# Computability Aspects of Differential Games in Euclidian Spaces

## TL;DR

This paper explores the computability limits of pursuit-evasion differential games in Euclidean spaces, demonstrating undecidability and constructing computable strategies, thus advancing understanding of algorithmic aspects in continuous game theory.

## Contribution

It establishes undecidability results for differential games and constructs computable strategies, including a non-deterministic winning strategy for the Man in pursuit-evasion scenarios.

## Key findings

- Proves basic undecidability of differential games.
- Constructs a computable non-deterministic winning strategy for Man.
- Raises open questions about deterministic computable winning strategies.

## Abstract

We study computability-theoretic aspects of differential games. Our focus is on pursuit and evasion games played in Euclidean spaces in the tradition of Rado's "Lion versus Man" game. In some ways, these games can be viewed as continuous versions of reachability games. We prove basic undecidability of differential games, and study natural classes of pursuit-evasion games in Euclidean spaces where the winners can win via computable strategies. The winning strategy for Man found by Besicovitch for the traditional "Lion versus Man" is not computable. We show how to modify it to yield a computable non-deterministic winning strategy, and raise the question whether Man can win in a computable and deterministic way.