Verifier Theory and Unverifiability

📝 Abstract

Despite significant developments in Proof Theory, surprisingly little attention has been devoted to the concept of proof verifier. In particular, the mathematical community may be interested in studying different types of proof verifiers (people, programs, oracles, communities, superintelligences) as mathematical objects. Such an effort could reveal their properties, their powers and limitations (particularly in human mathematicians), minimum and maximum complexity, as well as self-verification and self-reference issues. We propose an initial classification system for verifiers and provide some rudimentary analysis of solved and open problems in this important domain. Our main contribution is a formal introduction of the notion of unverifiability, for which the paper could serve as a general citation in domains of theorem proving, as well as software and AI verification.

💡 Analysis

Despite significant developments in Proof Theory, surprisingly little attention has been devoted to the concept of proof verifier. In particular, the mathematical community may be interested in studying different types of proof verifiers (people, programs, oracles, communities, superintelligences) as mathematical objects. Such an effort could reveal their properties, their powers and limitations (particularly in human mathematicians), minimum and maximum complexity, as well as self-verification and self-reference issues. We propose an initial classification system for verifiers and provide some rudimentary analysis of solved and open problems in this important domain. Our main contribution is a formal introduction of the notion of unverifiability, for which the paper could serve as a general citation in domains of theorem proving, as well as software and AI verification.

📄 Content

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Verifier Theory and Unverifiability

Roman V. Yampolskiy Computer Engineering and Computer Science University of Louisville roman.yampolskiy@louisville.edu

Abstract Despite significant developments in Proof Theory, surprisingly little attention has been devoted to the concept of proof verifier. In particular, the mathematical community may be interested in studying different types of proof verifiers (people, programs, oracles, communities, superintelligences) as mathematical objects. Such an effort could reveal their properties, their powers and limitations (particularly in human mathematicians), minimum and maximum complexity, as well as self-verification and self-reference issues. We propose an initial classification system for verifiers and provide some rudimentary analysis of solved and open problems in this important domain. Our main contribution is a formal introduction of the notion of unverifiability, for which the paper could serve as a general citation in domains of theorem proving, as well as software and AI verification.

Keywords: Verifier Theory, Proof Theory, Observer, Verified Verifier, Verifiability.

1. On Observers and Verifiers The concept of an ‘observer’ shows up in contexts as diverse as physics (particularly quantum and relativity), biophysics, neuroscience, cognitive science, artificial intelligence, philosophy of consciousness, and cosmology [1], but what is an equivalent idea in mathematics? We believe it is the notion of the proof verifier. Consequently, the majority of open questions recently raised [1] by the Foundational Questions Institute related to the physics of the observer could be asked about proof verifiers. In particular, the mathematical community may be interested in studying different types of proof verifiers (people, programs, oracles, communities, superintelligences, etc.) as mathematical objects, ways they can be formalized, their power and limitations (particularly in human mathematicians), minimum and maximum complexity, as well as self-verification and self- reference in verifiers.

Proof Theory has been developed to study proofs as formal mathematical objects consisting of axioms from which, by rules of inference, one can arrive at theorems [2]. However, the indispensable concept of the verifier has been conspicuously absent from the discussion, particularly with regards to its formalization and practical manifestation. A verifier in the context of mathematics is an agent capable of checking a given proof, step-by-step, starting from axioms to make sure that all intermediate deductions are indeed warranted, that the final conclusion follows, and consequently, that the claimed theorem is indeed true. In this work we present an overview of different types of verifiers currently relied on by the mathematical community, as well as a few novel types of verifiers which we suggest be added to the repertoire of mathematicians at least as theoretical tools of Verifier Theory. Our general analysis should be equally applicable to different types of proofs (induction, contradiction, exhaustion, enumeration, refinement, nonconstructive, probabilistic, holographic, experiment, picture, etc.) and to computer software.
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2. Historical Perspective The field of mathematics progresses by proving theorems, which in turn serve as building blocks for future proofs of yet more interesting and useful theorems. To avoid introduction of costly errors in the form of incorrect theorems, proofs typically undergo an examination process, usually as a part of a peer-review. Traditionally, human mathematicians have been employed as proof verifiers; however, history is full of examples of undetected errors and important omissions even in the most widely examined proofs [3-7]. It has been estimated that at least a third of all mathematical publications contain errors [8]. To avoid errors and make the job of human verifiers as easy as possible “a single step in a deduction has been required … [t]o be simple enough, broadly speaking, to be apprehended as correct by a human being in a single intellectual act. No doubt this custom originated in the desire that each single step of a deduction should be indubitable, even though the deduction as a whole may consist of a long chain of such steps” [9].

Despite such stringent requirements, it has long been realized that a single human verifier is not reliable enough to ascertain validity of a proof with a sufficient degree of reliability. In fact, it is known that humans are subject to hundreds of well-known “bugs”1, and probably many more unknown ones. To reduce the number of potential mistakes and to increase our confidence in the validity of a proof, a number of independent human mathematicians should examine an important mathematical claim. As Calude puts it “A theorem is a statement which could be checked individually by a mathematician and confirmed also individua

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