LINKED EXPRESSIONS

"At the fundamental level things have no intrinsic properties; all properties are relations between things" (Leo Smolin).

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Simple Linked Expressions
Definition

Two expressions, x and y, are simply linked from x to y when:

1. There exists a dependency relationship between them such that y (wholly or partially) is a function of x (also wholly or partially).
   
   
2. The relationship is permanent, i.e., at all times if x changes, y automatically changes.

This functional relationship is defined by a generic expression:

• If the generic expression is parameterized, the link is generic (between categories of expressions). If it is not parameterized, the link is specific.
  
  
• The generic expression may appear independently of the related x and y expressions and/or appear directly (locally) in the target expression, y.

Examples

1. Specific binding of x to y. The relation is defined by an independent generic expression.
   
   ⟨( y\2 = (x\1 + x\2) )⟩
   (component 2 of y is always the sum of the first two components of x)
   
   ⟨( (y\4 = (x3 + x\4) )⟩
   (component 4 of y is always the sum of components 3 and 4 of x)
   
   (y = (a b c d e))
(x = (1 2 3 4))
y // ev. (a 3 c 7 e)

(x = (2 3 4 5))
   (new value of x)
   y // ev. (a 5 c 9 e)
   (y changes automatically)
   
   
2. Specific direct linkage from x to y. The relationship is specified in the target expression.
   
   (y = (1 2 3))
(x = (a ⟨y\2⟩ b)) // ev. x=(a 2 b)

(y = (4 5 6))
   (new value of y)
   x // ev. (a 5 b)
   (x changes automatically)
   
   
3. Generic linkage: in all sequences of length greater than 1, the first component must always be the first element of u.
   
   (u = (a b c d))
⟨( (x# > 1) → (x\1 = u\1) )⟩ // generic relation

(v = (7 2 3 4)) // ev. (v = (a 2 3 4))
(w = (8 5 6 6 9)) // ev. (w = (a 5 6 9))

(u = (d c b a)) // redefine u
v // ev. (d 2 3 4)
w // ev. (d 5 6 9)
   
   
4. Local linkage using a previously defined function.
   
   Calculates the minimum value between n1 and n2.
   ⟨( (minimo(n1 n2) = (n1 ← n1≤n2) →' n2) )⟩

(u = (3 7 ⟨ min(u\1 u\2) ⟩)) // ev. (u = (3 7 3))

(u\1 = 9) // we change the first element.
u // ev. (9 7 7)

Interlaced Expressions
Definition

Two interlaced expressions, x and y, are mutually linked expressions. That is, there are dependency relationships from x to y, and vice versa. Therefore, the content of one expression is a function of the other and vice versa, and any change in one of them immediately affects the other. The entanglement can be partial or total, depending on the extent of the affected expressions.


Example

Specific entanglement between x and y.

(x = (1 2 3 4))
(y = (a b c d))

⟨( x\1 = (y\1 + y\2) )⟩
(ratio of y to x)

⟨( y\3 = (x\2 + x3) )⟩
(ratio of x to y)

x // ev. (a+b 2 3 4)
y // ev. (a b 5 d)


Self-Linked Expressions
Definition

A self-linked expression is an expression linked to itself.


Examples

1. Self-linking specific to x:
   
   ⟨( x\3 = (x\1 + x\2) )⟩
   (component 3 of x is always the sum of the first two).
   
   (x = (10 11 12 13)) // ev. (x = (10 11 21 13))

(x = 0) // we change the second element.
x // ev. (10 0 10 13)
   
   
2. Generic self-linking: in all sequences of length greater than 2, the third component must always be the sum of the first two.
   
   ⟨( (x# > 2) → (x3 = (x1 + x2)) )⟩

(u = (10 11 12 13)) // ev. (u = (10 11 21 13))

(v = (a b c)) // ev. (v = (a b a+b))

(v = ( abc )) // ev. (v = ( abc ) (the sequence is of length 1)

Higher Order Linked Expressions
Definition

They are those in which the link is made to already linked expressions.

In general, we can have a network of functional relations in the form of linked, simple, interlinked, self-linked or self-interlinked expressions. And using or combining generic and specific expressions.


Examples

1. ⟨( x\1 = (u\1 + v\1) )⟩
⟨( x\2 = (u\2 + v\2) )⟩
⟨( y\1 = x\2 )⟩
⟨( y\2 = x\1 )⟩
(u = (1 2 3 4))
(v = (11 12 13 14))
x // ev. (12 14)
y // ev. (14 12)
   
   
2. ⟨( v\1 = u\1 )⟩
⟨( u\2 = (v\1 + 1) )⟩
⟨( v\2 = (u\2)2 )⟩
⟨( u\3 = (v\2 + 1) )⟩
⟨( v\3 = (u\3)*2 )⟩
⟨( u\4 = (v\3 + 1) )⟩
⟨( v\4 = (u\4)*2 )⟩
(u = 1)
u // ev. (1 2 5 11)
v // ev. (1 4 10 22)
   
   
3. The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, ...) is an example of a higher-level self-linked expression, since each element of the sequence is successively deduced from the previous two.
   
   ( fibo = (1 1) ) // initial Fibonacci sequence.
⟨ ((fibo =: (fibo(i−1) + fibo(i−2)))←((i>2) ⟩
   
   By defining the function as a potential substitution, we have the potential Fibonacci sequence:
   (1 1 2 3 5 8 ...).
   
   We could also have defined it with actual substitution, but limiting it to a certain length (number of elements n):
   
   ( fibo = (1 1) ) // initial Fibonacci sequence.
(n = 8) // 8 elements
⟨ ((fibo = (fibo(i−1) + fibo(i−2)))) ← (i>2 ∧ i≤n) ⟩
fibo // ev. (1 1 1 2 3 5 8 13 21)