial credit 5 final credit 6 compute optimal delays t opt in l to l n ; compute optimal possible delays t in l to l n ; compute other points on the energy function curve. 5/9

66 Computing optimal delays Linear observer Example x= x= t opt : 3 t : initial credit 5 + δ final credit 6 + δ compute optimal delays t opt in l to l n ; compute optimal possible delays t in l to l n ; compute other points on the energy function curve. 5/9

67 Computing optimal delays Linear observer Example Original automaton: c= 3 {c} c= {c} 4 Normal-form automaton: c= 3 {c} c= {c} 4 w out 6 γ 4 β α point w in w out α / 8 β γ w in 6/9

68 Outline of the presentation Exponential observer. Introduction. Solving the problem (and even more) along a unit path Linear observer Exponential observer 3. Solving the general problem 4. Conclusion 7/9

69 Restricted unit path Exponential observer x= 3 4 x= /9

70 Restricted unit path Exponential observer x= 3 4 x= starting with initial credit, it is not possible to reach the final location; 8/9

71 Restricted unit path Exponential observer x= 3 4 x= starting with initial credit, it is not possible to reach the final location; starting with credit and spending t.u. in, we have credit exp() 7.39 when exiting final location;, which is not sufficient to reach the 8/9

72 Restricted unit path Exponential observer x= 3 4 x= starting with initial credit, it is not possible to reach the final location; starting with credit and spending t.u. in, we have credit exp() 7.39 when exiting final location; starting with credit,, which is not sufficient to reach the spending.8 t.u. in, we have credit exp(.8) 4.95; 8 we reach with credit around.95; spending the remaining. t.u. there, we exit the path with credit approx..7. 8/9

73 Normal form for exponential observers Exponential observer A restricted unit path x= p p p p n p n x= r r r n is in normal form if one of the following cases holds: n = (trivial normal form); the rates r i are positive and increasing, and p i r i r i r i r i < p i r i r i+ r i r i+ for all i n (positive normal form); 9/9

74 Normal form for exponential observers Exponential observer A restricted unit path x= p p p p n p n x= r r r n is in normal form if one of the following cases holds: Lemma n = (trivial normal form); the rates r i are positive and increasing, and p i r i r i r i r i < p i r i r i+ r i r i+ for all i n (positive normal form); Any restricted unit path can be transformed into an equivalent (w.r.t. maximal final credit) normal form path. 9/9

75 Normal form for exponential observers Intuition Exponential observer c in r i p i r i+ c out We have t opt i c out = (c in e r i t opt i t opt i+ + p i ) e r i+t opt i+ /9

76 Normal form for exponential observers Intuition Exponential observer c in r i p i r i+ c out We have t opt i +δ t opt i+ δ c out = (c in e r i (t opt i +δ) + p i ) e r i+(t opt i+ δ) /9

77 Normal form for exponential observers Intuition Exponential observer c in r i p i r i+ c out We have t opt i +δ t opt i+ δ c out = (c in e r i (t opt i +δ) + p i ) e r i+(t opt i+ δ) c out δ = r i c in e r i (t opt i +δ) e r i+(t opt i+ δ) (r i+ (c in e r i (t opt i +δ) + p i ) e r i+(t opt i+ δ) /9

78 Normal form for exponential observers Intuition Exponential observer c in r i p i r i+ c out We have t opt i +δ t opt i+ δ c out = (c in e r i (t opt i +δ) + p i ) e r i+(t opt i+ δ) c out δ = r i c in e r i t opt i = e r i+t opt i+ (ri+ (c in e r i t opt i + p i ) e r i+t opt i+ /9

79 Normal form for exponential observers Intuition Exponential observer c in r i p i r i+ c out We have t opt i +δ t opt i+ δ c out = (c in e r i (t opt i +δ) + p i ) e r i+(t opt i+ δ) Hence c in e r i t opt i = p i r i+ r i r i+ /9

80 Normal form for exponential observers Intuition Exponential observer c in r i p i r i+ c out We have t opt i +δ t opt i+ δ c out = (c in e r i (t opt i +δ) + p i ) e r i+(t opt i+ δ) Hence c in e r i t opt i = p i r i+ r i r i+ Lemma The optimal credit with which to exit l i is p i r i+ r i r i+. /9

81 Normal form for exponential observers Intuition Exponential observer c in r i p i r i+ c out We have t opt i +δ t opt i+ δ c out = (c in e r i (t opt i +δ) + p i ) e r i+(t opt i+ δ) Hence c in e r i t opt i = p i r i+ r i r i+ Lemma Optimal runs spend no time in l i if p i r i r i r i + p i p i r i+ r i r i+. /9

82 Computing optimal delays Exponential observer Example x= 3 4 x= /9

83 Computing optimal delays Exponential observer Example x= 3 4 x= c opt /9

84 Computing optimal delays Exponential observer Example x= 3 4 x= c opt t opt : /9

85 Computing optimal delays Exponential observer Example x= 3 4 x= c opt t opt : 8/3 5 ln( ) /9

86 Computing optimal delays Exponential observer Example x= 3 4 x= c opt t opt : ln( 5 c in ) 8/3 5 ln( ) /9

87 Computing optimal delays Exponential observer Example x= 3 4 x= c opt t opt : ln( 5 c in ) 8/3 5 ln( ) Lemma The optimal strategy is to delay t opt i as long as possible. /9

88 Computing optimal delays Exponential observer Example x= 3 4 x= c opt t opt : t min : ln( 5 c in ) 8/3 5 ln( ) 8 ln( 4 5/3 ) /9

89 Computing optimal delays Exponential observer Example x= 3 4 x= c opt t opt : t min : ln( 5 c in ) 8/3 5 ln( ) ln( 5 c in ) 8/3 5 ln( ) 8 ln( 4 5/3 ) /9

90 Computing optimal delays Exponential observer Example x= 3 4 x= c opt t opt : t min : ln( 5 c in ) 8/3 5 ln( ) ln( 5 c in ) 8/3 5 ln( ) 8 ln( 4 5/3 ) The minimal initial credit to reach the final location is c min = 5 e ( ) /4 5 ( ) / /9

91 Computing optimal delays Exponential observer Example x= 3 4 x= c opt t opt : t min : ln( 5 c in ) 8/3 5 ln( ) ln( 5 c in ) 8/3 5 ln( ) 8 ln( 4 5/3 ) Starting with credit k c min (between c min and 5): /9

92 Computing optimal delays Exponential observer Example x= 3 4 x= c opt t opt : t min : ln( 5 c in ) 8/3 5 ln( ) ln( 5 c in ) 8/3 5 ln( ) 8 ln( 4 5/3 ) Starting with credit k c min (between c min and 5): we spend ln( 5 k c min ) in location ; /9

93 Computing optimal delays Exponential observer Example x= 3 4 x= c opt t opt : t min : ln( 5 c in ) 8/3 5 ln( ) ln( 5 c in ) 8/3 5 ln( ) 8 ln( 4 5/3 ) Starting with credit k c min (between c min and 5): we spend ln( 5 k c min ) = t min ln(k) in location ; /9

94 Computing optimal delays Exponential observer Example x= 3 4 x= c opt t opt : t min : ln( 5 c in ) 8/3 5 ln( ) ln( 5 c in ) 8/3 5 ln( ) 8 ln( 4 5/3 ) Starting with credit k c min (between c min and 5): we spend ln( 5 k c min ) = t min ln(k) in location ; we transfer ln(k) t.u. to location 8 ; /9

95 Computing optimal delays Exponential observer Example x= 3 4 x= c opt t opt : t min : ln( 5 c in ) 8/3 5 ln( ) ln( 5 c in ) 8/3 5 ln( ) 8 ln( 4 5/3 ) Starting with credit k c min (between c min and 5): we spend ln( 5 k c min ) = t min ln(k) in location ; we transfer ln(k) t.u. to location 8 ; the final credit is 4 e 8 ln(k) 4. /9

96 Computing optimal delays Exponential observer Example x= 3 4 x= c opt t opt : t min : ln( 5 c in ) 8/3 5 ln( ) ln( 5 c in ) 8/3 5 ln( ) 8 ln( 4 5/3 ) Starting with credit k c min (between c min and 5): we spend ln( 5 k c min ) = t min ln(k) in location ; we transfer ln(k) t.u. to location 8 ; the final credit is 4 e 8 ln(k) 4 = 4 ( cin c min ) 4 4. /9

97 Computing optimal delays Exponential observer Theorem Given a restricted unit path and an initial credit, we can compute in polynomial time the optimal final credit (in closed form). /9

98 Computing optimal delays Exponential observer Theorem Given a restricted unit path and an initial credit, we can compute in polynomial time the optimal final credit (in closed form). Moreover the energy function: is piecewise of the form α (c in β) r i /r j + γ, with r i r j ; has continuous derivative. /9

99 Computing optimal delays ginal automaton: Exponential β observer l-form automaton: {c} Example c= c= {c} wout c= {c} c= {c} α win point win wout γ α e 5 (/5) /4 (4/3) / β 5 γ 7/3 e 8 5/3 (3/4) 8/ e 8 5/ β interval equation of the curve α β wout = 5 ( ) 4 3 win 4 e 5 (4/3) /5 β γ wout = 5 ( ) 8/5 3 win 3 e 5 4 8/3 γ + wout = (win 4) e 8 4 α win point win wout α e 5 (/5) /4 (4/3) / e 8 5/3 (3/4) 8/5 4 3/9

100 Outline of the presentation Exponential observer. Introduction. Solving the problem (and even more) along a unit path Linear observer Exponential observer 3. Solving the general problem 4. Conclusion 4/9

101 Basic simplifications Solving the general problem Remark For the sake of simplicity, we restrict to closed timed automata. 5/9

102 Basic simplifications Solving the general problem Lemma We can assume that there is a global invariant x, and that there are only three kind of transitions: x or x= or x= x:= 5/9

103 Basic simplifications Solving the general problem Lemma We can assume that there is a global invariant x, and that there are only three kind of transitions: x or x= or x= x:= Example x x l y y y= y= y= y= l, l, l, l,3 y:= y:= y:= y y y= y:= 5/9

104 Basic simplifications Solving the general problem Lemma We can assume that there is a global invariant x, and that there are only three kind of transitions: x or x= or x= x:= Example x p i x x= p i r r r r x:= x:= 5/9

105 Handling non-resetting cycles Solving the general problem Lemma For each location l, we can compute a value w Zeno (l) such that there is an infinite non-resetting feasible run from l with initial credit w iff w w Zeno (l). 6/9

106 Handling non-resetting cycles Solving the general problem Lemma For each location l, we can compute a value w Zeno (l) such that there is an infinite non-resetting feasible run from l with initial credit w iff w w Zeno (l). Lemma From an automaton A, we can compute an equivalent automaton A labelled with w Zeno and not containing any non-resetting cycle. 6/9

107 Main result Solving the general problem Theorem Optimization, reachability and existence of infinite runs satisfying the constraint can be decided in EXPTIME in single-clock PTA either with a linear observer; or with an exponential observer with non-positive discrete costs. 7/9

108 Main result Solving the general problem Theorem Optimization, reachability and existence of infinite runs satisfying the constraint can be decided in EXPTIME in single-clock PTA either with a linear observer; or with an exponential observer with non-positive discrete costs. transform the automaton into an automaton with energy functions; x:= x:= x:= x:= 7/9

109 Main result Solving the general problem Theorem Optimization, reachability and existence of infinite runs satisfying the constraint can be decided in EXPTIME in single-clock PTA either with a linear observer; or with an exponential observer with non-positive discrete costs. transform the automaton into an automaton with energy functions; x:= f x:= x:= x:= 7/9

110 Main result Solving the general problem Theorem Optimization, reachability and existence of infinite runs satisfying the constraint can be decided in EXPTIME in single-clock PTA either with a linear observer; or with an exponential observer with non-positive discrete costs. transform the automaton into an automaton with energy functions; f x:= f x:= x:= x:= 7/9

111 Main result Solving the general problem Theorem Optimization, reachability and existence of infinite runs satisfying the constraint can be decided in EXPTIME in single-clock PTA either with a linear observer; or with an exponential observer with non-positive discrete costs. transform the automaton into an automaton with energy functions; f x:= f x:= x:= x:= 7/9

112 Main result Solving the general problem Theorem Optimization, reachability and existence of infinite runs satisfying the constraint can be decided in EXPTIME in single-clock PTA either with a linear observer; or with an exponential observer with non-positive discrete costs. transform the automaton into an automaton with energy functions; f x:= f x:= x:= x:= check if simple cycles can be iterated, or if a Zeno cycle can be reached (use of w Zeno ). 7/9

113 Outline of the presentation Conclusion. Introduction. Solving the problem (and even more) along a unit path Linear observer Exponential observer 3. Solving the general problem 4. Conclusion 8/9

114 Conclusion & future work Conclusion Results: computation of (infinite) schedules satisfying some simple energy constraints (energy should remain ) surprisingly decidable for exponential observers? required an optimization algorithm along unit paths 9/9

115 Conclusion & future work Conclusion Results: computation of (infinite) schedules satisfying some simple energy constraints (energy should remain ) surprisingly decidable for exponential observers? required an optimization algorithm along unit paths Many open problems: general case for exponential observers? can we go beyond linear and exponential observers? (In particular can we handle observers that mix linear and exponential evolutions?) what if there are upper bounds on the observer variable? what if there are more than one clock? what if there is an interacting environment (games)?... 9/9