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A computing machinery using a continuous memory tape

Yigit Oktar

TL;DR

The paper tackles the NP-complete subset sum problem by proposing a frequency-domain computing framework that treats memory as a continuous analogue of a Turing tape. It introduces a discrete, pseudo-polynomial algorithm with time $O(SN)$ and extends this to a continuous setting where sums are encoded in a product form $S_N(t)$ and read via spectral methods, though reading a specific sum remains challenging in software. The work also develops a rich theoretical viewpoint using generating functions and polynomial representations, and discusses dedicated hardware paths (circuit and wireless) to realize reading efficiently. While the approach does not resolve whether $P=NP$, it offers a compelling avenue for combining symbolic computation, hardware specialization, and pattern-recognition tasks in both theoretical and practical contexts.

Abstract

By considering a discrete tape where each cell corresponds to an integer, thus to a possible sum, a pseudo-polynomial solution can be given to subset sum problem, which is an NP-complete problem and a cornerstone application for this study, using shifts and element-wise summations. This machinery can be extended symbolically to continuous case by thinking each possible sum as a single frequency impulse on the frequency band. Multiplication with a cosine in this case corresponds to the shifting operation as modulation in communication systems. Preliminary experimentation suggests that signal generation thus solution space calculation can be done in polynomial time. However, reading the value at a specific frequency (sum value) is problematic, namely cannot be simulated in polynomial time currently. Dedicated hardware implementation might be a solution, where both circuit-based and wireless versions might be tried out. A polynomial representation is also given that is claimed to be analogous to a tape of a Turing machine. Both rational and real number versions of the subset sum problem are also discussed, where the rational version of the problem is mapped to 0-1 range with specific patterns of True values. Although this machinery may not be totally equivalent to a non-deterministic Turing machine, it may be helpful for non-deterministic universal Turing machine actualization. It may pave way to both theoretical and practical considerations that can help computing machinery, information processing, and pattern recognition domains in various ways.

A computing machinery using a continuous memory tape

TL;DR

The paper tackles the NP-complete subset sum problem by proposing a frequency-domain computing framework that treats memory as a continuous analogue of a Turing tape. It introduces a discrete, pseudo-polynomial algorithm with time and extends this to a continuous setting where sums are encoded in a product form and read via spectral methods, though reading a specific sum remains challenging in software. The work also develops a rich theoretical viewpoint using generating functions and polynomial representations, and discusses dedicated hardware paths (circuit and wireless) to realize reading efficiently. While the approach does not resolve whether , it offers a compelling avenue for combining symbolic computation, hardware specialization, and pattern-recognition tasks in both theoretical and practical contexts.

Abstract

By considering a discrete tape where each cell corresponds to an integer, thus to a possible sum, a pseudo-polynomial solution can be given to subset sum problem, which is an NP-complete problem and a cornerstone application for this study, using shifts and element-wise summations. This machinery can be extended symbolically to continuous case by thinking each possible sum as a single frequency impulse on the frequency band. Multiplication with a cosine in this case corresponds to the shifting operation as modulation in communication systems. Preliminary experimentation suggests that signal generation thus solution space calculation can be done in polynomial time. However, reading the value at a specific frequency (sum value) is problematic, namely cannot be simulated in polynomial time currently. Dedicated hardware implementation might be a solution, where both circuit-based and wireless versions might be tried out. A polynomial representation is also given that is claimed to be analogous to a tape of a Turing machine. Both rational and real number versions of the subset sum problem are also discussed, where the rational version of the problem is mapped to 0-1 range with specific patterns of True values. Although this machinery may not be totally equivalent to a non-deterministic Turing machine, it may be helpful for non-deterministic universal Turing machine actualization. It may pave way to both theoretical and practical considerations that can help computing machinery, information processing, and pattern recognition domains in various ways.
Paper Structure (24 sections, 4 theorems, 11 equations, 2 figures, 1 table, 2 algorithms)

This paper contains 24 sections, 4 theorems, 11 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Previous work
  3. Proposed Methodology
  4. Pseudo-polynomial algorithm
  5. Extension to continuous case
  6. Signal generation
  7. Reading the sum value using Fourier transform
  8. Reading the sum value through convolution
  9. Experimentation
  10. Notes on dedicated hardware implementation
  11. Circuit-based implementation
  12. Wireless implementation
  13. More theoretical viewpoint
  14. Polynomial representation of a tape of a Turing Machine
  15. Polynomial Representations for the Subset Sum Problem
  16. ...and 9 more sections

Key Result

Lemma 6

Let $F(x)$ designate the current status of the polynomial tape. Then, the value of $n^{th}$ cell can be read by performing $\frac{d^{n}{F(x)}}{dx^{n}}\rvert_{x=0}\frac{1}{n!}$, namely taking the $n^{th}$ derivative at $x=0$ and dividing by $n$ factorial.

Figures (2)

  • Figure 1: For the case of 3 values (X=3, Y=4, Z=1), by simple shifts and element-wise summation all combinations are calculated on the tape in 3 steps
  • Figure 2: A neural network architecture consisting of three input and two output nodes and that includes a continuous memory tape as its hidden layer

Theorems & Definitions (14)

  • Definition 1: SUBSET SUM
  • Definition 2: SUBSET SUM: INTEGER VARIANT (IV)
  • Definition 3: SUBSET SUM: ZERO-SUM VARIANT (ZSV)
  • Definition 4
  • Definition 5
  • Lemma 6
  • proof : Proof of Lemma \ref{['reading']}:
  • Lemma 7
  • proof : Proof of Lemma \ref{['shifting']}:
  • Lemma 8
  • ...and 4 more