SPECIAL TECHNIQUES

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

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Conversion from sequence to set and vice versa

Conversion of sequence s to set: {s↓}

Conversion of set c in sequence: ( c↓ )

Examples:

1. (s = (a b c))
{s↓} // ev. {a b c}
   
   
2. (c = {a b c})
( c↓ ) // ev. (a b c)

Access to the integer and decimal parts of a number

Examples:

1. (r = 123.456) // positive number equivalent to (123 . 456)
r\1 // ev. 123 (integer part)
r\3 // ev. 456 (decimal part)
   
   
2. (r = −123.456) // number equivalent to (− (123 . 456))
r\2\1 // ev. 123 (whole part)
r\2\3 // ev. 456 (decimal part)

Access to the content of a text

Although a text is always self-evaluating, it is possible to access its content, as a sequence that is, even change it.

It should be noted that the double quotation marks act as parentheses, but prevent against the evaluation of its components.

Example:

x = "abcde"
x\1 // ev. a
x2 // ev. b
...
( x↓ ) // ev. abcde

Removal of repeated components of a sequence

The process is as follows:

• The sequence is converted as a whole.
• As this set is evaluated, repeated elements are removed.
• The resulting expression is converted back to sequence.

The expression is: ( {x↓}↓ )

Example:

(x = (a a b c c c c))
{x↓} // ev. {a a a b c c c c} ev. {a b c}
( {x↓}↓ ) // ev. ( a b c ) ev. abc

Hierarchical hyperaccess

These are hierarchical accesses with a variable number of levels. They are realized by the expression

(x ☆n) or (x ↑star;n)

where n is an integer equal to or greater than zero.

Examples:

1. (x = (a (b (b (c d))))
⟨( v(n) = (x ↓☆n) )⟩
v(0) // ev. (a (a (b (c (c d))))
v(1) // ev. a (b (c d))
v(2) // ev. b (c d)
v(3) // ev. c d
v(4) // ev. c d
   
   
2. (x = (a*b + a*c + b*c))
⟨( v(n) = (x ↓☆n) )⟩
v(0) // ev. (a*b + a*c + b*c)
v(1) // ev. a*b + a*c + b*c
v(2) // ev. a * b a * c b * c
v(3) // ev. a * b a * c b * c

Distribution of repetitions

Corresponds to expressions in which ★n is distributed among the components of an expression. Examples:

1. [[a b c]★2] // rep. a★2 b★2 c★2 rep. a a b b b c c
   
   
2. [[a b c]★⌊1...3⌋] // rep. a★1 b★2 c★3 rep. a b b b c c c
   
   
3. [[a b]★[1 2]] //rep. a★1 a★2 b★1 b★2 rep. a a a a b b b

Operator substitution

In an expression it is possible to perform substitutions of expressions in general and, therefore, also of operators. Examples:

1. ((a+b+c)/(+° = *°) // ev. a*b*c
2. (1/5)/(/° = ...°) // ev. 1...5 rep. 12345
3. (x/3)/(/° = = =°) // ev. x=3

Insertion

Here are some illustrative examples of inserting an element into a sequence.

1. (x = (a b c d))
x = x/(b = (b u)↓) // insertion of element u after element b
x // ev. (a b u c d)
   
   
2. (x = (a b c d b))
x = x/(b = (b u)↓) // insertion of element u after elements b
x // ev. (a b u c c d b u)
   
   
3. (x = (a b c d))
x = x/(b = (b u v)↓) // insertion of elements u and v after elem. b // ev. (a b u v c d)
   
   
4. (x = (a b c d))
x = (u x↓) // insertion of element u in front of the first element of x
x // ev. (u a b c d)
   
   
5. (x = (a b c d))
x = (x↓ u) // insertion of element u after the last element of x
x // ev. (a b c d u)

Ranges and distribution

You can simplify the specification of expressions with ranges by a distribution operation.

Examples:

1. (1...3 1 1...9 1...5) eq. ([1...[3 9 5]]) This is a sequence of three ranges, which share the same initial element.
   
   
2. (3...1 4...1 5...1) eq. ([[3 4 5]...1]) eq. ([[3...5]...1])
   
   In this case the three ranks share the final element, with the initial elements forming another rank.
   
   
3. (1...3 1...4 1...5) eq. ([1...[3 4 5]]) eq. ([1...[3...5]]) This is a sequence of three ranges that share the same initial element and whose final elements form, in turn, another range.

Ranges of variables

The specification of a range of variables x1 ... xn, can be done as follows:
x1...xn, since (x1+1 = x2), (x2+1 = x3), etc.

Example:

((x1 = ab) (x2 = 12) (x3 = 44) (x4 = cd))
( ( ( x1...x4 ) ) ) // rep. (ab 12 44 cd)


Operators as arguments

Operators can be arguments of generic expressions. Examples:

1. ⟨( r(p) = (1p5) )⟩
r(...) // rep. (1...5) rep. 1 2 3 4 5
r(999) // ev. (1 999 5)
   
   
2. ⟨( z(p) = 1p2p3p4p5 )⟩
z(θ) // ev. 12345
r(+) // ev. 1+2+3+4+4+5 ev. 15
r(44) // ev. (1 44 2 44 3 44 4 44 5)

How to know if an expression is part of another

If we want to know if the expression x contains another y:

("x contains y" ←' (x/(y=θ) = (x/(y=θ)°) → "x does not contain y")

For example, if x=(a b c):

x/(b=θ) // ev. (a c) ≠ x (b is contained in x)
x/(r=θ) // (self-evaluates, r is not contained in x)


Parameterized variables vs. sequences

An alternative to sequences is to use parameterized variables. For example, the sequence x=(a b c d) could be replaced by

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

To refer to the complete sequence you would have to do

( [x([1...4]) = a)] )

This technique has the advantage that the argument can be of any type and not just numeric.