IEEE Systems, Man and Cybernetics Magazine - July 2018 - 19

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

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