A Circuit-Level Description of 2-Adic Computing​

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▣ The Hensel CPU architecture performs exact arithmetic in the field of 2-adic numbers. Because the coefficients of 2-adic expansions are either 0 or 1, operands can be encoded in bits as finite lists of 2-adic expansion coefficients, and computations can be performed with standard MOSFET technology. Moreover, with SciSci Research and Future Computing having already developed an encoding scheme for 2-adic operands and a processor architecture for efficient 2-adic computing, the circuit design proved to be elementary and amenable to scaling. 

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The Combinational Logic of Exact Arithmetic

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▣ The Hensel CPU circuit has a tree-like structure, and can be thought of as a tree of 2AALUs. Each 2AALU is simple in design. It consists of no more than a XOR gate, an AND gate, and a Φ-input. 

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▣ Each 2AALU in the circuit tree at a given level receives an input from its parent vertex in the tree, as well as an input from its own Φ. These two inputs are then fed to its XOR gate and AND gate, and the outputs from each are then fed to the two 2AALUs that are its children vertices in the circuit tree. The sequence of 2AALUs that feed an output of 1 downstream all the way down to a sink vertex determines the path in the circuit tree. 

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▣ Just as cluster processor registers (CPRs) reside at level ℓ=1 within the processor cluster, so too do they reside at the end of each distinct path in the circuit tree. Moreover, the address of each CPR (e.g., (1,0,1)) is the same as the list of XOR-gate values in the circuit tree path terminating at that CPR. The Hensel load-store architecture dictates that χ-IDs be loaded to CPRs with matching addresses, and the circuit design actually gives one χ-address matching for free: one can load a χ-ID to the address-matching CPR  simply by sending a π-sequence to the CPR at the end of the relevant circuit tree path. 

▣ Given an operand with χ-IDs loaded to various CPRs, χ-modification is performed by changing Φ-input values, which in turn change the AND and XOR gate outputs, giving a new path in the circuit tree, terminating at a new CPR; this is termed Φ-perturbation. Φ-inputs are encoded by FC-2-2025, the encoding standard for arithmetic instructions. Thus, instructions for exact arithmetic in the Hensel processor are no more than specifications of Φ-inputs which perturb the circuit tree path to the CPR whose address matches the χ-ID for the arithmetic output.

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▣ 2AALU carriers are packaged within one another recurrently, with a given 2AALU packaged within the carrier of the 2AALU that is its parent vertex in the circuit tree (alongside the other 2AALU sharing the same parent vertex), with all 2AALUs still being directly surface mounted. This gives the nested structure often discussed in Hensel reports. 

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Computational Advantages

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▣ Matching between χ-IDs and CPRs is automatic, due to the positioning of CPRs in the circuit tree.

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▣ Given any pair of FC-3-2025 encoded input operands and an arithmetic operation, one can generate a FC-2-2025 encoding for the arithmetic computation instructions, which is simply a list of Φ-inputs, which can be introduced in parallel. The only remaining computation is the running of the path in the circuit tree, which introduces a mild O(n) term, where n is one less than the nest depth. (In the case of the Virtual Hensel, n=6).

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▣ The circuit tree, in which 2AALUs feed outputs to their children nodes, lends itself readily to nested packaging, and thus gives rise to the nested structure described in previous reports, which is highly compatible with the nature of 2-adic distance and thus realizes the many efficiency and optimality properties for 2-adic computing discussed in the original report. 

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▣ Because all FC encodings are in bits, the circuit design employs standard transistor technology; no exotic hardware is required. Moreover, because the Hensel CPU performs arithmetic as χ-modification according to stored instructions, 2AALU operations are very simple, and thus 2AALU design is elementary. It is expected that the light cost footprint of 2AALUs and CPRs will allow for high-nest-depth designs with broad operand capacity and arithmetic reach in order to realize an exact computing capability for the accelerated computing revolution.

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Executive Summary