The Universal Fragment of Presburger Arithmetic with Unary Uninterpreted Predicates is Undecidable

The first-order theory of addition over the natural numbers, known as Presburger arithmetic, is decidable in double exponential time. Adding an uninterpreted unary predicate to the language leads to an undecidable theory. We sharpen the known boundary between decidable and undecidable in that we show that the purely universal fragment of the extended theory is already undecidable. Our proof is based on a reduction of the halting problem for two-counter machines to unsatisfiability of sentences in the extended language of Presburger arithmetic that does not use existential quantification. On the other hand, we argue that a single $\forall\exists$ quantifier alternation turns the set of satisfiable sentences of the extended language into a $\Sigma^1_1$-complete set. Some of the mentioned results can be transfered to the realm of linear arithmetic over the ordered real numbers. This concerns the undecidability of the purely universal fragment and the $\Sigma^1_1$-hardness for sentences with at least one quantifier alternation. Finally, we discuss the relevance of our results to verification. In particular, we derive undecidability results for quantified fragments of separation logic, the theory of arrays, and combinations of the theory of equality over uninterpreted functions with restricted forms of integer arithmetic. In certain cases our results even imply the absence of sound and complete deductive calculi

Orbit-finite linear programming

An infinite set is orbit-finite if, up to permutations of the underlying structure of atoms, it has only finitely many elements. We study a generalisation of linear programming where constraints are expressed by an orbit-finite system of linear inequalities. As our principal contribution we provide a decision procedure for checking if such a system has a real solution, and for computing the minimal/maximal value of a linear objective function over the solution set. We also show undecidability of these problems in case when only integer solutions are considered. Therefore orbit-finite linear programming is decidable, while orbit-finite integer linear programming is not.Comment: Full version of LICS 2023 pape

Analytic Combinatorics in Several Variables: Effective Asymptotics and Lattice Path Enumeration

The field of analytic combinatorics, which studies the asymptotic behaviour of sequences through analytic properties of their generating functions, has led to the development of deep and powerful tools with applications across mathematics and the natural sciences. In addition to the now classical univariate theory, recent work in the study of analytic combinatorics in several variables (ACSV) has shown how to derive asymptotics for the coefficients of certain D-finite functions represented by diagonals of multivariate rational functions. We give a pedagogical introduction to the methods of ACSV from a computer algebra viewpoint, developing rigorous algorithms and giving the first complexity results in this area under conditions which are broadly satisfied. Furthermore, we give several new applications of ACSV to the enumeration of lattice walks restricted to certain regions. In addition to proving several open conjectures on the asymptotics of such walks, a detailed study of lattice walk models with weighted steps is undertaken.Comment: PhD thesis, University of Waterloo and ENS Lyon - 259 page

Sovelluskohtaiset tyyppijärjestelmät

Type system tailored for specific domain could radically improve quality of the program. Many domains have natural types, yet they are difficult to encode in mainstream languages’ type systems. If the encoding is possible, it’s luckily to be very troublesome to work with. We investigate a single approach of developing domain specific type systems and type checking and inference algorithms for them, and apply it to MATLAB and JavaScript programs.Tyyppijärjestelmä, joka on suunniteltu tiettyä sovellusaluetta varten, voi mer- kittävästi parantaa sovelluksen laatua. Monilla sovellusalueilla on luonnollisia tyyppejä, joita ei ole helppo ilmaista yleiskäyttöisten ohjelmointikielten tyyp- pijärjestelmissä. Ja vaikka se olisikin mahdollista, koodaus ei ole välttämättä luonteva. Tässä tutkielmassa tarkastellaan erästä tapaa suunnitella ja toteuttaa sovelluskohtaisia tyyppijärjestelmiä ja niiden tyyppitarkistus- ja tyyppipäättely- algoritmeja. Sovellamme menetelmää Matlab- ja JavaScript-kielillä kirjoitettuihin ohjelmiin

Journées Francophones des Langages Applicatifs 2018

National audienceLes 29èmes journées francophones des langages applicatifs (JFLA) se déroulent en 2018 à l'observatoire océanographique de Banyuls-sur-Mer. Les JFLA réunissent chaque année, dans un cadre convivial, concepteurs, développeurs et utilisateurs des langages fonctionnels, des assistants de preuve et des outils de vérification de programmes en présentant des travaux variés, allant des aspects les plus théoriques aux applications industrielles.Cette année, nous avons sélectionné 9 articles de recherche et 8 articles courts. Les thématiques sont variées : preuve formelle, vérification de programmes, modèle mémoire, langages de programmation, mais aussi théorie de l'homotopieet blockchain

Beyond Logic. Proceedings of the Conference held in Cerisy-la-Salle, 22-27 May 2017

The project "Beyond Logic" is devoted to what hypothetical reasoning is all about when we go beyond the realm of "pure" logic into the world where logic is applied. As such extralogical areas we have chosen philosophy of science as an application within philosophy, informatics as an application within the formal sciences, and law as an application within the field of social interaction. The aim of the conference was to allow philosophers, logicians and computer scientists to present their work in connection with these three areas. The conference took place 22-27 May, 2017 in Cerisy-la-Salle at the Centre Culturel International de Cerisy. The proceedings collect abstracts, slides and papers of the presentations given, as well as a contribution from a speaker who was unable to attend