Approximations

Module Approximations.jl – polygonal approximation of sets.

decompose(S::LazySet{N},
          partition::AbstractVector{<:AbstractVector{Int}},
          block_options) where {N}

Decompose a high-dimensional set into a Cartesian product of overapproximations of the projections over the specified subspaces.

Input

• S – set
• partition – vector of blocks (i.e., of vectors of integers) (see the Notes below)
• block_options – mapping from block indices in partition to a corresponding overapproximation option; we only require access via [⋅] (but see also the Notes below)

Output

A CartesianProductArray containing the low-dimensional approximated projections.

Algorithm

For each block a specific project method is called, dispatching on the corresponding overapproximation option.

Notes

The argument partition deserves some explanation. Typically, the list of blocks should form a partition of the set $\{1, \dots, n\}$ represented as a list of consecutive blocks, where $n$ is the ambient dimension of set S.

However, technically there is no problem if the blocks are not consecutive, blocks are missing, blocks occur more than once, or blocks are overlapping. The resulting set must be interpreted with care in such cases (e.g., it will not necessarily be an overapproximation of S).

For convenience, the argument block_options can also be given as a single option instead of a mapping, which is then interpreted as the option for all blocks.

Examples

The argument block_options supports different options: one can specify the target set, the degree of accuracy, and template directions. These options are exemplified below, where we use the following example.

julia> S = Ball2(zeros(4), 1.0);  # set to be decomposed (4D 2-norm unit ball)

julia> P2d = [1:2, 3:4];  # a partition with two blocks, each of size two

julia> P1d = [[1], [2], [3], [4]];  # a partition with four blocks, each of size one

Different set types

We can decompose using polygons in constraint representation:

julia> Y = decompose(S, P2d, HPolygon);

julia> all(ai isa HPolygon for ai in array(Y))
true

For decomposition into 1D subspaces, we can use Interval:

julia> Y = decompose(S, P1d, Interval);

julia> all(ai isa Interval for ai in array(Y))
true

However, if you need to specify different set types for different blocks, the interface presented so far does not apply. See the paragraph Advanced input for different block approximations below for how to do that.

Refining the decomposition: $ε$-close approximation

The $ε$ option can be used to refine a decomposition, i.e., obtain a more accurate result. We use the Iterative refinement algorithm from the Approximations module.

To illustrate this, consider again the set S from above. We decompose into two 2D polygons. Using smaller $ε$ implies a better precision, thus more constraints in each 2D decomposition. In the following example, we look at the number of constraints in the first block.

julia> d(ε, bi) = array(decompose(S, P2d, (HPolygon => ε)))[bi]
d (generic function with 1 method)

julia> [length(constraints_list(d(ε, 1))) for ε in [Inf, 0.1, 0.01]]
3-element Vector{Int64}:
  4
  8
 32

Refining the decomposition: template polyhedra

Another way to refine a decomposition is by using template polyhedra. The idea is to specify a set of template directions and then compute on each block the polytopic overapproximation obtained by evaluating the support function of the given input set over the template directions.

For example, octagonal 2D approximations of the set S are obtained with:

julia> B = decompose(S, P2d, OctDirections);

julia> length(B.array) == 2 && all(dim(bi) == 2 for bi in B.array)
true

See Template directions for the available template directions. Note that, in contrast to the polygonal $ε$-close approximation from above, this method can be applied to blocks of any size.

julia> B = decompose(S, [1:4], OctDirections);

julia> length(B.array) == 1 && dim(B.array[1]) == 4
true

Advanced input for different block approximations

Instead of defining the approximation option uniformly for each block, we can define different approximations for different blocks. For this purpose, the argument block_options can also be a mapping from block index (in the partition) to the corresponding approximation option.

For example, we can approximate the first block with a Hyperrectangle and the second block with $ε$-close approximation for $ε = 0.1$:

julia> res = array(decompose(S, P2d, Dict(1 => Hyperrectangle, 2 => 0.1)));

julia> typeof(res[1]), typeof(res[2])
(Hyperrectangle{Float64, Vector{Float64}, Vector{Float64}}, HPolygon{Float64, Vector{Float64}})

decompose(S::LazySet, block_options; [block_size]::Int=1)

Decompose a high-dimensional set into a Cartesian product of overapproximations of the projections over uniformly-sized subspaces.

Input

• S – set
• block_options – overapproximation option or mapping from block indices to a corresponding overapproximation option
• block_size – (optional; default: 1) size of the blocks

Output

A CartesianProductArray containing the low-dimensional approximated projections.

overapproximate(X::S, ::Type{S}, args...) where {S<:LazySet}

Overapproximating a set of type S with type S is a no-op.

Input

• X – set
• Type{S} – target set type
• args – further arguments (ignored)
• kwargs – further keyword arguments (ignored)

Output

The input set.

overapproximate(S::LazySet)

Alias for overapproximate(S, Hyperrectangle) resp. box_approximation(S).

overapproximate(S::LazySet, ::Type{<:Hyperrectangle})

Alias for box_approximation(S).

overapproximate(S::LazySet, ::Type{<:BallInf})

Alias for ballinf_approximation(S).

overapproximate(S::LazySet{N},
                ::Type{<:HPolygon},
                [ε]::Real=Inf) where {N}

Overapproximate a given 2D set using iterative refinement.

Input

• S – two-dimensional bounded set
• HPolygon – target set type
• ε – (optional, default: Inf) error tolerance
• prune – (optional, default: true) flag for removing redundant constraints in the end

Output

A polygon in constraint representation.

Notes

The result is always a convex overapproximation of the input set.

If no error tolerance ε is given, or is Inf, the result is a box-shaped polygon. For convex input sets, the result is an ε-close polygonal overapproximation with respect to the Hausdorff distance.

overapproximate(S::LazySet, ε::Real)

Alias for overapproximate(S, HPolygon, ε).

overapproximate(X::LazySet{N}, dirs::AbstractDirections;
                [prune]::Bool=true) where {N}

Overapproximate a (possibly unbounded) set with template directions.

Input

• X – set
• dirs – directions
• prune – (optional, default: true) flag for removing redundant constraints

Output

A polyhedron overapproximating the set X with the directions from dirs. The overapproximation is computed using the support function. The result is an HPolytope if it is bounded and otherwise an HPolyhedron.

overapproximate(X::LazySet{N}, dirs::Type{<:AbstractDirections}) where {N}

Overapproximate a set with template directions.

Input

• X – set
• dirs – type of direction representation

Output

A polyhedron overapproximating the set X with the directions from dirs. The result is an HPolytope if it is bounded and otherwise an HPolyhedron.

overapproximate(cap::Intersection{N, <:LazySet, <:AbstractPolyhedron},
                dirs::AbstractDirections;
                kwargs...
               ) where {N}

Overapproximate the intersection between a set and a polyhedron given a set of template directions.

Input

• cap – intersection of a set and a polyhedron
• dirs – template directions
• kwargs – additional arguments that are passed to the support function algorithm

Output

A polytope or polyhedron in H-representation such that the normal direction of each half-space is given by an element of dirs.

Algorithm

Let di be a direction drawn from the set of template directions dirs. Let X be the set and let P be the polyhedron. We overapproximate the set X ∩ H with a polytope or polyhedron in constraint representation using a given set of template directions dirs.

The idea is to solve the univariate optimization problem ρ(di, X ∩ Hi) for each half-space of the set P and then take the minimum. This gives an overapproximation of the exact support function.

This algorithm is inspired from [1].

Notes

This method relies on having available the constraints_list of the polyhedron P.

This method may return a non-empty set even if the original set is empty.

[1] G. Frehse, R. Ray. Flowpipe-Guard Intersection for Reachability Computations with Support Functions. ADHS 2012.

overapproximate(cap::Intersection{N, <:HalfSpace, <:AbstractPolytope},
                dirs::AbstractDirections;
                [kwargs]...
               ) where {N}

Overapproximate the intersection between a half-space and a polytope given a set of template directions.

Input

• cap – intersection of a half-space and a polytope
• dirs – template directions
• kwargs – additional arguments that are passed to the support function algorithm

Output

A polytope in H-representation such that the normal direction of each half-space is given by an element of dirs.

overapproximate(Z::AbstractZonotope, ::Type{<:HParallelotope},
                [indices]=1:dim(Z))

Overapproximate a zonotopic set with a parallelotope in constraint representation.

Input

• Z – zonotopic set
• HParallelotope – target set type
• indices – (optional; default: 1:dim(Z)) generator indices selected when constructing the parallelotope

Output

An overapproximation of the given zonotopic set using a parallelotope.

Algorithm

The algorithm is based on Proposition 8 discussed in Section 5 of [1].

[1] Althoff, M., Stursberg, O., & Buss, M. (2010). Computing reachable sets of hybrid systems using a combination of zonotopes and polytopes. Nonlinear analysis: hybrid systems, 4(2), 233-249.

overapproximate(X::Intersection{N, <:AbstractZonotope, <:Hyperplane},
                dirs::AbstractDirections) where {N}

Overapproximate the intersection between a zonotopic set and a hyperplane with a polyhedron or polytope using the given directions.

Input

• X – intersection between a zonotopic set and a hyperplane
• dirs – type of direction representation

Output

An overapproximation of the intersection between a zonotopic set and a hyperplane. If the directions are bounding, the result is an HPolytope, otherwise the result is an HPolyhedron.

Algorithm

This function implements [Algorithm 8.1, 1].

[1] Colas Le Guernic. Reachability Analysis of Hybrid Systems with Linear continuous dynamics (Doctoral dissertation). 2009.

overapproximate(QM::QuadraticMap{N, <:SparsePolynomialZonotope},
                ::Type{<:SparsePolynomialZonotope}) where {N}

Overapproximate a quadratic map of a sparse polynomial zonotope with a sparse polynomial zonotope.

Input

• QM – quadratic map of a sparse polynomial zonotope
• SparsePolynomialZonotope – target type

Output

A sparse polynomial zonotope overapproximating the quadratic map of a sparse polynomial zonotope.

Algorithm

This method implements Proposition 13 of [1].

[1] N. Kochdumper and M. Althoff. Sparse Polynomial Zonotopes: A Novel Set Representation for Reachability Analysis. Transactions on Automatic Control

overapproximate(S::LazySet, ::Type{<:Interval})

Return the overapproximation of a set with an interval.

Input

• S – one-dimensional set
• Interval – target type

Output

An interval.

Algorithm

We use the extrema function.

overapproximate(cap::Intersection, ::Type{<:Interval})

Return the overapproximation of a lazy intersection with an interval.

Input

• cap – one-dimensional intersection
• Interval – target type

Output

An interval.

Algorithm

The algorithm recursively overapproximates the two intersected sets with intervals and then intersects these. (Note that this fails if the sets are unbounded.)

For convex sets this algorithm is precise.

overapproximate(cap::IntersectionArray, ::Type{<:Interval})

Return the overapproximation of an intersection array with an interval.

Input

• cap – one-dimensional intersection array
• Interval – target type

Output

An interval.

Algorithm

The algorithm recursively overapproximates the intersected sets with intervals and then intersects these. (Note that this fails if the sets are unbounded.)

For convex sets this algorithm is precise.

overapproximate(Z::Zonotope, ::Type{<:Zonotope}, r::Union{Integer, Rational})

Reduce the order of a zonotope.

Input

• Z – zonotope
• Zonotope – target set type
• r – desired order

Output

A new zonotope with r generators, if possible.

Algorithm

This method falls back to reduce_order with the default algorithm.

overapproximate(X::ConvexHull{N, <:AbstractZonotope, <:AbstractZonotope},
                ::Type{<:Zonotope}) where {N}

Overapproximate the convex hull of two zonotopic sets.

Input

• X – convex hull of two zonotopic sets
• Zonotope – target set type
• algorithm – (optional; default: "mean") choice of algorithm; possible values are "mean" and "join"

Output

A zonotope $Z$ such that $X ⊆ Z$.

Algorithm

The algorithm can be controlled by the parameter algorithm. Note that the results of the two implemented algorithms are generally incomparable.

'mean' method

If algorithm == "mean", we choose the method proposed in [1]. The convex hull of two zonotopic sets $Z₁$ and $Z₂$ of the same order, which we write

\[Z_j = ⟨c^{(j)}, g^{(j)}_1, …, g^{(j)}_p⟩\]

for $j = 1, 2$, can be overapproximated as follows:

\[CH(Z_1, Z_2) ⊆ \frac{1}{2}⟨c^{(1)}+c^{(2)}, g^{(1)}_1+g^{(2)}_1, …, g^{(1)}_p+g^{(2)}_p, c^{(1)}-c^{(2)}, g^{(1)}_1-g^{(2)}_1, …, g^{(1)}_p-g^{(2)}_p⟩.\]

If the zonotope order is not the same, this algorithm calls reduce_order to reduce the order to the minimum of the arguments.

It should be noted that the output zonotope is not necessarily the minimal enclosing zonotope, which is in general expensive to obtain in high dimensions. This is further investigated in [2].

'join' method

If algorithm == "join", we choose the method proposed in [3, Definition 1]. The convex hull $X$ of two zonotopic sets $Z₁$ and $Z₂$ is overapproximated by a zonotope $Z₃$ such that the box approximation of $X$ is identical with the box approximation of $Z₃$. Let $□(X)$ denote the box approximation of $X$. The center of $Z₃$ is the center of $□(X)$.

The generator construction consists of two phases. In the first phase, we construct generators $g$ as a combination of one generator from $Z₁$, say, $g₁$, with another generator from $Z₂$, say, $g₂$. The entry of $g$ in the $i$-th dimension is given as

\[ g[i] = \arg\min_{\min(g₁[i], g₂[i]) ≤ x ≤ \max(g₁[i], g₂[i])} |x|.\]

If $g$ is the zero vector, it can be omitted.

In the second phase, we construct another generator for each dimension. These generators are scaled unit vectors. The following formula defines the sum of all those generators.

\[ \sup(□(X)) - c - ∑_g |g|\]

where $c$ is the center of the new zonotope and the $g$s are the generators constructed in the first phase.

References

[1] Reachability of Uncertain Linear Systems Using Zonotopes. A. Girard. HSCC 2005.

[2] Zonotopes as bounding volumes. L. J. Guibas, A. T. Nguyen, L. Zhang. SODA 2003.

[3] The zonotope abstract domain Taylor1+. K. Ghorbal, E. Goubault, S. Putot. CAV 2009.

overapproximate(lm::LinearMap{N, <:AbstractZonotope},
                ::Type{<:Zonotope}) where {N}

Overapproximate a lazy linear map of a zonotopic set with a zonotope.

Input

• lm – lazy linear map of a zonotopic set
• Zonotope – target set type

Output

The tight zonotope corresponding to lm.

overapproximate(P::SimpleSparsePolynomialZonotope, ::Type{<:Zonotope};
                [nsdiv]=1, [partition]=nothing)

Overapproximate a simple sparse polynomial zonotope with a zonotope.

Input

• P – simple sparse polynomial zonotope
• Zonotope – target set type
• nsdiv – (optional, default: 1) size of uniform partitioning grid
• partition – (optional, default: nothing) tuple of integers indicating the number of blocks in each dimensino; the length should match nparams(P)

Output

A zonotope.

overapproximate(P::SimpleSparsePolynomialZonotope, ::Type{<:Zonotope},
                dom::IntervalBox)

Overapproximate a simple sparse polynomial zonotope over the parameter domain dom with a zonotope.

Input

• P – simple sparse polynomial zonotope
• Zonotope – target set type
• dom – parameter domain, which should be a subset of [-1, 1]^q, where q = nparams(P)

Output

A zonotope.

overapproximate(P::SparsePolynomialZonotope, ::Type{<:Zonotope})

Overapproximate a sparse polynomial zonotope with a zonotope.

Input

• P – sparse polynomial zonotope
• Zonotope – target set type

Output

A zonotope.

overapproximate(X::LazySet, ZT::Type{<:Zonotope},
                dir::Union{AbstractDirections, Type{<:AbstractDirections}};
                [algorithm]="vrep", kwargs...)

Overapproximate a set with a zonotope.

Input

• X – set
• Zonotope – target set type
• dir – directions used for the generators
• algorithm – (optional, default: "vrep") algorithm used to compute the overapproximation
• kwargs – further algorithm-specific options

Output

A zonotope that overapproximates X and uses at most the generator directions provided in dir (redundant directions will be ignored).

Notes

Two algorithms are available:

• "vrep" – Overapproximate a polytopic set with a zonotope of minimal total generator sum using only generators in the given directions. Under this constraint, the zonotope has the minimal sum of generator vectors. See the docstring of _overapproximate_zonotope_vrep for further details.

• "cpa" – Overapproximate a polytopic set with a zonotope using a Cartesian decomposition into two-dimensional blocks. See the docstring of _overapproximate_zonotope_cpa for further details.

overapproximate(r::Rectification{N, <:AbstractZonotope},
                ::Type{<:Zonotope}) where {N}

Overapproximate the rectification of a zonotopic set with a zonotope.

Input

• r – lazy rectification of a zonotopic set
• Zonotope – target set type

Output

A zonotope overapproximation of the set obtained by rectifying Z.

Algorithm

This method implements [Theorem 3.1, 1].

[1] Singh, G., Gehr, T., Mirman, M., Püschel, M., & Vechev, M. Fast and effective robustness certification. NeurIPS 2018.

overapproximate(CHA::ConvexHullArray{N, <:AbstractZonotope},
                ::Type{<:Zonotope}) where {N}

Overapproximate the convex hull of a list of zonotopic sets with a zonotope.

Input

• CHA – convex hull array of zonotopic sets
• Zonotope – target set type

Output

A zonotope overapproximation of the convex hull array of zonotopic sets.

Algorithm

This method iteratively applies the overapproximation algorithm to the convex hull of two zonotopic sets from the given array of zonotopic sets.

overapproximate(QM::QuadraticMap{N, <:AbstractZonotope},
                ::Type{<:Zonotope}) where {N}

Overapproximate a quadratic map of a zonotopic set with a zonotope.

Input

• QM – quadratic map of a zonotopic set
• Zonotope – target set type

Output

A zonotope overapproximating the quadratic map of a zonotopic set.

Notes

Mathematically, a quadratic map of a zonotope with matrices $Q$ is defined as:

\[Z_Q = \right\{ \lambda | \lambda_i = x^T Q\^{(i)} x,~i = 1, \ldots, n,~x \in Z \left\}\]

Algorithm

This method implements [Lemma 1, 1].

[1] Matthias Althoff and Bruce H. Krogh. Avoiding geometric intersection operations in reachability analysis of hybrid systems. HSCC 2012.

overapproximate(X::Intersection{N, <:AbstractZonotope, <:Hyperplane},
                ::Type{<:Zonotope})

Overapproximate the intersection of a zonotopic set and a hyperplane with a zonotope.

Input

• X – intersection of a zonotopic set and a hyperplane
• Zonotope – target set type

Output

A zonotope overapproximating the intersection.

Algorithm

This method implements Algorithm 3 in [1].

[1] Moussa Maïga, Nacim Ramdani, Louise Travé-Massuyès, Christophe Combastel: A CSP versus a zonotope-based method for solving guard set intersection in nonlinear hybrid reachability. Mathematics in Computer Science (8) 2014.

overapproximate(lm::LinearMap{N, <:CartesianProductArray},
                ::Type{<:CartesianProductArray{N, S}}
               ) where {N, S<:LazySet}

Decompose a lazy linear map of a Cartesian product array while keeping the original block structure.

Input

• lm – lazy linear map of Cartesian product array
• CartesianProductArray – target set type

Output

A CartesianProductArray representing the decomposed linear map.

overapproximate(lm::LinearMap{N, <:CartesianProductArray},
                ::Type{<:CartesianProductArray},
                dir::Type{<:AbstractDirections}) where {N}

Decompose a lazy linear map of a Cartesian product array with template directions while keeping the original block structure.

Input

• lm – lazy linear map of a Cartesian product array
• CartesianProductArray – target set type
• dir – template directions for overapproximation

Output

A CartesianProductArray representing the decomposed linear map.

overapproximate(lm::LinearMap{N, <:CartesianProductArray},
                ::Type{<:CartesianProductArray},
                set_type::Type{<:LazySet}) where {N}

Decompose a lazy linear map of a Cartesian product array with a given set type while keeping the original block structure.

Input

• lm – lazy linear map of a Cartesian product array
• CartesianProductArray – target set type
• set_type – set type for overapproximation

Output

A CartesianProductArray representing the decomposed linear map.

overapproximate(rm::ResetMap{N, <:CartesianProductArray},
                ::Type{<:CartesianProductArray}, oa) where {N}

Overapproximate a reset map (that only resets to zero) of a Cartesian product with a new Cartesian product.

Input

• rm – reset map
• CartesianProductArray – target set type
• oa – overapproximation option

Output

A Cartesian product with the same block structure.

Notes

This implementation currently only supports resets to zero.

Algorithm

We convert the ResetMap into a LinearMap and then call the corresponding overapproximate method.

overapproximate(cap::Intersection{N,
                                  <:CartesianProductArray,
                                  <:AbstractPolyhedron},
                ::Type{<:CartesianProductArray}, oa) where {N}

Overapproximate the intersection of a Cartesian product of a finite number of sets and a polyhedron with a new Cartesian product.

Input

• cap – lazy intersection of a Cartesian product array and a polyhedron
• CartesianProductArray – target set type
• oa – overapproximation option

Output

A CartesianProductArray that overapproximates the intersection of cpa and P.

Algorithm

The intersection only needs to be computed in the blocks of cpa that are constrained in P. Hence we first collect those constrained blocks in a lower-dimensional Cartesian product array and then convert to an HPolytope X. Then we take the intersection of X and the projection of Y onto the corresponding dimensions. (This projection is purely syntactic and exact.) Finally we decompose the result again and plug together the unaffected old blocks and the newly computed blocks. The result is a CartesianProductArray with the same block structure as in X.

overapproximate(vTM::Vector{TaylorModel1{T, S}}, ::Type{<:Zonotope};
                [remove_zero_generators]::Bool=true
                [normalize]::Bool=true) where {T, S}

Overapproximate a Taylor model in one variable with a zonotope.

Input

• vTM – vector of TaylorModel1
• Zonotope – target set type
• remove_zero_generators – (optional; default: true) flag to remove zero generators of the resulting zonotope
• normalize – (optional; default: true) flag to skip the normalization of the Taylor models

Output

A zonotope that overapproximates the range of the given Taylor model.

Examples

If the polynomials are linear, this method exactly transforms to a zonotope. The nonlinear case necessarily introduces overapproximation error. Consider the linear case first:

julia> using LazySets, TaylorModels

julia> const IA = IntervalArithmetic;

julia> I = IA.Interval(-0.5, 0.5) # interval remainder
[-0.5, 0.5]

julia> x₀ = IA.Interval(0.0) # expansion point
[0, 0]

julia> D = IA.Interval(-3.0, 1.0)
[-3, 1]

julia> p1 = Taylor1([2.0, 1.0], 2) # define a linear polynomial
 2.0 + 1.0 t + 𝒪(t³)

julia> p2 = Taylor1([0.9, 3.0], 2) # define another linear polynomial
 0.9 + 3.0 t + 𝒪(t³)

julia> vTM = [TaylorModel1(pi, I, x₀, D) for pi in [p1, p2]]
2-element Vector{TaylorModel1{Float64, Float64}}:
  2.0 + 1.0 t + [-0.5, 0.5]
  0.9 + 3.0 t + [-0.5, 0.5]

Here, vTM is a Taylor model vector, since each component is a Taylor model in one variable (TaylorModel1). Using overapproximate(vTM, Zonotope) we can compute its associated zonotope in generator representation:

julia> Z = overapproximate(vTM, Zonotope);

julia> center(Z)
2-element Vector{Float64}:
  1.0
 -2.1

julia> Matrix(genmat(Z))
2×3 Matrix{Float64}:
 2.0  0.5  0.0
 6.0  0.0  0.5

Note how the generators of this zonotope mainly consist of two pieces: one comes from the linear part of the polynomials, and another one corresponds to the interval remainder. This conversion gives the same upper and lower bounds as the range evaluation using interval arithmetic:

julia> X = box_approximation(Z)
Hyperrectangle{Float64, Vector{Float64}, Vector{Float64}}([1.0, -2.1], [2.5, 6.5])

julia> Y = evaluate(vTM[1], vTM[1].dom) × evaluate(vTM[2], vTM[2].dom)
[-1.5, 3.5] × [-8.60001, 4.40001]

julia> H = convert(Hyperrectangle, Y) # this IntervalBox is the same as X
Hyperrectangle{Float64, StaticArraysCore.SVector{2, Float64}, StaticArraysCore.SVector{2, Float64}}([1.0, -2.1000000000000005], [2.5, 6.500000000000001])

However, the zonotope returns better results if we want to approximate the Taylor model because it is not axis-aligned:

julia> d = [-0.35, 0.93];

julia> ρ(d, Z) < ρ(d, X)
true

This method also works if the polynomials are non-linear; for example suppose that we add a third polynomial with a quadratic term:

julia> p3 = Taylor1([0.9, 3.0, 1.0], 3)
 0.9 + 3.0 t + 1.0 t² + 𝒪(t⁴)

julia> vTM = [TaylorModel1(pi, I, x₀, D) for pi in [p1, p2, p3]]
3-element Vector{TaylorModel1{Float64, Float64}}:
           2.0 + 1.0 t + [-0.5, 0.5]
           0.9 + 3.0 t + [-0.5, 0.5]
  0.9 + 3.0 t + 1.0 t² + [-0.5, 0.5]

julia> Z = overapproximate(vTM, Zonotope);

julia> center(Z)
3-element Vector{Float64}:
  1.0
 -2.1
  2.4

julia> Matrix(genmat(Z))
3×4 Matrix{Float64}:
 2.0  0.5  0.0  0.0
 6.0  0.0  0.5  0.0
 6.0  0.0  0.0  5.0

The last generator corresponds to the addition of the interval remainder and the box overapproximation of the nonlinear part of p3 over the domain.

Algorithm

Let $\text{vTM} = (p, I)$ be a vector of $m$ Taylor models, where $I$ is the interval remainder in $\mathbb{R}^m$. Let $p_{lin}$ (resp. $p_{nonlin}$) correspond to the linear (resp. nonlinear) part of each scalar polynomial.

The range of $\text{vTM}$ can be enclosed by a zonotope with center $c$ and matrix of generators $G$, $Z = ⟨c, G⟩$, by performing a conservative linearization of $\text{vTM}$:

\[ vTM' = (p', I') := (p_{lin} − p_{nonlin} , I + \text{Int}(p_{nonlin})).\]

This algorithm proceeds in two steps:

1- Conservatively linearize $\text{vTM}$ as above and compute a box overapproximation of the nonlinear part. 2- Transform the linear Taylor model to a zonotope exactly through variable normalization onto the symmetric intervals $[-1, 1]$.

overapproximate(vTM::Vector{TaylorModelN{N, T, S}}, ::Type{<:Zonotope};
                [remove_zero_generators]::Bool=true
                [normalize]::Bool=true) where {N, T, S}

Overapproximate a multivariate Taylor model with a zonotope.

Input

• vTM – vector of TaylorModelN
• Zonotope – target set type
• remove_zero_generators – (optional; default: true) flag to remove zero generators of the resulting zonotope
• normalize – (optional; default: true) flag to skip the normalization of the Taylor models

Output

A zonotope that overapproximates the range of the given Taylor model.

Examples

Consider a vector of two 2-dimensional Taylor models of order 2 and 4, respectively.

julia> using LazySets, TaylorModels

julia> const IA = IntervalArithmetic;

julia> x₁, x₂ = set_variables(Float64, ["x₁", "x₂"], order=8)
2-element Vector{TaylorN{Float64}}:
  1.0 x₁ + 𝒪(‖x‖⁹)
  1.0 x₂ + 𝒪(‖x‖⁹)

julia> x₀ = IntervalBox(0..0, 2) # expansion point
[0, 0]²

julia> Dx₁ = IA.Interval(0.0, 3.0) # domain for x₁
[0, 3]

julia> Dx₂ = IA.Interval(-1.0, 1.0) # domain for x₂
[-1, 1]

julia> D = Dx₁ × Dx₂ # take the Cartesian product of the domain on each variable
[0, 3] × [-1, 1]

julia> r = IA.Interval(-0.5, 0.5) # interval remainder
[-0.5, 0.5]

julia> p1 = 1 + x₁^2 - x₂
 1.0 - 1.0 x₂ + 1.0 x₁² + 𝒪(‖x‖⁹)

julia> p2 = x₂^3 + 3x₁^4 + x₁ + 1
 1.0 + 1.0 x₁ + 1.0 x₂³ + 3.0 x₁⁴ + 𝒪(‖x‖⁹)

julia> vTM = [TaylorModelN(pi, r, x₀, D) for pi in [p1, p2]]
2-element Vector{TaylorModelN{2, Float64, Float64}}:
            1.0 - 1.0 x₂ + 1.0 x₁² + [-0.5, 0.5]
  1.0 + 1.0 x₁ + 1.0 x₂³ + 3.0 x₁⁴ + [-0.5, 0.5]

julia> Z = overapproximate(vTM, Zonotope);

julia> center(Z)
2-element Vector{Float64}:
   5.5
 124.0

julia> Matrix(genmat(Z))
2×4 Matrix{Float64}:
 0.0  -1.0  5.0    0.0
 1.5   0.0  0.0  123.0

Algorithm

We refer to the algorithm description for the univariate case.

_overapproximate_zonotope_vrep(X::LazySet{N},
                               dir::AbstractDirections;
                               solver=default_lp_solver(N)) where {N}

Overapproximate a polytopic set with a zonotope of minimal total generator sum using only generators in the given directions.

Input

• X – polytopic set
• dir – directions used for the generators
• solver – (optional, default: default_lp_solver(N)) the backend used to solve the linear program

Output

A zonotope that overapproximates X and uses at most the directions provided in dir (redundant directions will be ignored). Under this constraint, the zonotope has the minimal sum of generator vectors.

Notes

The algorithm only requires one representative of each generator direction and their additive inverse (e.g., only one of [1, 0] and [-1, 0]) and assumes that the directions are normalized. We preprocess the directions in that respect.

Algorithm

We solve a linear program parametric in the vertices $v_j$ of X and the directions $d_k$ in dir presented in Section 4.2 in [1], adapting the notation to the one used in this library.

\[ \min \sum_{k=1}^l α_k \ s.t. \ c + \sum_{k=1}^l b_{kj} * d_k = v_j \quad \forall j \ -α_k ≤ b_{kj} ≤ α_k \quad \forall k, j \ α_k ≥ 0 \quad \forall k\]

The resulting zonotope has center c and generators α_k · d_k.

Note that the first type of side constraints is vector-based and that the nonnegativity constraints (last type) are not stated explicitly in [1].

[1] Zonotopes as bounding volumes. L. J. Guibas, A. T. Nguyen, L. Zhang. SODA 2003.

_overapproximate_zonotope_cpa(X::LazySet, dir::Type{<:AbstractDirections})

Overapproximate a polytopic set with a zonotope using Cartesian decomposition.

Input

• X – polytopic set
• dir – directions used for the generators

Output

A zonotope that overapproximates X.

Notes

The algorithm decomposes X into 2D sets and overapproximates those sets with zonotopes, and finally converts the Cartesian product of the sets to a zonotope.

Algorithm

The implementation is based on Section 8.2.4 in [1].

[1] Le Guernic, C. Reachability analysis of hybrid systems with linear continuous dynamics (Doctoral dissertation). 2009.

underapproximate(X::S, dirs::AbstractDirections;
                [apply_convex_hull]::Bool=false) where {N, S<:LazySet{N}}

Compute the underapproximation of a convex set by sampling support vectors.

Input

• X – convex set
• dirs – directions
• apply_convex_hull – (optional, default: false) if true, post-process the support vectors with a convex hull operation

Output

The VPolytope obtained by taking the convex hull of support vectors of X along the directions determined by dirs.

Notes

Since the support vectors are not always unique, this algorithm may return a strict underapproximation even if the set can be exactly approximated using the given template.

underapproximate(X::LazySet, ::Type{<:Hyperrectangle};
                 solver=nothing) where {N}

Underapproximate a convex polygon with a hyperrectangle of maximal area.

Input

• X – convex polygon
• Hyperrectangle – type for dispatch
• solver – (optional; default: nothing) nonlinear solver; if nothing, default_nln_solver(N) will be used

Output

A hyperrectangle underapproximation with maximal area.

Notes

The implementation only works for 2D sets, but the algorithm can be generalized.

Due to numerical issues, the result may be slightly outside the set.

Algorithm

The algorithm is taken from [1, Theorem 17] and solves a convex program (in fact a linear program with nonlinear objective). (The objective is modified to an equivalent version due to solver issues.)

[1] Mehdi Behroozi - Largest inscribed rectangles in geometric convex sets. arXiv:1905.13246.

underapproximate(X::LazySet, ::Type{<:Ball2})

Compute the largest inscribed Euclidean ball in a set X.

Input

• X – set
• Ball2 – target type

Output

A largest Ball2 contained in X.

Algorithm

We use chebyshev_center_radius(X).

approximate(R::Rectification; apply_convex_hull::Bool=false)

Approximate a rectification of a polytopic set with a convex polytope.

Input

• R – rectification of a polytopic set
• apply_convex_hull – (optional; default: false) option to remove redundant vertices

Output

A polytope in vertex representation (VPolygon in 2D, VPolytope otherwise). There is no guarantee that the result over- or underapproximates R.

Algorithm

Let $X$ be the set that is rectified. We compute the vertices of $X$, rectify them, and return the convex hull of the result.

Notes

Let $X$ be the set that is rectified and let $p$ and $q$ be two vertices on a facet of $X$. Intuitively, an approximation may occur if the line segment connecting these vertices crosses a coordinate hyperplane and if the line segment connecting the rectified vertices has a different angle.

As a corollary, the approximation is exact for the special cases that the original set is contained in either the positive or negative orthant or is axis-aligned.

box_approximation(S::LazySet{N}) where {N}

Overapproximate a set by a tight hyperrectangle.

Input

• S – set

Output

A tight hyperrectangle.

Notes

An alias for this function is interval_hull.

Algorithm

The center and radius of the hyperrectangle are obtained by averaging the low and high coordinates of S computed with the extrema function.

box_approximation(S::CartesianProductArray{N, <:AbstractHyperrectangle}) where {N}

Overapproximate the Cartesian product of a finite number of hyperrectangular sets by a tight hyperrectangle.

Input

• S– Cartesian product of a finite number of hyperrectangular sets

Output

A hyperrectangle.

Algorithm

This method falls back to the convert method. Since the sets wrapped by the Cartesian product array are hyperrectangles, this can be done without overapproximation.

box_approximation(S::CartesianProduct{N, <:AbstractHyperrectangle, <:AbstractHyperrectangle}) where {N}

Overapproximate the Cartesian product of two hyperrectangular sets by a tight hyperrectangle.

Input

• S– Cartesian product of two hyperrectangular sets

Output

A hyperrectangle.

Algorithm

This method falls back to the convert method. Since the sets wrapped by the Cartesian product array are hyperrectangles, this can be done without overapproximation.

box_approximation(lm::LinearMap{N, <:AbstractHyperrectangle}) where {N}

Return a tight overapproximation of the linear map of a hyperrectangular set using a hyperrectangle.

Input

• lm– linear map of a hyperrectangular set

Output

A hyperrectangle.

Algorithm

If c and r denote the center and vector radius of a hyperrectangle H, a tight hyperrectangular overapproximation of M * H is obtained by transforming c ↦ M*c and r ↦ abs.(M) * r, where abs.(⋅) denotes the element-wise absolute-value operator.

box_approximation(R::Rectification{N}) where {N}

Overapproximate the rectification of a set by a tight hyperrectangle.

Input

• R– rectification of a set

Output

A hyperrectangle.

Algorithm

Box approximation and rectification distribute. We first check whether the wrapped set is empty. If so, we return the empty set. Otherwise, we compute the box approximation of the wrapped set, rectify the resulting box (which is simple), and finally convert the resulting set to a box.

box_approximation(Z::AbstractZonotope)

Return a tight overapproximation of a zonotope with an axis-aligned box.

Input

• Z – zonotope

Output

A hyperrectangle.

Algorithm

This function implements the method in [Section 5.1.2, 1]. A zonotope $Z = ⟨c, G⟩$ can be tightly overapproximated by an axis-aligned hyperrectangle such that its center is $c$ and the radius along dimension $i$ is the column-sum of the absolute values of the $i$-th row of $G$ for $i = 1,…, p$, where $p$ is the number of generators of $Z$.

[1] Althoff, M., Stursberg, O., & Buss, M. (2010). Computing reachable sets of hybrid systems using a combination of zonotopes and polytopes. Nonlinear analysis: hybrid systems, 4(2), 233-249.

box_approximation(am::AbstractAffineMap{N, <:AbstractHyperrectangle}) where {N}

Overapproximate the affine map of a hyperrectangular set by a tight hyperrectangle.

Input

• am – affine map of a hyperrectangular set

Output

A hyperrectangle.

Algorithm

If c and r denote the center and vector radius of a hyperrectangle H and v is the translation vector, a tight hyperrectangular overapproximation of M * H + v is obtained by transforming c ↦ M*c+v and r ↦ abs.(M) * r, where abs.(⋅) denotes the element-wise absolute-value operator.

box_approximation(ch::ConvexHull; [algorithm]::String="box")

Overapproximate a convex hull with a tight hyperrectangle.

Input

• ch – convex hull
• algorithm – (optional; default: "box") algorithm choice

Output

A hyperrectangle.

Algorithm

Let X and Y be the two sets of ch. We make use of the following property:

\[\square(CH(X, Y)) = \square\left( X \cup Y \right) = \square\left( \square(X) \cup \square(Y) \right)\]

If algorithm == "extrema", we compute the low and high coordinates of X and Y via extrema.

If algorithm == "box", we instead compute the box approximations of X and Y via box_approximation.

In both cases we then take the box approximation of the result.

The "extrema" algorithm is more efficient if extrema is efficient because it does not need to allocate the intermediate hyperrectangles.

box_approximation(ms::MinkowskiSum)

Overapproximate the Minkowski sum of two sets with a tight hyperrectangle.

Input

• ms – Minkowski sum

Output

A hyperrectangle.

Algorithm

The box approximation distributes over the Minkowski sum:

\[\square(X \oplus Y) = \square(X) \oplus \square(Y)\]

It suffices to compute the box approximation of each summand and then take the concrete Minkowski sum for hyperrectangles.

interval_hull

Alias for box_approximation.

symmetric_interval_hull(S::LazySet{N}) where {N}

Overapproximate a set by a tight hyperrectangle centered in the origin.

Input

• S – set

Output

A tight hyperrectangle that is centrally symmetric wrt. the origin.

Notes

An alias for this function is box_approximation_symmetric.

Algorithm

The center of the box is the origin, and the radius is obtained via the extrema function.

box_approximation_symmetric

Alias for symmetric_interval_hull.

ballinf_approximation(S::LazySet)

Overapproximate a set by a tight ball in the infinity norm.

Input

• S – set

Output

A tight ball in the infinity norm.

Algorithm

The center and radius of the ball are obtained by averaging the low and high coordinates of S computed with the extrema function.

overapproximate_hausdorff(X::S, ε::Real) where {N<:AbstractFloat, S<:LazySet{N}}

Return an ε-close overapproximation of the given 2D convex set (in terms of the Hausdorff distance) in the form of a polygon in constraint representation.

Input

• X – 2D convex set
• ε – error bound

Output

A polygon in constraint representation.

LocalApproximation{N, VN<:AbstractVector{N}}

Type that represents a local approximation in 2D.

Fields

• p1 – first inner point
• d1 – first direction
• p2 – second inner point
• d2 – second direction
• q – intersection of the lines l1 ⟂ d1 at p1 and l2 ⟂ d2 at p2
• refinable – flag stating whether this approximation is refinable
• err – error upper bound

Notes

The criteria for being refinable are determined in new_approx.

PolygonalOverapproximation{N, SN<:LazySet{N}, VN<:AbstractVector{N}}

Type that represents a polygonal overapproximation of a convex set.

Fields

• S – convex set
• approx_stack – stack of local approximations that still need to be examined
• constraints – vector of half-spaces that are already finalized (i.e., they satisfy the given error bound)

new_approx(S::LazySet, p1::VN, d1::VN,
           p2::VN, d2::VN) where {N<:AbstractFloat, VN<:AbstractVector{N}}

Create a LocalApproximation instance for the given excerpt of a polygonal overapproximation.

Input

• S – convex set
• p1 – first inner point
• d1 – first direction
• p2 – second inner point
• d2 – second direction

Output

A local approximation of S in the given directions.

addapproximation!(Ω::PolygonalOverapproximation, p1::VN, d1::VN,
                  p2::VN, d2::VN) where {N, VN<:AbstractVector{N}}

Input

• Ω – polygonal overapproximation of a convex set
• p1 – first inner point
• d1 – first direction
• p2 – second inner point
• d2 – second direction

Output

The list of local approximations in Ω of the set Ω.S is updated in-place and the new approximation is returned by this function.

refine(approx::LocalApproximation, S::LazySet)

Refine a given local approximation of the polygonal overapproximation of a convex set by splitting along the normal direction of the approximation.

Input

• approx – local approximation to be refined
• S – 2D convex set

Output

The tuple consisting of the refined right and left local approximations.

tohrep(Ω::PolygonalOverapproximation)

Convert a polygonal overapproximation into a polygon in constraint representation.

Input

• Ω – polygonal overapproximation of a convex set

Output

A polygon in constraint representation.

Algorithm

Internally, the constraints of Ω are already sorted.

convert(::Type{HalfSpace}, approx::LocalApproximation)

Convert a local approximation to a half-space.

Input

• approx – local approximation

Output

A half-space.

AbstractDirections{N, VN}

Abstract type for representations of direction vectors.

Notes

This type is parameterized by N and VN, where:

• N stands for the numeric type
• VN stands for the vector type with coefficients of type N

Each implementing subtype is an iterator over a set of directions. For that they implement the standard iterator methods from Base, namely Base.length (returns the number of directions) and Base.iterate. Moreover, the following methods should be implemented:

• dim – return the ambient dimension of the vectors
• eltype – return the type of each vector

Optionally, subtypes may implement:

• isbounding – (defaults to false) return true if an overapproximation with the direction vectors results in a bounded set, given a bounded input set, and false otherwise
• isnormalized – (defaults to false) is true if each direction vector has norm one w.r.t. the usual vector 2-norm

isbounding(ad::AbstractDirections)
isbounding(ad::Type{<:AbstractDirections})

Check whether an overapproximation with a set of direction vectors results in a bounded set, given a bounded input set.

Input

• ad – direction vectors or a subtype of AbstractDirections

Output

Given a bounded set $X$, we can construct an outer polyhedral approximation of $X$ by using the direction vectors ad as normal vectors of the facets. If this function returns true, then the result is again guaranteed to be a bounded set (i.e., a polytope). Note that the result does not depend on the specific shape of $X$, as long as $X$ is bounded.

Notes

By default, this function returns false in order to be conservative. Custom subtypes of AbstractDirections should hence add a method for this function.

The function can be applied to an instance of an AbstractDirections subtype or to the subtype itself. By default, the check on the instance falls back to the check on the subtype.

isnormalized(ad::AbstractDirections)
isnormalized(ad::Type{<:AbstractDirections})

Check whether the given direction vectors are normalized with respect to the 2-norm.

Input

• ad – direction vectors or a subtype of AbstractDirections

Output

true if the 2-norm of each element in ad is one and false otherwise.

Notes

By default, this function returns false in order to be conservative. Custom subtypes of AbstractDirections should hence add a method for this function.

The function can be applied to an instance of an AbstractDirections subtype or to the subtype itself. By default, the check on the instance falls back to the check on the subtype.

project(S::LazySet,
        block::AbstractVector{Int},
        directions::Type{<:AbstractDirections},
        [n]::Int;
        [kwargs...]
       )

Project a high-dimensional set to a given block using direction vectors.

Input

• S – set
• block – block structure - a vector with the dimensions of interest
• directions – direction vectors
• n – (optional, default: dim(S)) ambient dimension of the set S

Output

The polyhedral overapproximation of the projection of S in the given directions.

BoxDirections{N, VN} <: AbstractDirections{N, VN}

Box directions representation.

Fields

• n – dimension

Notes

Box directions can be seen as the vectors where only one entry is ±1, and all other entries are 0. In dimension $n$, there are $2n$ such directions.

The default vector representation used in this template is a ReachabilityBase.Arrays.SingleEntryVector, although other implementations can be used such as a regular Vector and a SparseVector.

Examples

The template can be constructed by passing the dimension. For example, in dimension two:

julia> dirs = BoxDirections(2)
BoxDirections{Float64, ReachabilityBase.Arrays.SingleEntryVector{Float64}}(2)

julia> length(dirs)
4

By default, each direction is represented as a SingleEntryVector, i.e., a vector with only one non-zero element,

julia> eltype(dirs)
ReachabilityBase.Arrays.SingleEntryVector{Float64}

In two dimensions, the directions defined by BoxDirections are normal to the facets of a box.

julia> collect(dirs)
4-element Vector{ReachabilityBase.Arrays.SingleEntryVector{Float64}}:
 [1.0, 0.0]
 [0.0, 1.0]
 [0.0, -1.0]
 [-1.0, 0.0]

The numeric type can be specified as well:

julia> BoxDirections{Rational{Int}}(10)
BoxDirections{Rational{Int64}, ReachabilityBase.Arrays.SingleEntryVector{Rational{Int64}}}(10)

julia> length(ans)
20

DiagDirections{N, VN} <: AbstractDirections{N, VN}

Diagonal directions representation.

Fields

• n – dimension

Notes

Diagonal directions are vectors where all entries are ±1. In dimension $n$, there are in total $2^n$ such directions.

Examples

The template can be constructed by passing the dimension. For example, in dimension two:

julia> dirs = DiagDirections(2)
DiagDirections{Float64, Vector{Float64}}(2)

julia> length(dirs) # number of directions
4

By default, each direction is represented as a regular Vector:

julia> eltype(dirs)
Vector{Float64} (alias for Array{Float64, 1})

In two dimensions, the directions defined by DiagDirections are normal to the facets of a ball in the 1-norm.

julia> collect(dirs)
4-element Vector{Vector{Float64}}:
 [1.0, 1.0]
 [-1.0, 1.0]
 [1.0, -1.0]
 [-1.0, -1.0]

The numeric type can be specified as well:

julia> DiagDirections{Rational{Int}}(10)
DiagDirections{Rational{Int64}, Vector{Rational{Int64}}}(10)

julia> length(ans)
1024

OctDirections{N, VN} <: AbstractDirections{N, VN}

Octagon directions representation.

Fields

• n – dimension

Notes

Octagon directions consist of all vectors that are zero almost everywhere except in two dimensions $i$, $j$ (possibly $i = j$) where it is $±1$. In dimension $n$, there are $2n^2$ such directions.

Examples

The template can be constructed by passing the dimension. For example, in dimension two:

julia> dirs = OctDirections(2)
OctDirections{Float64, SparseArrays.SparseVector{Float64, Int64}}(2)

julia> length(dirs) # number of directions
8

By default, the directions are represented as sparse vectors:

julia> eltype(dirs)
SparseArrays.SparseVector{Float64, Int64}

In two dimensions, the directions are normal to the facets of an octagon.

julia> first(dirs)
2-element SparseArrays.SparseVector{Float64, Int64} with 2 stored entries:
  [1]  =  1.0
  [2]  =  1.0

julia> Vector.(collect(dirs))
8-element Vector{Vector{Float64}}:
 [1.0, 1.0]
 [1.0, -1.0]
 [-1.0, 1.0]
 [-1.0, -1.0]
 [1.0, 0.0]
 [0.0, 1.0]
 [0.0, -1.0]
 [-1.0, 0.0]

The numeric type can be specified as well:

julia> OctDirections{Rational{Int}}(10)
OctDirections{Rational{Int64}, SparseArrays.SparseVector{Rational{Int64}, Int64}}(10)

julia> length(ans)
200

BoxDiagDirections{N, VN} <: AbstractDirections{N, VN}

Box-diagonal directions representation.

Fields

• n – dimension

Notes

Box-diagonal directions can be seen as the union of diagonal directions (all entries are ±1) and box directions (one entry is ±1, all other entries are 0). The iterator first enumerates all diagonal directions, and then all box directions. In dimension $n$, there are in total $2^n + 2n$ such directions.

Examples

The template can be constructed by passing the dimension. For example, in dimension two:

julia> dirs = BoxDiagDirections(2)
BoxDiagDirections{Float64, Vector{Float64}}(2)

julia> length(dirs) # number of directions
8

By default, each direction is represented as a regular vector:

julia> eltype(dirs)
Vector{Float64} (alias for Array{Float64, 1})

In two dimensions, the directions are normal to the facets of an octagon, i.e., the template coincides with OctDirections.

julia> collect(dirs)
8-element Vector{Vector{Float64}}:
 [1.0, 1.0]
 [-1.0, 1.0]
 [1.0, -1.0]
 [-1.0, -1.0]
 [1.0, 0.0]
 [0.0, 1.0]
 [0.0, -1.0]
 [-1.0, 0.0]

The numeric type can be specified as well:

julia> BoxDiagDirections{Rational{Int}}(10)
BoxDiagDirections{Rational{Int64}, Vector{Rational{Int64}}}(10)

julia> length(ans)
1044

PolarDirections{N<:AbstractFloat, VN<:AbstractVector{N}} <: AbstractDirections{N, VN}

Polar directions representation.

Fields

• Nφ – length of the partition of the polar angle
• stack – list of computed directions

Notes

The PolarDirections constructor computes a sample of the unit sphere in $\mathbb{R}^2$, which is parameterized by the polar angle $φ ∈ Dφ := [0, 2π]$; see the Wikipedia entry on the polar coordinate system for details. The resulting directions are stored in stack.

The integer argument $Nφ$ defines how many samples of $Dφ$ are taken. The Cartesian components of each direction are obtained with

\[[cos(φᵢ), sin(φᵢ)].\]

Examples

The integer passed as an argument is used to discretize $φ$:

julia> pd = PolarDirections(2);

julia> pd.stack
2-element Vector{Vector{Float64}}:
 [1.0, 0.0]
 [-1.0, 1.2246467991473532e-16]

julia> length(pd)
2

SphericalDirections{N<:AbstractFloat, VN<:AbstractVector{N}} <: AbstractDirections{N, VN}

Spherical directions representation.

Fields

• Nθ – length of the partition of the azimuthal angle
• Nφ – length of the partition of the polar angle
• stack – list of computed directions

Notes

The SphericalDirections constructor provides a sample of the unit sphere in $\mathbb{R}^3$, which is parameterized by the azimuthal and polar angles $θ ∈ Dθ := [0, π]$ and $φ ∈ Dφ := [0, 2π]$ respectively; see the Wikipedia entry on the spherical coordinate system for details.

The integer arguments $Nθ$ and $Nφ$ define how many samples along the domains $Dθ$ and $Dφ$ are respectively taken. The Cartesian components of each direction are obtained with

\[[sin(θᵢ)*cos(φᵢ), sin(θᵢ)*sin(φᵢ), cos(θᵢ)].\]

The north and south poles are treated separately so that those points are not considered more than once.

Examples

The template can be built in different ways. If you pass only one integer, the same value is used to discretize both $θ$ and $φ$:

julia> sd = SphericalDirections(3);

julia> sd.Nθ, sd.Nφ
(3, 3)

julia> length(sd)
4

Pass two integers to control the discretization in $θ$ and in $φ$ separately:

julia> sd = SphericalDirections(4, 5);

julia> length(sd)
10

julia> sd = SphericalDirections(4, 8);

julia> length(sd)
16

CustomDirections{N, VN<:AbstractVector{N}} <: AbstractDirections{N, VN}

User-defined direction vectors.

Fields

• directions – list of direction vectors
• n – (optional; default: computed from directions) dimension
• check_boundedness – (optional; default: true) flag to check boundedness
• check_normalization – (optional; default: true) flag to check whether all directions are normalized

Notes

This struct is a wrapper for a list of user-defined directions. There are fields for the list of directions, their dimension, and (boolean) cache fields for the boundedness and normalization properties. The latter are checked by default upon construction.

To check boundedness, we construct the polyhedron with constraints $d·x <= 1$ for each direction $d$ and check if this set is bounded. (Note that the bound $1$ is arbitrary and that this set may be empty, which however implies boundedness.)

The dimension will also be determined automatically, unless the empty vector is passed (in which case the optional argument n needs to be specified).

Examples

Create a template with box directions in dimension two:

julia> dirs = CustomDirections([[1.0, 0.0], [-1.0, 0.0], [0.0, 1.0], [0.0, -1.0]]);

julia> dirs.directions
4-element Vector{Vector{Float64}}:
 [1.0, 0.0]
 [-1.0, 0.0]
 [0.0, 1.0]
 [0.0, -1.0]

julia> LazySets.Approximations.isbounding(dirs)
true

julia> LazySets.Approximations.isnormalized(dirs)
true

hausdorff_distance(X::LazySet{N}, Y::LazySet{N}; [p]::N=N(Inf),
                   [ε]=N(1e-3)) where {N}

Compute the Hausdorff distance between two convex sets up to a given threshold.

Input

• X – convex set
• Y – convex set
• p – (optional, default: Inf) norm parameter of the Hausdorff distance
• ε – (optional, default: 1e-3) precision threshold; the true Hausdorff distance is allowed to diverge from the result by at most this value

Output

A value from the $ε$-neighborhood of the Hausdorff distance between $X$ and $Y$.

Notes

Given a $p$-norm, the Hausdorff distance $d_H^p(X, Y)$ between sets $X$ and $Y$ is defined as follows:

\[ d_H^p(X, Y) = \inf\{δ ≥ 0 \mid Y ⊆ X ⊕ δ 𝐵_p^n \text{ and } X ⊆ Y ⊕ δ 𝐵_p^n\}\]

Here $𝐵_p^n$ is the $n$-dimensional unit ball in the $p$-norm.

The implementation may internally rely on the support function of $X$ and $Y$; hence any imprecision in the implementation of the support function may affect the result. At the time of writing, the only convex set type with imprecise support function is the lazy Intersection.

Algorithm

We perform binary search for bounding the Hausdorff distance in an interval $[l, u]$, where initially $l$ is $0$ and $u$ is described below. The binary search terminates when $u - l ≤ ε$, i.e., the interval becomes sufficiently small.

To find an upper bound $u$, we start with the heuristics of taking the biggest distance in the axis-parallel directions. As long as this bound does not work, we increase the bound by $2$.

Given a value $δ$, to check whether the sets are within Hausdorff distance $δ$, we simply check the inclusions given above, where on the right-hand side we use a lazy Bloating.