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Gilles Dowek

Publications and source records attributed to Gilles Dowek.

At least 73 records · Page 4Linked to original sources
arXiv2023

Teaching G{ö}del's incompleteness theorems

The basic notions of logic-predicate logic, Peano arithmetic, incompleteness theorems, etc.-have for long been an advanced topic. In the last decades, they became more widely taught, inphilosophy, mathematics, and computer science departments, to graduate and to undergraduate students. Many textbooks now present these notions, in particular the incompleteness theorems. Having taught these notions for several decades, our community can now stand back and analyze the choices faced when designing such a course. In this note, we attempt to analyze the choices faced when teaching the incompleteness theorems.

math.HO↗
arXiv2023

A New Connective in Natural Deduction, and its Application to Quantum Computing

We investigate an unsuspected connection between logical connectives with non-harmonious deduction rules, such as Prior's tonk, and quantum computing. We argue that these connectives model the information-erasure, the non-reversibility, and the non-determinism that occur, among other places, in quantum measurement. We introduce an intuitionistic propositional logic with a non-harmonious logical connective sup and two interstitial rules, and show that the proof language of this logic forms the core of a quantum programming language.

cs.LO↗
arXiv2023

A modular construction of type theories

The lambda-Pi-calculus modulo theory is a logical framework in which many type systems can be expressed as theories. We present such a theory, the theory U, where proofs of several logical systems can be expressed. Moreover, we identify a sub-theory of U corresponding to each of these systems, and prove that, when a proof in U uses only symbols of a sub-theory, then it is a proof in that sub-theory.

cs.LO↗
arXiv2021

Interacting Safely with an Unsafe Environment

We give a presentation of Pure type systems where contexts need not be well-formed and show that this presentation is equivalent to the usual one. The main motivation for this presentation is that, when we extend Pure type systems with computation rules, like in the logical framework Dedukti, we want to declare the constants before the computation rules that are needed to check the well-typedness of their type.

cs.LO↗
arXiv2019

Two linearities for quantum computing in the lambda calculus

We propose a way to unify two approaches of non-cloning in quantum lambda-calculi: logical and algebraic linearities. The first approach is to forbid duplicating variables, while the second is to consider all lambda-terms as algebraic-linear functions. We illustrate this idea by defining a quantum extension of first-order simply-typed lambda-calculus, where the type is linear on superposition, while allows cloning base vectors. In addition, we provide an interpretation of the calculus where superposed types are interpreted as vector spaces and non-superposed types as their basis.

cs.LO↗
arXiv2019

Proof Normalisation in a Logic Identifying Isomorphic Propositions

We define a fragment of propositional logic where isomorphic propositions, such as $A\land B$ and $B\land A$, or $A\Rightarrow (B\land C)$ and $(A\Rightarrow B)\land(A\Rightarrow C)$ are identified. We define System I, a proof language for this logic, and prove its normalisation and consistency.

cs.LO↗
arXiv2017

SCTL: Towards Combining Model Checking and Proof Checking

Model checking and automated theorem proving are two pillars of formal methods. This paper investigates model checking from an automated theorem proving perspective, aiming at combining the expressiveness of automated theorem proving and the complete automaticity of model checking. The focus of this paper is on the verification of temporal logic properties of Kripke models. The main contributions of this paper are: first the definition of an extended computation tree logic that allows polyadic predicate symbols, then a proof system for this logic, taking Kripke models as parameters, then, the design of a proof-search algorithm for this calculus and a new automated theorem prover to implement it. The verification process is completely automatic, and produces either a counterexample when the property does not hold, or a certificate when it does. The experimental result compares well to existing state-of-the-art tools on some benchmarks, including an application to air traffic control and the design choices that lead to this efficiency are discussed.

cs.LO↗
arXiv2017

Models and termination of proof reduction in the $λ$$Π$-calculus modulo theory

We define a notion of model for the $λ$$Π$-calculus modulo theory and prove a soundness theorem. We then define a notion of super-consistency and prove that proof reduction terminates in the $λ$$Π$-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 .

cs.LO↗
arXiv2017

Lineal: A linear-algebraic Lambda-calculus

We provide a computational definition of the notions of vector space and bilinear functions. We use this result to introduce a minimal language combining higher-order computation and linear algebra. This language extends the Lambda-calculus with the possibility to make arbitrary linear combinations of terms alpha.t + beta.u. We describe how to "execute" this language in terms of a few rewrite rules, and justify them through the two fundamental requirements that the language be a language of linear operators, and that it be higher-order. We mention the perspectives of this work in the field of quantum computation, whose circuits we show can be easily encoded in the calculus. Finally, we prove the confluence of the entire calculus.

quant-ph↗
arXiv2016

Free fall and cellular automata

Three reasonable hypotheses lead to the thesis that physical phenomena can be described and simulated with cellular automata. In this work, we attempt to describe the motion of a particle upon which a constant force is applied, with a cellular automaton, in Newtonian physics, in Special Relativity, and in General Relativity. The results are very different for these three theories.

nlin.CG↗
arXiv2016

Rules and derivations in an elementary logic course

When teaching an elementary logic course to students who have a general scientific background but have never been exposed to logic, we have to face the problem that the notions of deduction rule and of derivation are completely new to them, and are related to nothing they already know, unlike, for instance, the notion of model, that can be seen as a generalization of the notion of algebraic structure. In this note, we defend the idea that one strategy to introduce these notions is to start with the notion of inductive definition [1]. Then, the notion of derivation comes naturally. We also defend the idea that derivations are pervasive in logic and that defining precisely this notion at an early stage is a good investment to later define other notions in proof theory, computability theory, automata theory, ... Finally, we defend the idea that to define the notion of derivation precisely, we need to distinguish two notions of derivation: labeled with elements and labeled with rule names. This approach has been taken in [2].

cs.LO↗
arXiv2016

Decidability, Introduction Rules and Automata

We present a method to prove the decidability of provability in several well-known inference systems. This method generalizes both cut-elimination and the construction of an automaton recognizing the provable propositions.

cs.LO↗
arXiv2015

Discrete geodesics and cellular automata

This paper proposes a dynamical notion of discrete geodesics, understood as straightest trajectories in discretized curved spacetime. The notion is generic, as it is formulated in terms of a general deviation function, but readily specializes to metric spaces such as discretized pseudo-riemannian manifolds. It is effective: an algorithm for computing these geodesics naturally follows, which allows numerical validation---as shown by computing the perihelion shift of a Mercury-like planet. It is consistent, in the continuum limit, with the standard notion of timelike geodesics in a pseudo-riemannian manifold. Whether the algorithm fits within the framework of cellular automata is discussed at length. KEYWORDS: Discrete connection, parallel transport, general relativity, Regge calculus.

cs.DM↗
arXiv2014

Cut-elimination and the decidability of reachability in alternating pushdown systems

We give a new proof of the decidability of reachability in alternating pushdown systems, showing that it is a simple consequence of a cut-elimination theorem for some natural-deduction style inference systems. Then, we show how this result can be used to extend an alternating pushdown system into a complete system where for every configuration $A$, either $A$ or $\neg A$ is provable.

cs.LO↗
arXiv2014

The probability of non-confluent systems

We show how to provide a structure of probability space to the set of execution traces on a non-confluent abstract rewrite system, by defining a variant of a Lebesgue measure on the space of traces. Then, we show how to use this probability space to transform a non-deterministic calculus into a probabilistic one. We use as example Lambda+, a recently introduced calculus defined through type isomorphisms.

cs.LO↗