[Paper Review] Limits to measurement in experiments governed by algorithms

Edwin Beggs, José Félix Costa+1 more|ArXiv.org|Nov 19, 2009

Computability, Logic, AI Algorithms17 references4 citations

TL;DR

This paper investigates how algorithmic control of physical experiments imposes fundamental limits on measurement precision in Newtonian mechanics. By modeling experimental procedures as Turing machines, the authors demonstrate that certain physical quantities—like mass—cannot be measured with arbitrary accuracy due to inherent computational constraints, revealing a new form of uncertainty principle rooted in computability theory rather than experimental error.

ABSTRACT

We pose the following question: If a physical experiment were to be completely controlled by an algorithm, what effect would the algorithm have on the physical measurements made possible by the experiment? In a programme to study the nature of computation possible by physical systems, and by algorithms coupled with physical systems, we have begun to analyse (i) the algorithmic nature of experimental procedures, and (ii) the idea of using a physical experiment as an oracle to Turing Machines. To answer the question, we will extend our theory of experimental oracles in order to use Turing machines to model the experimental procedures that govern the conduct of physical experiments. First, we specify an experiment that measures mass via collisions in Newtonian Dynamics; we examine its properties in preparation for its use as an oracle. We start to classify the computational power of polynomial time Turing machines with this experimental oracle using non-uniform complexity classes. Second, we show that modelling an experimenter and experimental procedure algorithmically imposes a limit on what can be measured with equipment. Indeed, the theorems suggest a new form of uncertainty principle for our knowledge of physical quantities measured in simple physical experiments. We argue that the results established here are representative of a huge class of experiments.

Motivation & Objective

• To investigate how algorithmically controlling physical experiments affects the precision and feasibility of physical measurements.
• To extend the theory of physical oracles by modeling the experimenter and experimental procedure as a Turing machine, introducing a new principle (Principle 6) for analyzing measurement control.
• To demonstrate that computational limits—specifically, undecidability and complexity constraints—impose fundamental bounds on what can be measured, even in idealized Newtonian systems.
• To show that this limitation is not due to experimental error but arises from the logical and computational structure of the algorithmic procedure itself.
• To argue that this form of indeterminacy is a general feature of physical measurement, applicable across the physical sciences.

Proposed method

• Model the experimental procedure—specifically, a collision-based mass measurement experiment (CME) in Newtonian dynamics—as a Turing machine that controls the sequence of measurements and data collection.
• Use the CME as a physical oracle for polynomial-time Turing machines to analyze its computational power in terms of non-uniform complexity classes such as $P/poly$ and $BPP//\log^*$.
• Formulate the time required for measurement as a function of desired accuracy, showing that higher precision demands exponentially more time.
• Establish that not all masses can be measured due to undecidability in the algorithmic control of the experimental sequence, even with perfect equipment.
• Apply the theory to a general class of experiments, arguing that the results are representative of a broad range of physical measurements.
• Use computability theory to show that the inability to measure certain values is not due to noise or error, but to the logical limits of algorithmic procedures.

Experimental results

Research questions

• RQ1 What are the computational limits imposed on physical measurements when the experimental procedure is fully algorithmically controlled?
• RQ2 Can a Turing machine controlling a physical experiment measure all physical quantities with arbitrary precision?
• RQ3 How does the time required for measurement scale with the desired accuracy in algorithmically controlled experiments?
• RQ4 Does the algorithmic nature of experimental procedures introduce a new kind of uncertainty principle in classical physics?
• RQ5 To what extent are measurement limitations due to computational undecidability rather than experimental error?

Key findings

• The CME experiment, when controlled by a Turing machine, cannot measure all masses with arbitrary precision due to inherent computational undecidability.
• The time required for measurement grows exponentially with the desired accuracy, making high-precision measurement infeasible within polynomial time.
• The computational power of polynomial-time Turing machines with the CME oracle is bounded by non-uniform complexity classes such as $P/poly$ and $BPP//\log^*$, depending on precision assumptions.
• The results suggest a new form of uncertainty principle in classical mechanics, where measurement limits arise from computability constraints, not from physical noise or error.
• This limitation is not specific to the CME but is representative of a wide class of physical experiments, indicating a general epistemic constraint in physics.
• The study reveals that not all physical quantities can be measured, even with perfect equipment, because the algorithmic control process itself imposes logical limits on what can be observed.