11/15/2012, 11:10 PM
FinBetaPaper3.pdf (Size: 861.65 KB / Downloads: 3,557)
This is the third paper on the topic of nept and nopt structures, that continues the ideas contained in the papers “Hyperoperations and Nopt Structures” and “Hierarchies and Nopt Structures”.The paper is called “Transitional Sequences and Nopt Structures”
Abstract (Beta version)
Variations of Nopt structures. Computational pathways. Transitional sequences from exponentiation to tetration. A canonical transitional sequence that moves through the hyperoperator hierarchy. Laddered exponents. Complexity based definitions of finite and infinite. Some Bowers array numbers compared with Naropt structures.
Keywords:
Natural numbers, hereditary base, transitional sequences, laddered exponents, Nopt structures, long repdigit numbers, binary sequences in other bases.
Note:
Nopt structures have 2 levels of interpretation:
(1) as geometric, recursive tiling patterns following a prescribed folding pattern and overlayed on a regular geometric tesselation of the plane. In this case, the term “noptiles” may be more suitable than “nopt structures”. Usually the underlying tesselation is a two dimensional grid of squares, but it may be possible to use other shapes such as regular hexagons as the background tesselation.
(2) nopt structures are noptiles where a recursion-based operation, and computational pathway are additionally specified. The canonical example is nested exponential power towers corresponding to the hyperoperators. Other important interpretations are possible such as what happens when you continue PVN (or SPN) numbers into nopt structures. Philosophers interested in “finite” and “infinite” should take note, since it is a natural generalisation of the familiar counting numbers in base10 using Place Value Notation into recursive structures of finite ordertype.
There are 4 important caveats that are needed in order to accept and understand a recursion interpretation of geometric noptiles:
(1) The idea of minimal symbolic notation and “formal” power towers
(2) The mapping from minimal symbolic notation to Coloured Square Diagrams, using some kind of appropriate colour scheme to represent the symbols or glyphs of a formula. No maths software can adequately represent formulae as complicated as those associated with the NEPT-form of arbitrary hyperoperators in the hyperoperator hierarchy.
(3) The “wonky-H-fractal” to “uniform-H-fractal” assumption is acceptable.
The uniform-H-fractal assumption is needed in order to create extensible noptiles that act as schematic representations of the computational pathways.
(4) The pseudocode for the folding patterns works (see “Hyperoperations and Nopt Structures” and “Transitional Sequences and Nopt Structures”), is extensible, and is, in principle, programmable.
Animations (using excel CSDs, powerpoint and mediafile) of the 3 aspects of nopt structures can be made. The 3 aspects are folding patterns, computational pathway and transitional sequence.
The Base(n) Ackermann function (aka Seed(n) Ackermann function) is defined as:
n^^n, n^^^n, n^^^^n, ... (where n>=3 is a constant natural number)
This is a little different from the “Ackermann number sequence” in Wikipedia:
1^1, 2^^2, 3^^^3, 4^^^^4, .... (using Knuth arrows)
A smooth canonical transition sequence using noptiles can be described and animated, the animations show a "smooth" quality, and the NEPT-form recursion interpretation of the noptiles is then mathematically accurate for the transitional sequence through the Base(n) Ackermann numbers described in the paper "Transitional Sequences and Nopt Structures".
In some ways, nopt structures provide a relatively simple and intuitive unified framework for discussing various kinds of recursion in maths and computer science.
Alister (“Mike Smith”)