REVERSAL

"Symmetry is a characteristic of the human mind" (Alexander Pushkin).

"Symmetry underlies the laws of the universe" (Brian Greene).

"Symmetry implies dynamics" (Eugene Wigner).

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Semantics

Reveert an expression. If the expression is a sequence or an open expression, it is equivalent to evaluating the sequence starting from the right end. If the expression is a set, the operation has no effect, since all components are evaluated in parallel.


Syntax

x∼ // evaluate sequence x on the right-hand side


Definition

⟨( (x∼ = ([x/[x#… 1]])) ← (x=( x↓ )⟩ // case x sequence or open expression;

⟨( (x∼ = x) ← (x={x↓} )⟩ // case x set


Justification

This is a very useful operation, especially in arithmetic expressions.


Examples

1. (a b c d)∼ // ev. (d c b a)
   
   
2. (r1 ^ r2 ^ r3 ^ r4)∼ // ev. (r4 ^ r3 ^ r2 ^ r1)
   
   This expression would evaluate as ((r4^r3)^r2)^r1
   
   
3. (x ← y ← z ← u)∼ // ev. (u → z → y → x) (are equivalent)
   
   
4. (a/b/c/d)∼ // ev. d/c/b/a
   
   
5. (xy/(x=3)/(y=2))∼ // ev. (y=2)/(x=3)/xy
   
   
6. ∼(2^3^5) // ev. (2^3^5) ev. (2^3)^5 ev. 8^5 ev. 32768
   
   
7. (2^3^5)∼ // ev. (5^3^2) ev. (5^3)^2 ev. 125^2 ev. 15625
   
   
8. {a b c}∼ // ev. {a b c}

Remarks

• By defiinition, ⟨( ∼x = x )⟩, i.e. is the default evaluation
  
  
• Following the principle of top-down hierarchical evaluation, the reversal takes place before the evaluation of the expression. That is, the expression is first reversed and then evaluated.
  
  
• The operation only affects the first level of the expression hierarchy. That is, if an expression contains, in turn, expressions, the latter are not reversed.
  
  ((a b) (c d))∼ // ev. ((c d) (a b))
  
  
• If the expression is a descriptive sequence, the reversal will normally be descriptive as well.
  
  ( 1 ... 5 )∼ // ev. ( 5 ... 1 )
  
  
• If the sequence has only one component, the result is invariant.
  
  2∼ // ev. 2

Properties

1. ⟨(x∼∼ = x)⟩ // by the definition of reversal
   
   
2. ⟨( (x∼ = x) ← (x# = 1) )⟩ // atomic expression reversal
   
   
3. ⟨( ( ( r1 ... r2 )∼ = ( r2 ... r1 )⟩ // range reversal