Initial Semantics for higher-order typed syntax in Coq

Initial Semantics aims at characterizing the syntax associated to a signature as the initial object of some category. We present an initial semantics result for typed higher-order syntax together with its formalization in the Coq proof assistant. The main theorem was first proved on paper in the second author's PhD thesis in 2010, and verified formally shortly afterwards. To a simply-typed binding signature S over a fixed set T of object types we associate a category called the category of representations of S. We show that this category has an initial object Sigma(S). From its construction it will be clear that the object Sigma(S) merits the name abstract syntax associated to S. Our theorem is implemented and proved correct in the proof assistant Coq through heavy use of dependent types. The approach through monads gives rise to an implementation of syntax where both terms and variables are intrinsically typed, i.e. where the object types are reflected in the meta-level types. This article is to be seen as a research article rather than about the formalization of a classical mathematical result. The nature of our theorem - involving lengthy, technical proofs and complicated algebraic structures - makes it particularly interesting for formal verification. Our goal is to promote the use of computer theorem provers as research tools, and, accordingly, a new way of publishing mathematical results: a parallel description of a theorem and its formalization should allow the verification of correct transcription of definitions and statements into the proof assistant, and straightforward but technical proofs should be well-hidden in a digital library. We argue that Coq's rich type theory, combined with its various features such as implicit arguments, allows a particularly readable formalization and is hence well-suited for communicating mathematics.Comment: Article as published in JFR (cf. Journal ref). Features some more example

Coinductive Formal Reasoning in Exact Real Arithmetic

In this article we present a method for formally proving the correctness of the lazy algorithms for computing homographic and quadratic transformations -- of which field operations are special cases-- on a representation of real numbers by coinductive streams. The algorithms work on coinductive stream of M\"{o}bius maps and form the basis of the Edalat--Potts exact real arithmetic. We use the machinery of the Coq proof assistant for the coinductive types to present the formalisation. The formalised algorithms are only partially productive, i.e., they do not output provably infinite streams for all possible inputs. We show how to deal with this partiality in the presence of syntactic restrictions posed by the constructive type theory of Coq. Furthermore we show that the type theoretic techniques that we develop are compatible with the semantics of the algorithms as continuous maps on real numbers. The resulting Coq formalisation is available for public download.Comment: 40 page

Hybrid Type-Logical Grammars, First-Order Linear Logic and the Descriptive Inadequacy of Lambda Grammars

In this article we show that hybrid type-logical grammars are a fragment of first-order linear logic. This embedding result has several important consequences: it not only provides a simple new proof theory for the calculus, thereby clarifying the proof-theoretic foundations of hybrid type-logical grammars, but, since the translation is simple and direct, it also provides several new parsing strategies for hybrid type-logical grammars. Second, NP-completeness of hybrid type-logical grammars follows immediately. The main embedding result also sheds new light on problems with lambda grammars/abstract categorial grammars and shows lambda grammars/abstract categorial grammars suffer from problems of over-generation and from problems at the syntax-semantics interface unlike any other categorial grammar

Efficient Normal-Form Parsing for Combinatory Categorial Grammar

Under categorial grammars that have powerful rules like composition, a simple n-word sentence can have exponentially many parses. Generating all parses is inefficient and obscures whatever true semantic ambiguities are in the input. This paper addresses the problem for a fairly general form of Combinatory Categorial Grammar, by means of an efficient, correct, and easy to implement normal-form parsing technique. The parser is proved to find exactly one parse in each semantic equivalence class of allowable parses; that is, spurious ambiguity (as carefully defined) is shown to be both safely and completely eliminated.Comment: 8 pages, LaTeX packaged with three .sty files, also uses cgloss4e.st

Generalized Bhattacharyya and Chernoff upper bounds on Bayes error using quasi-arithmetic means

Bayesian classification labels observations based on given prior information, namely class-a priori and class-conditional probabilities. Bayes' risk is the minimum expected classification cost that is achieved by the Bayes' test, the optimal decision rule. When no cost incurs for correct classification and unit cost is charged for misclassification, Bayes' test reduces to the maximum a posteriori decision rule, and Bayes risk simplifies to Bayes' error, the probability of error. Since calculating this probability of error is often intractable, several techniques have been devised to bound it with closed-form formula, introducing thereby measures of similarity and divergence between distributions like the Bhattacharyya coefficient and its associated Bhattacharyya distance. The Bhattacharyya upper bound can further be tightened using the Chernoff information that relies on the notion of best error exponent. In this paper, we first express Bayes' risk using the total variation distance on scaled distributions. We then elucidate and extend the Bhattacharyya and the Chernoff upper bound mechanisms using generalized weighted means. We provide as a byproduct novel notions of statistical divergences and affinity coefficients. We illustrate our technique by deriving new upper bounds for the univariate Cauchy and the multivariate $t$-distributions, and show experimentally that those bounds are not too distant to the computationally intractable Bayes' error.Comment: 22 pages, include R code. To appear in Pattern Recognition Letter

Demystifying Fixed k-Nearest Neighbor Information Estimators

Estimating mutual information from i.i.d. samples drawn from an unknown joint density function is a basic statistical problem of broad interest with multitudinous applications. The most popular estimator is one proposed by Kraskov and St\"ogbauer and Grassberger (KSG) in 2004, and is nonparametric and based on the distances of each sample to its $k^{\rm th}$ nearest neighboring sample, where $k$ is a fixed small integer. Despite its widespread use (part of scientific software packages), theoretical properties of this estimator have been largely unexplored. In this paper we demonstrate that the estimator is consistent and also identify an upper bound on the rate of convergence of the bias as a function of number of samples. We argue that the superior performance benefits of the KSG estimator stems from a curious "correlation boosting" effect and build on this intuition to modify the KSG estimator in novel ways to construct a superior estimator. As a byproduct of our investigations, we obtain nearly tight rates of convergence of the $\ell_2$ error of the well known fixed $k$ nearest neighbor estimator of differential entropy by Kozachenko and Leonenko.Comment: 55 pages, 8 figure

Finite Device-Independent Extraction of a Block Min-Entropy Source against Quantum Adversaries

The extraction of randomness from weakly random seeds is a problem of central importance with multiple applications. In the device-independent setting, this problem of quantum randomness amplification has been mainly restricted to specific weak sources of Santha-Vazirani type, while extraction from the general min-entropy sources has required a large number of separated devices which is impractical. In this paper, we present a device-independent protocol for amplification of a single min-entropy source (consisting of two blocks of sufficiently high min-entropy) using a device consisting of two spatially separated components and show a proof of its security against general quantum adversaries.Comment: 17 page