Related papers: The computational content of Nonstandard Analysis
This paper explores the connection between two central results in the proof theory of classical logic: Gentzen's cut-elimination for the sequent calculus and Herbrands "fundamental theorem". Starting from Miller's expansion-tree-proofs, a…
Kleene's computability theory based on his S1-S9 computation schemes constitutes a model for computing with objects of any finite type and extends Turing's `machine model' which formalises computing with real numbers. A fundamental…
Reverse Mathematics is a program in the foundations of mathematics. It provides an elegant classification in which the majority of theorems of ordinary mathematics fall into only five categories, based on the 'Big Five' logical systems.…
In classical information theory, the Doeblin coefficient of a classical channel provides an efficiently computable upper bound on the total-variation contraction coefficient of the channel, leading to what is known as a strong…
We introduce an experimental test for ruling out classical explanations for the statistics obtained when measuring arbitrary observables at arbitrary times using individual detectors. This test requires some trust in the measurements,…
In this paper, we highlight a new computational aspect of Nonstandard Analysis relating to higher-order computability theory. In particular, we prove that the Gandy-Hyland functional equals a primitive recursive functional involving…
Interactive theorem proving software is typically designed around a trusted proof-checking kernel, the sole system component capable of authenticating theorems. Untrusted automation procedures reside outside of the kernel, and drive it to…
Kleene's computability theory based on the S1-S9 computation schemes constitutes a model for computing with objects of any finite type and extends Turing's 'machine model' which formalises computing with real numbers. A fundamental…
We initiate the study of parallel quantum programming by defining the operational and denotational semantics of parallel quantum programs. The technical contributions of this paper include: (1) find a series of useful proof rules for…
We introduce constructive and classical systems for nonstandard arithmetic and show how variants of the functional interpretations due to Goedel and Shoenfield can be used to rewrite proofs performed in these systems into standard ones.…
Recent astonishing experiments with quantum computers have demonstrated unambiguously the existence of a quantum multiverse, where calculations of mind-boggling complexity are effortlessly computed in just a few minutes. Here, we…
Quantum advantage is notoriously hard to find and even harder to prove. For example the class of functions computable with classical physics actually exactly coincides with the class computable quantum-mechanically. It is strongly believed,…
The uncountability of $\mathbb{R}$ is one of its most basic properties, known far outside of mathematics. Cantor's 1874 proof of the uncountability of $\mathbb{R}$ even appears in the very first paper on set theory, i.e. a historical…
Given a first-order sentence, a model-checking computation tests whether the sentence holds true in a given finite structure. Data provenance extracts from this computation an abstraction of the manner in which its result depends on the…
Consider a fixed universe of $N=2^n$ elements and the uniform distribution over elements of some subset of size $K$. Given samples from this distribution, the task of complement sampling is to provide a sample from the complementary subset.…
Auditing is an increasingly important operation for computer programming, for example in security (e.g. to enable history-based access control) and to enable reproducibility and accountability (e.g. provenance in scientific programming).…
These are notes on discrete mathematics for computer scientists. The presentation is somewhat unconventional. Indeed I begin with a discussion of the basic rules of mathematical reasoning and of the notion of proof formalized in a natural…
We discuss models of computing that are beyond classical. The primary motivation is to unearth the cause of nonclassical advantages in computation. Completeness results from computational complexity theory lead to the identification of very…
Quantum models of computation are widely believed to be more powerful than classical ones. Efforts center on proving that, for a given problem, quantum algorithms are more resource efficient than any classical one. All this, however,…
It is a commonplace to say that `one can search through the natural numbers', by which is meant the following: For a property, decidable in finite time and which is not false for all natural numbers, checking said property starting at zero,…