The Church-Turing thesis as a guiding principle for physics

TL;DR

The paper analyzes how physical principles constrain the Church-Turing thesis, arguing that the thesis must reflect real-world computation's physical limits. It presents two concrete explorations: a Zeno-squeezed oracle that attempts infinite-speed computation in finite proper time, yielding a diagonalization paradox for the halting problem; and a framework of one-to-one reversible computation where measurement and information erasure are nontrivial, modeled with reversible automata and permutation matrices. The results suggest that classical irreversibility is not fundamental in bijective dynamics, while quantum information introduces a probabilistic resolution to certain paradoxes. Overall, the work highlights that the interaction between computation and physics—especially reversibility and quantum effects—shapes the practical and foundational scope of computational theories.

Abstract

Two aspects of the physical side of the Church-Turing thesis are discussed. The first issue is a variant of the Eleatic argument against motion, dealing with Zeno squeezed time cycles of computers. The second argument reviews the issue of one-to-one computation, that is, the bijective (unique and reversible) evolution of computations and its relation to the measurement process.

Figures (2)

• Figure 1: The lowest "root" represents the initial state of the computer. Forward computation represents upwards motion through a sequence of states represented by open circles. Different symbols $p_i$ correspond to different initial computer states. a) One-to-one computation. b) Many-to-one junction which is information discarding. Several computational paths, moving upwards, merge into one. c) One-to-many computation is allowed only if no information is created and discarded; e.g., in copy-type operations on blank memory. From Landauer landauer-94.
• Figure 2: A system in a black box with an input interface and an output interface.