Related papers: On low for speed oracles
The logical depth of a {\em reversible} Turing machine equals the shortest running time of a shortest program for it. This is applied to show that the result in L.F. Antunes, A. Souto, and P.M.B. Vit\'anyi, On the Rate of Decrease in…
This paper is an extended version of our work in \cite{Ca2025}. We extend the concept of effective reducibility between statements of set theory with ordinal Turing machines (OTMs) explored in \cite{Ca2018} for $\Pi_{2}$-statements to…
We investigate infinitary wellfounded systems for linear logic with fixed points, with transfinite branching rules indexed by some closure ordinal $\alpha$ for fixed points. Our main result is that provability in the system for some…
Coalgebraic bisimilarity minimization generalizes classical automaton minimization to a large class of automata whose transition structure is specified by a functor, subsuming strong, weighted, and probabilistic bisimilarity. This offers…
We exhibit a family of computably enumerable sets which can be learned within polynomial resource bounds given access only to a teacher, but which requires exponential resources to be learned given access only to a membership oracle. In…
The concept of ``countable set'' is attributed to Georg Cantor, who set the boundary between countable and uncountable sets in 1874. The concept of ``computable set'' arose in the study of computing models in the 1930s by the founders of…
Recently, a new paradigm was introduced in automata theory. The main idea is to classify regular languages according to their propensity to be sorted, establishing a deep connection between automata theory and data compression [J. ACM…
Large language models (LLMs) have rapidly progressed into general-purpose agents capable of solving a broad spectrum of tasks. However, current models remain inefficient at reasoning: they apply fixed inference-time compute regardless of…
Human language and logic abilities are computationally quantified within the well-studied grammar-automata hierarchy. We identify three hierarchical tiers and two corresponding transitions and show their correspondence to specific abilities…
This work introduces a relative diffusion transformation (RDT) - a simple unitary transformation which acts in a subspace, localized by an oracle. Such a transformation can not be fulfilled on quantum Turing machines with this oracle in…
Adversarial computations are a widely studied class of computations where resource-bounded probabilistic adversaries have access to oracles, i.e., probabilistic procedures with private state. These computations arise routinely in several…
From the existence of an efficient quantum algorithm for factoring, it is likely that quantum computation is intrinsically more powerful than classical computation. At present, the best upper bound known for the power of quantum computation…
An approximation of a real is a sequence of rational numbers that converges to the real. An approximation is left-c.e. if it is computable and nondecreasing and is d.c.e. if it is computable and has bounded variation. A real is computably…
Computability logic (CoL) provides a semantic foundation in which formulas represent interactive computational problems and validity corresponds to uniform algorithmic solvability. Building on this foundation, clarithmetics -- CoL-based…
We show how to compute efficiently with nominal sets over the total order symmetry, by developing a direct representation of such nominal sets and basic constructions thereon. In contrast to previous approaches, we work directly at the…
Nfer is a rule-based language for abstracting event streams into a hierarchy of intervals with data. Nfer has multiple implementations and has been applied in the analysis of spacecraft telemetry and autonomous vehicle logs. This work…
A major part of computability theory focuses on the analysis of a few structures of central importance. As a tool, the method of coding with first-order formulas has been applied with great success. For instance, in the c.e. Turing degrees,…
The probability distribution P from which the history of our universe is sampled represents a theory of everything or TOE. We assume P is formally describable. Since most (uncountably many) distributions are not, this imposes a strong…
BSS RAMs over first-order structures help to characterize algorithms for processing objects by means of useful operations and relations. They are the result of a generalization of several types of abstract machines. We want to discuss…
We introduce an analog of the theory of Borel equivalence relations in which we study equivalence relations that are decidable by an infinite time Turing machine. The Borel reductions are replaced by the more general class of infinite time…