## 2 Sculpture Garden Problem and Applications
Since we are interested in more than simply observing the inside and outside of a polygon, solutions to the art gallery or prison yard problems would not change into solutions to the sculpture garden problem. In other words, we intend to establish the time when a point is inside a polygon using only the guards as witnesses. Indeed, any polygon \(P\) can be triangulated and two angle guards can be used to define each of the resulting \(n+2(h-1)\) triangles, where \(h\) is the number of holes in \(P\). This would give rise to a concise formula \(F\) defining \(P\). However, it uses at least \(2n+4(h-1)\) angle guards, which is a more constant-depth formula. Upper and Lower BoundsThe sculpture garden problem has different types due to the different restrictions as guards could be observed in varied forms including vertices, edges, interior or exterior of the polygon, and the SGP can be manifested in different types as well. However, in each case, finding the upper and lower bound is a problem that has already been investigated. An angle guard \(g\) with angle \(\alpha\in(0,360)\) is a pair \((a,\omega_{a})\) of a point \(a\) and an infinitive wedge \(\omega_{a}\) of aperture \(\alpha\) at apex \(a\) and views \(\omega_{\alpha}\). It can be shown as a Boolean predicate \(B_{g}(p)\), in which for a point \(p\) in the plane, \(B_{g}(p)\) is _True_ if \(p\) is inside the angle associated with \(g\), otherwise it is _False_. Given a polygon \(P\) with \(n\) vertices, we intend to allocate the minimum number of angle guards with arbitrary angles at vertices of \(P\). Thus, a monotone Boolean formula, \(F_{P}(. )\), based on the angle guard predicates, \(B_{g}(. )\), is obtained as follows:\[F_{P}(p)=\left\{ \begin{array}{ll}True&\forall p\in int(P)\\ False&\text{otherwise}\end{array} \right. \tag{1}\]  
It is worth mentioning that \(int(P)\) is the interior of polygon \(P\). If \(F_{P}(p)\) is a solution of the sculpture garden problem for a given \(P\), \(P\) is defined by \(F_{P}\). The complement of an angle guard \(g=(a,\omega_{a})\) is an angle guard \(g^{\prime}\) at the point \(a\) with angle \(2\pi-\alpha\). Hence, the wedge associated with \(g^{\prime}\) is the _complement_ of \(\omega_{\alpha}\) in the plane. If formula \(F\) is a solution for the sculpture garden problem for polygon \(P\), then the complement of \(F\) which is denoted by \(F^{\prime}\) defines the exterior of \(P\). To obtain \(F^{\prime}\), initially, we replace every angle guard \(g\) by its complement, (i.e., \(g^{\prime}\)), and then swap the operations \(AND\) and \(OR\). In addition, we define, a _pocket_ of a simple polygon as the areas outside of the polygon and inside of its convex hull. As [PERSON] et al. [2; 3] reported, for any polygon \(P\), a set of \(n+2(h-1)\) angle guards and an associated concise formula \(F\) are present, solving the sculpture garden problem where \(h\) is the number of holes in \(P\). So, a simple polygon can be defined with \(n-2\) guards. They have conjectured a class of simple \(n\)-gons that require at least \(n-2\) vertex guards. The main goal of this paper is to solve a special case of this open problem for natural vertex guards. They showed that at least \(\lceil n/2\rceil\) guards are required to solve the sculpture garden problem for any polygon with no two edges lying on the same line. Furthermore, for any convex polygon \(P\), a natural angle guard vertex placement is present whose \(\lceil n/2\rceil\) guards are sufficient. They showed that \(\lceil n/2\rceil+\mathcal{O}(1)\) angle guards suffice to solve the sculpture garden problem for pseudo-triangles. Moreover, for any orthogonal polygon \(P\) (which is probably the most likely real-world application), a set of \(\lfloor 3(n-2)/4\rfloor\) angle guards and an associated concise formula \(F\) are available to solve the sculpture garden problem using \(\lceil n/2\rceil\) natural angle guards. They gave an example of a class of simple polygons containing sculpture garden solutions that used \(\mathcal{O}(\sqrt{n})\) guards.