Computer Aided Synthesis: A Game-Theoretic Approach

Abstract

In this invited contribution, we propose a comprehensive introduction to game theory applied in computer aided synthesis. In this context, we give some classical results on two-player zero-sum games and then on multi-player non zero-sum games. The simple case of one-player games is strongly related to automata theory on infinite words. All along the article, we focus on general approaches to solve the studied problems, and we provide several illustrative examples as well as intuitions on the proofs.

Appendix

In this appendix, we give a sketch of proof for Muller games in Theorem 31. Recall that each player i has the objective $\varOmega _i = \{\rho \in Plays\mid inf(\rho ) \in \mathcal{F}_i \}$ with $\mathcal{F}_i \subseteq 2^V$, and that the values $val_i(v)$, $v \in V$, in each game $G_i$ can be computed in polynomial time (Theorem 21). To prove P membership for the constraint problem with bounds $(\mu _i)_{i \in \varPi }, (\nu _i)_{i \in \varPi }$, we apply the approach (3). Notice that for the required play $\rho \in Plays(v_0)$ in (3), the set $U = inf(\rho )$ must be a strongly connected component that is reachable from the initial vertex $v_0$. Moreover if for some i, $f_i(\rho ) = 0$ then ${val}_i(\rho _k) = 0$ for all $\rho _k \in V_i$, and if $\nu _i = 0$, then $f_i(\rho ) = 0$. We thus proceed as follows. (i) For each i such that $\nu _i = 1$, for each $U \in \mathcal{F}_i$ (seen as a potential $U = inf(\rho )$), the following computations are done in polynomial time for all $j\in \varPi$:

• if $\mu _j = 1$ ((3) imposes $f_j(\rho ) = 1$), test whether $U \in \mathcal{F}_j$,

• if $\nu _j = 0$ ((3) imposes $f_j(\rho ) = 0$), test whether $U \not \in \mathcal{F}_j$ and whether each $v \in U \cap V_j$ has value ${val}_j(v) = 0$,

• if $\mu _j = 0$ and $\nu _j = 1$ ((3) allows either $f_j(\rho ) = 0$ or $f_j(\rho ) = 1$), then if $U \not \in \mathcal{F}_j$, test whether each $v \in U \cap V_j$ has value ${val}_j(v) = 0$.

Finally, construct in polynomial time the game $G'$ from G such that each $V_j$ is limited to $\{v \in V_j \mid {val}_j(v) = 0 \}$ whenever $U \not \in \mathcal{F}_j$, and test whether U is a strongly connected component that is reachable from $v_0$ in $G'$. As soon as this sequence of tests is positive, there exists $\rho$ satisfying (3). (ii) It may happen that step (i) cannot be applied (because there is no j such that $\mu _j = 1$, and for j such that $\mu _j = 0$ and $\nu _j = 1$, there is no potential $U = inf(\rho )$). In this case, we construct in polynomial time a two-player game $G'$ from G such that each $V_i$ is limited to $\{v \in V_i \mid {val}_i(v) = 0 \}$, player 1 controls no vertex and player 2 is formed by the coalition of all $i \in \varPi$, and the objective is a Muller objective with $\mathcal{F} = \cup _{i\in \varPi } \mathcal{F}_i$. We then test in polynomial time whether player 1 has no winning strategy from $v_0$ in this Muller game.