Type Safety of Rewrite Rules in Dependent Types

The expressiveness of dependent type theory can be extended by identifying types modulo some additional computation rules. But, for preserving the decidability of type-checking or the logical consistency of the system, one must make sure that those user-defined rewriting rules preserve typing. In this paper, we give a new method to check that property using Knuth-Bendix completion

Type safety of rewrite rules in dependent types

International audienceThe expressiveness of dependent type theory can beextended by identifying types modulo some additional computation rules. But, forpreserving the decidability of type-checking or the logicalconsistency of the system, one must make sure that those user-definedrewriting rules preserve typing. In this paper, we give a newmethod to check that property using Knuth-Bendix completion

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

Modeling and Analyzing Adaptive User-Centric Systems in Real-Time Maude

Pervasive user-centric applications are systems which are meant to sense the presence, mood, and intentions of users in order to optimize user comfort and performance. Building such applications requires not only state-of-the art techniques from artificial intelligence but also sound software engineering methods for facilitating modular design, runtime adaptation and verification of critical system requirements. In this paper we focus on high-level design and analysis, and use the algebraic rewriting language Real-Time Maude for specifying applications in a real-time setting. We propose a generic component-based approach for modeling pervasive user-centric systems and we show how to analyze and prove crucial properties of the system architecture through model checking and simulation. For proving time-dependent properties we use Metric Temporal Logic (MTL) and present analysis algorithms for model checking two subclasses of MTL formulas: time-bounded response and time-bounded safety MTL formulas. The underlying idea is to extend the Real-Time Maude model with suitable clocks, to transform the MTL formulas into LTL formulas over the extended specification, and then to use the LTL model checker of Maude. It is shown that these analyses are sound and complete for maximal time sampling. The approach is illustrated by a simple adaptive advertising scenario in which an adaptive advertisement display can react to actions of the users in front of the display.Comment: In Proceedings RTRTS 2010, arXiv:1009.398

Expressiveness of Generic Process Shape Types

Shape types are a general concept of process types which work for many process calculi. We extend the previously published Poly* system of shape types to support name restriction. We evaluate the expressiveness of the extended system by showing that shape types are more expressive than an implicitly typed pi-calculus and an explicitly typed Mobile Ambients. We demonstrate that the extended system makes it easier to enjoy advantages of shape types which include polymorphism, principal typings, and a type inference implementation.Comment: Submitted to Trustworthy Global Computing (TGC) 2010

Translating HOL to Dedukti

Dedukti is a logical framework based on the lambda-Pi-calculus modulo rewriting, which extends the lambda-Pi-calculus with rewrite rules. In this paper, we show how to translate the proofs of a family of HOL proof assistants to Dedukti. The translation preserves binding, typing, and reduction. We implemented this translation in an automated tool and used it to successfully translate the OpenTheory standard library.Comment: In Proceedings PxTP 2015, arXiv:1507.0837

Encoding TLA+ set theory into many-sorted first-order logic

We present an encoding of Zermelo-Fraenkel set theory into many-sorted first-order logic, the input language of state-of-the-art SMT solvers. This translation is the main component of a back-end prover based on SMT solvers in the TLA+ Proof System