The Comprehensibility Theorem and the Foundations of Artificial Intelligence

Abstract

Problem-solving software that is not-necessarily infallible is central to AI. Such software whose correctness and incorrectness properties are deducible by agents is an issue at the foundations of AI. The Comprehensibility Theorem, which appeared in a journal for specialists in formal mathematical logic, might provide a limitation concerning this issue and might be applicable to any agents, regardless of whether the agents are artificial or natural. The present article, aimed at researchers interested in the foundations of AI, addresses many questions related to that theorem, including differences between it and results of Gödel and Turing that have sometimes played key roles in Minds and Machines articles. This study also suggests that-if one is willing to assume a thesis due to Donald Knuth-the Comprehensibility Theorem is the first mathematical theorem implying the impossibility of any AI agent or natural agent-including a not-necessarily infallible human agent-satisfying a rigorous and deductive interpretation of the self-comprehensibility challenge. Some have pointed out the difficulty of self-comprehensibility, even according to presumably a less rigorous interpretation. This includes Socrates, who considered it to be among the most important of intellectual tasks. Selfcomprehensibility in some form might be essential for a kind of self-reflection useful for self-improvement that might enable some agents to increase their success. We use the methods of applied mathematics, rather than philosophy, although some topics considered could be of interest to philosophers.

A Theorem about Computationalism and “Absolute” Truth

“…The single natural number code for a board configuration would include information about the value of n for the instance of generalized chess. (Of course, the ten digits suffice to construct the numeral for any n.) Although the halting problem game the two recent theorems in Charlesworth (2006Charlesworth ( , 2014 are about seems quite different from generalized chess, it is the case that for both kinds of games-unlike for usual chess-there is an infinite number of possible natural number inputs (and outputs).…”

“…See pp. 465-466 of Charlesworth (2014) for an explanation of additional ways infallibility assumptions on an agent are avoided. 8 One might ask: How important are halting problems?…”

“…8 One might ask: How important are halting problems? Their wide importance is elucidated from p. 442 through the first full paragraph on p. 447 of Charlesworth (2014).…”

“…We employ a mathematical framework that supported two recent theorems related to mathematical logic found in Charlesworth (2006). The first of the theorems is explicated in Charlesworth (2014), and that first theorem justifies the terminology used in the second theorem. Using that framework we prove a new theorem that avoids infallibility assumptions, in contrast to Gödel's argument for the conjecture in his Gibbs lecture.…”

Modelling of the relation of implication with use of the directed relational networks

“…Приводится обзор методов распределенного обучения для работы с "очень большими" наборами данных. В работе [14] высказывается сомнение в том, что решение проблемы программного обеспечения занимает центральное место в ИИ.…”