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Quantum Structures as Generative Scores: Partition Logic, Generative Logic, and Aesthetic Form

Christian Jendreiko, Karl Svozil

Abstract

We connect partition logic with Generative Logic by translating finite partition logics into Prolog-based Simple Generative Logic Grammars. As a proof of concept, we use the five-atom V-logic L_{12} to generate a modular visual artifact, the \emph{Quantum Square}. The approach separates logical structure from its visual, textual, or sonic realization. This makes partition logic useful both as a generative design resource and as a tool for communicating complementarity.

Quantum Structures as Generative Scores: Partition Logic, Generative Logic, and Aesthetic Form

Abstract

We connect partition logic with Generative Logic by translating finite partition logics into Prolog-based Simple Generative Logic Grammars. As a proof of concept, we use the five-atom V-logic L_{12} to generate a modular visual artifact, the \emph{Quantum Square}. The approach separates logical structure from its visual, textual, or sonic realization. This makes partition logic useful both as a generative design resource and as a tool for communicating complementarity.
Paper Structure (17 sections, 1 theorem, 29 equations, 2 figures, 2 tables)

This paper contains 17 sections, 1 theorem, 29 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Partition logic between formal structure and design material
  3. Partitions and partition logics
  4. Formal Example A: a horizontal sum of binary partitions
  5. Formal Example B: the V-logic $L_{12}$
  6. A quantum realization of the same logical form
  7. Generative Logic and SGLGs
  8. Generative Logic
  9. A Simple Generative Logic Grammar
  10. From partition logic to grammar
  11. Formal Example C: the horizontal sum as a grammar
  12. Formal Example D: the V-logic as a grammar
  13. The Quantum Square: hypergraph, schema, and output
  14. Why this object is aesthetically interesting
  15. A triangle-logic generalization of the Quantum Square
  16. ...and 2 more sections

Key Result

Proposition 1

Let $L$ be a finite partition logic with separating two-valued states. The grammar $G_L$ constructed from Eqs. eq:tfsets--eq:rowrule2 preserves the incidence relation between atoms and two-valued states. In the generated row for $x_j$, the symbol $s_i$ occurs to the left of the separator iff $v_i(x_

Figures (2)

  • Figure 1: Three representations of the V-logic $L_{12}$. (a) Greechie-style hypergraph. (b) Atom-state incidence schema, with colored cells indicating $v_i(x)=1$ and gray cells indicating $v_i(x)=0$. (c) The grammar-generated output, the Quantum Square. In each row, the states assigning value $1$ to the atom are listed first, then a black separator, and then the states assigning value $0$.
  • Figure 2: Triangle-logic generalization of the Quantum Square. (a) Greechie-style depiction of the triangle logic associated with Eq. \ref{['eq:trianglepartitions']}. The intertwining atoms $a$, $c$, and $e$ are shown by concentric circles because each belongs to two contexts. (b) Atom-state incidence schema. (c) Tile rendering obtained from the grammar in Eq. \ref{['eq:trianglegrammar']}, using four recurring state colors and a black separator between support and complement. Each row lists the states with value $1$, then the separator, and then the states with value $0$.

Theorems & Definitions (2)

  • Proposition 1
  • proof