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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.iteror.org/big/Retro/expo/binary-variation.html?single-row

Page link: www.iteror.org/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. + 1 = 2 < + 2 = 3
  3. + 2 = 4 < + 4 = 7
  4. + 4 = 8 < + 8 = 15
  5. + 10 = 18 < + 16 = 31
  6. + 26 = 44 < + 32 = 63
  7. + 70 = 114 ~ + 64 = 127
  1. 1 = 1
  2. + 1 = 2 < + 2 = 3
  3. + 2 = 4 < + 4 = 7
  4. + 5 = 9 < + 8 = 15
  5. + 14 = 23 < + 16 = 31
  6. + 42 = 65 < + 32 = 63
  7. + 126 = 191 ~ + 64 = 127

Of course brackets contain algorithmic information. It was foolish of me to doubt this.

— Gerard van Novaloka
The Hague, 28 January 2011
www.iteror.org