The Python library continuoussets provides classes for the representation of continuous sets and the evaluation of set operations on them.

• Installation
• Set Representations
  • Intervals
  • Zonotopes
• Set Operations
• References

Installation

The library continuoussets requires Python 3.9 or higher. The only direct dependencies are numpy, scipy, and matplotlib. Installation via PyPI is recommended:

pip install continuoussets

Proficient users may choose their own installation path.

Set Representations

The implemented classes inherit from the abstract base class ConvexSet. They represent continuous sets of n-dimensional vectors.

Tip

Many operations, notably including the constructors, also support scalar types, such as int and float, as well as vectors defined using the type list. However, these are internally converted to numpy arrays, which may slow down the computation.

Important

The usage of keyword arguments for constructors is mandatory.

Intervals

An interval I is defined using a lower bound lb and an upper bound ub:

I = { x | lb <= x <= ub }

where lb and ub are vectors of equal length, and the inequality holds elementwise.

The Interval class allows to instantiate such objects:

I = Interval(lb = numpy.array([-2., 0.]), ub = numpy.array([2., 1.]))

Zonotopes

A zonotopes Z is defined using a center c and a generator matrix G:

Z = { c + sum_i G_i a_i | -1 <= a_i <= 1 }

where G_i are the columns of G. In contrast to intervals, zonotopes can represent dependencies between different dimensions.

The Zonotope class allows to instantiate such objects:

Z = Zonotope(c = numpy.array([1., 0.]), G = numpy.array([[1., 0., 2.], [-1., 1., 1.]]))

Set Operations

Many standard set operations are implemented:

• boundary_point: Computation of boundary points
• cartesian_product: Cartesian product
• compact: Reduction to the minimal set representation size without changing the set
• convex_hull: Convex hull
• matmul: Linear map
• minkowski_sum: Minkowski sum
• minkowski_difference: Minkowski/Pontryagin difference
• project: Projection of the set onto an axis-aligned subspace
• reduce: Reduction of the set representation size, possibly incurring enlargement
• support_function: Support function evaluation
• vertices: Vertex enumeration
• volume: Volume computation

Important

All operations return new class instances.

Note

Operands for binary operations can also be vectors, represented by 1D numpy arrays.

Tip

Many operations support various evaluation modes via the keyword argument mode. These detail whether an exact solution, an outer approximation or an inner approximation should be computed. Not all modes are supported for each operation, some operations cannot be evaluated exactly, and runtime may differ strongly between modes.

Furthermore, the following checks are supported:

• contains: Containment of one set or vector in another set
• intersects: Intersection between a set and another set or vector
• __eq__: Equality of a set and another set or vector
• represents: Equivalent representation of a set by another set representation

The Interval class additionally supports interval arithmetic for range bounding purposes. This includes

• basic operations: addition (__add__), subtraction (__sub__), multiplication (__mul__), division (__truediv__), exponentiation (__pow__)
• trigonometric functions: sin, cos, tan, arcsin, arccos, arctan
• other standard functions: sqrt, log, log10

References

This library is heavily inspired by the MATLAB toolbox CORA. However, this implementation is based on original sources, e.g.,

• G. Alefeld and G. Mayer. “Interval analysis: Theory and applications”. In: Computational and Applied Mathematics 121.1-2 (2000), pp. 421–464. doi: 10.1016/S0377-0427(00)00342-3
• M. Althoff. “Reachability analysis and its application to the safety assessment of autonomous cars”. Dissertation. Technische Universität München, 2010.