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Gilles Dowek

Publications and source records attributed to Gilles Dowek.

At least 55 records · Page 3Linked to original sources
arXiv2023

Eigenvariables, bracketing and the decidability of positive minimal predicate logic

We give a new proof of a theorem of Mints that the positive fragment of minimal predicate logic is decidable. The idea of the proof is to replace the eigenvariable condition of sequent calculus by an appropriate scoping mechanism. The algorithm given by this proof seems to be more practical than that given by the original proof. A naive implementation is given at the end of the paper. Another contribution is to show that this result extends to a large class of theories, including simple type theory (higherorder logic) and second-order propositional logic. We obtain this way a new proof of the decidability of the inhabitation problem for positive types in system F.

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arXiv2023

Truth values algebras and proof normalization

We extend the notion of Heyting algebra to a notion of truth values algebra and prove that a theory is consistent if and only if it has a B-valued model for some non trivial truth values algebra B. A theory that has a B-valued model for all truth values algebras B is said to be super-consistent. We prove that super-consistency is a model-theoretic sufficient condition for strong normalization.

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arXiv2023

The Undecidability of Unification Modulo $σ$ Alone

The rewriting system sigma is the set of rules propagating explicit substitutions in the lambda-calculus with explicit substitutions. In this note, we prove the undecidability of unification modulo sigma.

cs.LO↗
arXiv2023

Skolemization in Simple Type Theory: the Logical and the Theoretical Points of View

Peter Andrews has proposed, in 1971, the problem of finding an analog of the Skolem theorem for Simple Type Theory. A first idea lead to a naive rule that worked only for Simple Type Theory with the axiom of choice and the general case has only been solved, more than ten years later, by Dale Miller. More recently, we have proposed with Th{é}r{è}se Hardin and Claude Kirchner a new way to prove analogs of the Miller theorem for different, but equivalent, formulations of Simple Type Theory. In this paper, that does not contain new technical results, I try to show that the history of the skolemization problem and of its various solutions is an illustration of a tension between two points of view on Simple Type Theory: the logical and the theoretical points of view.

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arXiv2023

Enumerating proofs of positive formulae

We provide a semi-grammatical description of the set of normal proofs of positive formulae in minimal predicate logic, i.e. a grammar that generates a set of schemes, from each of which we can produce a finite number of normal proofs. This method is complete in the sense that each normal proof-term of the formula is produced by some scheme generated by the grammar. As a corollary, we get a similar description of the set of normal proofs of positive formulae for a large class of theories including simple type theory and System F.

cs.LO↗
arXiv2023

The physical Church thesis and the sensitivity to initial conditions

The physical Church thesis is a thesis about nature that expresses that all that can be computed by a physical system-a machine-is computable in the sense of computability theory. At a first look, this thesis seems contradictory with the existence, in nature, of chaotic dynamical systems, that is systems whose evolution cannot be ''computed'' because of their sensitivity to initial conditions. The goal of this note is to show that there exist dynamical systems that are both computable and chaotic, and thus that the existence in nature of chaotic dynamical system is not, per se, a refutation of the physical Church thesis. Thus, chaos seems to be compatible with computability, in the same way as it is compatible with determinism.

cs.LO↗
arXiv2023

Specifying programs with propositions and with congruences

We give a presentation of Krivine and Parigot's Second-order functional arithmetic in Deduction modulo. Expressing this theory in Deduction modulo sheds light on an original aspect of this theory: the fact that programs are specified, not with propositions, but with congruences.

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arXiv2023

How can we prove that a proof search method is not an instance of another?

We introduce a method to prove that a proof search method is not an instance of another. As an example of application, we show that Polarized resolution modulo, a method that mixes clause selection restrictions and literal selection restrictions, is not an instance of Ordered resolution with selection.

cs.LO↗
arXiv2023

On the Expressive Power of Schemes

We present a calculus, called the scheme-calculus, that permits to express natural deduction proofs in various theories. Unlike $λ$-calculus, the syntax of this calculus sticks closely to the syntax of proofs, in particular, no names are introduced for the hypotheses. We show that despite its non-determinism, some typed scheme-calculi have the same expressivity as the corresponding typed $λ$-calculi.

cs.LO↗
arXiv2023

A Simple Proof That Super-Consistency Implies Cut Elimination

We give a simple and direct proof that super-consistency implies the cut elimination property in deduction modulo. This proof can be seen as a simplification of the proof that super-consistency implies proof normalization. It also takes ideas from the semantic proofs of cut elimination that proceed by proving the completeness of the cut-free calculus. As an application, we compare our work with the cut elimination theorems in higher-order logic that involve V-complexes.

cs.LO↗
arXiv2023

The principle of a finite density of information

The possibility to describe the laws of the Universe in a computational way seems to be correlated to a principle that the density of information is bounded. This principle, that is dual to that of a finite velocity of information, has already been investigated in Physics, and is correlated to the old idea that there is no way to know a magnitude with an infinite precision. It takes different forms in classical Physics and in quantum Physics.

cs.LO↗
arXiv2023

A theory independent Curry-De Bruijn-Howard correspondence

Instead of developing a customized typed lambda-calculus for each theory, we attempt to design a general parametric calculus that permits to express the proofs of any theory. This way, the problem of expressing proofs in the lambda-calculus is separated from that of choosing a theory.

cs.LO↗
arXiv2023

Execution traces and reduction sequences

In this note, we defend that the notion of algorithm as a set of execution traces is somewhat independent of the notion of abstract state machine. It can be reformulated in the more general framework of small step operational semantics.

cs.LO↗
arXiv2023

Complementation: a bridge between finite and infinite proofs

When a proposition has no proof in an inference system, it is sometimes useful to build a counter-proof explaining, step by step, the reason of this non-provability. In general, this counter-proof is a (possibly) infinite co-inductive proof in a different inference system. In this paper, we show that, for some decidable inference systems, this (possibly) infinite proof has a representation as a finite proof in yet another system, equivalent to the previous one.

cs.LO↗
arXiv2023

Logipedia: a multi-system encyclopedia of formal proofs

Libraries of formal proofs are an important part of our mathematical heritage, but their usability and sustainability is poor. Indeed, each library is specific to a proof system, sometimes even to some version of this system. Thus, a library developed in one system cannot, in general, be used in another and when the system is no more maintained, the library may be lost. This impossibility of using a proof developed in one system in another has been noted for long and a remediation has been proposed: as we have empirical evidence that most of the formal proofs developed in one of these systems can also be developed in another, we can develop a standard language, in which these proofs can be translated, and then used in all systems supporting this standard. Logipedia is an attempt to build such a multi-system online encyclopedia of formal proofs expressed in such as standard language. It is based on two main ideas: the use of a logical framework and of reverse mathematics.

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