The ILLTP Library for Intuitionistic Linear Logic

Benchmarking automated theorem proving (ATP) systems using standardized problem sets is a well-established method for measuring their performance. However, the availability of such libraries for non-classical logics is very limited. In this work we propose a library for benchmarking Girard's (propositional) intuitionistic linear logic. For a quick bootstrapping of the collection of problems, and for discussing the selection of relevant problems and understanding their meaning as linear logic theorems, we use translations of the collection of Kleene's intuitionistic theorems in the traditional monograph "Introduction to Metamathematics". We analyze four different translations of intuitionistic logic into linear logic and compare their proofs using a linear logic based prover with focusing. In order to enhance the set of problems in our library, we apply the three provability-preserving translations to the propositional benchmarks in the ILTP Library. Finally, we generate a comprehensive set of reachability problems for Petri nets and encode such problems as linear logic sequents, thus enlarging our collection of problems

Classical and Intuitionistic Subexponential Logics are Equally Expressive

International audienceIt is standard to regard the intuitionistic restriction of a classical logic as increasing the expressivity of the logic because the classical logic can be adequately represented in the intuitionistic logic by double-negation, while the other direction has no truth-preserving propositional encodings. We show here that subexponential logic, which is a family of substructural refinements of classical logic, each parametric over a preorder over the subexponential connectives, does not suffer from this asymmetry if the preorder is systematically modified as part of the encoding. Precisely, we show a bijection between synthetic (i.e., focused) partial sequent derivations modulo a given encoding. Particular instances of our encoding for particular subexponential preorders give rise to both known and novel adequacy theorems for substructural logics

On the unity of logic

AbstractNous présentons un calcul des séquents unifié, commun aux logiques classique, intuitionniste et linéaire. La principale nouveautéest que les logiques classique, intuitionniste et linéaire apparaissent comme des fragments, c'estádire comme des classes particuliéres de formules et de séquents. Par exemple la démonstration d'unénoncéintuitionniste pourra utiliser des lemmes classiques ou intuitionnistes sans limitation: simplement aprèsélimination des coupures, la démonstration se fera entièrement dans le fragment intuitionniste, ce qui est superficiellement assurépar la propriétéde la sous-formule (seulement des formules intuitionnistes sont utilisées) et plus profondément par un traitement très rigoureux des règles structurelles. Cette approche est radicalement différente de l'approche habituelle qui consiste tout bonnementàchanger la règle du jeu quand on veut changer de logique, c'estàdire de style de séquent: ici il n'y a plus qu'une seule logique, qui au grédes utilisation peut apparaître classique, intuitionniste ou linéaire

Explorations in Subexponential Non-associative Non-commutative Linear Logic

In a previous work we introduced a non-associative non-commutative logic extended by multimodalities, called subexponentials, licensing local application of structural rules. Here, we further explore this system, exhibiting a classical one-sided multi-succedent classical analogue of our intuitionistic system, following the exponential-free calculi of Buszkowski, and de Groote, Lamarche. A large fragment of the intuitionistic calculus is shown to embed faithfully into the classical fragment

Explorations in Subexponential non-associative non-commutative Linear Logic

In a previous work we introduced a non-associative non-commutative logic extended by multimodalities, called subexponentials, licensing local application of structural rules. Here, we further explore this system, considering a classical one-sided multi-succedent classical version of the system, following the exponential-free calculi of Buszkowski's and de Groote and Lamarche's works, where the intuitionistic calculus is shown to embed faithfully into the classical fragment