Untyping Typed Algebras and Colouring Cyclic Linear Logic

We prove "untyping" theorems: in some typed theories (semirings, Kleene algebras, residuated lattices, involutive residuated lattices), typed equations can be derived from the underlying untyped equations. As a consequence, the corresponding untyped decision procedures can be extended for free to the typed settings. Some of these theorems are obtained via a detour through fragments of cyclic linear logic, and give rise to a substantial optimisation of standard proof search algorithms.Comment: 21

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

The Finite Embeddability Property for Some Noncommutative Knotted Varieties of RL and DRL

Residuated lattices, although originally considered in the realm of algebra providing a general setting for studying ideals in ring theory, were later shown to form algebraic models for substructural logics. The latter are non-classical logics that include intuitionistic, relevance, many-valued, and linear logic, among others. Most of the important examples of substructural logics are obtained by adding structural rules to the basic logical calculus FL. We denote by � the varieties of knotted residuated lattices. Examples of these knotted rules include integrality and contraction. The extension of �� by the rules corresponding to these two equations is equivalent to Gentzen’s original system �� for intuitionism. Apart from applications to logic and to abstract ring theory, residuated lattices are connected to mathematical linguistics, computer science, and quantum mechanics, among other areas. Even thought the connections to other disciplines are abundant, the current document is of purely algebraic nature. Results in [17] establish the finite model property (FMP) for the implicational fragment of ��� extended by some knotted rules. The finite embeddability property (FEP) is known to hold for commutative ��^�_� (�� = ��); the strong finite model property follows for the corresponding logics. Recent results by Horčík show that the word problem is undecidable for the varieties ��^�_� when 1 ≤ � \u3c � or 2 ≤ � \u3c �. Therefore these varieties do not have the FEP. We refer the reader to [16] for details on how this is connected to the Burnside problems in group theory and to regularity of languages in automata theory. In the present document, using purely algebraic methods, we prove the FEP for subvarieties of ��^�_� and ���^�_� that satisfy properties weaker than commutativity. The proof uses the theory of residuated frames introduced by Galatos and Jipsen . In Chapter 1, we present the basic definitions and constructions that will be used throughout the full document. We point the reader towards Section 1.4, where we list a relevant list of varieties for which the FEP holds or not. Chapter 2 presents a proof of the FEP for subvarieties of ��^�_� that satisfy the identity ��� = �^2�. The proof of this case relies on finding the free object over the class of pomonoids that satisfy the previous equality and �� ≤ ��. Chapter 3 focuses on the study of the noncommutative equation that we use to define the varieties studied in the following two chapters. This equation arises as a natural generalization of the basic equation ��� = �^2�. Chapter 4 presents the FEP for ��^�_�. In the general case, the free object in the class is fairly complicated, so we identify instead an object outside the class, which is both free and structured enough to allow us to prove the result. In the last section, we extend our result to cover some other subvarieties of knotted residuated lattices. These subvarieties include the cyclic, cyclic-involutive, and representable ones. Chapter 5 details a proof for the fully distributive case. Here we enrich the free object discovered in Chapter 4 by creating the meet semilattice generated by it. We remark that the FEP for a variety � is equivalent to the condition that all finitely presented algebras in � are residually finite. Varieties of semigroups with this property have been fully characterized in [20]. In particular, the variety of monoids axiomatized by ��� = �^2� has been studied and has the FEP. These results do not imply the FEP for the corresponding variety of residuated lattices, which also serves as the simplest case of our analysis. 1. Science, the magazine itself is not showing up at the top, but at the botto

On the Algebra of Structural Contexts

Article dans revue scientifique avec comité de lecture.We discuss a general way of defining contexts in linear logic, based on the observation that linear universal algebra can be symmetrized by assigning an additional variable to represent the output of a term. We give two approaches to this, a syntactical one based on a new, reversible notion of term, and an algebraic one based on a simple generalization of typed operads. We relate these to each other and to known examples of logical systems, and show new examples, in particular discussing the relationship between intuitionistic and classical systems. We then present a general framework for extracting deductive system from a given theory of contexts, and prove that all these systems have cut-elimination by the means of a generic argument

Bunched logics: a uniform approach

Bunched logics have found themselves to be key tools in modern computer science, in particular through the industrial-level program verification formalism Separation Logic. Despite this—and in contrast to adjacent families of logics like modal and substructural logic—there is a lack of uniform methodology in their study, leaving many evident variants uninvestigated and many open problems unresolved. In this thesis we investigate the family of bunched logics—including previously unexplored intuitionistic variants—through two uniform frameworks. The first is a system of duality theorems that relate the algebraic and Kripke-style interpretations of the logics; the second, a modular framework of tableaux calculi that are sound and complete for both the core logics themselves, as well as many classes of bunched logic model important for applications in program verification and systems modelling. In doing so we are able to resolve a number of open problems in the literature, including soundness and completeness theorems for intuitionistic variants of bunched logics, classes of Separation Logic models and layered graph models; decidability of layered graph logics; a characterisation theorem for the classes of bunched logic model definable by bunched logic formulae; and the failure of Craig interpolation for principal bunched logics. We also extend our duality theorems to the categorical structures suitable for interpreting predicate versions of the logics, in particular hyperdoctrinal structures used frequently in Separation Logic

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