Related papers: Generalized Effective Reducibility
The famous G\"odel incompleteness theorem says that for every sufficiently rich formal theory (containing formal arithmetic in some natural sense) there exist true unprovable statements. Such statements would be natural candidates for being…
We initiate the effective metric structure theory of Keisler randomizations. We show that a classical countable structure $\mathcal{M}$ has a decidable presentation if and only if its Borel randomization $\mathcal{M}^{[0,1)}$ has a…
We reprove the countable splitting lemma by adapting Nawrotzki's algorithm which produces a sequence that converges to a solution. Our algorithm combines Nawrotzki's approach with taking finite cuts. It is constructive in the sense that…
We investigate different notions of "computable topological base" for represented spaces. We show that several non-equivalent notions of bases become equivalent when we consider computably enumerable bases. This indicates the existence of a…
A new characterization of provably recursive functions of first-order arithmetic is described. Its main feature is using only terms consisting of 0, the successor S and variables in the quantifier rules, namely, universal elimination and…
Manifold learning and effective model building are generally viewed as fundamentally different types of procedure. After all, in one we build a simplified model of the data, in the other, we construct a simplified model of the another…
We develop techniques to investigate relativized hierarchical unambiguous computation. We apply our techniques to generalize known constructs involving relativized unambiguity based complexity classes (UP and \mathcal{UP}) to new constructs…
We introduce a first-order theory of finite full binary trees and then identify decidable and undecidable fragments of this theory. We show that the analogue of Hilbert`s 10th Problem is undecidable by constructing a many-to-one reduction…
The notion of computable reducibility between equivalence relations on the natural numbers provides a natural computable analogue of Borel reducibility. We investigate the computable reducibility hierarchy, comparing and contrasting it with…
We investigate the theory of finite observables, i.e., resolutions of the finite-dimensional identity by means of positive operators, that have a physical interpretation in terms of measurement schemes. We focus on extremal and rank-one…
For those of us who generally live in the world of syntax, semantic proof techniques such as reducibility, realizability or logical relations seem somewhat magical despite -- or perhaps due to -- their seemingly unreasonable effectiveness.…
We consider a randomised version of Kleene's realisability interpretation of intuitionistic arithmetic in which computability is replaced with randomised computability with positive probability. In particular, we show that (i) the set of…
The celebrated Trakhtenbrot's theorem states that the set of finitely valid sentences of first-order logic is not computably enumerable. In this note we will extend this theorem by proving that the finite satisfiability problem of any…
We initiate a formal study of reproducibility in optimization. We define a quantitative measure of reproducibility of optimization procedures in the face of noisy or error-prone operations such as inexact or stochastic gradient computations…
We prove the Extended Church-Turing Thesis: Every effective algorithm can be efficiently simulated by a Turing machine. This is accomplished by emulating an effective algorithm via an abstract state machine, and simulating such an abstract…
Automatic (i.e., computer-assisted) theorem proving (ATP) can come in many flavors. This document presents early steps in our effort towards defining object-oriented theorem proving (OOTP) as a new style of ATP. Traditional theorem proving…
Nonuniformity is a central concept in computational complexity with powerful connections to circuit complexity and randomness. Nonuniform reductions have been used to study the isomorphism conjecture for NP and completeness for larger…
We prove a number of results motivated by global questions of uniformity in computability theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of…
A prototype system is described whose core functionality is, based on propositional logic, the elimination of second-order operators, such as Boolean quantifiers and operators for projection, forgetting and circumscription. This approach…
We introduce a reducibility on classes of structures, essentially a uniform enumeration reducibility. This reducibility is inspired by the Friedman-Stanley paper on using Borel reductions to compare classes of countable structures. This…