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

Linking Focusing and Resolution with Selection

Focusing and selection are techniques that shrink the proof search space for respectively sequent calculi and resolution. To bring out a link between them, we generalize them both: we introduce a sequent calculus where each occurrence of an atom can have a positive or a negative polarity; and a resolution method where each literal, whatever its sign, can be selected in input clauses. We prove the equivalence between cut-free proofs in this sequent calculus and derivations of the empty clause in that resolution method. Such a generalization is not semi-complete in general, which allows us to consider complete instances that correspond to theories of any logical strength. We present three complete instances: first, our framework allows us to show that ordinary focusing corresponds to hyperresolution and semantic resolution; the second instance is deduction modulo theory; and a new setting, not captured by any existing framework, extends deduction modulo theory with rewriting rules having several left-hand sides, which restricts even more the proof search space

Efficiently Simulating Higher-Order Arithmetic by a First-Order Theory Modulo

In deduction modulo, a theory is not represented by a set of axioms but by a congruence on propositions modulo which the inference rules of standard deductive systems---such as for instance natural deduction---are applied. Therefore, the reasoning that is intrinsic of the theory does not appear in the length of proofs. In general, the congruence is defined through a rewrite system over terms and propositions. We define a rigorous framework to study proof lengths in deduction modulo, where the congruence must be computed in polynomial time. We show that even very simple rewrite systems lead to arbitrary proof-length speed-ups in deduction modulo, compared to using axioms. As higher-order logic can be encoded as a first-order theory in deduction modulo, we also study how to reinterpret, thanks to deduction modulo, the speed-ups between higher-order and first-order arithmetics that were stated by G\"odel. We define a first-order rewrite system with a congruence decidable in polynomial time such that proofs of higher-order arithmetic can be linearly translated into first-order arithmetic modulo that system. We also present the whole higher-order arithmetic as a first-order system without resorting to any axiom, where proofs have the same length as in the axiomatic presentation

Linking Focusing and Resolution with Selection

International audienceFocusing and selection are techniques that shrink the proof-search space for respectively sequent calculi and resolution. To bring out a link between them, we generalize them both: we introduce a sequent calculus where each occurrence of an atomic formula can have a positive or a negative polarity; and a resolution method where each literal, whatever its sign, can be selected in input clauses. We prove the equivalence between cut-free proofs in this sequent calculus and derivations of the empty clause in that resolution method. Such a generalization is not semi-complete in general, which allows us to consider complete instances that correspond to theories of any logical strength. We present three complete instances: first, our framework allows us to show that ordinary focusing corresponds to hyperresolution and semantic resolution; the second instance is deduction modulo theory and the related framework called superdeduction; and a new setting, not captured by any existing framework, extends deduction modulo theory with rewriting rules having several left-hand sides, which restricts even more the proof-search space

Systèmes Canoniques Abstraits : Application à la Déduction Naturelle et à la Complétion

Canonical Systems and Inference (ACSI) were introduced by N. Dershowitz and C. Kirchner to formalize the intuitive notion of good proof and good inference appearing typically in first-order logic or in Knuth-Bendix like completion procedures. Since this abstract framework, based on proof orderings, is intended to be generic, it is of fundamental interest to show its adequacy to represent the main systems of interest. It is here done for minimal propositional natural deduction, and for the standard (Knuth-Bendix) completion, closing an open question. For the first proof system, a generalisation of the ACSI is needed. We provide here a conservative one, in the sense that all results of the original framework still hold. For the second one, two proof representations, proof terms and proofs by replacement, are compared to built a proof ordering that provides an instantiation adapted to the ACSI framework

Automating Theories in Intuitionistic Logic

International audienceDeduction modulo consists in applying the inference rules of a deductive system modulo a rewrite system over terms and formulae. This is equivalent to proving within a so-called compatible theory. Conversely, given a first-order theory, one may want to internalize it into a rewrite system that can be used in deduction modulo, in order to get an analytic deductive system for that theory. In a recent paper, we have shown how this can be done in classical logic. In intuitionistic logic, however, we show here not only that this may be impossible, but also that the set of theories that can be transformed into a rewrite system with an analytic sequent calculus modulo is not co-recursively enumerable. We nonetheless propose a procedure to transform a large class of theories into compatible rewrite systems. We then extend this class by working in conservative extensions, in particular using Skolemization

A First-Order Representation of Pure Type Systems Using Superdeduction

International audienceSuperdeduction is a formalism closely related to deduction modulo which permits to enrich a deduction system (especially a first-order one such as natural deduction or sequent calculus) with new inference rules automatically computed from the presentation of a theory. We give a natural encoding from every functional Pure Type System (PTS) into superdeduction by defining an appropriate first-order theory. We prove that this translation is correct and conservative, showing a correspondence between valid typing judgments in the PTS and provable sequents in the corresponding superdeductive system. As a byproduct, we also introduce the superdeductive sequent calculus for intuitionistic logic, which was until now only defined for classical logic. We show its equivalence with the superdeductive natural deduction. This implies that superdeduction can be easily used as a logical framework. These results lead to a better understanding of the implementation and the automation of proof search for PTS, as well as to more cooperation between proof assistants

Logique Equationnelle et probabilités selon Halpern

Stage Magistère 1ière Année. Rapport de stage.L'introduction de probabilités en logique est motivée par le besoin de faire des raisonnements prenant en compte l'incertitude, pour lesquels on aimerait à quel point on peut se fier à une affirmation particulière. De tels raisonnements trouveraient des applications dans des domaines variés, comme par exemple l'intelligence artificielle ou le domaine médical

Detection of First Order Axiomatic Theories

International audienceAutomated theorem provers for first-order logic with equality have become very powerful and useful, thanks to both advanced calculi --- such as superposition and its refinements --- and mature implementation techniques. Nevertheless, dealing with some axiomatic theories remains a challenge because it gives rise to a search space explosion. Most attempts to deal with this problem have focused on specific theories, like AC (associative commutative symbols) or ACU (AC with neutral element). Even detecting the presence of a theory in a problem is generally solved in an ad-hoc fashion. We present here a generic way of describing and recognizing axiomatic theories in clausal form first-order logic with equality. Subsequently, we show some use cases for it, including a redundancy criterion that can be applied to some equational theories, extending some work that has been done by Avenhaus, Hillenbrand and Löchne

Analysis of biofilm-nanoparticles interaction using microscopy (fluorescence, MEB, STEM, MET, EDS)

International audienceAmong biofilm's properties, the ability to interact with/catch pollutants can have applications in bioremediation. Here, biofilm interactions with metals (as iron nanoparticles (NanoFer 25S)) was evaluated using various approaches in microscopy. For this, biofilm growth, sampling, labelling and treatment were developed for each type of microscopy to access the surface or inside of the biofilm, biofilm composition, and metal location. Multispecies biofilms were grown on sand or in PVC tubes inoculated with aquifer water spiked with a nutritive solution to enhance denitrification, and then put in contact with nanoparticles. According to the targeted microscopy, biofilms were (i) sampled as flocs or attached biofilm, (ii) submitted to cells (DAPI) and/or lectins (PNA and ConA coupled to FITC or Au nanoparticles) labelling, and (iii) prepared for observation (fixation, cross-section, freezing…). Fluorescent microscopy revealed that nanoparticles were embedded in the biofilm structure as 0.5-5µm size aggregates. SEM observations also showed NP aggregates closed to microorganisms but it was not possible to conclude a potential interaction between nanoparticles and the biological membranes. STEM-in-SEM analysis showed NP aggregates could enter inside the biofilm over a depth of 7-11µm. Moreover, microorganisms were circled by an EPS ring that prevented the direct interaction between NP and membrane. TEM(STEM)/EDS revealed that NP aggregates were co-localized with lectins suggesting a potential role of exopolysaccharides in NP embedding. The combination of several approaches in microscopy is thus a good tool to better understandi and characterize biofilm/pollutant interaction