Related papers: Generalized Effective Reducibility
We analyze the degree-structure induced by large reducibilities under the Axiom of Determinacy. This generalizes the analysis of Borel reducibilities given in references [1], [6] and [5] e.g. to the projective levels.
We prove, for stably computably enumerable formal systems, direct analogues of the first and second incompleteness theorems of G\"odel. A typical stably computably enumerable set is the set of Diophantine equations with no integer…
In this paper, we inquire the key concept P-reduction in Cook's theorem and reveal that there exists the fallacy of definition in P-reduction caused by the disguised displacement of NDTM from Oracle machine to Turing machine. The definition…
We construct examples of finitely generated decidable group presentations that satisfy certain combinations of solvability for the word problem, solvability for the bounded word problem, and computablity for the Dehn function. We prove that…
Length generalization refers to the ability to extrapolate from short training sequences to long test sequences and is a challenge for current large language models. While prior work has proposed some architecture or data format changes to…
For any class of operators which transform unary total functions in the set of natural numbers into functions of the same kind, we define what it means for a real function to be uniformly computable or conditionally computable with respect…
We first partly develop a mathematical notion of stable consistency intended to reflect the actual consistency property of human beings. Then we give a generalization of the first and second G\"odel incompleteness theorem to stably…
The history of computability theory and and the history of analysis are surprisingly intertwined since the beginning of the twentieth century. For one, \'Emil Borel discussed his ideas on computable real number functions in his introduction…
Three recent arguments seek to show that the universal applicability of unitary quantum theory is inconsistent with the assumption that a well-conducted measurement always has a definite physical outcome. In this paper I restate and analyze…
Recent research (arXiv:2310.11453, arXiv:2402.17764) has proposed binary and ternary transformer networks as a way to significantly reduce memory and improve inference speed in Large Language Models (LLMs) while maintaining accuracy. In…
The purposes of this note are the following two; we first generalize Okada-Takeuti's well quasi ordinal diagram theory, utilizing the recent result of Dershowitz-Tzameret's version of tree embedding theorem with gap conditions. Second, we…
Contrary to the classical case, the relation between quantum programming languages and quantum Turing Machines (QTM) has not being fully investigated. In particular, there are features of QTMs that have not been exploited, a notable example…
In this paper, we study the power and limitations of computing effectively generic sequences using effectively random oracles. Previously, it was known that every 2-random sequence computes a 1-generic sequence (as shown by Kautz) and every…
Inspired by Quantum Mechanics, we reformulate Hilbert's tenth problem in the domain of integer arithmetics into problems involving either a set of infinitely-coupled non-linear differential equations or a class of linear Schr\"odinger…
Large language models (LLMs) have proven to be highly effective for solving complex reasoning tasks. Surprisingly, their capabilities can often be improved by iterating on previously generated solutions. In this context, a reasoning plan…
We identify computability-theoretic properties enabling us to separate various statements about partial orders in reverse mathematics. We obtain simpler proofs of existing separations, and deduce new compound ones. This work is part of a…
This paper demonstrates the relativity of Computability and Nondeterministic; the nondeterministic is just Turing's undecidable Decision rather than the Nondeterministic Polynomial time. Based on analysis about TM, UM, DTM, NTM, Turing…
We consider the question whether there is an infinitary analogue of the Church-Turing-thesis. To this end, we argue that there is an intuitive notion of transfinite computability and build a canonical model, called Idealized Agent Machines…
We examine the connections between deterministic, complete, and general global optimisation of continuous functions and a general concept of regression from the perspective of constructive type theory via the concept of 'searchability'. We…
Martin-L\"of's Intuitionistic Theory of Types is becoming popular for formal reasoning about computer programs. To handle recursion schemes other than primitive recursion, a theory of well-founded relations is presented. Using primitive…