Contents
1 Introduction 11
1.1 Reactive Systems . . . 11
1.2 Synthesis and Games . . . 12
1.3 Contributions . . . 15
2 Preliminaries 19 2.1 Notations and Conventions . . . 19
2.2 Languages, Automata, and Topology . . . 21
2.3 Quantitative PayoffFunctions . . . 24
2.4 Computational Complexity . . . 25
2.5 Recurrent Problems . . . 26
2.5.1 Quantified Boolean formulas . . . 26
2.5.2 Counter machines . . . 26
3 Quantitative Games 29 3.1 Games Played on Graphs . . . 29
3.1.1 Winning condition . . . 32
3.1.2 Values of a game . . . 33
3.1.3 Classical games . . . 35
3.2 Games Played on Automata . . . 40
3.3 Non-Zero-Sum Games . . . 43
I Regret 45
4 Background I: Non-Zero-Sum Solution Concepts 47 4.1 Regret Definition . . . 484.2 Examples . . . 49
4.3 Prefix Independization . . . 51
4.4 Contributions . . . 52
5 Minimizing Regret Against an Unrestricted Adversary 55 5.1 Additional Preliminaries for Regret . . . 55
5.2 Lower Bounds . . . 56
5.3 Upper Bounds for Prefix-Independent Functions . . . 58
5.4 Upper Bounds for Discounted Sum . . . 61
5.4.1 Deciding 0-regret . . . 62
5.4.2 Deciding r-regret . . . 65 7
8 CONTENTS
5.4.3 Simple regret-minimizing behaviors . . . 69
6 Minimizing Regret Against Positional Adversaries 73 6.1 Lower Bounds . . . 74
6.2 Upper Bounds for Prefix-Independent Functions . . . 80
6.3 Upper Bounds for Discounted Sum . . . 86
6.3.1 Deciding 0-regret . . . 87
6.3.2 Deciding r-regret . . . 90
7 Minimizing Regret Against Eloquent Adversaries 99 7.1 Lower Bounds . . . 102
7.2 Upper Bound for 0-Regret . . . 114
7.2.1 Existence of regret-free strategies . . . 114
7.2.2 Regular words suffice for Adam . . . 116
7.3 Upper Bounds for Prefix-Independent Functions . . . 120
7.4 Upper Bounds for Discounted Sum . . . 123
7.4.1 Deciding r-regret: determinizable cases . . . 123
7.4.2 TheÁ-gap promise problem . . . 124
II Partial Observability 127
8 Background II: Partial-Observation Games are Hard 129 8.1 Observable Determinacy . . . 1318.2 Contributions . . . 132
9 Partial-Observation Energy Games 135 9.1 The Energy Objective . . . 135
9.2 Undecidability of the Unknown Initial Credit Problem . . . 136
9.3 The Fixed Initial Credit Problem . . . 139
9.3.1 Upper bound . . . 139
9.3.2 Lower bound . . . 143
10 Partial-Observation Mean-PayoffGames 151 10.1 Undecidability of Mean-PayoffGames with Partial Observation . 154 10.2 Strategy Transfer from the Unfolding . . . 155
10.2.1 Strategy transfer for Eve . . . 159
10.2.2 Strategy transfer for Adam . . . 160
10.3 Decidable Classes of MPGs with Limited Observation . . . 162
10.3.1 Forcibly terminating games . . . 162
10.3.2 Forcibly first abstract cycle games . . . 169
10.3.3 First abstract cycle games . . . 173
10.4 Decidable Classes of MPGs with Partial Observation . . . 174
11 Partial-Observation Window Mean-PayoffGames 181 11.1 Window Mean-PayoffObjectives . . . 183
11.1.1 Relations among objectives . . . 184
11.1.2 Lower bounds . . . 185
11.2 DirFix games . . . 188
11.2.1 A symbolic algorithm forDirFix games . . . 192
11.3 Fixgames . . . 195
CONTENTS 9 11.4 UFixgames . . . 197
12 Conclusion and Future Work 201
12.1 Summary . . . 201 12.2 Conclusion . . . 202 12.3 Future Work . . . 202