arXiv subjects

Valeria de Paiva

Publications and source records attributed to Valeria de Paiva.

28 records · Page 2Linked to original sources

arXiv2021

The Gödel Fibration

We introduce the notion of a Gödel fibration, which is a fibration categorically embodying both the logical principle of traditional Skolemization (we can exchange the order of quantifiers paying the price of a functional) and the existence of a prenex normal form presentation for every logical formula. Building up from Hofstra's earlier fibrational characterization of the de Paiva's categorical Dialectica construction, we show that a fibration is an instance of the Dialectica construction if and only if it is a Gödel fibration. This result establishes an internal presentation of the dialectica construction. Then we provide a deep structural analysis of the Dialectica construction producing a full description of which categorical structure behaves well with respect to this construction, focusing on (weak) finite products and coproducts. We conclude describing the applications we envisage for this generalized fibrational version of the Dialectica construction.

math.CT↗

arXiv2020

Deriving Theorems in Implicational Linear Logic, Declaratively

The problem we want to solve is how to generate all theorems of a given size in the implicational fragment of propositional intuitionistic linear logic. We start by filtering for linearity the proof terms associated by our Prolog-based theorem prover for Implicational Intuitionistic Logic. This works, but using for each formula a PSPACE-complete algorithm limits it to very small formulas. We take a few walks back and forth over the bridge between proof terms and theorems, provided by the Curry-Howard isomorphism, and derive step-by-step an efficient algorithm requiring a low polynomial effort per generated theorem. The resulting Prolog program runs in O(N) space for terms of size N and generates in a few hours 7,566,084,686 theorems in the implicational fragment of Linear Intuitionistic Logic together with their proof terms in normal form. As applications, we generate datasets for correctness and scalability testing of linear logic theorem provers and training data for neural networks working on theorem proving challenges. The results in the paper, organized as a literate Prolog program, are fully replicable. Keywords: combinatorial generation of provable formulas of a given size, intuitionistic and linear logic theorem provers, theorems of the implicational fragment of propositional linear intuitionistic logic, Curry-Howard isomorphism, efficient generation of linear lambda terms in normal form, Prolog programs for lambda term generation and theorem proving.

cs.LO↗

arXiv2020

Dialectica Fuzzy Petri Nets

Brown and Gurr have introduced a model of Petri Nets that is based on de~Paiva's Dialectica categories. This model was refined in an unpublished technical report, where Petri nets with multiplicities, instead of {\em elementary} nets (i.e., nets with multiplicities zero and one only) were considered. In this note we expand this modelling to deal with {\em fuzzy} petri nets. The basic idea is to use as the dualizing object in the Dialectica categories construction, the unit interval that has all the properties of a {\em lineale} structure.

cs.LO↗

arXiv2019

On the Lambek Calculus with an Exchange Modality

In this paper we introduce Commutative/Non-Commutative Logic (CNC logic) and two categorical models for CNC logic. This work abstracts Benton's Linear/Non-Linear Logic by removing the existence of the exchange structural rule. One should view this logic as composed of two logics; one sitting to the left of the other. On the left, there is intuitionistic linear logic, and on the right is a mixed commutative/non-commutative formalization of the Lambek calculus. Then both of these logics are connected via a pair of monoidal adjoint functors. An exchange modality is then derivable within the logic using the adjunction between both sides. Thus, the adjoint functors allow one to pull the exchange structural rule from the left side to the right side. We then give a categorical model in terms of a monoidal adjunction, and then a concrete model in terms of dialectica Lambek spaces.

cs.LO↗

arXiv2019

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.

cs.LO↗

arXiv2019

Proceedings Joint International Workshop on Linearity & Trends in Linear Logic and Applications

This volume contains a selection of papers presented at Linearity/TLLA 2018: Joint Linearity and TLLA workshops (part of FLOC 2018) held on July 7-8, 2018 in Oxford. Linearity has been a key feature in several lines of research in both theoretical and practical approaches to computer science. On the theoretical side there is much work stemming from linear logic dealing with proof technology, complexity classes and more recently quantum computation. On the practical side there is work on program analysis, expressive operational semantics for programming languages, linear programming languages, program transformation, update analysis and efficient implementation techniques. Linear logic is not only a theoretical tool to analyse the use of resources in logic and computation. It is also a corpus of tools, approaches, and methodologies (proof nets, exponential decomposition, geometry of interaction, coherent spaces, relational models, etc.) that were originally developed for the study of linear logic's syntax and semantics and are nowadays applied in several other fields.

cs.LO↗

arXiv2018

Linguistic Legal Concept Extraction in Portuguese

This work investigates legal concepts and their expression in Portuguese, concentrating on the "Order of Attorneys of Brazil" Bar exam. Using a corpus formed by a collection of multiple-choice questions, three norms related to the Ethics part of the OAB exam, language resources (Princeton WordNet and OpenWordNet-PT) and tools (AntConc and Freeling), we began to investigate the concepts and words missing from our repertory of concepts and words in Portuguese, the knowledge base OpenWordNet-PT. We add these concepts and words to OpenWordNet-PT and hence obtain a representation of these texts that is "contained" in the lexical knowledge base.

cs.CL↗

arXiv2018

Dialectica Categories for the Lambek Calculus

We revisit the old work of de Paiva on the models of the Lambek Calculus in dialectica models making sure that the syntactic details that were sketchy on the first version got completed and verified. We extend the Lambek Calculus with a κmodality, inspired by Yetter's work, which makes the calculus commutative. Then we add the of-course modality !, as Girard did, to re-introduce weakening and contraction for all formulas and get back the full power of intuitionistic and classical logic. We also present the categorical semantics, proved sound and complete. Finally we show the traditional properties of type systems, like subject reduction, the Church-Rosser theorem and normalization for the calculi of extended modalities, which we did not have before.

cs.LO↗

arXiv2016

Proceedings of the LexSem+Logics Workshop 2016

Lexical semantics continues to play an important role in driving research directions in NLP, with the recognition and understanding of context becoming increasingly important in delivering successful outcomes in NLP tasks. Besides traditional processing areas such as word sense and named entity disambiguation, the creation and maintenance of dictionaries, annotated corpora and resources have become cornerstones of lexical semantics research and produced a wealth of contextual information that NLP processes can exploit. New efforts both to link and construct from scratch such information - as Linked Open Data or by way of formal tools coming from logic, ontologies and automated reasoning - have increased the interoperability and accessibility of resources for lexical and computational semantics, even in those languages for which they have previously been limited. LexSem+Logics 2016 combines the 1st Workshop on Lexical Semantics for Lesser-Resources Languages and the 3rd Workshop on Logics and Ontologies. The accepted papers in our program covered topics across these two areas, including: the encoding of plurals in Wordnets, the creation of a thesaurus from multiple sources based on semantic similarity metrics, and the use of cross-lingual treebanks and annotations for universal part-of-speech tagging. We also welcomed talks from two distinguished speakers: on Portuguese lexical knowledge bases (different approaches, results and their application in NLP tasks) and on new strategies for open information extraction (the capture of verb-based propositions from massive text corpora).

cs.CL↗

arXiv2011

Fuzzy Topological Systems

Dialectica categories are a very versatile categorical model of linear logic. These have been used to model many seemingly different things (e.g., Petri nets and Lambek's calculus). In this note, we expand our previous work on fuzzy petri nets to deal with fuzzy topological systems. One basic idea is to use as the dualizing object in the Dialectica categories construction, the unit real interval [0,1], which has all the properties of a {\em lineale}. The second basic idea is to generalize Vickers's notion of a topological system.

cs.LO↗