E-Cyclist: Implementation of an Efficient Validation of FOL ID Cyclic Induction Reasoning (System Description)

Checking the soundness of cyclic induction reasoning for first-order logic with inductive definitions (FOLID) is decidable but the standard checking method is based on an exponential complement operation for Büchi automata. Recently, we introduced a polynomial checking method whose most expensive steps recall the comparisons done with multiset path orderings. We describe the implementation of our method in the Cyclist prover. Referred to as E-Cyclist, it successfully checked all the proofs included in the original distribution of Cyclist. Heuristics have been devised to automatically define from the analysis of the proof derivations the ordering measures that satisfy the ordering constraints. FOLID cyclic proof derivations may also be hard to certify. E-Cyclist witnesses a strong relation between the two cyclic and well-founded induction reasonings. This opens the perspective of using the known certification methods that work for well-founded induction proofs

ARC: An Educational Project on Automated Reasoning in the Class

International audienceThe international Erasmus+ European Project: "ARC-Automated Reasoning in the Class", running from 2019 to 2022 is a partnership of universities from Austria, France, Germany, Hungary, and Romania, and has the purpose of developing advanced material for teaching subjects related to Computational Logic by using Automated Reasoning. The material includes a comprehensive textbook treating the necessary theoretical background (selected topics in Mathematical Logic), but mostly the practical methods from Automated Theorem Proving, as well as the description of the basic programming paradigms and the associated languages, in relation to their logical aspects. Furthermore, we address the most important applications, like program verification and testing, semantic representation of information, algorithm synthesis, etc. One of the main goals of the approach is to improve the logical background of the software professionals in order to motivate them to use formal methods for certification of complex systems and thus to avoid costly failures

SPIKE, an automatic theorem prover -- revisited

International audienceSPIKE, an induction-based theorem prover built to reason on conditional theories with equality, is one of the few formal tools able to perform automatically mutual and lazy induction. Designed at the beginning of 1990s, it has been successfully used in many non-trivial applications and served as a prototype for different proof experiments and extensions. The first paper introducing SPIKE is [14], published shortly after the tool was created. The goal of this paper is to highlight and bring together in one spot the major changes supported by SPIKE since then

Validating Back-links of FOLID Cyclic Pre-proofs

International audienceCyclic pre-proofs can be represented as sets of finite tree derivations with back-links. In the frame of the first-order logic with inductive definitions (FOLID), the nodes of the tree derivations are labelled by sequents and the back-links connect particular terminal nodes, referred to as buds, to other nodes labelled by a same sequent. However, only some back-links can constitute sound pre-proofs. Previously , it has been shown that special ordering and derivability conditions, defined along the minimal cycles of the digraph representing a particular normal form of the cyclic pre-proof, are sufficient for validating the back-links. In that approach, a same constraint could be checked several times when processing different minimal cycles, hence one may require additional recording mechanisms to avoid redundant computation in order to downgrade the time complexity to polynomial. We present a new approach that does not need to process minimal cycles. It based on a normal form that allows to define the validation conditions by taking into account only the root-bud paths from the non-singleton strongly connected components of its digraph

Récurrence noethérienne pour le raisonnement de premier ordre

National audienceLa récurrence nœthérienne est un des principes les plus généraux de raisonnement formel. Dans le cadre du raisonnement de premier ordre, nous présentons une classification de ses instances pouvant être partagées en instances basées sur des termes et des formules. Nous donnons un aperçu du raisonnement par récurrence nœthérienne basée sur des termes et sur des formules, puis nous établissons des relations entre eux. Enfin, nous présentons une méthodologie pour la certification du raisonnement basé sur des formules à l’aide de l’assistant de preuve Coq

Обґрунтування теорії структури конструкцій технологічних машин і приладів

The object of research is the theory of the design of technological machines and devices. The main idea of this theory is the possibility of a priori quantitative evaluation of the design of machine and instrument constructions for manufacturability, unification and other indicators based on the identification of structural design features and the creation of quantitative criteria.An in-depth study of this work has shown that the weaknesses of the research are related to the rather high influence of the heuristic component on the process of scientific and engineering search. One of the most problematic places is that the complete formalization of this process can lead to a halt in the development of scientific and technical thought. There is a risk of reducing the construction process to the reproduction of an infinite number of similar structures that differ in certain characteristics, but lie within the same constructive series.To prevent the negative impact of the proposed theory on the rate of technical progress, it is necessary to strictly limit the field of its use, using it to optimize the design of entire products or their individual components and automate the process of searching for a better prototype for improvement in a combination of features.To prove the main idea of the theory, methods of analysis and induction are used in the course of the research, on the basis of which, using the lemma and existence theorems, basic axiomatic theories are formulated with the subsequent completion of theorems, although the content of the considered theory is not mathematical, but technical objects.The structure theory of design is unique, since being a model of logical computation, it makes it possible to reveal regularities in classes and series of a set of designs on the basis of which to propose ways of optimizing and improving the productivity of the design process by establishing a connection between the features of the designs and their coding according to the chain of successive transformations. Coding of structures in place in the chain of transformations opens up wide prospects for automation of the design process.Рассматривается теория структуры конструкций технологических машин и приборов с получением на ее основе количественных критериев и примерами использования представленной теории. Обосновываются критерии собираемости, ремонтопригодности, унификации и функциональной насыщенности, позволяющие давать количественную оценку конструкциям на стадии их разработки. Доказано, что они объективно отражают техническое совершенство конструкций, поскольку не зависят от изменяющихся во времени факторов, таких как себестоимость.Розглядається теорія структури конструкцій технологічних машин та приладів з отриманням кількісних критеріїв на її основі та приклади застосування представленої теорії. Обґрунтовуються критерії складання, ремонтопридатності, уніфікації та функціональної насиченості, які дозволяють давати кількісну оцінку конструкціям на стадії їх розробки. Доведено, що вони об’єктивно відображають технічну досконалість конструкцій, оскільки не залежать від факторів, що змінюються з часом, наприклад, таких як собівартість

Обґрунтування теорії структури конструкцій технологічних машин і приладів

The object of research is the theory of the design of technological machines and devices. The main idea of this theory is the possibility of a priori quantitative evaluation of the design of machine and instrument constructions for manufacturability, unification and other indicators based on the identification of structural design features and the creation of quantitative criteria.An in-depth study of this work has shown that the weaknesses of the research are related to the rather high influence of the heuristic component on the process of scientific and engineering search. One of the most problematic places is that the complete formalization of this process can lead to a halt in the development of scientific and technical thought. There is a risk of reducing the construction process to the reproduction of an infinite number of similar structures that differ in certain characteristics, but lie within the same constructive series.To prevent the negative impact of the proposed theory on the rate of technical progress, it is necessary to strictly limit the field of its use, using it to optimize the design of entire products or their individual components and automate the process of searching for a better prototype for improvement in a combination of features.To prove the main idea of the theory, methods of analysis and induction are used in the course of the research, on the basis of which, using the lemma and existence theorems, basic axiomatic theories are formulated with the subsequent completion of theorems, although the content of the considered theory is not mathematical, but technical objects.The structure theory of design is unique, since being a model of logical computation, it makes it possible to reveal regularities in classes and series of a set of designs on the basis of which to propose ways of optimizing and improving the productivity of the design process by establishing a connection between the features of the designs and their coding according to the chain of successive transformations. Coding of structures in place in the chain of transformations opens up wide prospects for automation of the design process.Рассматривается теория структуры конструкций технологических машин и приборов с получением на ее основе количественных критериев и примерами использования представленной теории. Обосновываются критерии собираемости, ремонтопригодности, унификации и функциональной насыщенности, позволяющие давать количественную оценку конструкциям на стадии их разработки. Доказано, что они объективно отражают техническое совершенство конструкций, поскольку не зависят от изменяющихся во времени факторов, таких как себестоимость.Розглядається теорія структури конструкцій технологічних машин та приладів з отриманням кількісних критеріїв на її основі та приклади застосування представленої теорії. Обґрунтовуються критерії складання, ремонтопридатності, уніфікації та функціональної насиченості, які дозволяють давати кількісну оцінку конструкціям на стадії їх розробки. Доведено, що вони об’єктивно відображають технічну досконалість конструкцій, оскільки не залежать від факторів, що змінюються з часом, наприклад, таких як собівартість

12th International Workshop on Termination (WST 2012) : WST 2012, February 19–23, 2012, Obergurgl, Austria / ed. by Georg Moser

This volume contains the proceedings of the 12th International Workshop on Termination (WST 2012), to be held February 19–23, 2012 in Obergurgl, Austria. The goal of the Workshop on Termination is to be a venue for presentation and discussion of all topics in and around termination. In this way, the workshop tries to bridge the gaps between different communities interested and active in research in and around termination. The 12th International Workshop on Termination in Obergurgl continues the successful workshops held in St. Andrews (1993), La Bresse (1995), Ede (1997), Dagstuhl (1999), Utrecht (2001), Valencia (2003), Aachen (2004), Seattle (2006), Paris (2007), Leipzig (2009), and Edinburgh (2010). The 12th International Workshop on Termination did welcome contributions on all aspects of termination and complexity analysis. Contributions from the imperative, constraint, functional, and logic programming communities, and papers investigating applications of complexity or termination (for example in program transformation or theorem proving) were particularly welcome. We did receive 18 submissions which all were accepted. Each paper was assigned two reviewers. In addition to these 18 contributed talks, WST 2012, hosts three invited talks by Alexander Krauss, Martin Hofmann, and Fausto Spoto

Mechanically Certifying Formula-based Noetherian Induction Reasoning

International audienceIn first-order logic, the formula-based instances of the Noetherian induction principle allow to perform effectively simultaneous, mutual and lazy induction reasoning. Compared to the term- based Noetherian induction instances, they are not directly supported by the current proof assistants.We provide general formal tools for certifying formula-based Noetherian induction proofs by the Coq proof assistant, then show how to apply them to certify proofs of conjectures about conditional specifications, built with: i) a reductive rewrite-based induction system, and ii) a reductive-free cyclic induction system. The generation of reductive proofs and their certification process can be easily automatised, without requiring additional definitions or proof transforma- tions, but may involve many ordering constraints to be checked during the certification process. On the other hand, the reductive-free proofs generate fewer ordering constraints, may involve more general specifications and the certification process is more effective. However, their proof generation is less automatic and the generated proofs need to be normalised before being certi- fied. The methodology for certifying reductive-free cyclic induction proofs related to conditional specifications extends a previous approach used for implicit induction proofs and it can be easily adapted to certify any formula-based Noetherian induction reasoning.In practice, the methodology has been implemented to automatically certify implicit induction proofs generated by the SPIKE theorem prover as well as reductive-free cyclic proofs built by the same system but in a less automatic way