Related papers: Every Computably Enumerable Random Real Is Provabl…
In a recent historical overview, Cristian S. Calude, Elena Calude, and Solomon Marcus identify eight stages in the development of the concept of a mathematical proof in support of an ambitious conjecture: we can express classical…
We describe a "slow" version of the hierarchy of uniform reflection principles over Peano Arithmetic ($\mathbf{PA}$). These principles are unprovable in Peano Arithmetic (even when extended by usual reflection principles of lower…
Based on the MRDP theorem concerning the Hilbert tenth problem, there is a corresponding Diophantine equation called proof equation for every formula of the First-order Peano Arithmetic (PA). A formula is provable in PA, if and only if the…
We show that there exists a universal quantum Turing machine (UQTM) that can simulate every other QTM until the other QTM has halted and then halt itself with probability one. This extends work by Bernstein and Vazirani who have shown that…
The aim of this paper is to present an elementary computable theory of probability, random variables and stochastic processes. The probability theory is baed on existing approaches using valuations and lower integrals. Various approaches to…
We introduce a notion of realizability with ordinal Turing machines based on recognizability rather than computability, i.e., the ability to uniquely identify an object. We show that the arising concept of $r$-realizabilty has the property…
We introduce a protocol through which a pair of quantum mechanical devices may be used to generate n bits of true randomness from a seed of O(log n) uniform bits. The bits generated are certifiably random based only on a simple statistical…
Based on the MRDP theorem, we introduce the ideas of the proof equation of a formula and universal proof equation of Peano Arithmetic (PA); and then, combining universal proof equation and G\"odel's Second Incompleteness Theorem, it is…
We show that when certain statements are provable in subsystems of constructive analysis using intuitionistic predicate calculus, related sequential statements are provable in weak classical subsystems. In particular, if a $\Pi^1_2$…
There are several forms of irreducibility in computing systems, ranging from undecidability to intractability to nonlinearity. This paper is an exploration of the conceptual issues that have arisen in the course of investigating speed-up…
We study the computational power of randomized computations on infinite objects, such as real numbers. In particular, we introduce the concept of a Las Vegas computable multi-valued function, which is a function that can be computed on a…
We prove that, assuming $\mathrm{ZF}$, and restricted to any pointed set, Chaitin's $\Omega_U:x\mapsto \Omega_U^x=\sum_{U^x(\sigma)\downarrow}2^{-|\sigma|}$ is not injective for any universal prefix-free Turing machine $U$, and that…
This paper presents a new representation of natural numbers and discusses its consequences for computability and computational complexity. The paper argues that the introduction of the first Peano axiom in the traditional definition of…
We revisit the question (most famously) initiated by Turing: can human intelligence be completely modeled by a Turing machine? We show that the answer is \emph{no}, assuming a certain weak soundness hypothesis. More specifically we show…
We give upper bound for several highness properties in computability randomness theory. First, we prove that discrete covering property does not imply the ability to compute a 1-random real, answering a question of Greenberg, Miller and…
We show that if a set $A$ is computable from every superlow 1-random set, then $A$ is strongly jump-traceable. This theorem shows that the computably enumerable (c.e.) strongly jump-traceable sets are exactly the c.e.\ sets computable from…
This paper develops an assume-guarantee (AG) framework for the compositional verification of probabilistic automata (PAs) with uncertain transition probabilities. We study parametric probabilistic automata (pPAs), where probabilities are…
We distinguish finitarily between algorithmic verifiability, and algorithmic computability, to show that Goedel's 'formally' unprovable, but 'numeral-wise' provable, arithmetical proposition [(Ax)R(x)] can be finitarily evidenced as:…
The pseudoinverse of a matrix, a generalized notion of the inverse, is of fundamental importance in linear algebra and, thereby, in many different fields. Despite its proven existence, an algorithmic approach is typically necessary to…
We study the implications of model completeness of a theory for the effectiveness of presentations of models of that theory. It is immediate that for a computable model $\mathcal A$ of a computably enumerable, model complete theory, the…