A binary variation with brackets
To be certain that bracket characters count as algorithmic information we set up a mathematical experiment. The table below captures the resolution of a structure for a binary character algorithm plus brackets.
As the length of the algorithmic words grows, eventually the number
of different expressions per extra character approaches that of (a fraction of)
radix 2, so that the algorithm's resolution becomes almost maximal.
When brackets can be inserted freely (without counting them as characters), the total
of all potentially different expressions becomes higher than radix 2 after 7 characters.
The algorithm is some function with an arbitrary number of parameters.
Simple next-placement of number values is a commutative operation,
taken to be addition as usual.
The order is by: 1. number of characters (without brackets), 2. conjectured size of the
resulting number (after reduction), given that parameter values on the right
consistently dominate every result (as usual in an operatorial function
for Big numbers, as if 1↑ω
).
About the function of the separators ,
nothing is known.
In the
green table
the arithmetical names
on the right suggest an interpretation which reduces the resolution
considerably, different names often expressing the same number. But when the word length
increases these overlaps will become rarer, so even here radix 2 can be beaten
(helped by minimal `
brackets).
In the
blue table
there are version names for the numbers instead.
These help us maintain the same order among numbers
while we increase the expression size and make inventory after every next
1,
character.
Here when the names change shape
this announces a new (sub)class of algorithmic expressions.
Multiple separators ,..
(for array dimensions) are
not yet
already
implemented.
To display the numbers
that go beyond
covered just by
regular functions,
click here!
Page link:
www.allergrootste.com/big/Retro/expo/binary-variation.html?single-row
Page link:
www.allergrootste.com/big/Retro/expo/binary-variation.html?multi-dimensional
char # | word | name |
---|---|---|
1 | ||
1 | 1 | 1 |
2 | ||
2 | 11 | 2 |
3 | ||
3 | 111 | 3 |
4 | 1,1 | 1*1 |
4 | ||
5 | 1111 | 4 |
6 | (1,1)1 | 1*1+1 |
7 | 11,1 | 2*1 |
8 | 1,11 | 1*2 |
5 | ||
9 | 11111 | 5 |
10 | (1,1)11 | 1*1+2 |
11 | (11,1)1 | 2*1+1 |
12 | 111,1 | 3*1 |
13 | (1,1),1 | 1*1`*1 |
14 | (1,11)1 | 1*2+1 |
15 | 11,11 | 2*2 |
16 | 1,111 | 1*3 |
17 | 1,(1,1) | 1*`1*1 |
18 | 1,1,1 | 1^1 |
6 | ||
19 | 111111 | 6 |
20 | (1,1)111 | 1*1+3 |
21 | (1,1)(1,1) | 1*1+.. [#2] |
22 | (11,1)11 | 2*1+2 |
23 | (111,1)1 | 3*1+1 |
24 | 1111,1 | 4*1 |
25 | ((1,1),1)1 | 1*1`*1+1 |
26 | (1,1)1,1 | 1*1+1`*1 |
27 | (11,1),1 | 2*1`*1 |
28 | (1,11)11 | 1*2+2 |
29 | (1,11),1 | 1*2`*1 |
30 | (11,11)1 | 2*2+1 |
31 | 111,11 | 3*2 |
32 | (1,1),11 | 1*1`*2 |
33 | (1,111)1 | 1*3+1 |
34 | 11,111 | 2*3 |
35 | 1,1111 | 1*4 |
36 | (1,(1,1))1 | 1*1*1`+1 |
37 | 11,(1,1) | 2*`1*1 |
38 | 1,(1,1)1 | 1*`1*1+1 |
39 | 1,(11,1) | 1*`2*1 |
40 | 1,(1,11) | 1*`1*2 |
41 | (1,1,1)1 | 1^1+1 |
42 | 11,1,1 | 2^1 |
43 | 1,11,1 | 1^2 |
44 | 1,1,11 | 1^^1 |
7 | ||
45 | 1111111 | 7 |
46 | (1,1)1111 | 1*1+4 |
47 | (1,1)(1,1)1 | 1*1+..1 [#2] |
48 | (11,1)111 | 2*1+3 |
49 | (11,1)(1,1) | 2*1+1*1 |
50 | (111,1)11 | 3*1+2 |
51 | (1111,1)1 | 4*1+1 |
52 | 11111,1 | 5*1 |
53 | ((1,1),1)11 | 1*1`*1+2 |
54 | ((1,1)1,1)1 | 1*1+1`*1+1 |
55 | (1,1)11,1 | 1*1+2`*1 |
56 | ((11,1),1)1 | 2*1`*1+1 |
57 | (11,1)1,1 | 2*1+1`*1 |
58 | (111,1),1 | 3*1`*1 |
59 | ((1,1),1),1 | 1*1`*1`*1 |
60 | (1,11)111 | 1*2+3 |
61 | (1,11)(1,1) | 1*2+1*1 |
62 | ((1,11),1)1 | 1*2`*1+1 |
63 | (1,11)1,1 | 1*2+1`*1 |
64 | (11,11)11 | 2*2+2 |
65 | (11,11),1 | 2*2`*1 |
66 | (111,11)1 | 3*2+1 |
67 | 1111,11 | 4*2 |
68 | ((1,1),11)1 | 1*1`*2+1 |
69 | (1,1)1,11 | 1*1+1`*2 |
70 | (11,1),11 | 2*1`*2 |
71 | (1,11),11 | 1*2`*2 |
72 | (1,111)11 | 1*3+2 |
73 | (1,111),1 | 1*3`*1 |
74 | (11,111)1 | 2*3+1 |
75 | 111,111 | 3*3 |
76 | (1,1),111 | 1*1`*3 |
77 | (1,1111)1 | 1*4+1 |
78 | 11,1111 | 2*4 |
79 | 1,11111 | 1*5 |
80 | (1,(1,1))11 | 1*1*1`+2 |
81 | (1,(1,1)),1 | 1*`1*1`*1 |
82 | (11,(1,1))1 | 2*1*1`+1 |
83 | 111,(1,1) | 3*`1*1 |
84 | (1,1),(1,1) | 1*1`*1*1` |
85 | (1,(1,1)1)1 | 1*`1*1+1`+1 |
86 | 11,(1,1)1 | 2*`1*1+1 |
87 | 1,(1,1)11 | 1*`1*1+2 |
88 | (1,(11,1))1 | 1*2*1`+1 |
89 | 11,(11,1) | 2*`2*1 |
90 | 1,(11,1)1 | 1*`2*1+1 |
91 | 1,(111,1) | 1*`3*1 |
92 | 1,((1,1),1) | 1*1*1`*1` |
93 | (1,(1,11))1 | 1*1*2`+1 |
94 | 11,(1,11) | 2*`1*2 |
95 | 1,(1,11)1 | 1*`1*2+1 |
96 | 1,(11,11) | 1*`2*2 |
97 | 1,(1,111) | 1*`1*3 |
98 | 1,(1,(1,1)) | 1*`1*`1*1 |
99 | (1,1,1)11 | 1^1+2 |
100 | (1,1,1),1 | 1^1*1 |
101 | 1,(1,1,1) | 1*1^1 |
102 | (11,1,1)1 | 2^1+1 |
103 | 111,1,1 | 3^1 |
104 | (1,1),1,1 | 1*1`^1 |
105 | (1,11,1)1 | 1^2+1 |
106 | 11,11,1 | 2^2 |
107 | 1,111,1 | 1^3 |
108 | 1,(1,1),1 | 1^`1*1 |
109 | (1,1,11)1 | 1^^1+1 |
110 | 11,1,11 | 2^^1 |
111 | 1,11,11 | 1^^2 |
112 | 1,1,111 | 1^^^1 |
113 | 1,1,(1,1) | 1^..1 [^#1*1] |
114 | 1,1,1,1 | 1st Ackermann |
char # | word | name |
---|---|---|
1 | ||
1 | 1 | 0.1 |
2 | ||
2 | 11 | 0.2 |
3 | ||
3 | 111 | 0.3 |
8 | 1,1 | 1.0 |
4 | ||
4 | 1111 | 0.4 |
9 | (1,1)1 | 1.1 |
15 | 11,1 | 2.0 |
37 | 1,11 | 3.0 |
115 | 1,,1 | 4.0 |
5 | ||
5 | 11111 | 0.5 |
10 | (1,1)11 | 1.2 |
16 | (11,1)1 | 2.1 |
20 | 111,1 | 2.2 |
26 | (1,1),1 | 2.3 |
38 | (1,11)1 | 3.1 |
45 | 11,11 | 3.2 |
57 | 1,111 | 3.3 |
69 | 1,(1,1) | 3.4 |
94 | 1,1,1 | 3.5 |
116 | (1,,1)1 | 4.1 |
128 | 11,,1 | 5.0 |
146 | 1,,11 | 6.0 |
171 | 1,,,1 | 7.0 |
6 | ||
6 | 111111 | 0.6 |
11 | (1,1)111 | 1.3 |
13 | (1,1)(1,1) | 1.4 |
17 | (11,1)11 | 2.1.1 |
21 | (111,1)1 | 2.2.1 |
23 | 1111,1 | 2.2.2 |
27 | ((1,1),1)1 | 2.3.1 |
29 | (1,1)1,1 | 2.4 |
32 | (11,1),1 | 2.5 |
39 | (1,11)11 | 3.1.1 |
42 | (1,11),1 | 3.1.2 |
46 | (11,11)1 | 3.2.1 |
49 | 111,11 | 3.2.2 |
52 | (1,1),11 | 3.2.3 |
58 | (1,111)1 | 3.3.1 |
61 | 11,111 | 3.3.2 |
65 | 1,1111 | 3.3.3 |
70 | (1,(1,1))1 | 3.4.1 |
73 | 11,(1,1) | 3.4.2 |
77 | 1,(1,1)1 | 3.4.3 |
81 | 1,(11,1) | 3.4.4 |
87 | 1,(1,11) | 3.4.5 |
95 | (1,1,1)1 | 3.5.1 |
99 | 11,1,1 | 3.6 |
103 | 1,11,1 | 3.7 |
108 | 1,1,11 | 3.8 |
117 | (1,,1)11 | 4.2 |
120 | (1,,1),1 | 4.3 |
124 | 1,(1,,1) | 4.4 |
129 | (11,,1)1 | 5.1 |
133 | 111,,1 | 5.2 |
136 | (1,1),,1 | 5.3 |
141 | 1,1,,1 | 5.4 |
147 | (1,,11)1 | 6.1 |
151 | 11,,11 | 6.2 |
155 | 1,,111 | 6.3 |
159 | 1,,(1,1) | 6.4 |
166 | 1,,1,1 | 6.5 |
172 | (1,,,1)1 | 7.1 |
176 | 11,,,1 | 8.0 |
181 | 1,,,11 | 9.0 |
187 | 1,,,,1 | 10.0 |
7 | ||
7 | 1111111 | 0.7 |
12 | (1,1)1111 | 1.3.1 |
14 | (1,1)(1,1)1 | 1.4.1 |
18 | (11,1)111 | 2.1.2 |
19 | (11,1)(1,1) | 2.1.3 |
22 | (111,1)11 | 2.2.1.1 |
24 | (1111,1)1 | 2.2.1.2 |
25 | 11111,1 | 2.2.1.3 |
28 | ((1,1),1)11 | 2.3.2 |
30 | ((1,1)1,1)1 | 2.4.1 |
31 | (1,1)11,1 | 2.4.2 |
33 | ((11,1),1)1 | 2.5.1 |
34 | (11,1)1,1 | 2.6 |
35 | (111,1),1 | 2.7 |
36 | ((1,1),1),1 | 2.8 |
40 | (1,11)111 | 3.1.1.1 |
41 | (1,11)(1,1) | 3.1.1.2 |
43 | ((1,11),1)1 | 3.1.2.1 |
44 | (1,11)1,1 | 3.1.3 |
47 | (11,11)11 | 3.2.1.1 |
48 | (11,11),1 | 3.2.1.2 |
50 | (111,11)1 | 3.2.2.1 |
51 | 1111,11 | 3.2.2.2 |
53 | ((1,1),11)1 | 3.2.3.1 |
54 | (1,1)1,11 | 3.2.4 |
55 | (11,1),11 | 3.2.5 |
56 | (1,11),11 | 3.2.6 |
59 | (1,111)11 | 3.3.1.1 |
60 | (1,111),1 | 3.3.1.2 |
62 | (11,111)1 | 3.3.2.1 |
63 | 111,111 | 3.3.2.2 |
64 | (1,1),111 | 3.3.2.3 |
66 | (1,1111)1 | 3.3.3.1 |
67 | 11,1111 | 3.3.4 |
68 | 1,11111 | 3.3.5 |
71 | (1,(1,1))11 | 3.4.1.1 |
72 | (1,(1,1)),1 | 3.4.1.2 |
74 | (11,(1,1))1 | 3.4.2.1 |
75 | 111,(1,1) | 3.4.2.2 |
76 | (1,1),(1,1) | 3.4.2.3 |
78 | (1,(1,1)1)1 | 3.4.3.1 |
79 | 11,(1,1)1 | 3.4.3.2 |
80 | 1,(1,1)11 | 3.4.3.3 |
82 | (1,(11,1))1 | 3.4.4.1 |
83 | 11,(11,1) | 3.4.4.2 |
84 | 1,(11,1)1 | 3.4.4.3 |
85 | 1,(111,1) | 3.4.4.4 |
86 | 1,((1,1),1) | 3.4.4.5 |
88 | (1,(1,11))1 | 3.4.5.1 |
89 | 11,(1,11) | 3.4.6 |
90 | 1,(1,11)1 | 3.4.7 |
91 | 1,(11,11) | 3.4.8 |
92 | 1,(1,111) | 3.4.9 |
93 | 1,(1,(1,1)) | 3.4.10 |
96 | (1,1,1)11 | 3.5.2 |
97 | (1,1,1),1 | 3.5.3 |
98 | 1,(1,1,1) | 3.5.4 |
100 | (11,1,1)1 | 3.6.1 |
101 | 111,1,1 | 3.6.2 |
102 | (1,1),1,1 | 3.6.3 |
104 | (1,11,1)1 | 3.7.1 |
105 | 11,11,1 | 3.7.2 |
106 | 1,111,1 | 3.7.3 |
107 | 1,(1,1),1 | 3.7.4 |
109 | (1,1,11)1 | 3.8.1 |
110 | 11,1,11 | 3.9 |
111 | 1,11,11 | 3.10 |
112 | 1,1,111 | 3.11 |
113 | 1,1,(1,1) | 3.12 |
114 | 1,1,1,1 | 3.13 |
118 | (1,,1)111 | 4.2.1 |
119 | (1,,1)(1,1) | 4.2.2 |
121 | ((1,,1),1)1 | 4.3.1 |
122 | (1,,1)1,1 | 4.3.2 |
123 | (1,,1),11 | 4.3.4 |
125 | (1,(1,,1))1 | 4.4.1 |
126 | 11,(1,,1) | 4.5 |
127 | 1,(1,,1)1 | 4.6 |
130 | (11,,1)11 | 5.1.1 |
131 | (11,,1),1 | 5.1.2 |
132 | 1,(11,,1) | 5.1.3 |
134 | (111,,1)1 | 5.2.1 |
135 | 1111,,1 | 5.2.2 |
137 | ((1,1),,1)1 | 5.3.1 |
138 | (1,1)1,,1 | 5.3.2 |
139 | (11,1),,1 | 5.3.3 |
140 | (1,11),,1 | 5.3.4 |
142 | (1,1,,1)1 | 5.4.1 |
143 | 11,1,,1 | 5.5 |
144 | 1,11,,1 | 5.6 |
145 | (1,,1),,1 | 5.7 |
148 | (1,,11)11 | 6.1.1 |
149 | (1,,11),1 | 6.1.2 |
150 | 1,(1,,11) | 6.1.3 |
152 | (11,,11)1 | 6.2.1 |
153 | 111,,11 | 6.2.2 |
154 | (1,1),,11 | 6.2.3 |
156 | (1,,111)1 | 6.3.1 |
157 | 11,,111 | 6.3.2 |
158 | 1,,1111 | 6.3.3 |
160 | (1,,(1,1))1 | 6.4.1 |
161 | 11,,(1,1) | 6.4.2 |
162 | 1,,(1,1)1 | 6.4.3 |
163 | 1,,(11,1) | 6.4.4 |
164 | 1,,(1,11) | 6.4.5 |
165 | 1,,(1,,1) | 6.4.6 |
167 | (1,,1,1)1 | 6.5.1 |
168 | 11,,1,1 | 6.6 |
169 | 1,,1,11 | 6.7 |
170 | 1,,1,,1 | 6.8 |
173 | (1,,,1)11 | 7.2 |
174 | (1,,,1),1 | 7.3 |
175 | 1,(1,,,1) | 7.4 |
177 | (11,,,1)1 | 8.1 |
178 | 111,,,1 | 8.2 |
179 | (1,1),,,1 | 8.3 |
180 | 1,1,,,1 | 8.4 |
182 | (1,,,11)1 | 9.1 |
183 | 11,,,11 | 9.2 |
184 | 1,,,111 | 9.3 |
185 | 1,,,(1,1) | 9.4 |
186 | 1,,,1,1 | 9.5 |
188 | (1,,,,1)1 | 10.1 |
189 | 11,,,,1 | 11.0 |
190 | 1,,,,11 | 12.0 |
191 | 1,,,,,1 | 13.0 |
Conclusion
The table shows that the number of different algorithmic expressions (words with brackets) increases faster with each added character – matching binary exponential resolution after 7 at 6 characters:
- 1 = 1
- + 1 = 2 < + 2 = 3
- + 2 = 4 < + 4 = 7
- + 4 = 8 < + 8 = 15
- + 10 = 18 < + 16 = 31
- + 26 = 44 < + 32 = 63
- + 70 = 114 ~ + 64 = 127
- 1 = 1
- + 1 = 2 < + 2 = 3
- + 2 = 4 < + 4 = 7
- + 5 = 9 < + 8 = 15
- + 14 = 23 < + 16 = 31
- + 42 = 65 < + 32 = 63
- + 126 = 191 ~ + 64 = 127
Of course brackets contain algorithmic information. It was foolish of me to doubt this.
— Giga Gerard
The Hague, 28 January 2011