Given a policy of a Markov Decision Process, we define a SAFEZONE as a subset of states, such that most of the policy's trajectories are confined to this subset. The quality of a SAFEZONE is parameterized by the number of states and the escape probability, i.e., the probability that a random trajectory will leave the subset. SAFEZONES are especially interesting when they have a small number of states and low escape probability. We study the complexity of finding optimal SAFEZONES, and show that in general, the problem is computationally hard. Our main result is a bi-criteria approximation learning algorithm with a factor of almost 2 approximation for both the escape probability and SAFEZONE size, using a polynomial size sample complexity.

Given a policy of a Markov Decision Process, we define a SafeZone as a subset of states, such that most of the policy's trajectories are confined to this subset. The quality of a SafeZone is parameterized by the number of states and the escape probability, i.e., the probability that a random trajectory will leave the subset. SafeZones are especially interesting when they have a small number of states and low escape probability. We study the complexity of finding optimal SafeZones, and show that in general, the problem is computationally hard. For this reason, we concentrate on finding approximate SafeZones. Our main result is a bi-criteria approximation learning algorithm with a factor of almost $2$ approximation for both the escape probability and \newprob size, using a polynomial size sample complexity.

Reactive synthesis is classically modeled as a game, though often applied to domains such as arbiter circuits and communication protocols [1]. We aim to show how reactive synthesis can be applied to a literal game - cops and robbers - to generate strategies for agents in the game. We propose a game that requires the coordination of multiple agents in a space of datatypes and operations that are richer than is easily captured by the traditional Linear Temporal Logic (LTL) approach of synthesis over Boolean streams [2]. In particular, we draw inspiration from prior work on Coordination Synthesis [3], LTL moduluo theories (LTLt) [4], and Temporal Stream Logic Moduluo theories (TSL-MT) [5, 6] to describe our problem and potential solution spaces. The traditional game [7] asks whether K cops can catch a single robber on a graph. In a temporal logic setting, this amounts to a safety condition on the robbers (they are never caught by the cops), and the dual liveness condition for the cops (they eventually catch the robbers). We modify the traditional graph theory focused version of the game to have a more visual game on a grid system, allowing for various configurations, including: An environment with various node types such as walls, safe zones, and open spaces.

Given a policy of a Markov Decision Process, we define a SafeZone as a subset of states, such that most of the policy's trajectories are confined to this subset. The quality of a SafeZone is parameterized by the number of states and the escape probability, i.e., the probability that a random trajectory will leave the subset. SafeZones are especially interesting when they have a small number of states and low escape probability. We study the complexity of finding optimal SafeZones, and show that in general, the problem is computationally hard. For this reason, we concentrate on finding approximate SafeZones.