Related papers: Every Computably Enumerable Random Real Is Provabl…
Quantum theory (QT) has been confirmed by numerous experiments, yet we still cannot fully grasp the meaning of the theory. As a consequence, the quantum world appears to us paradoxical. Here we shed new light on QT by having it follow from…
In much discussed work Artemov has recently shown that, for $\mathrm{PA}$, the consistency schema admits a form of uniform verification via selector proofs, despite the unprovability of the corresponding uniform consistency sentence…
This paper constructively proves the existence of an effective procedure generating a computable (total) function that is not contained in any given effectively enumerable set of such functions. The proof implies the existence of machines…
We show in this article that uncomputability is also a relative property of subrecursive classes built on a recursive relative incompressible function, which acts as a higher-order "yardstick" of irreducible information for the respective…
In this paper, we introduce a foundation for computable model theory of rational Pavelka logic (an extension of {\L}ukasiewicz logic) and continuous logic, and prove effective versions of some theorems in model theory. We show how to reduce…
Voltage peaks on a conventional computer's power lines allow for the well-known dangerous DPA attacks. We show that measurement of a quantum computer's transient state during a computational step reveals information about a complete…
We prove various extensions of the Tennenbaum phenomenon to the case of computable quotient presentations of models of arithmetic and set theory. Specifically, no nonstandard model of arithmetic has a computable quotient presentation by a…
The classic model of computable randomness considers martingales that take real or rational values. Recent work by Bienvenu et al. (2012) and Teutsch (2014) shows that fundamental features of the classic model change when the martingales…
The Church-Turing thesis states that any sufficiently powerful computational model which captures the notion of algorithm is computationally equivalent to the Turing machine. This equivalence usually holds both at a computability level and…
In his seminal paper from 1936, Alan Turing introduced the concept of non-computable real numbers and presented examples based on the algorithmically unsolvable Halting problem. We describe a different, analytically natural mechanism for…
A certain mathematician M, considering some hypothesis H, conclusion C and text P, can arrive at one of the following judgments: (1) P does not convince M of the fact that since H, it follows that C; (2) P is the proof that since H, it…
Turing's famous 'machine' framework provides an intuitively clear conception of 'computing with real numbers'. A recursive counterexample to a theorem shows that the theorem does not hold when restricted to computable objects. These…
We propose a definition of quantum computable functions as mappings between superpositions of natural numbers to probability distributions of natural numbers. Each function is obtained as a limit of an infinite computation of a quantum…
Chaitin [G. J. Chaitin, J. Assoc. Comput. Mach., vol.22, pp.329-340, 1975] introduced \Omega number as a concrete example of random real. The real \Omega is defined as the probability that an optimal computer halts, where the optimal…
This work studies the problem of constructing capacity-achieving codes from an algorithmic perspective. Specifically, we prove that there exists a Turing machine which, given a discrete memoryless channel $p_{Y|X}$, a target rate $R$ less…
We consider the issue of computability at the most fundamental level of physical reality: the Planck scale. To this aim, we consider the theoretical model of a quantum computer on a non commutative space background, which is a computational…
We study the computational power of parsing expression grammars (PEGs). We begin by constructing PEGs with unexpected behaviour, and surprising new examples of languages with PEGs, including the language of palindromes whose length is a…
We propose a new generalisation of Cayley automatic groups, varying the time complexity of computing multiplication, and language complexity of the normal form representatives. We first consider groups which have normal form language in the…
In this paper we introduce the notion of $e$-computability as a method of finding the Waring rank of forms. We use this notion to find infinitely many new examples which satisfy Strassen's Conjecture.
SAT is not in P, is true and provable in a simply consistent extension B' of a first order theory B of computing, with a single finite axiom characterizing a universal Turing machine. Therefore, P is not equal to NP, is true and provable in…