Functionality Locality, Mixture & Control = Logic = Memory (2024)

1.1 Spatial Locality and Temporal Locality

To the best of the knowledge from the author, the concept of locality shall be originally credited to physics. A quotation can be delivered as follows. “In physics, the principle of locality states that an object is influenced directly only by its immediate surroundings. A theory that includes the principle of locality is said to be a “local theory"." These concepts are used in computer science, and rephrased as “Locality of Reference". This work quotes the following classification, and supplies the evolved types of locality for recent developments.

With the evolve of theory and practice in computer architecture, there are also following types of locality to be mentioned.

Based on the above classifications, we can conclude that: all concepts can be derived from Spatial Locality. Therefore, the central of our focus lies on Spatial Locality, and it is expected to be consistent with new concepts.

1.2 Functionality Locality

The key insight of this work is that: all existing understanding on locality only considers how references shall be taken advantage of, without the consideration on the access order of the information. A recent work([4]) demonstrates that it is feasible to deliver polymorphic functionalities within a single piece of information. Therefore, it is expected to revisit the local theory (from physics), and how such a revisit plays a role computer architecture.

Functionality Locality. It can be defined as: the access order of a single piece of information can determine different functionalities, though the information has no spatial changes. Figure1 demonstrates an example from([4], and it is very straightforward to understand: ➊ from top to bottom, the binary system can be used for computation; and ➋ from bottom to top, the bitmap is used for queries. Also, it can be understood as: a single piece of information can be used in more-than-one systems, which are originally oriented for different functionalities.

1.3 Practical Examples of Functionality Locality

The discovery of Functionality Locality motivates a retrospection on computer designs, in terms of the bit-parallel (i.e., horizontal) and bit-serial (i.e., vertical) data layouts. To date, modern processors leverage bit-parallel layouts, to maximize the computational power. This substantially causes the issues of data movements between the processor and memory. Therefore, such an issue ignites the trend of Processing Using Memory in DRAM (i.e., the modern memory chips) by([6]), and serves as the basis for the recently-discussed paradigm - Memory-Centric Computing([3])¹¹1Note that, though the identified idea can be used within the processors, the benefits can not be fully unleashed as explained..

This work revisits the idea in a short manner, to serve as the foundation for the further discussion. Figure2 gives out the example: after the three steps, The final state can be written as AB + BC + AC. With some proper changes, the final state is C(A + B) + $\neg$C(AB). Therefore, one can view A and B as the operand bit, and C as the control bit.

2.1 Definition

Mixture is a bit sequence with the arbitrary length, and its functionalities include compute²²2Note that any computations can be formalized using logic operations due to the logic completeness., query and move. Therefore, the following guidelines can be derived, based on the length of this sequence, and how the sequence is layouted.

The above definitions shall cover all existing quantifiers (such as scalar and vector) and bitmaps. However, following the “Indexes $\approx$ Values" principle([4]), a Mixture can compute, query and move without any additional layout transformations.

2.2 Synergy with an Analytic Framework

[5] provides an analytic framework to connect (ordered) values, representations (if needed) and (disordered) information, in a consistently-recursive manner. This substantially motivates a question: can there be the minimal set of them be stored, so that the rest can be fully restored? The definition of Mixture naturally complements with the above requirements: only those ones, which can meet the full functionalities, shall be kept so that the rest can be fully restored. This conjecture can be mathematically explained via Mandelbrot set and/or Julia set, which are recursively used for the formation.

3.1 The “Control = Logic = Memory" Principle

As described in Section1.3, exploiting analog capabilities of three cells delivers a formation as C(A + B) + $\neg$C(AB). This work leverages the perspectives from([5]), and claims that: the denotations of A, B and C are simply the representations, due to the temporality. In fact, the actual fact is there are interactions between A, B and C at a time slot. Therefore, this work makes a conjecture with this special case, Control can be equalized with Logic, and Logic can be equalized with Memory: namely Control = Logic = Memory.

3.2 Everything is Memory

A revisit to existing architectures is delivered in Figure3.

With the identified principle, Control = Logic = Memory, it is clear that Harvard model can be understood as a form of specialization of Von Neumann Model, with different levels of the granularity for the memory. Therefore, the conceptual reasoning in this work expands Memory-Centric Computing into Everything is Memory. More importantly, this expansion, along with Mixture (particularly Mandelbrot and/or Julia sets), provides a completely different perspective on self-replication theory([8]) (which solely focused on representations).

3.3 Memory-Centric Architectures

This work conjectures that, the computer organization shall be used to store all Mixtures, formalized as an analog with the recursively-formalized analytic framework([5]). Therefore, all components, as memory, are recursively organized and the hierarchy is expected to be asynchronous, in terms of the granularity. This is because: the purposes of recursively-organized memory are to store ((sub)sets of) Mixtures, and they are not necessarily symmetry. It would be an interesting discussion, to see whether the functionalities of such a computer shall be determined in advance. Clearly, it depends on whether the artificial intelligence shall play a god, though nobody can guarantee that the god (if any) truly prefers that human beings shall exist or not.

4.1 Mixture Meets Other Concepts

4.2 (In)finity, and Leibniz Binary System

Though the author assumes the infinity can be beneficial if there is perpetual motion, this work considers the determination of a finite/infinite can be used for whether a successful formation of Mandelbrot/Julia set can be achieved, under the formalization of([5]). This is mainly due to its characteristics on the self-similarity, in the context of fractal geometry. If it is finite, it shall be capable to be scaled for the encoding within the Leibniz Binary System([1]).