# The Computational Complexity of Angry Birds

## TL;DR

This paper rigorously analyzes the computational complexity of Angry Birds, proving it to be NP-hard, PSPACE-hard, PSPACE-complete, or EXPTIME-hard, highlighting the game's inherent difficulty for AI agents.

## Contribution

It provides the first proofs that Angry Birds levels are computationally hard in multiple complexity classes, including the novel result of EXPTIME-hardness for a single-player game.

## Key findings

- NP-hardness proven via reduction from 3-SAT
- PSPACE-hardness proven via reduction from TQBF
- First demonstration of EXPTIME-hardness in a single-player game

## Abstract

The physics-based simulation game Angry Birds has been heavily researched by the AI community over the past five years, and has been the subject of a popular AI competition that is currently held annually as part of a leading AI conference. Developing intelligent agents that can play this game effectively has been an incredibly complex and challenging problem for traditional AI techniques to solve, even though the game is simple enough that any human player could learn and master it within a short time. In this paper we analyse how hard the problem really is, presenting several proofs for the computational complexity of Angry Birds. By using a combination of several gadgets within this game's environment, we are able to demonstrate that the decision problem of solving general levels for different versions of Angry Birds is either NP-hard, PSPACE-hard, PSPACE-complete or EXPTIME-hard. Proof of NP-hardness is by reduction from 3-SAT, whilst proof of PSPACE-hardness is by reduction from True Quantified Boolean Formula (TQBF). Proof of EXPTIME-hardness is by reduction from G2, a known EXPTIME-complete problem similar to that used for many previous games such as Chess, Go and Checkers. To the best of our knowledge, this is the first time that a single-player game has been proven EXPTIME-hard. This is achieved by using stochastic game engine dynamics to effectively model the real world, or in our case the physics simulator, as the opponent against which we are playing. These proofs can also be extended to other physics-based games with similar mechanics.