Name and definition was inspired by Moon & Spencer's work: "Theory of Holors".
A holor can be understood as generalization of a Tensor. Likewise it can be seen as canonical extension of a matrix to n dimensions. My definition actually represents a specific sub-class of the M&S holor. I have chosen different semantics and nomenclature to describe its properties as it seemed better suited to the realm of intended use cases.
In the scope of this library the holor is physically represented as multidimensional array. (Jagged arrays are excluded.)
A given holor has an order n, with n being a non-negative whole number. I use the specifying term n-holor.
A: The 0-holor, is a scalar. It can be represented as real (or complex) number. All possible values are allowed.
B: The (n+1)-holor is an array of n-holors. I call the array-size: Leading Dimension. Each element is an n-holor and each has the the same leading dimension. Values across elements are independent.
Above definition implies that an n-holor is a hierarchical composition of n sub-holors: { n-holor, (n-1)-holor, ... , 1-holor }, each with a specified leading dimension.
I call the ordered set (or sequence) of 'n' leading dimensions shape. The shape of the 0-holor is the empty set: { }.
It follows from the definition that all elements of a holor must have the same shape.
One can mentalize the holor as n-orthotoype (or n-hypercuboid), with the 0-holor being an 'atom' (, which is an orthotoype of unit-size).
Constructing the (n+1)-holor means stacking multiple n-orthotoypes on each other in a new dimension. Since each sub-orthotoype has a fixed and finite shape, its atoms can be folded into linear space (e.g. linear addressable memory) such that all sub-holors also retain a valid representation.
I define volume as the product of all elements of shape.
The 0-holor has volume 1 (empty product).
Note: 'volume' is the related to Moon & Spencer's 'Number of Merates'
I call the composite of all individual scalars of a given holor the value of that holor. The value is stored as array of a specified data type. Currently the types f2_t (float) and f3_t (double) are supported.
The holor-value can be interpreted as vector. Its dimension is equal to the volume of the holor. For a given shape all scalars have a well defined order.
I define a unique order by adding axiom 'C' to a holor's definition:
C: The value of an (n+1)-holor is the concatenated array of the associated values of n-holors as specified in axiom 'B'.
I extend the definition of holor to include a objects with shape but no associated value.
D: I define a holor with shape but no value and call it vacant.
I call a holor with value determined.
| Name | Description | Shape | Volume |
|---|---|---|---|
| 0-Holor | Scalar | {} | 1 |
| 1-Holor | Vector of size m | {m} | m |
| 2-Holor | (n x m)-Matrix | {n,m} | n * m |
| 3-Holor | Array of k elements; each being a (n x m)-Matrix | {k,n,m} | k * n * m |
A holor has well defined operations like addition, multiplication, convolution, transposition, etc. Where suitable, operations are expressed as operators, which are dedicated objects.
We define the HVM as virtual machine with holors as data elements holor-operands as principal instructions. A HVM contains at least the following components:
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A database for storing holors where each holor is addressable by an integer index.
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A collection of virtual machine operations (vop). Each vop representing an operation of arity n on holors and containing an array of n+1 indices, where the first n indices address the input holors and the last one the output holor of the given operation. Normally, a vop can handle holors of all data types (f2_t or f3_t) in any combination.
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An executable array of vop, called track. Executing a track means executing all vop in the track from first to last in given sequence.
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An database of tracks, called library.
Additional components are allowed but not mandatory. Typically such components would include operations such as conditional execution, jump, loop, branch, call-subroutine, etc (turning a HVM into a Turing Machine).
© 2019, 2020 Johannes B. Steffens