The borders and corner tiles are also rotated as follows, relative to the three tiles on top e 0,0, e 0,1, e 0,2: Rot90 (x), for x = e 0,3, e 1,3, e 2,3 Rot180 (x), for x = e 3,1, e 3,2, e 3,3 Rot270 (x), for x = e 1,0, e 2,0, e 3,0 . The relationships between the elements are as follows: OnLeft ^e i,j, e i,j + 1 h, for i = 0, f, 3, j = 0, f, 2. Below ^e i + 1,j, e i,j h, for i = 0, f, 2, j = 0, f, 3. / 6j ! " 1, f, m , · ^ x 0,j .Image = x 0,1 .Image / x n + 1,j .Image = x 0,1 .Image / Rot180 ^ x n + 1,j hh / 6i ! " 1, f, n , · ^ x i,0 .Image = x 0,1 .Image / Rot270 ^ x i,0 h / Rot90 ^ x i,m + 1 h / x i,m + 1 .Image = x 0,1 .Image h . The properties of the corner tiles are x 0,0 .Image = x 0,m + 1 .Image = x n + 1,0 .Image = x n + 1,m + 1 .Image, x 0,0 .Width = x 0,0 .length = x 1,1 .Width, Rot90 ^ x 0,m + 1 h, Rot180 ^ x n + 1,m + 1 h, Rot270 ^ x n + 1,0 h . Patterns Definition 3 (pattern): A pattern P in a domain D = T, R, Ax is an ordered pair V, P , where ◆ V = " v 1 : TE 1, f, v k : TE k , is a collection of elements v i of type TE i, which is a type expression; V defines the elements in the pattern ◆ P is a well-formed logic formula (i.e., a predicate) with " v 1 : TE 1, f, v k : TE k , as its free variables. It specifies the constraints on the elements and the relationships between the elements in the pattern. The semantics of pattern P = V, P , written Sem (P), is that 7v 1 ! TE 1, f, 7v k ! TE k · P. A type expression TE can be a type T ! T or a power set type P (TE) if TE is a type expression. Here, the values of a power set type P (T) are nonempty finite sets of elements of type T. Example 3: An Example of a Pattern (Bordered Center). A common pattern in a floor tile layout is the so-called bordered center (BC), in which the center is an n # m matrix of tiles surrounded by matching borders and corner tiles. Figure 3 is an instance of such a pattern. We can formally specify it by defining the set of variables with T Var (BC) = " X : P ^Tile h, and Pred ^ BC h as follows. The layout is an (n + 2) # (m + 2) matrix (with n, m 2 0 ) in which the tiles are laid one next to another. It is easy to see that the tile layout given in Figure 3 is an instance of the BC pattern, because the conditions of the BC pattern are all true when we assign element e i,j in the instance to variables x i,j in the pattern. Definition 4 (satisfaction): Let D be a given domain, z = E, R be a given phenomenon, and P = V, P be a pattern in domain D. If there is a type-preserving assignment a of the variables in V to elements in E, such that [[P]] a,R = true, we say that phenomenon z is an instance of pattern P and write z t P. Note that [[P]] a,R = true means that the evaluation of P under assignment a with the conditions of R is true. The detailed definition is omitted for the sake of space. The aforementioned definitions of pattern and satisfaction enable us to deal with the recognition of an instance of a pattern as the evaluation of logic formulas in a finite structure. For example, to show that the floor layout in Figure 3 is an instance of the BC pattern, we simply need to assign variables in the pattern to particular elements in the instance and thereby prove that all of the conditions of the pattern are true. X = " x i,j | i = 0, f, n + 1, j = 0, f, m + 1 , . Informally, x i,j is the tile laid on row i, column j. OnLeft ^ x i, j, x i,j + 1 h, for i = 0, f, n + 1, j = 0, f, m, Below ^ x i + 1,j, x i,j h, for i = 0, f, n, j = 0, f, m + 1. The properties of the central tiles are x i,j .Image = x 1,1 . Image, for i = 1, f, n, j = 1, fm. The properties of the noncorner border tiles are x 0,1 .VCon / ^ x 0,1 .Width = x 1,1 .Width h Figure 3. an example of a tile-laid floor. Ju ly 2018 IEEE SyStEmS, man, & CybErnEtICS magazInE 19

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