IEEE Systems, Man and Cybernetics Magazine - July 2018 - 18

◆ Width, Length : Integer, whose value is the width (or

length) of the tile in millimeters
◆ Image : JPEG, whose value is the image of the tile in
JPEG format
◆ CSym, HSym, VSym, DSym : Boolean, which indicates
whether the image is centrally, horizontally, vertically,
or diagonally symmetrical, with centrally symmetrical
meaning that every line through the center is symmetrical about that center
◆ HCon, VCon : Boolean, which indicates whether or
not two tiles of that image can be laid out next to each
other horizontally (or vertically); we then say the image
of the tile is horizontally (or vertically) connectable.
Figure 2 shows a number of tiles with different attributes. Tiles (a)-(g) are in the shape of a square, while (h)
and (i) are rectangles. Tiles (a) and (h) have centrally symmetrical images; (b) is both horizontally and vertically
symmetrical, but not centrally symmetrical; (c) is vertically symmetrical, but not horizontally symmetrical. Tiles (d)
and (e) are diagonally symmetrical; (f) and (g) are not symmetrical at all. Tiles (a), (b), (f), (h), and (i) are connectable
both horizontally and vertically, but (c) is only horizontally
connectable, while tiles (d), (e), and (g) are not connectable at all.
The tile relations important for layout are as follows:
◆ OnLeft : Tile # Tile, where OnLeft ^ x, y h means that
tile x is placed on the left of y
◆ Below : Tile # Tile, where Below ^ x, y h means that tile
x is placed below y
◆ Rot90, Rot180, Rot270 : Tile, where Rot90 (x) means
that tile x is laid with the image rotated 90 o clockwise, and Rot180 and Rot270 are similar.
Example axioms about tiles include
6x ! Tile. ^ x.Shape = Square + x.Width = x.Lengthh
6x ! Tile · ^ x.Shape = Square / x.CSymh
& ^ x.HSym / x.VSym h

These axioms state, in order, that square tiles have equal
width and length, centrally symmetrical square tiles are
also vertically and horizontally symmetrical, vertically
symmetrical tiles are horizontally connectable, and horizontally symmetrical tiles are vertically connectable.
Axioms like these represent knowledge that is universally applicable in the subject domain. Element types,
attributes, and relations also represent knowledge that is
often found in the form of ontology and that can often be
obtained from data through feature extraction and data
mining. Image processing could be used to deduce the
symmetry and connectability of tiles. The pattern-oriented
research methodology we promote in this article uses all
of these techniques alongside human knowledge engineering to create reusable, validatable, testable, comprehensible, and transferable knowledge.
Instances of Phenomena
In a given subject domain, it is possible to observe certain
phenomena that consist of a number of elements, i.e., artifacts and events of the domain, related to each other in a
particular way. To recognize a phenomenon as an instance
of a pattern is to recognize its constituent elements, noting
their types and the values of their attributes, and acknowledging how the elements relate to each other.
Definition 2 (instance): Given a subject domain
D = T, R, Ax , an instance z is an ordered pair E, R ,
where
◆ E = " e 1, f, e n , is a collection of particular elements
e i, i = 1, f, k, of type Ti ! T in the domain D, and
◆ R = " r1, f, rm , is a set of relations that hold between
the elements in E.
Example 2: An Instance of a Floor Layout.
Figure 3 shows an instance of a floor tile layout. In this
instance, the elements are E = " e i,j | i, j = 0, f, 3 ,, where
e i,j is the tile laid in row i, column j. They have the following attributes:
6x ! " e 0,0, e 0,3, e 3,0, e 3,3 , · ^ x.Image = Corner.jpeg h

6x ! Tile. ^ x.VSym & x.HCon h

6x = " e 0,1, e 0,2, e 1,0, e 1,3, e 2,0, e 2,3, e 3,1, e 3,2 ,

6x ! Tile. ^ x.HSym & x.VCon h .

· ^ x.Image = Border.jpeg h
6x ! " e 1,1, e 1,2, e 2,1, e 2,2 , · ^ x.Image = Center.jpeg h .
The following rules determine the other attributes:
6x ! Tile. ^ x.Image = Corner.jpeg & x.DSymh

(a)

(b)

(c)

(d)

(e)

6x ! Tile. ^ x.Image = Border.jpeg & x.VSymh
6x ! Tile. ^ x.Image = Center.jpeg & x.DSym h
6x ! Tile. ^ x.Shape = Square h .

(f)

(g)

(h)

(i)

Figure 2. (a-i) Various tiles with different attributes.
18

IEEE SyStEmS, man, & CybErnEtICS magazInE July 2 0 1 8

The central part is laid out with the tiles rotated, i.e.,
Rot90 ^e 1,2 h, Rot180 ^e 2,2 h, Rot270 ^e 2,1h .



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