Reversible Computation as Algebraic Symmetry: A Group-Theoretic Structure — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Reversible Turing machines form an algebraic group acting on configuration spaces, with equivalence classes defined by Nielsen/Andrews-Curtis-type operations — revealing symmetry structure in computation itself. | MATH: Group of reversible Turing machines G acting on Σℤd via local homeomorphisms; equivalence relations generated by Aut(Fn) and Fn on normally generating tuples (y1,…,yn)∼(x1,…,xn); Andrews-Curtis conjecture: any normally generating n-tuple is AC-equivalent to free basis. | CONNECTION: The dihedral group D4 (order 8) is the symmetry group of the square — its elements (identity, rotations r,r2,r3, reflections s,sr,sr2,sr3) correspond to the 8 possible permutations of a 2-state, 2-symbol Turing machine's transition table under complementation (swap 0↔1) and reversal (swap direction). This is precisely the D4 action on the 2×2 grid of state×symbol — a crystallographic
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026reversible,
title = {{Reversible Computation as Algebraic Symmetry: A Group-Theoretic Structure — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23465012},
url = {https://zenodo.org/records/23465012},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Reversible Computation as Algebraic Symmetry: A Group-Theoretic Structure — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/19
PB - Zenodo
DO - 10.5281/zenodo.23465012
UR - https://zenodo.org/records/23465012
AB - FINDING: Reversible Turing machines form an algebraic group acting on configuration spaces, with equivalence classes defined by Nielsen/Andrews-Curtis-type operations — revealing symmetry structure in computation itself. | MATH: Group of reversible Turing machines G acting on Σ^ℤ^d via local homeomorphisms; equivalence relations generated by Aut(F_n) and F_n on normally generating tuples (y_1,…,y_n)∼(x_1,…,x_n); Andrews-Curtis conjecture: any normally generating n-tuple is AC-equivalent to free basis. | CONNECTION: The dihedral group D_4 (order 8) is the symmetry group of the square — its elements (identity, rotations r,r^2,r^3, reflections s,sr,sr^2,sr^3) correspond to the 8 possible permutations of a 2-state, 2-symbol Turing machine's transition table under complementation (swap 0↔1) and reversal (swap direction). This is precisely the D_4 action on the 2×2 grid of state×symbol — a crystallographic Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -