Models and termination of proof reduction in the λ\lambdaΠ\Pi-calculus modulo theory

We define a notion of model for the $\lambda$$\Pi$-calculus modulo theory and prove a soundness theorem. We then define a notion of super-consistency and prove that proof reduction terminates in the $\lambda$$\Pi$-calculus modulo any super-consistent theory. We prove this way the termination of proof reduction in several theories including Simple type theory and the Calculus of constructions

An Open Challenge Problem Repository for Systems Supporting Binders

A variety of logical frameworks support the use of higher-order abstract syntax in representing formal systems; however, each system has its own set of benchmarks. Even worse, general proof assistants that provide special libraries for dealing with binders offer a very limited evaluation of such libraries, and the examples given often do not exercise and stress-test key aspects that arise in the presence of binders. In this paper we design an open repository ORBI (Open challenge problem Repository for systems supporting reasoning with BInders). We believe the field of reasoning about languages with binders has matured, and a common set of benchmarks provides an important basis for evaluation and qualitative comparison of different systems and libraries that support binders, and it will help to advance the field.Comment: In Proceedings LFMTP 2015, arXiv:1507.0759

Classical Mathematics for a Constructive World

Interactive theorem provers based on dependent type theory have the flexibility to support both constructive and classical reasoning. Constructive reasoning is supported natively by dependent type theory and classical reasoning is typically supported by adding additional non-constructive axioms. However, there is another perspective that views constructive logic as an extension of classical logic. This paper will illustrate how classical reasoning can be supported in a practical manner inside dependent type theory without additional axioms. We will see several examples of how classical results can be applied to constructive mathematics. Finally, we will see how to extend this perspective from logic to mathematics by representing classical function spaces using a weak value monad.Comment: v2: Final copy for publicatio

Induction and Deduction in Baysian Data Analysis

The classical or frequentist approach to statistics (in which inference is centered on significance testing), is associated with a philosophy in which science is deductive and follows Popperis doctrine of falsification. In contrast, Bayesian inference is commonly associated with inductive reasoning and the idea that a model can be dethroned by a competing model but can never be directly falsified by a significance test. The purpose of this article is to break these associations, which I think are incorrect and have been detrimental to statistical practice, in that they have steered falsificationists away from the very useful tools of Bayesian inference and have discouraged Bayesians from checking the fit of their models. From my experience using and developing Bayesian methods in social and environmental science, I have found model checking and falsification to be central in the modeling process.philosophy of statistics, decision theory, subjective probability, Bayesianism, falsification, induction, frequentism

Existential witness extraction in classical realizability and via a negative translation

We show how to extract existential witnesses from classical proofs using Krivine's classical realizability---where classical proofs are interpreted as lambda-terms with the call/cc control operator. We first recall the basic framework of classical realizability (in classical second-order arithmetic) and show how to extend it with primitive numerals for faster computations. Then we show how to perform witness extraction in this framework, by discussing several techniques depending on the shape of the existential formula. In particular, we show that in the Sigma01-case, Krivine's witness extraction method reduces to Friedman's through a well-suited negative translation to intuitionistic second-order arithmetic. Finally we discuss the advantages of using call/cc rather than a negative translation, especially from the point of view of an implementation.Comment: 52 pages. Accepted in Logical Methods for Computer Science (LMCS), 201