From Timed Automata to Stochastic Hybrid Games
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1 From Timed Automata to Stochastic Hybrid Games Model Checking, Performance Analysis, Optimization, Synthesis, and Machine Learning Kim G. Larsen Aalborg University, DENMARK
2 Topics Timed Automata Decidability (regions) Symbolic Verification (zones) Priced Timed Automata Decidability (priced regions) Symbolic Verification (priced zones) Stochastic Timed Automata Stochastic Semantics Statistical Model Checking Stochastic Hybrid Automata Timed Games & Interfaces Strategies, Symbolic Synthesis Refinement Stochastic Priced Timed Games Strategies Symbolic Synthesis (zones) Stochastic Strategies Reinforcement Learning Verification Optimization Synthesis Component Testing Performance Analysis Optimal Synthesis CLASSIC CORA TIGA ECDAR TRON SMC STRATEGO 1995 TU Graz, May 2017 Kim Larsen [2]
3 Priced Timed Automata Optimal Scheduling Kim G. Larsen Aalborg University DENMARK
4 Overview Timed Automata Scheduling Priced Timed Automata Optimal Reachability Optimal Infinite Scheduling Multi Objectives Verification Optimization Synthesis Component CLASSIC CORA TIGA ECDAR Energy Automata Testing Performance Analysis Optimal Synthesis TRON SMC STRATEGO TU Graz, May 2017 Kim Larsen [4]
5 Real Time Scheduling Only 1 Pass Cheat is possible (drive close to car with Pass ) UNSAFE 5 10 Pass SAFE CAN The THEY Car & MAKE Bridge IT Problem TO SAFE WITHIN 70 MINUTES??? TU Graz, May 2017 Kim Larsen [5]
6 Let us play! TU Graz, May 2017 Kim Larsen [6]
7 Real Time Scheduling RIO12/DAY3/ Solve Scheduling Problem using UPPAAL UNSAFE 5 10 SAFE TU Graz, May 2017 Kim Larsen [7]
8 Resources & Tasks Resource Synchronization Task Shared variable TU Graz, May 2017 Kim Larsen [8]
9 Task Graph Scheduling Example A 1 B C D 2 + * Compute : (D * ( C * ( A + B )) + (( A + B ) + ( C * D )) using 2 processors 4 C 3 * 4 + P1 (fast) + 2ps P2 (slow) + 5ps D 5 * 6 + * 3ps * 7ps P1 P TU Graz, May 2017 Kim Larsen [9] time
10 Task Graph Scheduling Example A 1 B C D 2 + * Compute : (D * ( C * ( A + B )) + (( A + B ) + ( C * D )) using 2 processors C 3 * 4 + P1 (fast) + 2ps P2 (slow) + 5ps D 5 * 6 + * 3ps * 7ps P1 P TU Graz, May 2017 Kim Larsen [10] time
11 Task Graph Scheduling Example A 1 B C D 2 + * Compute : (D * ( C * ( A + B )) + (( A + B ) + ( C * D )) using 2 processors C 3 * 4 + P1 (fast) + 2ps P2 (slow) + 5ps D 5 * 6 + * 3ps * 7ps P1 P TU Graz, May 2017 Kim Larsen [11] time
12 Task Graph Scheduling Example A 1 B C D 2 + * Compute : (D * ( C * ( A + B )) + (( A + B ) + ( C * D )) using 2 processors C D 3 5 * * P1 P P1 (fast) 3ps TU Graz, May 2017 RIO12/DAY3/taskgraph-AVACS2010 Kim Larsen [12] + * 2ps P2 (slow) time + * 5ps 7ps E<> (Task1.End and and Task6.End)
13 Experimental Results Symbolic A* Branch-&-Bound 60 sec Abdeddaïm, Kerbaa, Maler TU Graz, May 2017 Kim Larsen [13]
14 Priced Timed Automata
15 EXAMPLE: Optimal rescue plan for cars with different subscription rates for city driving! 5 Golf SAFE Citroen BMW Datsun 3 10 OPTIMAL PLAN HAS ACCUMULATED COST=195 and TOTAL TIME=65! TU Graz, May 2017 Kim Larsen [15]
16 Experiments COST-rates G C B D SCHEDULE COST TIME #Expl #Pop d Min Time CG> G< BD> C< CG> CG> G< BG> G< GD> GD> G< CG> G< BG> CG> G< BD> C< CG> CD> C< CB> C< CG> BD> B< CB> C< CG> time< TU Graz, May 2017 RIO12/DAY3/carcost Kim Larsen [16]
17 Task Graph Scheduling Revisited A 1 B C D 2 + * Compute : (D * ( C * ( A + B )) + (( A + B ) + ( C * D )) using 2 processors C 3 * 4 + P1 (fast) + 2ps P2 (slow) + 5ps D 5 * 6 + ENERGY: * Idle In use 3ps 1oW Idle * In use 7ps 20W W 20 30W 25 P1 P TU Graz, May 2017 Kim Larsen [17] time
18 Task Graph Scheduling Revisited A 1 B C D 2 + * Compute : (D * ( C * ( A + B )) + (( A + B ) + ( C * D )) using 2 processors C 3 * 4 + P1 (fast) + 2ps P2 (slow) + 5ps D 5 * 6 + ENERGY: * Idle In use 3ps 10W * Idle In use 7ps 20W W 20 30W 25 P1 P TU Graz, May 2017 RIO12/DAY3/taskgraph-AVACS2010-cost Kim Larsen [18] time
19 Task Graph Scheduling Revisited A 1 B C D 2 + * Compute : (D * ( C * ( A + B )) + (( A + B ) + ( C * D )) using 2 processors C 3 * 4 + P1 (fast) + 2ps P2 (slow) + 5ps D 5 * 6 + ENERGY: * Idle In use 3ps 10W * Idle In use 7ps 20W W 20 30W 25 P1 P TU Graz, May 2017 Kim Larsen [19] time
20 A simple example TU Graz, May 2017 Kim Larsen [20]
21 A simple example Q: What is cheapest cost for reaching? TU Graz, May 2017 Kim Larsen [21]
![for PTA THM [Bouyer, Brojaue, Briuere, Raskin 07] Optimal](/docs-images/93/112638955/images/22-1.jpg "reachability is PSPACE-complete for PTA 3 0 3 0 0 0 0 0 TU")
22 Corner Point Regions THM [Behrmann, Fehnker..01] [Alur,Torre,Pappas 01] Optimal reachability is decidable for PTA THM [Bouyer, Brojaue, Briuere, Raskin 07] Optimal reachability is PSPACE-complete for PTA TU Graz, May 2017 Kim Larsen [22]
23 Priced Zones [CAV01] A zone Z: 1 x 2 Æ 0 y 2 Æ x - y 0 A cost function C C(x,y)= 2 x - 1 y + 3 TU Graz, May 2017 Kim Larsen [23]
![Priced Zones Reset [CAV01] Z[x=0]: x=0 Æ 0 y 2 A zone Z: 1 x 2 Æ 0 y 2 Æ x - y 0 C = 1 y](/docs-images/93/112638955/images/24-1.jpg "+ 3 C= -1 y + 5 A cost function C C(x,y) = 2 x - 1 y + 3 TU Graz, May 2017 Kim Larsen")
24 Priced Zones Reset [CAV01] Z[x=0]: x=0 Æ 0 y 2 A zone Z: 1 x 2 Æ 0 y 2 Æ x - y 0 C = 1 y + 3 C= -1 y + 5 A cost function C C(x,y) = 2 x - 1 y + 3 TU Graz, May 2017 Kim Larsen [24]
25 Symbolic Branch & Bound Algorithm Z' Z Z is bigger & cheaper than Z is a well-quasi ordering which guarantees termination! TU Graz, May 2017 Kim Larsen [25]
26 Example: Aircraft Landing cost e*(t-t) E d+l*(t-t) T L t E earliest landing time T target time L latest time e cost rate for being early l cost rate for being late d fixed cost for being late Planes have to keep separation distance to avoid turbulences caused by preceding planes TU Graz, May 2017 Kim Larsen [26] Runway
27 Example: Aircraft Landing x <= 5 x >= 4 x=5 land! cost+=2 x <= 5 x <= 9 cost =3 cost =1 x=5 land! 4 earliest landing time 5 target time 9 latest time 3 cost rate for being early 1 cost rate for being late 2 fixed cost for being late Planes have to keep separation distance to avoid turbulences caused by preceding planes TU Graz, May 2017 RIO12/DAY3/aircraft Kim Larsen [27] Runway
28 Aircraft Landing Source of examples: Baesley et al 2000 TU Graz, May 2017 Kim Larsen [28]
29 Symbolic Branch & Bound Algorithm Zone based Linear Programming Problems (dualize) Min Cost Flow TU Graz, May 2017 Kim Larsen [29]
30 Zone LP Min Cost Flow Exploiting duality minimize 3x 1-2x 2 +7 when x 1 -x x 2 3 x 2 1 minimize 3y 2,0 -y 0,2 +y 1,2 y 0,1 when y 2,0 -y 0,1 -y 0,2 =1 y 0,2 +y 1,2 =2 y 0,1 -y 1,2 =-3 TU Graz, May 2017 Kim Larsen [30]
31 Zone LP Min Cost Flow Exploiting duality minimize 3x 1-2x 2 +7 when x 1 -x x 2 3 x 2 1 minimize 3y 2,0 -y 0,2 +y 1,2 y 0,1 when y 2,0 -y 0,1 -y 0,2 =1 y 0,2 +y 1,2 =2 y 0,1 -y 1,2 =-3 TU Graz, May 2017 Kim Larsen [31]
32 Aircraft Landing (revisited) Using MCF/Netsimplex [TACAS04] TU Graz, May 2017 Kim Larsen [32]
33 NOT IMPLEMENTED w ZONES TU Graz, May 2017 Kim Larsen [33]
34 Optimal Infinite Schedule TU Graz, May 2017 Kim Larsen [34]
35 Informationsteknologi EXAMPLE: Optimal WORK plan for cars with different subscription rates for city driving! 5 Golf Citroen maximal 100 min. at each location BMW Datsun 3 10 UCb
36 Informationsteknologi Workplan I Golf U Citroen U e(25) Golf U Citroen U e(25) Golf U Citroen U BMW U Datsun U BMW S Datsun S BMW U Datsun U e(25) 300 Golf S Citroen U e(20) Golf U Citroen U e(25) Golf U Citroen S BMW S Datsun U 300 BMW U Datsun U 300 BMW U Datsun S e(20) 300 e(25) 300 Golf U BMW U Citroen U Datsun U e(25) 300 Golf U BMW U Citroen S Datsun S Value of workplan: (4 x 300) / 90 = UCb
37 Informationsteknologi Workplan II Golf Citroen 25/125 Golf Citroen Golf Citroen 5/25 20/180 Golf Citroen BMW Datsun BMW Datsun BMW Datsun BMW Datsun 10/90 Golf Citroen 10/130 Golf Citroen 25/125 Golf Citroen 5/10 Golf Citroen BMW Datsun BMW Datsun BMW Datsun BMW Datsun 5/65 Golf BMW Citroen Datsun 25/225 Golf BMW Citroen Datsun Golf Citroen 10/90 10/0 BMW Datsun Golf BMW Citroen Datsun Value of workplan: 25/50 Golf Citroen 5/10 Golf 10/0 Citroen BMW Datsun BMW Datsun UCb 560 / 100 = 5.6
38 Optimal Infinite Scheduling Maximize throughput: i.e. maximize Reward / Time in the long run! TU Graz, May 2017 Kim Larsen [38]
39 Optimal Infinite Scheduling Minimize Energy Consumption: i.e. minimize Cost / Time in the long run TU Graz, May 2017 Kim Larsen [39]
40 Optimal Infinite Scheduling Maximize throughput: i.e. maximize Reward / Cost in the long run TU Graz, May 2017 Kim Larsen [40]
41 Mean Pay-Off Optimality Bouyer, Brinksma, Larsen: HSCC04,FMSD07 Accumulated cost s c 1 c 2 c 3 c n r 1 r 2 r 3 r n : BAD Accumulated reward Value of path s: val(s) = lim n!1 c n /r n Optimal Schedule s * : val(s * ) = inf s val(s) TU Graz, May 2017 Kim Larsen [41]
42 Discount Optimality < 1 : discounting factor Larsen, Fahrenberg: INFINITY 08 Cost of time t n s c(t 1 ) c(t 2) c(t 3) c(t n ) t 1 t 2 t 3 t n : BAD Time of step n Value of path s: val(s) = Optimal Schedule s * : val(s * ) = inf s val(s) TU Graz, May 2017 Kim Larsen [42]
43 Soundness of Corner Point Abstraction TU Graz, May 2017 Kim Larsen [43]
44 Optimal Conditional Reachability with Jacob I. Rasmussen
45 Informationsteknologi EXAMPLE: Optimal rescue plan for cars with different subscription rates for city driving! UNSAFE 5 Golf SAFE Citroen BMW 3 10 My CAR! Minimizes Cost MYCAR subject to Cost Citroen 60 Datsun Cost BMW 90 Cost Datsun 10 UCb min Cost MYCAR = 270 time = 70
46 Informationsteknologi UCb Optimal Conditional Reachability Dual-priced TA l l 1 2 x 2 l 3 d+=1 y 1 c = 1 d = 4 x 2 y:=0 c = 2 d = 1 x 3 y 2 y:=0 PROBLEM: Reach l 3 in a way which minimizes c subject to d 4 SOLUTION: c = 11/3 wait 1/3 in l 1 ; goto l 2 ; wait 5/3 in l 2 ; goto l 3
47 Informationsteknologi Discrete Trajectories l 1,0,0 0,1 l l 2 1 x 2 l3 y 1 d+=1 c = 1 d = 4 x 2 y:=0 c = 2 d = 1 x 3 y 2 l 2,0,0 l 2,1,0 l 2,2,0 y:=0 1,4 l 1,1,1 0,1 l 2,1,1 2,1 2,1 l 2,2,1 l 2,3,1 2,1 1,4 2,1 0,1 l 1,2,2 l 2,2,2 2,1 l 2,3,2 0,0 0,0 0,0 UCb 0,0
48 Informationsteknologi Discrete Trajectories l 1,0,0 0,1 l l 2 1 x 2 l3 y 1 d+=1 c = 1 d = 4 x 2 y:=0 c = 2 d = 1 x 3 y 2 l 2,0,0 l 2,1,0 l 2,2,0 y:=0 1,4 l 1,1,1 0,1 l 2,1,1 2,1 2,1 l 2,2,1 l 2,3,1 2,1 1,4 2,1 0,1 l 1,2,2 l 2,2,2 2,1 l 2,3,2 d 10 0,0 0,0 8 6 UCb 0,0 4 0, c
49 Informationsteknologi UCb Dual Priced Zones
50 Informationsteknologi UCb Dual Priced Zones
51 Informationsteknologi UCb Dual Priced Zones
52 Informationsteknologi UCb Reset
53 Informationsteknologi UCb Reset
54 Informationsteknologi UCb Reset
55 Informationsteknologi UCb Reset
56 Informationsteknologi UCb Exploration l l 2 1 x 2 l3 y 1 d+=1 c = 1 d = 4 y:=0 c = 2 d = 1 y:=0 x 2 x 3 y 2
57 Informationsteknologi UCb Exploration l l 2 1 x 2 l3 y 1 d+=1 c = 1 d = 4 y:=0 c = 2 d = 1 y:=0 x 2 x 3 y 2
58 Informationsteknologi UCb Exploration l l 2 1 x 2 l3 y 1 d+=1 c = 1 d = 4 y:=0 c = 2 d = 1 y:=0 x 2 x 3 y 2
59 Informationsteknologi UCb Termination
60 Multiple Objective Scheduling 16,10 P 2 P 1 2,3 2,3 cost 1 ==4 cost 2 ==3 cost 2 6,6 10,16 P 6 P 3 P 4 Pareto Frontier P 7 P 5 2,2 8,2 4W 3W cost 1 TU Graz, May 2017 Kim Larsen [60]
61 Applications
62 Experimental Results TU Graz, May 2017 Kim Larsen [62]
63 Experimental Results COP15 TU Graz, May 2017 Kim Larsen [63]
64 Informationsteknologi Informationsteknologi Informationsteknologi Informationsteknologi SIDMAR Production (2002) VHS UCb Steel Production Plant A. Fehnker, T. Hune, K. G. Larsen, P. Pettersson Case study of Esprit-LTR project VHS Physical plant of SIDMAR located in Gent, Belgium. Part between blast furnace and hot rolling mill. Objective: model the plant, obtain schedule and control program for plant. Crane A Machine 1 Machine 2 Machine 3 Machine 4 Machine 5 Crane B Lane 2 Buffer Storage Place Continuos Casting Machine Lane 1 UCb Steel Production Plant Input: sequence of steel loads ( pigs ). Load follows Recipe to obtain certain quality, e.g: start; T1@10; T2@20; T3@10; T2@10; end within 120. Crane A Machine 1 Machine 2 Machine 2 2 Machine 4 Machine 5 =127 Crane B Output: sequence of higher quality steel Controller Synthesis for LEGO 2 Lane 2 Buffer Storage Continuos Casting Machine Lane 1 A single load (part of) Crane B LEGO RCX Mindstorms. Local controllers with control programs. IR protocol for remote invocation of programs. Central controller. TU Graz, May 2017 UPPAAL Kim Larsen [64] UCb UCb crane a m1 m2 m3 m4 buffer storage casting 1971 lines of RCX code (n=5), (n=60). m5 crane b central controller Synthesis
![Behrmann, Brinksma, Hendriks, Mader 16 th IFAC World Congress UCb Results Extended Case TU Graz, May 2017 Kim Larsen [65] UCb #jobs work max. min.](/docs-images/93/112638955/images/65-2.jpg "cost heuristic orders hrs found in 60 s Orders of magnitude 29 - es, no, nl -faster than 530,771 29 Competitive - es, no, g MILP, GAMS/CPLEX - 647,410 with 29 avail.")
65 Informationsteknologi Informationsteknologi Informationsteknologi AXXOM Case Study (2005) ARTEMIS AXXOM Case study Laquer Production Scheduling Recipes UPPAAL template for metal UCb 3 types of recipes for uni/metallic/bronce use of resources, processing times, timing 29 (73, 219) orders: start time, due date, recipe extensions: delay cost, storage cost, setup cost weekend, nights Behrmann, Brinksma, Hendriks, Mader 16 th IFAC World Congress UCb Results Extended Case TU Graz, May 2017 Kim Larsen [65] UCb #jobs work max. min. cost heuristic orders hrs found in 60 s Orders of magnitude 29 - es, no, nl -faster than 530, Competitive - es, no, g MILP, GAMS/CPLEX - 647,410 with 29 avail. Orion-pies, no, nl - 1,714,875 results 29 avail. es, no, g - 2,263, expl. no 4 192,881,129
66 Océ Datapath, 2012 Tilt-Tray Sorters Scheduling rules & Optimization, 2008 ASML, 2004: Wafer Scanners Optimization of Throughput Philips: Indoor Lighting systems, 2014 TU Graz, May 2017 Kim Larsen [66]
67 TU Graz, May 2017 NANO Satellites (2015) NanoSatellites Nanosatellite: 1 20 kg Smaller Cheaper Faster Nano/Microsatellite Launch History and Projection (1-50 kg) But also capable! Projections based on announced and future plans of developers and programs indicate between 2,000 and 2,750 nano/microsatellites will require a launch from 2014 through 2020 w Morten Bisgaard, GOMSPACE Holger Hermanns, Gilles Nies, Marvin Stenger, Saarbrücken 600 Number of Satellites (1-50 kg) Full Market Potential SpaceWorks Projection Historical Launches GomSpace Satellites Calendar Year The Full Market Potential dataset is a combination of publically announced launch intentions, market research, and qualitative/quantitative assessments to account for future activities and programs. The SpaceWorks Projection dataset reflects SpaceWorks expert value judgment on the likely market outcome. SENSATION, Review Meeting, April 11, 2016 * Please see End Notes 1, 2, 4, 5, and U satellite -> 2013 Tracking airplanes Simple mission 3U satellite -> major payloads Complex mission SENSATION, Review Meeting, April 11,
68 ay 2017 GOMX-3 Deployment
69 Scheduling Approach Planning GOMX-3 Behaviour 69 TU Graz, May 2017
70 ay 2017 Derived Schedule
71 ay 2017
72 Energy Automata
73 Managing Resources TU Graz, May 2017 Kim Larsen [73]
74 Consuming & Harvesting Energy Maximize throughput while respecting: 0 E MAX TU Graz, May 2017 Kim Larsen [74]
75 Energy Constrains Energy is not only consumed but may also be regained The aim is to continously satisfy some energy constriants TU Graz, May 2017 Kim Larsen [75]
76 One Weight Results undecidable decidable (flat) decidable decidable decidable decidable undecidable undecidable undecidable undecidable PSPACE-c PSPACE-c P Bouyer, U Fahrenberg, K Larsen, N Markey,... Infinite runs in weighted timed automata with energy constraints P. Bouyer, U. Fahrenberg, K. G. Larsen, N. Markey: Timed automata with observers under energy constraints. HSCC 2010 P. Bouyer, K. G. Larsen, and N. Markey. Lower-bound constrained runs in weighted timed automata. QEST 2012 TU Graz, May 2017 Kim Larsen [76]
77 L-Problem for 1-Clock Case Theorem: The L-problem is decidable in PTIME for 1-clock PTAs P Bouyer, U Fahrenberg, K Larsen, N Markey,... Infinite runs in weighted timed automata with energy constraints TU Graz, May 2017 Kim Larsen [77]
78 1½ Clocks = Discrete Updates TU Graz, May 2017 Kim Larsen [80]
79 New Approach: Energy Functions Maximize energy along paths Use this information to solve general problem P. Bouyer, U. Fahrenberg, K. G. Larsen, N. Markey: Timed automata with observers under energy constraints. HSCC 2010 TU Graz, May 2017 Kim Larsen [81]
80 Energy Function General Strategy Spend just enough time to survive the next negative update P. Bouyer, U. Fahrenberg, K. G. Larsen, N. Markey: Timed automata with observers under energy constraints. HSCC 2010 TU Graz, May 2017 Kim Larsen [82]
81 Exponential PTA General Strategy Spend just enough time to survive the next negative update so that after next negative update there is a certain positive amount! Minimal Fixpoint: P. Bouyer, U. Fahrenberg, K. G. Larsen, N. Markey: Timed automata with observers under energy constraints. HSCC 2010 TU Graz, May 2017 Kim Larsen [83]
![Exponential PTA Thm [BFLM09]: The L-problem is decidable for linear and](/docs-images/93/112638955/images/82-3.jpg "exponential 1-clock PTAs with negative discrete updates.")
82 Exponential PTA Thm [BFLM09]: The L-problem is decidable for linear and exponential 1-clock PTAs with negative discrete updates. Energy Function P. Bouyer, U. Fahrenberg, K. G. Larsen, N. Markey: Timed automata with observers under energy constraints. HSCC 2010 TU Graz, May 2017 Kim Larsen [84]
83 Multiple Costs & Clocks LU Problem Undecidable 2 clocks + 2 costs [1] 1 clock + 2 costs [2] 2 clocks + 1 cost [3] Update Test-Decrement c 1 Increment n=3 Decrement n=12 (1) Karin Quaas. On the interval-bound problem for weighted timed automata (2) Uli Fahrenberg, Line Juhl, Kim G. Larsen, and Jirı Srba. Energy games in multiweighted automata (3) Nicolas Markey. Verification of Embedded Systems Algorithms and Complexity TU Graz, May 2017 Kim Larsen [85]
84 Multiple Clocks & 1 Cost L problem is undecidable 4 clocks or more x=x 0 C=C 0 x=x 0 C=x 0 +C 0 x=x 0 C=C 0 x=x 0 C=(1-x 0 ) +C 0 P. Bouyer, K. G. Larsen, and N. Markey. Lower-bound constrained runs in weighted timed automata. QEST 2012 TU Graz, May 2017 Kim Larsen [86]
85 Upper Bounds LU Problem Undecidable 2 clocks + 2 costs [1] 1 clock + 2 costs [2] 2 clocks + 1 cost [3] Update c 1 Increment n=3 Decrement n=12 (1) Karin Quaas. On the interval-bound problem for weighted timed automata (2) Uli Fahrenberg, Line Juhl, Kim G. Larsen, and Jirı Srba. Energy games in multiweighted automata (3) Nicolas Markey. Verification of Embedded Systems Algorithms and Complexity TU Graz, May 2017 Kim Larsen [87]
86 Multiple Costs & 0 Clocks Uli Fahrenberg, Line Juhl, Kim G. Larsen, and Jirı Srba. Energy games in multiweighted automata TU Graz, May 2017 Kim Larsen [88]
87 Multiple Costs & 0 Clocks Uli Fahrenberg, Line Juhl, Kim G. Larsen, and Jirı Srba. Energy games in multiweighted automata TU Graz, May 2017 Kim Larsen [89]
88 Conclusion Priced Timed Automata a uniform framework for modeling and solving dynamic ressource allocation problems! Not mentioned here: Model Checking Issues (ext. of CTL and LTL). Future work: Zone-based algorithm for optimal infinite runs. Approximate solutions for priced timed games to circumvent undecidablity issues. Open problems for Energy Automata. Approximate algorithms for optimal reachability TU Graz, May 2017 Kim Larsen [90]
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