Initial functions are:
| Zero constant | Z(x) = 0 |
| One constant | 1(x) = 1 |
| Two constant | 2(x) = 2 |
| Successor function | S(x) = x + 1 |
| Double function | D(x) = 2x |
| Predecessor function | P(x) = x - 1, if x > 0 |
| | P(0) = 0 |
| Identity function | I(x) = x |
| Square function | Sq(x) = x*x |
| Square root function | Rt(x) = floor(sqrt(x)) |
| Power of zero | O(x) = 0, if x > 0 |
| | O(0) = 1 |
| Half function | Hf(x) = floor(x/2) |
| Alternation function | N(x) = 1, if x is odd |
| | N(x) = 0, otherwise |
| Power of two | Pw(x) = 2^x |
| Char. of squares | Q(x) = 1, if x is a square number |
| | Q(x) = 0, otherwise |
| Excess over a square | E(x) = x - Sq(Rt(x)) |
| M function | M(x) = x mod 3 |
| T function | T(x) = x + 2Rt(x)
(see
id:A112594) |
| Triangular numbers | A(x) = xth triangular number |
| V function | V(x) = inverse of A |
| K function | K(x) = first inverse of J |
| L function | L(x) = second inverse of J |
| W function | W(x) = too complicated to explain |
You can download it here:
Urfidesh.jar,
Urfidesh.jad.
|
Operators are:
| Addition | F+G(x) = F(x) + G(x) |
| Substraction | F-G(x) = F(x) - G(x), if F(x) >= G(x) |
| Composition** | F G(x) = F(G(x)) |
| Distance | |F-G|(x) = |F(x) - G(x)| |
| Arith. difference | F_G(x) = F(x) - G(x), if F(x) >= G(x) |
| | F_G(x) = 0, otherwise |
| Product | F.G(x) = F(x)G(x) |
| Division | F/G(x) = floor(F(x)/G(x)), if G(x) > 0 |
| Cantor function | J(F, G)(x) =
(see
cantor pairing function) |
| Iteration from 0 | F"(x) = F(F(...(F(F(0)))...)) x-times |
| Iteration from 1 | F^1(x) = F(F(...(F(F(1)))...)) x-times |
| Iteration from 2 | F^2(x) = F(F(...(F(F(2)))...)) x-times |
| F + I | F^+(x) = F(x) + x |
| F - I | F^-(x) = F(x) - x, if F(x) >= x |
| Inversion | F*(x) = min value of y such that F(y) = x holds |
| Summation | F#(x) = F(0) + F(1) + ... + F(x) |
(**) Spaces are used only for composition.
Precedence:
1) addition, substraction and arith. difference (left assoc.)
2) product, division (left assoc.)
3) composition
4) unary operators
5) parenthesis
Examples:
| Successor function | (K^1 K K^1)^+ |
| Square function | (S T)" |
| Half function | P Rt (E*-I) S |
| M function | (E S S)" |
| T function | (S+D Q S S S S)" |
| V function | Hf P Rt S D D D |
| K function | I-A V |
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