On the correctness of a branch displacement algorithm

The branch displacement problem is a well-known problem in assembler design. It revolves around the feature, present in several processor families, of having different instructions, of different sizes, for jumps of different displacements. The problem, which is provably NP-hard, is then to select the instructions such that one ends up with the smallest possible program. During our research with the CerCo project on formally verifying a C compiler, we have implemented and proven correct an algorithm for this problem. In this paper, we discuss the problem, possible solutions, our specific solutions and the proofs

General Recursion and Formal Topology.

Comment: In Proceedings PAR 2010, arXiv:1012.455

A Bi-Directional Refinement Algorithm for the Calculus of (Co)Inductive Constructions

The paper describes the refinement algorithm for the Calculus of (Co)Inductive Constructions (CIC) implemented in the interactive theorem prover Matita. The refinement algorithm is in charge of giving a meaning to the terms, types and proof terms directly written by the user or generated by using tactics, decision procedures or general automation. The terms are written in an "external syntax" meant to be user friendly that allows omission of information, untyped binders and a certain liberal use of user defined sub-typing. The refiner modifies the terms to obtain related well typed terms in the internal syntax understood by the kernel of the ITP. In particular, it acts as a type inference algorithm when all the binders are untyped. The proposed algorithm is bi-directional: given a term in external syntax and a type expected for the term, it propagates as much typing information as possible towards the leaves of the term. Traditional mono-directional algorithms, instead, proceed in a bottom-up way by inferring the type of a sub-term and comparing (unifying) it with the type expected by its context only at the end. We propose some novel bi-directional rules for CIC that are particularly effective. Among the benefits of bi-directionality we have better error message reporting and better inference of dependent types. Moreover, thanks to bi-directionality, the coercion system for sub-typing is more effective and type inference generates simpler unification problems that are more likely to be solved by the inherently incomplete higher order unification algorithms implemented. Finally we introduce in the external syntax the notion of vector of placeholders that enables to omit at once an arbitrary number of arguments. Vectors of placeholders allow a trivial implementation of implicit arguments and greatly simplify the implementation of primitive and simple tactics

SmartTools: a generator of interactive environments tools

SmartTools is a development environment generator that provides a structure editor and semantic tools as main features. The well-known visitor pattern technique is commonly used for designing semantic analysis, it has been automated and extended. SmartTools is easy to use thanks to its graphical user interface designed with the Java Swing APIs. It is built with an open architecture convinient for a partial or total integration of SmartTools in other environments. It makes the addition of new software components in SmartTools easy. As a result of the modular architecture, we built a distributed instance of SmartTools which required minimal effort. Being open to the XML technologies offers all the features of Smart Tools to any language defined with those technologies. But most of all, with its open architecture, SmartTools takes advantage of all the developments made around those technologies, like DOM, through the XML APIs. The fast development of SmartTools (which is a young project, one year old) validates our choices of being open and generic. The main goal of this tool is to provide help and support for designing software development environments for programming languages as well as application languages defined with XML technologies

Nominal Henkin Semantics: simply-typed lambda-calculus models in nominal sets

We investigate a class of nominal algebraic Henkin-style models for the simply typed lambda-calculus in which variables map to names in the denotation and lambda-abstraction maps to a (non-functional) name-abstraction operation. The resulting denotations are smaller and better-behaved, in ways we make precise, than functional valuation-based models. Using these new models, we then develop a generalisation of \lambda-term syntax enriching them with existential meta-variables, thus yielding a theory of incomplete functions. This incompleteness is orthogonal to the usual notion of incompleteness given by function abstraction and application, and corresponds to holes and incomplete objects.Comment: In Proceedings LFMTP 2011, arXiv:1110.668

Declarative Representation of Proof Terms

We present a declarative language inspired by the pseudo-natural language used in Matita for the explanation of proof terms. We show how to compile the language to proof terms and how to automatically generate declarative scripts from proof terms. Then we investigate the relationship between the two translations, identifying the amount of proof structure preserved by compilation and re-generation of declarative scripts

Reduction and Conversion Strategies for the Calculus of (co)Inductive Construtions: Part I

We compare several reduction and conversion strategies for the Calculus of (co)Inductive Constructions by running benchmarks on the library of the Coq proof assistant. All the strategies have been implemented in an independent verifier for the proof objects of Coq that is part of the Matita proof assistant

A User Interface for a Mathematical System that Allows Ambiguous Formulae

Mathematical systems that understand the usual ambiguous mathematical notation need well thought out user interfaces 1) to provide feedback on the way formulae are automatically interpreted, when a single best interpretation exists; 2) to dialogue with the user when human intervention is required because multiple best interpretations exist; 3) to present sets of errors to the user when no correct interpretation exists. In this paper we discuss how we handle ambiguity in the user interfaces of the Matita interactive theorem prover and the Whelp search engine

Explanation in Natural language of lambda-bar-mu-mu-tilde-terms

The lambda-bar-mu-mu-tilde-calculus, introduced by Curien and Herbelin, is a calculus isomorphic to (a variant of) the classical sequent calculus LK of Gentzen. As a proof format it has very remarkable properties that we plan to study in future works. In this paper we embed it with a rendering semantics that provides explanations in pseudo-natural language of its proof terms, in the spirit of the work of Yann Coscoy for the lambda-calculus. The rendering semantics unveils the richness of the calculus that allows to preserve several proof structures that are identified when encoded in the lambda-calculus