Behaviour-Aware Expression Distance (BED)

This repository implements the distance metric from the paper Quantifying Behavioural Distance Between Mathematical Expressions. Additionally, this repository contains reproducible experiments from the paper.

Quickstart Instructions

1. Install rust (instructions at https://www.rust-lang.org/tools/install)
2. Create a new (conda) environment
3. Install dependencies with the command: pip install -r requirements.txt
4. Use the BED class as in the example below

from bed import BED

expressions = [["2", "A", "*", "B"], ["C", "*", "B", "-", "A"]]
bed = BED(expressions, [(1, 5), (1, 5)], (0.2, 5), infix=True, seed=0)
print(bed.calculate_distances())

The example above can be found in the BED/example.py script.

Hyperparameter explanation

BED uses the following hyperparameters and their default values:

Parameter	Description	Type	Default Value
expressions	Expressions between which you want to calculate the distance	list[list[str]]
x_bounds	Lower and upper bounds for each variable that occurs in the expressions (Each variable can have a different domain)	list[(float, float)]
const_bounds	Lower and upper bounds for constant values that will be sampled during the computation of the distance	(float, float)	(0.2, 5)
points_sampled	Number of variable values sampled during the computation of the distance	int	64
consts_sampled	Number of constant values sampled during the computation of the distance	int	16
expressions2	If not None, the distance between expressions and expressions2 is computed instead of the distances between each pair of expressions in the expressions parameter	list[list[str]]	None
x	Points on which the distance will be computed. If None, points are sampled instead.	numpy.array	None
randomized	If True, new points and constants are sampled for each calculation of the distance (computationally expensive). If parameter x is not None, this parameter will be ignored.	bool	False
infix	By default, the input expressions should be in the postfix notation. If you input expressions in the infix notation, this parameter must be set to True	bool	False
cutoff_threshold	Threshold for the maximum value an expression can evaluate to. Evaluations with the maximum absolute value higher than this threshold will be ignored	float	1e20
default_distance	Distance between two expressions at a point where one expression produces valid evaluations and another invalid ones. Example point -1 for expressions $x$ and $\sqrt{x}$	float	1e10
seed	Random seed for reproducible results	int	None

Smoothness of the error landscape

You can evaluate the smoothness of the error landscape using BED/error_landscape_smoothness.py. An overview of the approach for calculating this smoothness can be seen in the image below.