Related papers: Relationships between computability-theoretic prop…
Computational pseudorandomness studies the extent to which a random variable $\bf{Z}$ looks like the uniform distribution according to a class of tests $\cal{F}$. Computational entropy generalizes computational pseudorandomness by studying…
A $\lambda$-calculus is introduced in which all programs can be evaluated in probabilistic polynomial time and in which there is sufficient structure to represent sequential cryptographic constructions and adversaries for them, even when…
Reversibility is a key issue in the interface between computation and physics, and of growing importance as miniaturization progresses towards its physical limits. Most foundational work on reversible computing to date has focussed on…
What does it mean to claim that a physical or natural system computes? One answer, endorsed here, is that computing is about programming a system to behave in different ways. This paper offers an account of what it means for a physical…
The present article introduces a reference framework for discussing resilience of computational systems. Rather than a property that may or may not be exhibited by a system, resilience is interpreted here as the emerging result of a dynamic…
Encodings, that is, injective functions from words to words, have been studied extensively in several settings. In computability theory the notion of encoding is crucial for defining computability on arbitrary domains, as well as for…
Computational complexity is examined using the principle of increasing entropy. To consider computation as a physical process from an initial instance to the final acceptance is motivated because many natural processes have been recognized…
Shapiro's notations for natural numbers, and the associated desideratum of acceptability - the property of a notation that all recursive functions are computable in it - is well-known in philosophy of computing. Computable structure theory,…
There are several forms of irreducibility in computing systems, ranging from undecidability to intractability to nonlinearity. This paper is an exploration of the conceptual issues that have arisen in the course of investigating speed-up…
We introduce the notion of finitary computable reducibility on equivalence relations on the natural numbers. This is a weakening of the usual notion of computable reducibility, and we show it to be distinct in several ways. In particular,…
We call an $\alpha \in \mathbb{R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha$ with $\alpha - a_n < 2^{-n}$ for infinitely many $n \in \mathbb{N}$. We…
The design of metaprogramming languages requires appreciation of the tradeoffs that exist between important language characteristics such as safety properties, expressive power, and succinctness. Unfortunately, such tradeoffs are little…
Let $F \subset S_k$ be a finite set of permutations and let $C_n(F)$ denote the number of permutations $\sigma$ in $S_n$ avoiding the set of patterns $F$. The Noonan-Zeilberger conjecture states that the sequence ${C_n(F)}$ is P-recursive.…
Computational complexity is a core theory of computer science, which dictates the degree of difficulty of computation. There are many problems with high complexity that we have to deal, which is especially true for AI. This raises a big…
We investigate partial functions and computability theory from within a constructive, univalent type theory. The focus is on placing computability into a larger mathematical context, rather than on a complete development of computability…
We prove that, for every theory $T$ which is given by an ${\mathcal L}_{\omega_1,\omega}$ sentence, $T$ has less than $2^{\aleph_0}$ many countable models if and only if we have that, for every $X\in 2^\omega$ on a cone of Turing degrees,…
Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational…
We introduce a notion of computable randomness for infinite sequences that generalises the classical version in two important ways. First, our definition of computable randomness is associated with imprecise probability models, in the sense…
The classical approach to inverse problems is based on the optimization of a misfit function. Despite its computational appeal, such an approach suffers from many shortcomings, e.g., non-uniqueness of solutions, modeling prior knowledge,…
Computational feasibility is a widespread concern that guides the framing and modeling of biological and artificial intelligence. The specification of cognitive system capacities is often shaped by unexamined intuitive assumptions about the…