The Tiy-Kouzen Papers, No. 1 :: Real-Valued Vector Computation

A New(ish) Model of Real-Valued Vector Computation

TiyKouzen Paper :: Introduction

The TiyKouzen Papers and indeed all papers published here are for pure entertainment value and for demonstration of research ability. Nevertheless, all rights of this work are reserved.

This series of essays constitutes a personal researc h project in mathematics that serves to both combine the models of Lambda Calculus and Blum-Shub-Smale (1989) Computation as well as expanding such a computational model to include the representations of instantaneous and accumulative change over vector spaces. Specifically, the main premise of this series will be to investigate the properties continuous vector computers when operating over 

The major premise of this inaugural paper is to formally define a Continuous Functional Computational Model (CFCM) that will act as the housing within which all of our further operations will be contained. Non-rigorously, a CFCM is an extension of the Blum-Shub-Smale model of computation that allows for the following parameters:

1. All operations (Computational and Otherwise) ought to be expressed in the language of Lambda Calculus.
2. Computations operate over tensors rather than simple scalars.
3. Computation ought to be expressed as any Lambda Calculus function involving boolean truths or real numbers.

Sidenote: A brief "point of order" is that I will often begin postulations with the phrase "Say for the sake of argumentation" (A phrasing borrowed from Ian Danskin's excellent series of The Alt-Right Playbook video essays). This phrasing may also be abbreviated as "SftsA :: " with the double-colon marking the initiation of the proposal.

Recall: In their excellent research on NP-Completeness problems when computing over subsets of the real numbers (R), Blum et al. defined the following as a mathematical framework for their machine.
A given BSS-Machine M that has \( N + 1 \) instructions is defined with its instructions in the list \( I \) from \( 0 \) to \( N \). The configuration of \( M \) is stored as a tuple, \( (k, r, w, x) \), where the following are defined thusly:

\( k \rightarrow \) The index of what instruction to next execute
\( r, w \rightarrow \) The read and write registers that each store some positive integer.
\( x \rightarrow \) An unending list of real numbers where no less than infinity of them are zero. (i.e., \( x = (x_0, x_1, x_2, \ldots ) \))

The default configuration for a BSS-Machine before any operation has been commenced (\( k = 0 \)) would be given as \( (0, 0, 0, x) \).

Definition 1: We can define a CFCM that is backward compatible with any BSS program and whose programs are expressible in the language of Lambda Calculus.

SftsA :: A BSS-Machine \( C \) such that \( C := (k, r, w, t) \).

The \( k \), \( r \), and \( w \) registers remain the same and for the sake of communication we substitute \( x \) for \( t \). If we expand the definition of \( C \) such that \( t \) is an infinite list of tensors of any given rank \( d \), it is still BSS-compatible with the given constraint that \( d = 0 \). Such a system would give us a CFCM, \( C \), that is defined such that
\[ C := (k, r, w, d, t) \]
\[ \rightarrow T = (t_0, t_1, \ldots) \]

Where \( t_n \) is a tensor of rank \( d \). The issue comes with the fact that for any \( d > 0 \), the size of the tensor is ambiguous. This problem can be alleviated by adding another metric, \( S \), to the configuration tuple that defines the maximum size for all \( t_n \) such that the size \( Z \) of the array is defined by \( Z = S^d \).

That would leave us with the following symbolic definition:

\[ C := (k, r, w, d, S, T) \]
\[ | \; T = (t_0, t_1, \ldots) \]
\[ \land \; \forall t_n \in T, \; t_n = \{t_n(u) : u \in \mathbb{N}^d, \forall i \in \{1, \ldots, d\} \, u_i \in [0, S - 1] \} \]

Expressing that definition semantically would be to say that every CFCM-Machine has an index, $k$, read and write registers, $r,w$, a rank (or dimension), $d$, a size, $S$, and an unending list of $d$-rank tensors, $T$. We further define that every element, $t_n \in T$ is an array with the size $S^d$ where each element has an index vector, $u$ that we can use to identify each particular element. (i.e. $t_n(\alpha,\beta,\gamma,\delta) = z $ to represent a that in register $t_n$, at the location $(\alpha,\beta,\gamma,\delta)$ the value $z$ is stored. 

This series of essays is first and foremost dedicated to my siblings, and to my first cousins ("Tiy Kouzen-ye")
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