\section{Vague Spatial Algebra}
\label{sec:VASA}

Vague Spatial Algebra (VASA)~\cite{PaulySch08QASDM,PaulySch10IS} is an exact model that defines vague spatial objects, which can be simple or complex. In addition, VASA defines a huge set of operations to handle these objects, and thus, this model is more extensive than other exact models. A vague spatial object in VASA is defined as a pair of crisp spatial objects of the same spatial data type, which must be disjoint or adjacent. The first object represents the \emph{kernel} part, while the second object represents the \emph{conjecture} part. Further, VASA is characterized by separate the portions of space of a spatial object by considering minimum and maximum approximations. As a result, a spatial object can comprise or expand according to a minimum limit (that is, the kernel part) and a maximum limit (that is, the conjecture part). Formally, let $\alpha \in \{\textit{crisp point}, \textit{crisp line}, \textit{crisp region}\}$. Then, a vague spatial data type in VASA is defined by $v(\alpha) = \alpha \times \alpha$, such that for an object $o = (o_k, o_c) \in v(\alpha)$, the property $\mathit{disjoint}(o_k, o_c) \logor \mathit{meet}(o_k, o_c)$ holds. The kernel and conjecture of $o$ are symbolized by $o_k$ and $o_c$, respectively.

VASA defines the following operations to handle vague spatial objects: vague geometric set operations, vague topological predicates, type-dependent vague spatial operations, and vague numerical operations. Vague geometric set operations are defined by reusing the crisp geometric set operations, that is, crisp geometric union, crisp geometric intersection, and crisp geometric difference.
Vague topological predicates are based on the three-valued logic, and thus, can return \emph{true}, \emph{false}, or \emph{maybe}. Let $A$ and $B$ be two vague spatial objects. A vague topological predicate is evaluated by using the well-known crisp 9-intersection matrix~\cite{SB06TODS} for the following combinations: $A_k \times B_k$, $A_k \times (B_c \cunion B_k)$, $(A_k \cunion A_c) \times B_k$, and $(A_k \cunion A_c) \times (B_c \cunion B_k)$, where $\cunion$ denotes the crisp geometric union.
Type-dependent vague spatial operations deal with a limited subset of possible types of input. For instance, the common border operation computes the shared boundary between two vague region objects and yields a vague line object.
Finally, vague numerical operations return a pair of numeric values corresponding to a minimum and a maximum value. For instance, the area of a vague region object has a minimum value that corresponds to the area of its kernel and a maximum value that corresponds to the area of the union between its kernel and its conjecture. Details of the formal definitions for vague spatial operations of VASA are given in~\citeN{PaulySch10IS}. 