INTERCALATION (Internal Distribution)

"All language communicates itself" (Walter Benjamin).

"Knowledge is language that names and describes" (Wittgenstein).

"To understand a language means to master a technique" (Wittgenstein).

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Semántica

Describes the result of intercalate an expression x between all components of a sequence y.


Sintaxis

x⊣y   or   y⊢x

Definición

It is performed recursively:

⟨( ((x⊣y =: ( y ←' (y# > 1) → ((y\1 x) ∪ (x⊣(y '∪ y\1))) )⟩

This definition is based on e.g.

x⊣(a b c d) =: ((a x) ∪ (x⊣(b c d))


Justification

The intercalation operation is very useful because it acts between components, which is the place where an infix operator is specified. It also simplifies the specification of expressions.


Ejemplos

1. a⊣(x and z) // rep. (x to y to z) eq. xayaz
   
   
2. +⊣(a b c) // rep. (a + b + c) eq. a+b+c
   
   
3. ^⊣(a b c) // rep. (a ^ b ^ c) eq. a^b^c
   
   
4. *⊣(1 3 5) // rep. 1*3*5 eq. 15
   
   
5. ←⊣(a b c) // rep. (a ← b ← c)
   
   
6. +⊣( 1...10 ) // rep. 1+...+10 rep. 55
   
   
7. *⊣( 1...10 ) // rep. 1*...*10 (factorial of 10)
   
   
8. ab⊣(xyz) // rep. (x ab y ab z)
   
   
9. ((+a+)↓ ⊣ xyz) // rep. x+a+y+a+a+z
   
   
10. abc⊣abc // rep. (a abc b abc c)
    
    
11. α⊣1234 // rep. 1α2α3α4
    
    
12. (u★3)⊣(a b c) // rep. (a u★3 b u★3 c) rep. auuubuuuc
    
    
13. (1...5)⊣(a b c) // rep. (a 1...5 b 1...5 c) rep. a12345b12345c
    
    
14. ((3+4)°)⊣(a b c) // rep. (a 3+4 b 3+4 c)
    
    
15. (a°°)⊣(1 2 3) // rep. (1 a° 2 a° 3)
    
    
16. (1...5)⊣( 1...3 ) // rep. (1 1 1...5 2 1...5 3) rep. 1123452123453
    
    
17. ()⊣(a b c) // rep. (a () b () c) (empty sequence interleaving)

Remarks

• Intercalation is also called "internal distribution". Since the evaluation is top-down hierarchical, the distribution operator takes precedence. Thus, for example,
  
  x⊣(θ θ θ θ) // rep. (θ x θ x θ) rep. (x x)
  
  
• It follows from the definition that when the sequence has only one component, the expression is self-evaluating. Examples:
  
  a⊣b a⊣() a⊣( abc ) abc⊣( abc )
  
  
• The derivative "intercaltion" has been defined as descriptive (like the distribution), but it could also be defined as operational, by immediate substitution.

Higher order intercalations

1. a⊣((b c)⊣(u v w)) // rep. a⊣(u bc v bc w) rep. (u a bc a v a bc a w)
   
   
2. (ab⊣ab)⊣ab // rep. (a ab b)⊣ab rep. (a ab ab ab ab b)
   
   
3. ab⊣(ab⊣ab) // rep. ab⊣(a ab b) rep. (a ab ab ab ab b)
   
   
4. x⊣(x⊣(a b c)) // rep. x⊣(a x b x c) rep. (a x x x b x x x c)
   
   Therefore, (x⊣(x⊣(a b c)) ≡ ((x★2)⊣(a b c))
   
   En general, ⟨( x⊣(x⊣y) ≡ ((x★2)⊣y) )⟩

Properties

1. ⟨( θ⊣x = x )⟩ // null expression interleaving
   
   
2. ⟨( (x⊣y)⊣z ≡ x⊣(y⊣z) )⟩ // asociatividad

Infinite type intercalations

Examples:

1. a⊣( 1... ) // rep. (1 to 2 to 3 to ...)
   
   
2. ( a★ )⊣(1 2 3) // rep. (1 (a a a a a ...) 2 (a a a a a ...) 3)