[Paper Review] Constraints on physical reality arising from a formalization of knowledge

David H. Wolpert|arXiv (Cornell University)|Nov 9, 2017

Computability, Logic, AI Algorithms18 references3 citations

TL;DR

This paper formalizes knowledge acquisition in physical agents through 'inference devices' (IDs), showing that constraints on joint knowledge among IDs impose fundamental limits on physical reality, independent of specific laws of physics. It extends IDs to overcome logical omniscience in epistemic logic, enabling a realistic formalization of physical knowledge with error bounds and probabilistic reasoning.

ABSTRACT

There are (at least) four ways that an agent can acquire information concerning the state of the universe: via observation, control, prediction, or via retrodiction, i.e., memory. Each of these four ways of acquiring information seems to rely on a different kind of physical device (resp., an observation device, a control device, etc.). However it turns out that certain mathematical structure is common to those four types of device. Any device that possesses a certain subset of that structure is known as an "inference device" (ID). Here I review some of the properties of IDs, including their relation with Turing machines, and (more loosely) quantum mechanics. I also review the bounds of the joint abilities of any set of IDs to know facts about the physical universe that contains them. These bounds constrain the possible properties of any universe that contains agents who can acquire information concerning that universe. I then extend this previous work on IDs, by adding to the definition of IDs some of the other mathematical structure that is common to the four ways of acquiring information about the universe but is not captured in the (minimal) definition of IDs. I discuss these extensions of IDs in the context of epistemic logic (especially possible worlds formalisms like Kripke structures and Aumann structures). In particular, I show that these extensions of IDs are not subject to the problem of logical omniscience that plagues many previously studied forms of epistemic logic.

Motivation & Objective

• To formalize how agents embedded in a physical universe acquire knowledge through observation, control, prediction, and memory.
• To identify universal constraints on the joint knowledge of multiple inference devices (IDs) in any physical universe, regardless of its underlying laws.
• To extend the ID framework to resolve the problem of logical omniscience in epistemic logic by incorporating structure from all four knowledge-acquisition modes.
• To develop a formalism for physical knowledge that supports probabilistic reasoning, error analysis, and distance metrics over possible states.
• To explore connections between inference complexity and algorithmic information theory, including analogues to the Heisenberg uncertainty principle.

Proposed method

• Defines an inference device (ID) as a physical system with a specific mathematical structure common to observation, control, prediction, and memory devices.
• Uses Kripke and Aumann structures to model knowledge and belief, embedding IDs within epistemic logic frameworks.
• Introduces a strengthened ID model that includes features from all four knowledge-acquisition modes, avoiding logical omniscience by restricting knowledge to computable, fallible inference.
• Applies probability distributions over universe states to model uncertainty and derive bounds on error products, analogous to quantum uncertainty principles.
• Introduces distance functions D(Γ(W), γ) to quantify claim error, enabling analysis of expected error, variance, and error propagation.
• Extends the formalism to consider physical knowledge in mathematical systems, modeling IDs as theorem-provers within a universe of true strings.

Experimental results

Research questions

• RQ1 What universal constraints arise on the joint knowledge of multiple inference devices embedded in a physical universe?
• RQ2 How can the formalism of inference devices overcome the problem of logical omniscience in epistemic logic?
• RQ3 Can a probabilistic error model for inference devices yield results formally similar to the Heisenberg uncertainty principle?
• RQ4 How can physical knowledge be axiomatized and extended to concepts like common knowledge, distributed knowledge, and semantic information?
• RQ5 What is the relationship between inference complexity (ID analogues of Kolmogorov complexity) and algorithmic information theory?

Key findings

• Any set of inference devices embedded in a physical universe is subject to fundamental impossibility theorems that constrain their joint knowledge, regardless of the universe's physical laws.
• The extended ID framework avoids logical omniscience by restricting knowledge to computable, fallible inference, making it suitable for modeling real agents.
• A product of error probabilities across two independent IDs is bounded below, formally resembling the Heisenberg uncertainty principle.
• Error in an ID’s claim about a universe state can be quantified using a distance function D(Γ(W), γ), enabling analysis of expected error and variance.
• The formalism supports a complete axiomatization of physical knowledge, analogous to Kripke structures, and can be extended to concepts like common knowledge and distributed knowledge.
• Physical knowledge can be applied to mathematical systems by modeling the universe as a set of true strings and IDs as theorem-proving algorithms, with results on their mutual knowledge limits.