The decision problem for a three-sorted fragment of set theory with restricted quantification and finite enumerations

We solve the satisfiability problem for a three-sorted fragment of set theory (denoted $3LQST_0^R$), which admits a restricted form of quantification over individual and set variables and the finite enumeration operator $\{\text{-}, \text{-}, \ldots, \text{-}\}$ over individual variables, by showing that it enjoys a small model property, i.e., any satisfiable formula $\psi$ of $3LQST_0^R$ has a finite model whose size depends solely on the length of $\psi$ itself. Several set-theoretic constructs are expressible by $3LQST_0^R$-formulae, such as some variants of the power set operator and the unordered Cartesian product. In particular, concerning the unordered Cartesian product, we show that when finite enumerations are used to represent the construct, the resulting formula is exponentially shorter than the one that can be constructed without resorting to such terms

Automatic modular abstractions for template numerical constraints

We propose a method for automatically generating abstract transformers for static analysis by abstract interpretation. The method focuses on linear constraints on programs operating on rational, real or floating-point variables and containing linear assignments and tests. In addition to loop-free code, the same method also applies for obtaining least fixed points as functions of the precondition, which permits the analysis of loops and recursive functions. Our algorithms are based on new quantifier elimination and symbolic manipulation techniques. Given the specification of an abstract domain, and a program block, our method automatically outputs an implementation of the corresponding abstract transformer. It is thus a form of program transformation. The motivation of our work is data-flow synchronous programming languages, used for building control-command embedded systems, but it also applies to imperative and functional programming

The Temporal Booleanization Theorem: realizability checking over numerical-LTL industrial requirements

Industrial systems are getting more complex every year, and due to that complexity growth, the languages to specify them are becoming increasingly more expressive: so that they can properly model both controllable parts and the environment, or even temporal events. As a result, a great effort has been made over the last few years to move forward in the area of realizability checking of complex requirements. However, there is a blocking problem in the state-of-the-art realizability checkers: they only accept requirements that only contain Boolean variables on them. Therefore, these checkers cannot handle many real-industrial requirements, like those requirements containing numerical variables. One approach is to Booleanize numeric requirements and convert them into equivalent Boolean requirements. This problem has been researched and it has been discovered that its solution is not trivial. Thus, the main contribution of this thesis is that we have proved a theorem which verifies that (1) a correct Booleanization exists for all requirements that use theories with a decidable Ǝ*∀* fragment; and, thus (2) we have a realizability checking procedure for numeric requirements based on safety two-player turn-based LTL games. In addition, an algorithm that performs this has been proposed and implemented in OCaml

Logic and Automata

Mathematical logic and automata theory are two scientific disciplines with a fundamentally close relationship. The authors of Logic and Automata take the occasion of the sixtieth birthday of Wolfgang Thomas to present a tour d'horizon of automata theory and logic. The twenty papers in this volume cover many different facets of logic and automata theory, emphasizing the connections to other disciplines such as games, algorithms, and semigroup theory, as well as discussing current challenges in the field

On the implicit learnability of knowledge

The deployment of knowledge-based systems in the real world requires addressing the challenge of knowledge acquisition. While knowledge engineering by hand is a daunting task, machine learning has been proposed as an alternative. However, learning explicit representations for real-world knowledge that feature a desirable level of expressiveness remains difficult and often leads to heuristics without robustness guarantees. Probably Approximately Correct (PAC) Semantics offers strong guarantees, however learning explicit representations is not tractable, even in propositional logic. Previous works have proposed solutions to these challenges by learning to reason directly, without producing an explicit representation of the learned knowledge. Recent work on so-called implicit learning has shown tremendous promise in obtaining polynomial-time results for fragments of first-order logic, bypassing the intractable step of producing an explicit representation of learned knowledge. This thesis extends these ideas to richer logical languages such as arithmetic theories and multi-agent logics. We demonstrate that it is possible to learn to reason efficiently for standard fragments of linear arithmetic, and we establish a general finding that provides an efficient reduction from the learning-to-reason problem for any logic to any sound and complete solver for that logic. We then extend implicit learning in PAC Semantics to handle noisy data in the form of intervals and threshold uncertainty in the language of linear arithmetic. We prove that our extended framework maintains existing polynomial-time complexity guarantees. Furthermore, we provide the first empirical investigation of this purely theoretical framework. Using benchmark problems, we show that our implicit approach to learning optimal linear programming objective constraints significantly outperforms an explicit approach in practice. Our results demonstrate the effectiveness of PAC Semantics and implicit learning for real-world problems with noisy data and provide a path towards robust learning in expressive languages. Development in reasoning about knowledge and interactions in complex multi-agent systems spans domains such as artificial intelligence, smart traffic, and robotics. In these systems, epistemic logic serves as a formal language for expressing and reasoning about knowledge, beliefs, and communication among agents, yet integrating learning algorithms within multi-agent epistemic logic is challenging due to the inherent complexity of distributed knowledge reasoning. We provide proof of correctness for our learning procedure and analyse the sample complexity required to assert the entailment of an epistemic query. Overall, our work offers a promising approach to integrating learning and deduction in a range of logical languages from linear arithmetic to multi-agent epistemic logics

Méthodes logico-numériques pour la vérification des systèmes discrets et hybrides

Cette thèse étudie la vérification automatique de propriétés de sûreté de systèmes logico-numériques discrets ou hybrides. Ce sont des systèmes ayant des variables booléennes et numériques et des comportements discrets et continus. Notre approche est fondée sur l'analyse statique par interprétation abstraite. Nous adressons les problèmes suivants : les méthodes d'interprétation abstraite numériques exigent l'énumération des états booléens, et par conséquent, ils souffrent du probléme d'explosion d'espace d'états. En outre, il y a une perte de précision due à l'utilisation d'un opérateur d'élargissement afin de garantir la terminaison de l'analyse. Par ailleurs, nous voulons rendre les méthodes d'interprétation abstraite accessibles à des langages de simulation hybrides. Dans cette thèse, nous généralisons d'abord l'accélération abstraite, une méthode qui améliore la précision des invariants numériques inférés. Ensuite, nous montrons comment étendre l'accélération abstraite et l'itération de max-stratégies à des programmes logico-numériques, ce qui aide à améliorer le compromis entre l'efficacité et la précision. En ce qui concerne les systèmes hybrides, nous traduisons le langage de programmation synchrone et hybride Zelus vers les automates hybrides logico-numériques, et nous étendons les méthodes d'analyse logico-numérique aux systèmes hybrides. Enfin, nous avons mis en oeuvre les méthodes proposées dans un outil nommé ReaVer et nous fournissons des résultats expérimentaux. En conclusion, cette thèse propose une approche unifiée à la vérification de systèmes logico-numériques discrets et hybrides fondée sur l'interprétation abstraite qui est capable d'intégrer des méthodes d'interprétation abstraite numériques sophistiquées tout en améliorant le compromis entre l'efficacité et la précision.This thesis studies the automatic verification of safety properties of logico-numerical discrete and hybrid systems. These systems have Boolean and numerical variables and exhibit discrete and continuous behavior. Our approach is based on static analysis using abstract interpretation. We address the following issues: Numerical abstract interpretation methods require the enumeration of the Boolean states, and hence, they suffer from the state space explosion problem. Moreover, there is a precision loss due to widening operators used to guarantee termination of the analysis. Furthermore, we want to make abstract interpretation-based analysis methods accessible to simulation languages for hybrid systems. In this thesis, we first generalize abstract acceleration, a method that improves the precision of the inferred numerical invariants. Then, we show how to extend abstract acceleration and max-strategy iteration to logico-numerical programs while improving the trade-off between efficiency and precision. Concerning hybrid systems, we translate the Zelus hybrid synchronous programming language to logico-numerical hybrid automata and extend logico-numerical analysis methods to hybrid systems. Finally, we implemented the proposed methods in ReaVer, a REActive System VERification tool, and provide experimental results. Concluding, this thesis proposes a unified approach to the verification of discrete and hybrid logico-numerical systems based on abstract interpretation, which is capable of integrating sophisticated numerical abstract interpretation methods while successfully trading precision for efficiency.SAVOIE-SCD - Bib.électronique (730659901) / SudocGRENOBLE1/INP-Bib.électronique (384210012) / SudocGRENOBLE2/3-Bib.électronique (384219901) / SudocSudocFranceF

The Logical Writings of Karl Popper

This open access book is the first ever collection of Karl Popper's writings on deductive logic. Karl R. Popper (1902-1994) was one of the most influential philosophers of the 20th century. His philosophy of science ("falsificationism") and his social and political philosophy ("open society") have been widely discussed way beyond academic philosophy. What is not so well known is that Popper also produced a considerable work on the foundations of deductive logic, most of it published at the end of the 1940s as articles at scattered places. This little-known work deserves to be known better, as it is highly significant for modern proof-theoretic semantics. This collection assembles Popper's published writings on deductive logic in a single volume, together with all reviews of these papers. It also contains a large amount of unpublished material from the Popper Archives, including Popper's correspondence related to deductive logic and manuscripts that were (almost) finished, but did not reach the publication stage. All of these items are critically edited with additional comments by the editors. A general introduction puts Popper's work into the context of current discussions on the foundations of logic. This book should be of interest to logicians, philosophers, and anybody concerned with Popper's work