Related papers: Infinite computations with random oracles
We initiate the study of computable presentations of real and complex C*-algebras under the program of effective metric structure theory. With the group situation as a model, we develop corresponding notions of recursive presentations and…
While it is well known that finding approximate optima of non-convex functions is computationally intractable, we show that the problem is, in fact, uncomputable in the oracle model. Specifically, we prove that no algorithm with access only…
A novel model of reversible computing, the $\aleph$-calculus, is introduced. It is declarative, reversible-Turing complete, and has a local term-rewriting semantics. Unlike previously demonstrated reversible term-rewriting systems, it does…
We show that in the setting of fair-coin measure on the power set of the natural numbers, each sufficiently random set has an infinite subset that computes no random set. That is, there is an almost sure event $\mathcal A$ such that if…
As inductive inference and machine learning methods in computer science see continued success, researchers are aiming to describe ever more complex probabilistic models and inference algorithms. It is natural to ask whether there is a…
We give several new examples of computable structures of high Scott rank. For earlier known computable structures of Scott rank $\omega_1^{CK}$, the computable infinitary theory is $\aleph_0$-categorical. Millar and Sacks asked whether this…
Term algebras are important objects in computer science and are correspondingly well-studied. A natural generalization is to quotient these algebras by finitely many ground term equations, obtaining what we call almost free algebras. One of…
If we define classical foundational concepts constructively, and introduce non-algorithmic effective methods into classical mathematics, then we can bridge the chasm between truth and provability, and define computational methods that are…
The paper considers quantitative versions of different randomness notions: algorithmic test measures the amount of non-randomness (and is infinite for non-random sequences). We start with computable measures on Cantor space (and Martin-Lof…
We prove that quantum computation is polynomially equivalent to classical probabilistic computation with an oracle for estimating the value of simple sums, quadratically signed weight enumerators. The problem of estimating these sums can be…
In classical computation, a "write-only memory" (WOM) is little more than an oxymoron, and the addition of WOM to a (deterministic or probabilistic) classical computer brings no advantage. We prove that quantum computers that are augmented…
Metric Temporal Logic (MTL) is a prominent specification formalism for real-time systems. In this paper, we show that the satisfiability problem for MTL over finite timed words is decidable, with non-primitive recursive complexity. We also…
We explore the class of real numbers that are computed in real time by deterministic chemical reaction networks that are integral in the sense that all their reaction rate constants are positive integers. We say that such a reaction network…
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…
This work studies the applicability of expensive external oracles such as large language models in answering top-k queries over predicted scores. Such scores are incurred by user-defined functions to answer personalized queries over…
We initiate a formal study of logical inferences in context of the measure problem in cosmology or what we call cosmic logic. We describe a simple computational model of cosmic logic suitable for analysis of, for example, discretized…
We consider notions of space complexity for Infinite Time Turing Machines (ITTMs) that were introduced by B. L\"owe and studied further by J. Winter. We answer several open questions about these notions, among them whether low space…
The elementary arithmetic operations $+,\cdot,\le$ on integers are well-known to be computable in the weak complexity class $\mathrm{TC}^0$, and it is a basic question what properties of these operations can be proved using only…
In this note we are concerned with the validity of an uncountable analogue of a combinatorial lemma due to Vlastimil Pt\'ak. We show that the validity of the result for $\omega_1$ can not be decided in ZFC alone. We also provide a…
Polynomial--time constant--space quantum Turing machines (QTMs) and logarithmic--space probabilistic Turing machines (PTMs) recognize uncountably many languages with bounded error (Say and Yakary\i lmaz 2014, arXiv:1411.7647). In this…