Decidable Exponentials in Nonassociative Noncommutative Linear Logic

The use of exponentials in linear logic greatly enhances its expressive power. In this paper we focus on nonassociative noncommutative multiplicative linear logic, and systematically explore modal axioms K, T, and 4 as well as the structural rules of contraction and weakening. We give sequent systems for each subset of these axioms; these enjoy cut elimination and have analogues in more structural logics. We then appeal to work of Bulinska extending work of Buszkowski to show that several of these logics are PTIME decidable and generate context free languages as categorial grammars. This contrasts associative systems where similar logics are known to generate all recursively enumerable languages, and are thus in particular undecidable

Non-associative, Non-commutative Multi-modal Linear Logic

Adding multi-modalities (called subexponentials) to linear logic enhances its power as a logical framework, which has been extensively used in the specification of e.g. proof systems, programming languages and bigraphs. Initially, subexponentials allowed for classical, linear, affine or relevant behaviors. Recently, this framework was enhanced so to allow for commutativity as well. In this work, we close the cycle by considering associativity. We show that the resulting system (acLLΣ ) admits the (multi)cut rule, and we prove two undecidability results for fragments/variations of acLLΣ

Full Nonassociative Lambek Calculus with Modalities and Its Applications in Type Grammars

Wydział Matematyki i InformatykiRozprawa jest poświęcona pełnemu niełącznemu rachunkowi Lambeka wzbogaconemu o różne modalności. Te systemy tworzą pewną rodzinę logik substrukturalnych. W rozprawie badamy rachunki NL (niełączny rachunek Lambeka), DFNL (pełny niełączny rachunek Lambeka z prawami dystrybutywności dla operacji kratowych) i BFNL (DFNL z negacją spełniającą prawa algebr Boole’a) oraz ich rozszerzenia o operatory modalne, tworzące parę rezyduacji i spełniające standardowe aksjomaty logik modalnych (T), (4) i (5). Rozważamy też gramatyki typów oparte na tych rachunkach. Główne wyniki: twierdzenie o eliminacji cięć dla modalnych rozszerzeń NL z założeniami, wielomianowa złożoność relacji konsekwencji dla tych systemów, lemat interpolacyjny dla modalnych rozszerzeń DFNL i BFNL z założeniami, silna własność skończonego modelu dla tych systemów, rozstrzygalność relacji konsekwencji dla tyc systemów, PSPACE-zupełność rachunku BFNL, bezkontekstowość języków generowanych przez gramatyki typów oparte na tych rachunkach. Rozprawa kontynuuje wcześniejsze badania W. Buszkowskiego, M. Farulewskiego, M. Moortgata, A.. Plummera, N. Kurtoniny i innych.The thesis is devoted to full nonassociative Lambek calculus enriched with different modalities. These systems form a family of substrutural logics. In this thesis we study systems NL (nonassociative Lambek calculus), DFNL (full nonassociative Lambek calculus with the distributive laws for lattice operations) and BFNL (DFNL with negation satisfying the laws of Boolean algebras) and their extensions by modal operators, being a residuation pair and fulfilling standard axioms of modal logics (T), (4), (5). We also consider the type grammars based on these calculi. Main results: the cut-elimination theorem for modal extensions of NL with assumptions, the polynomial-time complexity of the consequence relations for these systems, an interpolation lemma for modal extensions of DFNL and BFNL with assumptions, the strong finite model property of the latter systems, the decidability of the consequence relations for the latter systems, the PSPACE-completeness of BFNL, the context-freeness of the languages generated by the type grammars based on these systems. The thesis continues some research of W. Buszkowski, M. Farulewski, M. Moortgat, A. Plummer,, N. Kurtonina and others

Automated Reasoning

This volume, LNAI 13385, constitutes the refereed proceedings of the 11th International Joint Conference on Automated Reasoning, IJCAR 2022, held in Haifa, Israel, in August 2022. The 32 full research papers and 9 short papers presented together with two invited talks were carefully reviewed and selected from 85 submissions. The papers focus on the following topics: Satisfiability, SMT Solving,Arithmetic; Calculi and Orderings; Knowledge Representation and Jutsification; Choices, Invariance, Substitutions and Formalization; Modal Logics; Proofs System and Proofs Search; Evolution, Termination and Decision Prolems. This is an open access book