IEEE Systems, Man and Cybernetics Magazine - July 2018 - 21

VL =T Matrix 6Cols = 1 /

be defined. For example, let M.Cols and M.Rows denote
the numbers of columns and rows of a matrix M. The
width of a matrix M can be defined as follows:

6x, y ! b · ^ x.Image = y.Image / x.VConh@ .

M.Width =T x.Width # M.Cols, where x ! M.b.

The corners pattern specifies the constraints on the
tiles placed on four corners of a bordered central layout.

Here, note that we used the so-called dot notation M.x to
denote the variable x in an instance or pattern variable
M so that naming conflicts can be avoided without
renaming the variables.
A matrix of identical tiles that are connectable, called a
styled matrix, can be defined by applying the restriction
operator on the matrix pattern.
SM =T Matrix 66x ! b. ^^ x.Image = b 1,1 .Image h
/ x.VCon / x.HCon h@ .

(a) Laws on Subpattern Relation :
Θ) ∧ (Θ
Π Θ∧Θ

/ Rot90 (c ur) / Rot180 (c lr) / Rot270 (c ll).
Now we apply the pattern operators to define the BC
pattern. First, we rename the variables of the borders to
avoid naming conflicts in the pattern composition.

Left =T VL (l/bl); Right =T VL (l/br) .
The BC pattern can now be defined as follows:
BC =T ^Top ) Bottom ) Left ) Right ) SM ) Corners h
6Connection@,

(d) Laws on ⇓ and ⇑:
Π( X ⇓ x )( Y ⇓ y ) ≈ Π( Y ⇓ y )( X ⇓ x )

Ψ) ⇒ (Π Ψ)
Π⇒Π≈Θ

Π( x ⇑ X )( y ⇑ Y ) ≈ Π( y ⇑ Y )( x ⇑ X )

Π

TRUE

(b) Laws on Restriction [c]:
( c1 ⇒ c2 ) ⇒ Π[c1 ] Π[c2 ]
Π[c][c] ≈ Π[c]
Π[c1 ][c2 ] ≈ Π[c2 ][ c1]
Π[c1 ][c2 ] ≈ Π[c1 ∧ c2 ]
Π[true ] ≈ Π
Π[false ] ≈ FALSE
(c) Laws on Superposition ∗:
(Π ∗ Θ)
Θ

/ 6x ! cn · ^^ x.Image = c ul .Image) / x.DSymh

Π

Π

FALSE

Pred (Corners) =T ^cn = " cul, cur, cll, clr ,h

Top =T HL (l/bt); Bottom =T HL (l/bb);

The BC pattern with connectivity (BC-HVCon) can be
defined as being composed of top, bottom, left, and right
borders with a styled central matrix and four corners. The
left and right borders are vertical lines (VLs), which can be
defined as a matrix of one column with the condition that
the tiles are vertically connectable.

(Π

Var (Corners) =T " cn : P (Tile) ,,

Π

Π⇒ Π∗Θ ≈ Θ
Π∗Π≈ Π

Π ∗ TRUE ≈ TRUE ∗ Π ≈ Π
Π ∗ FALSE ≈ FALSE ∗ Π ≈ FALSE
Π∗Θ ≈ Θ∗Π
(Π ∗ Θ) ∗ Ψ ≈ Π ∗ (Θ ∗ Ψ)

(e) Laws connecting ∗ with others:
Π[c] ∗ Θ ≈ (Π ∗ Θ)[ c]
Π( x ⇑ X ) ∗ Θ ≈ (Π ∗ Θ)( x ⇑ X )
Π( X ⇓ x ) ∗ Θ ≈ (Π ∗ Θ)( X ⇓ x )
Π( X ↑ Xs ) ∗ Θ ≈
(Π ∗ Θ)( X ↑ Xs )( ( VΘ ) ⇓ VΘ )
(f) Laws connecting ⇑, ⇓ , and ↑:
Π( x ⇑ X )( X ⇓ x ) ≈ Π
Π( X ⇓ x )( x ⇑ X ) ≈ Π
Π( X ↑ Xs )( Xs ⇓ Var (Π)) ≈ Π
Π( x ⇑ X ) ≈ Π( X ↑ Xs )( Xs − X ↑ ⇓ X )
(g) Laws connecting [c ] with ⇑, ⇓ , and ↑:
Π(X ⇓ x ) ≈ Π[∃ x · ( X = { x } )]
Π( x ⇑ X ) ≈ Π[∃ X · (∀x ∈ X · p)]
Π( X ↑ Xs ) ≈ Π[∃ Xs · ( pX ↑Xs )]
Π[c]( x ⇑ X ) ≈ Π( x ⇑ X )[ cx ⇑ X ]
Π[c]( X ↑ Xs ) ≈ Π( X ↑ Xs )[ cX ↑Xs ]
Π[c]( X ⇓ x ) ≈ Π( X ⇓ x )[ cX ⇓ x ]

Figure 4. the laws of pattern operators, including on (a) subpattern relations; (b) restriction [ c ]; (c) superposition

*; (d) 0 and / ; (e) connecting * with others; (f) connecting / , 0 , and - ; and (g) connecting [ c ] with / , 0 ,
and - .

Ju ly 2018

IEEE SyStEmS, man, & CybErnEtICS magazInE

21



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