Related papers: PCF arithmetic without and with choice
This article presents a technique for proving problems hard for classes of the polynomial hierarchy or for PSPACE. The rationale of this technique is that some problem restrictions are able to simulate existential or universal quantifiers.…
This paper introduces a new simplified version of the countable branching recurrence of Computability Logic, proves its equivalence to the old one, and shows that the basic logic induced by it is a proper superset of the basic logic induced…
This paper presents two types of results related to hyperarithmetic analysis. First, we introduce new variants of the dependent choice axiom, namely $\mathrm{unique}~\Pi^1_0(\mathrm{resp.}~\Sigma^1_1)\text{-}\mathsf{DC}_0$ and…
We study a family of distance functions on rankings that allow for asymmetric treatments of alternatives and consider the distinct relevance of the top and bottom positions for ordered lists. We provide a full axiomatic characterization of…
We characterize Weihrauch reducibility in E-PA$^{\omega}$ + QF-AC$^{0,0}$ and all systems containing it by the provability in a linear variant of the same calculus using modifications of G\"odel's Dialectica interpretation that incorporate…
In considering the reliability of numerical programs, it is normal to "limit our study to the semantics dealing with numerical precision" (Martel, 2005). On the other hand, there is a great deal of work on the reliability of programs that…
Compound probabilistic context-free grammars (C-PCFGs) have recently established a new state of the art for unsupervised phrase-structure grammar induction. However, due to the high space and time complexities of chart-based representation…
Choice functions constitute a simple, direct and very general mathematical framework for modelling choice under uncertainty. In particular, they are able to represent the set-valued choices that appear in imprecise-probabilistic decision…
In reinforcement learning, the classic objectives of maximizing discounted and finite-horizon cumulative rewards are PAC-learnable: There are algorithms that learn a near-optimal policy with high probability using a finite amount of samples…
In the search for a logic capturing polynomial time the most promising candidates are Choiceless Polynomial Time (CPT) and rank logic. Rank logic extends fixed-point logic with counting by a rank operator over prime fields. We show that the…
We extend our previous work on sensitivity analysis for the risk ratio and difference contrasts under unmeasured confounding to any contrast. We prove that the bounds produced are still arbitrarily sharp, i.e. practically attainable. We…
In this paper, without the axiom of choice, we show that if a certain downward L\"owenheim-Skolem property holds then all grounds are uniformly definable. We also prove that the axiom of choice is forceable if and only if the universe is a…
During the last twenty years or so a wide range of realizability interpretations of classical analysis have been developed. In many cases, these are achieved by extending the base interpreting system of primitive recursive functionals with…
New identities and congruences involving the ranks and cranks of partitions are proved. The proof depends on a new partial differential equation connecting their generating functions.
We consider the problem of extending an acyclic binary relation that is invariant under a given family of transformations into an invariant preference. We show that when a family of transformations is commutative, every acyclic invariant…
We study very simple sorting algorithms based on a probabilistic comparator model. In our model, errors in comparing two elements are due to (1) the energy or effort put in the comparison and (2) the difference between the compared…
Non-convex optimization problems often arise from probabilistic modeling, such as estimation of posterior distributions. Non-convexity makes the problems intractable, and poses various obstacles for us to design efficient algorithms. In…
We examine so-called rank function equations and their solutions consisting of non-nilpotent matrices. Secondly, we present some geometrical properties of the set of solutions to certain rank function equations in the nilpotent case.
Let m>2 be an integer. We show that ZF + "For every integer n, Every countable family of non-empty sets of cardinality at most n has an infinite partial choice function" is not strong enough to prove that every countable set of m-element…
The aim of this paper is to give an alternative proof of Kac's theorem for weighted projective lines (\cite{W}) over the complex field. The geometric realization of complex Lie algebras arising from derived categories (\cite{XXZ}) is…