A case for weakening the Church-Turing Thesis
Bhupinder Singh Anand

TL;DR
This paper argues that the traditional Church-Turing Thesis should be weakened, based on Gödel's Theorem VII, suggesting that some recursive functions are verifiable but not computable, challenging the standard interpretation.
Contribution
It introduces a revised perspective on the Church-Turing Thesis, proposing a weaker version aligned with algorithmic verifiability rather than computability.
Findings
Recursive functions are verifiable but not necessarily computable.
The standard Church-Turing Thesis does not fully capture all effectively verifiable functions.
A weaker postulation of the Thesis is proposed, aligning with algorithmic verifiability.
Abstract
We conclude from Goedel's Theorem VII of his seminal 1931 paper that every recursive function f(x_{1}, x_{2}) is representable in the first-order Peano Arithmetic PA by a formula [F(x_{1}, x_{2}, x_{3})] which is algorithmically verifiable, but not algorithmically computable, if we assume that the negation of a universally quantified formula of the first-order predicate calculus is always indicative of the existence of a counter-example under the standard interpretation of PA. We conclude that the standard postulation of the Church-Turing Thesis does not hold if we define a number-theoretic formula as effectively computable if, and only if, it is algorithmically verifiable; and needs to be replaced by a weaker postulation of the Thesis as an equivalence.
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Taxonomy
TopicsComputability, Logic, AI Algorithms · Cellular Automata and Applications · DNA and Biological Computing
