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

From Double Pushout Grammars to Hypergraph Lambek Grammars With and Without Exponential Modality

We study how to relate well-known hypergraph grammars based on the double pushout (DPO) approach and grammars over the hypergraph Lambek calculus HL (called HL-grammars). It turns out that DPO rules can be naturally encoded by types of HL using methods similar to those used by Kanazawa for multiplicative-exponential linear logic. In order to generalize his reasonings we extend the hypergraph Lambek calculus by adding the exponential modality, which results in a new calculus HMEL0; then we prove that any DPO grammar can be converted into an equivalent HMEL0-grammar. We also define the conjunctive Kleene star, which behaves similarly to this exponential modality, and establish a similar result. If we add neither the exponential modality nor the conjunctive Kleene star to HL, then we can still use the same encoding and show that any DPO grammar with a linear restriction on the length of derivations can be converted into an equivalent HL-grammar.Comment: In Proceedings TERMGRAPH 2022, arXiv:2303.1421

Proof nets for linguistic analysis

This book investigates the possible linguistic applications of proof nets, redundancy free representations of proofs, which were introduced by Girard for linear logic. We will adapt the notion of proof net to allow the formulation of a proof net calculus which is soundand complete for the multimodal Lambek calculus. Finally, we will investigate the computational and complexity theoretic consequences of this calculus and give an introduction to a practical grammar development tool based on proof nets

A substructural logic for layered graphs

Complex systems, be they natural or synthetic, are ubiquitous. In particular, complex networks of devices and services underpin most of society's operations. By their very nature, such systems are difficult to conceptualize and reason about effectively. The concept of layering is widespread in complex systems, but has not been considered conceptually. Noting that graphs are a key formalism in the description of complex systems, we establish a notion of a layered graph. We provide a logical characterization of this notion of layering using a non-associative, non-commutative substructural, separating logic. We provide soundness and completeness results for a class of algebraic models that includes layered graphs, which give a mathematically substantial semantics to this very weak logic. We explain, via examples, applications in information processing and security

Da linguística gerativa à gramática categorial : sujeitos lexicais em infinitivos controlados

Orientadores: Marcelo Esteban Coniglio, Sonia Maria Lazzarini CyrinoTese (doutorado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciências HumanasResumo: A presente tese situa-se na interface da lógica e da linguística; o seu objeto de estudo são os pronomes lexicais em sentenças de controle em três línguas Românicas: Português, Italiano e Espanhol. Esse assunto tem recebido mais atenção na linguística gerativa, especialmente nos anos recentes, do que na gramática de cunho lógico. Talvez como consequência disso, há ainda muito a ser entendido sobre essas estruturas linguísticas e as suas propriedades lógicas. Essa tese tenta preencher as lacunas na literatura \--- ou, pelo menos, avançar nessa direção \--- colocando questões que não foram suficientemente exploradas até agora. Para tal efeito avançamos duas perguntas-chaves, uma linguística e a outra lógica. Elas são, respectivamente: Qual é o estatuto sintático dos pronomes lexicais em estruturas de controle? E: Quais são os mecanismos disponíveis, em uma gramática lógica livre de contração, para se reusar recursos semânticos? A tese divide-se, consequentemente, em duas partes: linguística gerativa e gramática categorial. Na Parte I revisamos algumas das principais teorias de controle gerativistas e a recente discussão acerca das cláusulas infinitivas com sujeito lexical. Na Parte II revisamos a literatura categorial, atendendo principalmente às propostas acerca das estruturas de controle e dos pronomes anafóricos. Em última instância, mostraremos que as propostas linguísticas e lógicas prévias precisam ser modificadas para se explicar o fenômeno linguístico em questão. Com efeito, nos capítulos finais de cada uma das partes avançamos propostas alternativas que, a nosso ver, resultam mais adequadas que as suas rivais. Mais específicamente, na Parte I avançamos uma proposta linguística na linha do cálculo de controle T/Agr de Landau. Na Parte II apresentamos duas propostas categoriais, uma na linha do cálculo categorial combinatório e a outra, na gramática lógica de tipos. Finalmente mostramos a implementação da última proposta em um analisador sintático e de demonstração categorialAbstract: The present thesis lies at the interface of logic and linguistics; its object of study are control sentences with overt pronouns in Romance languages (European and Brazilian Portuguese, Italian and Spanish). This is a topic that has received considerably more attention on the part of linguists, especially in recent years, than from logicians. Perhaps for this reason, much remains to be understood about these linguistic structures and their underlying logical properties. This thesis seeks to fill the lacunas in the literature \--- or at least take steps in this direction \--- by way of addressing a number of issues that have so far been under-explored. To this end we put forward two key questions, one linguistic and the other logical. These are, respectively: What is the syntactic status of the surface pronoun? And: What are the available mechanisms to reuse semantic resources in a contraction-free logical grammar? Accordingly, the thesis is divided into two parts: generative linguistics and categorial grammar. Part I starts by reviewing the recent discussion within the generative literature on infinitive clauses with overt subjects, paying detailed attention to the main accounts in the field. Part II does the same on the logical grammar front, addressing in particular the issues of control and of anaphoric pronouns. Ultimately, the leading accounts from both camps will be found wanting. The closing chapter of each of Part I and Part II will thus put forward alternative candidates, that we contend are more successful than their predecessors. More specifically, in Part I we offer a linguistic account along the lines of Landau's T/Agr theory of control. In Part II we present two alternative categorial accounts: one based on Combinatory Categorial Grammar, the other on Type-Logical Grammar. Each of these accounts offers an improved, more fine-grained perspective on control infinitives featuring overt pronominal subjects. Finally, we include an Appendix in which our type-logical proposal is implemented in a categorial parser/theorem-prover (categorial parser/theorem-prover)DoutoradoFilosofiaDoutora em Filosofia2013/08115-1, 2015/09699-2FAPESPCAPE