Related papers: Acceptable Complexity Measures of Theorems
Standard expositions of Goedel's 1931 paper on undecidable arithmetical propositions are based on two presumptions in Goedel's 1931 interpretation of his own, formal, reasoning - one each in Theorem VI and in Theorem XI - which do not meet…
The uncertainty principle bounds the uncertainties about incompatible measurements, clearly setting quantum theory apart from the classical world. Its mathematical formulation via uncertainty relations, plays an irreplaceable role in…
Little effort has been devoted to studying generalised notions or models of (un)predictability, yet is an important concept throughout physics and plays a central role in quantum information theory, where key results rely on the supposed…
The nature of concept learning is a core question in cognitive science. Theories must account for the relative difficulty of acquiring different concepts by supervised learners. For a canonical set of six category types, two distinct…
We survey results on the formalization and independence of mathematical statements related to major open problems in computational complexity theory. Our primary focus is on recent findings concerning the (un)provability of complexity…
Some quantum measurements can not be performed simultaneously, i.e. they are incompatible. Here we show that every set of incompatible measurements provides an advantage over compatible ones in a suitably chosen quantum state discrimination…
G\"odel's Incompleteness Theorems suggest that no single formal system can capture the entirety of one's mathematical beliefs, while pointing at a hierarchy of systems of increasing logical strength that make progressively more explicit…
Given a sound first-order p-time theory $T$ capable of formalizing syntax of first-order logic we define a p-time function $g_T$ that stretches all inputs by one bit and we use its properties to show that $T$ must be incomplete. We leave it…
A pair of quantum observables diagonal in the same "incoherent" basis can be measured jointly, so some coherence is obviously required for measurement incompatibility. Here we first observe that coherence in a single observable is linked to…
An accurate assessment of a model's complexity is crucial for topics such as interpretation, generalization, and model selection. However, most existing complexity measures either rely on heuristic assumptions or are computationally…
In Monoidal Computer I, we introduced a categorical model of computation where the formal reasoning about computability was supported by the simple and popular diagrammatic language of string diagrams. In the present paper, we refine and…
The literature dealing with G\"{o}del's legacy is largely preoccupied with challenging his philosophical views, regarding them as outdated. We believe that such an approach prevents us from seeing G\"{o}del's views in the right light and…
These five lectures on undecidability were given to students with a good level in mathematics but with no special knowledge on logic. The first conference presents the formalization of mathematics with a short historical survey, the…
The quantum measurement problem is formulated in the form of an insolubility theorem that states the impossibility of obtaining, for all available object preparations, a mixture of states of the compound object and apparatus system that…
We give examples of (i) a simple theory with a formula (with parameters) which does not fork over the empty set but has mu measure 0 for every automorphism invariant Keisler measure mu, and (ii) a definable group G in a simple theory such…
We introduce a new complexity measure of a path of (problems, solutions) pairs in terms of the length of the path in the condition metric which we define in the article. The measure gives an upper bound for the number of Newton steps…
Motivated by the problem of finding finite versions of classical incompleteness theorems, we present some conjectures that go beyond ${\bf NP\neq co NP}$. These conjectures formally connect computational complexity with the difficulty of…
We reveal a contradiction in measure-theoretic probability. The contradiction is an "equation" $1/2 = 0$ with its two sides representing probabilities. Unlike known paradoxes in mathematics, the revealed contradiction cannot be explained…
Goedel's Incompleteness Theorems have the same scientific status as Einstein's principle of relativity, Heisenberg's uncertainty principle, and Watson and Crick's double helix model of DNA. Our aim is to discuss some new faces of the…
As an approach to a Theory of Everything a framework for developing a coherent theory of mathematics and physics together is described. The main characteristic of such a theory is discussed: the theory must be valid and and sufficiently…