REPETITION

"But what has once been said can always be repeated" (Zeno of Elea).

"Repetition in space is harmony" (Jorge Wagensberg).

"Repetition is the only form of permanence that nature can achieve" (George Santayana).

"Order is repetition of units, Chaos is multiplicity without rhythm" (M.C. Escher).

---

Finite open-ended repetition
Semantics

Repeat n times an expression x, where n is an integer greater than or equal to zero. The result is an open expression.

• If n is zero, the result is the null expression θ.
  
  
• If n is not a number or is not a number greater than zero, the expression is self-evaluating.

Syntax

x☆n // repeat x n times

Definition (recursive)

⟨( x☆n =: ( (θ ← (n = 0) →'
(x ← (n = 1) →' (x (x☆(n−1))↓))↓ ) )⟩


Justification

Repetition is a simplified way of specifying an expression that is repeated a certain number of times.


Examples

1. a☆4 // rep. a a a a a a a a a
   
   
2. ( a☆4 ) // rep. ( a a a a a a a ) ev. aaaa
   
   
3. (ab)☆3 // rep. ab ab ab ab
   
   
4. abc☆0 // ev. θ
   
   
5. α☆3 // rep. α α α
   
   
6. (ab↓)☆3 // rep. a b a b a b a b a b
   
   
7. a☆(-3) // self-evaluates
   
   
8. a☆(3.5) // self-evaluate
   
   
9. 3☆xxx // self-evaluates

Remarks

• Repetition is a descriptive expression that is defined recursively.
  
  
• The result is an open expression to facilitate its connection with other operations.
  
  
• The * and ☆ operators, although related to repetition, are different. The * operator is associated with the operation of addition. The ☆ operator indicates repetition of an expression.

Properties

1. ⟨( ( (x☆0) = θ )⟩ // by definition.
   
   
2. ⟨( (x☆n)☆m ≡ x☆(n+m) )⟩
   
   Example: (ab☆2)☆3 // rep. ab ab ab ab ab ab ab ab ab ab ab ab ab eq. ab☆6
   
   
3. ⟨(( (x☆n x☆m)↓ = x☆(n+m) )⟩
   
   Example: (ab☆2 ab☆3) // rep. (ab ab ab ab ab ab ab ab) eq. ( ab☆5 )
   
   
4. ⟨( θ☆n = θ )⟩ // by the definition
   
   
5. ⟨( ({x☆n} = {x}) )⟩
   
   Example: {ab☆4} // rep. {ab ab ab ab} ev. {ab}
   
   
6. ⟨( ( ( ( x☆ n )# = n )⟩ // by the definition of repeat and length

Infinite open repetition
Semantics

Repeat indefinitely to the left or to the right. The result is an open descriptive expression.


Syntax

x☆ // infinite repetition to the right.

☆x // infinite repeat left


Definition (recursive)

⟨( x☆ =: (x x x☆)↓ )⟩ // infinite repeat to the right.

⟨( ☆x =: (☆x x x)↓ )↓ )⟩ // infinite repeat to the left


Examples

1. ab☆ // rep. ab ab ab ab ...
   
   
2. ( ab☆ ) // rep. ( ab ab ab ab ... )
   
   
3. ( 1☆ ) // rep. 111111...
   
   
4. ☆123 // rep. ... 123 123 123 123
   
   
5. sentence☆
   Equivalent to repeating statement infinitely many times. This has application in specifying potentially infinite loops, which terminate when a certain condition is met.

Properties

1. ⟨( ( ( ( x☆ )# = ∞ )⟩ // by definition of infinite repetition and length.
   
   
2. ⟨( ( ( ☆x )# = ∞ )⟩ // id.

Closed repetition
For practical reasons, we further define the repetition that produces a sequence, finite or infinite:


Syntax

x★n // repeat x n times

Definition

x★n = ( x☆n )

Examples

a★4 // rep. ( a a a a a a ) ev. aaaa
a★ // rep. ( a a a a a a ... )