Related papers: Some Mathematicians Are Not Turing Machines
This article answers two questions (posed in the literature), each concerning the guaranteed existence of proofs free of double negation. A proof is free of double negation if none of its deduced steps contains a term of the form n(n(t))…
It is commonly agreed that the success of future proof assistants will rely on their ability to incorporate computations within deduction in order to mimic the mathematician when replacing the proof of a proposition P by the proof of an…
We study the algebraic theory of computable functions, which can be viewed as arising from possibly non-halting computer programs or algorithms, acting on some state space, equipped with operations of composition, {\em if-then-else} and…
In order to study the axiomatization of the if-then-else construct over possibly non-halting programs and tests, this paper introduces the notion of $C$-sets by considering the tests from an abstract $C$-algebra. When the $C$-algebra is an…
There are growing uncertainties surrounding the classical model of computation established by G\"odel, Church, Kleene, Turing and others in the 1930s onwards. The mismatch between the Turing machine conception, and the experiences of those…
Can a probabilistic gambler get arbitrarily rich when all deterministic gamblers fail? We study this problem in the context of algorithmic randomness, introducing a new notion -- almost everywhere computable randomness. A binary sequence…
Mermin states that his nontechnical version of Bell's theorem stands and is not invalidated by time and setting dependent instrument parameters as claimed in one of our previous papers. We identify deviations from well-established protocol…
We give a precise definition of a formal mathematical object as any symbol for an individual constant, predicate letter, or a function letter that can be introduced through definition into a formal mathematical language without inviting…
Probability theory, epistemically interpreted, provides an excellent, if not the best available account of inductive reasoning. This is so because there are general and definite rules for the change of subjective probabilities through…
We prove that any Turing machine running on inputs of arbitrary length can be simulated by a constant bit-size transformer, as long as the context window is sufficiently long. This improves previous works, which require scaling up either…
There are many types of automata and grammar models that have been studied in the literature, and for these models, it is common to determine whether certain problems are decidable. One problem that has been difficult to answer throughout…
Mathematical proofs are both paradigms of certainty and some of the most explicitly-justified arguments that we have in the cultural record. Their very explicitness, however, leads to a paradox, because the probability of error grows…
In this paper, we present a generalized effective completeness theorem for continuous logic. The primary result is that any continuous theory is satisfied in a structure which admits a presentation of the same Turing degree. It then follows…
The paper puts into discussion the concept of universality, in particular for structures not of the power of Turing computability. The question arises if for such structures a universal structure of the same kind exists or not. For that the…
We discuss the possibility of constructing a function that validates the definition or not definition of the partial recursive functions of one variable. This is a topic in computability theory, which was first approached by Alan M. Turing…
As machine learning is increasingly used in essential systems, it is important to reduce or eliminate the incidence of serious bugs. A growing body of research has developed machine learning algorithms with formal guarantees about…
Interactive theorem provers have been used extensively to reason about various software/hardware systems and mathematical theorems. The key challenge when using an interactive prover is finding a suitable sequence of proof steps that will…
According to quantum theory, the outcomes of future measurements cannot (in general) be predicted with certainty. In some cases, even with a complete physical description of the system to be measured and the measurement apparatus, the…
We describe a "top down" approach for automated theorem proving (ATP). Researchers might usefully investigate the forms of the theorems mathematicians use in practice, carefully examine how they differ and are proved in practice, and code…
In an empirical logic, an experimentally verifiable proposition P relating to a quantum system is assigned the value of either true of false if the system is in the pure state that belongs or, respectively, does not belong to the Hilbert…