## 1 Introduction  
In the area of membrane computing [1; 2; 3], the novel computing model of \(P\) systems, introduced by [PERSON] [4], has served as a framework for dealing with problems in different areas of applications [5], such as optimization problems [6; 7; 8; 9], robots [10; 11; 12], power systems fault diagnosis [13; 14; 15], modelling complex market interactions [16], image process [17] and clustering [18], and for Turing computing power [19; 20; 21; 22] and computing efficiency [23; 24; 25; 26; 27; 28], both in its basic form and in its several variants. One such area of application of \(P\) systems is formal language theory with different models of \(P\) systems having been developed [29; 30; 31; 32] for handling the problem of generation of classes of languages, starting with the seminal work of [PERSON] [4]. In the extension of language theory to two dimensions, \(P\) systems have also played a significant role with different kinds of \(P\) systems with array objects and array evolving rules having been introduced (see, for example, [33; 34; 35]).  
The operation of insertion on words [36; 37; 38; 39] has been studied in string language theory in the context of DNA computing [40]. [PERSON] [41] considered this operation in two-dimensional picture arrays and introduced a picture array generating model. This model has a feature which is analogous to the pure 2D context-free grammar [42] where the operation is rewriting in parallel all symbols in a column or a row of a (rectangular) picture array by strings of equal length while in [41], insertion (instead of rewriting ) of strings of equal length is done in parallel between columns or between rows.  
Here we consider a restricted type of insertion rules with the \"contexts\" of length at most one, in the regions of a cell-like \(P\) system with the objects in the regions being rectangular picture arrays. The main objective of this study is to develop a new theoretical model of an array \(P\) system based on the bio-inspired computing model in the area of membrane computing, endowed with the capability to handle the problem of generation of two-dimensional patterns encoded as picture arrays. We compare the generative power of the resulting array \(P\) system model introduced here with other well-investigated picture array generating models. We also exhibit an application to generation of floor-design patterns, called \"kolam\" patterns [43]. A preliminary version of this paper was presented in the conference ACMC 2018 [44].  
In Section 2, needed notions and notations relevant to the study undertaken are briefly described, with Section 2.1 providing the basic notions and Section 2.2 recalling picture-insertion system in a restricted form. In Section 3, the new model of array \(P\) system with restricted picture insertion rules \((APRPIs)\) is introduced. In Section 4, \(APRPIs\) with one and two membranes are compared. In Section 5, comparison with Pure 2D context-free grammars is done while in Section 6, comparison with certain standard array generating models is carried out. Section 7 provides an application of the new model to generation of floor-designs. In the final Section 8, concluding remarks are given.