This work extends our research on fuzzy relational Galois connections, previously established in the context of complete Heyting algebras, to the broader framework of arbitrary residuated lattices. In this context, we study the properties of fuzzy closure relations and fuzzy closure systems, and the relationship with fuzzy relational Galois connections. The main result of the paper is the generalization of the necessary and sufficient conditions for the existence of a right adjoint for a fuzzy relation linking a fuzzy transitive directed graph to an unstructured set.

Simplification logic, a logic for attribute implications, was originally defined for Boolean sets. It was extended to distributive fuzzy sets by using a complete dual Heyting algebra. In this paper, we weaken this restriction in the sense that we prove that it is possible to define a simplification logic on fuzzy sets in which the membership value structure is not necessarily distributive. For this purpose, we replace the structure of the complete dual Heyting algebra by the so-called weak complete dual Heyting algebra. We demonstrate the soundness and completeness of this simplification logic, and provide a characterisation of the operations defining weak complete dual Heyting algebras.

We present a fuzzy version of the notion of relational Galois connection between fuzzy transitive directed graphs (fuzzy T-digraphs) on the specific setting in which the underlying algebra of truth values is a complete Heyting algebra. The components of such fuzzy Galois connection are fuzzy relations satisfying certain reasonable properties expressed in terms of the so-called full powering. Moreover, we provide a necessary and sufficient condition under which it is possible to construct a right adjoint for a given fuzzy relation between a fuzzy T-digraph and an unstructured set.

Galois connections seem to be ubiquitous in mathematics. They have been used to model solutions for both pure and application-oriented problems. Throughout the paper, the general framework is a complete fuzzy lattice over a complete residuated lattice. The existence of three fuzzy Galois connections (two antitone and one isotone) between three specific ordered sets is proved in this paper. The most interesting part is that fuzzy closure systems, fuzzy closure operators and strong fuzzy closure relations are formal concepts of these fuzzy Galois connections.

This paper is the natural extension of Fuzzy Closure Structures as Formal Concepts. In this paper we take into consideration the concept of closure system which is not dealt with in the previous one. Hence, a connection must be found between fuzzy ordered sets and a crisp ordered set. This problem is two-fold, the core of the fuzzy orders can be considered in order to complete the ensemble, or the crisp order can be fuzzified. Both ways are studied in the paper. The most interesting result is, similarly to the previous paper, that closure systems are formal concepts of these Galois connections as well.

This paper aims to contribute to the extension of classical Formal Concept Analysis (FCA), allowing the management of unknown information. In a preliminary paper, we define a new kind of attribute implications to represent the knowledge from the information currently available. The whole FCA framework has to be appropriately extended to manage unknown information. This paper introduces a new logic for reasoning with this kind of implications, which belongs to the family of logics with an underlying Simplification paradigm. Specifically, we introduce a new algebra, named weak dual Heyting Algebra, that allows us to extend the Simplification logic for these new implications. To provide a solid framework, we also prove its soundness and completeness and show the advantages of the Simplification paradigm. Finally, to allow further use of this extension of FCA in applications, an algorithm for automated reasoning, which is directly built from logic, is defined.

The concept of closure operator is key in several branches of mathematics. In this paper, closure operators are extended to relational structures, more specifically to fuzzy relations in the framework of complete fuzzy lattices. The core of the work is the search for a suitable definition of (strong) fuzzy closure relation, that is, a fuzzy relation whose relation with fuzzy closure systems is one-to-one. The study of the properties of fuzzy closure systems and fuzzy relations helps narrow down this exploration until an appropriate definition is settled.

The aim of this paper is to extend closure systems from being crisp sets with certain fuzzy properties to proper fuzzy sets. The presentation of the paper shows a thorough discussion on the different alternatives that could be taken to define the desired fuzzy closure systems. These plausible alternatives are discarded if they are proven impossible to be in a bijective correspondence with closure operators. Finally, a definition of fuzzy closure system is established and a one-to-one relation with closure operators is proved.

In this sequel to our previous article (Corderoet al., 2020) on general inference systems for reasoning with if–then dependencies, we study transformations of if–then rules to semantically equivalent collections of if–then rules suitable to solve several problems related to reasoning with data dependencies. We work in a framework of general lattice-based if–then rules whose semantics is parameterized by systems of isotone Galois connections. This framework allows us to obtain theoretical insight as well as algorithms on a general level and observe their special cases by choosing various types of parameterizations. This way, we study methods for automated reasoning with different types of if–then rules in a single framework that covers existing as well as novel types of rules. Our approach supports a large family of if–then rules, including fuzzy if–then rules with various types of semantics. The main results in this article include new observations on the syntactic inference of if–then rules, complete collections of rules, reduced normal forms of collections of rules, and automated reasoning methods. We demonstrate the generality of the framework and the results by examples of their particular cases focusing on fuzzy if–then rules.

Two alternative definitions of closure system in complete fuzzy lattices are introduced, first as a crisp set and then as a fuzzy one. It is valuated in a complete Heyting algebra and follows the classical definition on complete lattices. The classical bijection between closure systems and fuzzy closure operators is preserved. Then, the notion is compared with the most used definition given by Bělohlávek on the fuzzy powerset lattice.

In propositional temporal logic, the combination of the connectives "tomorrow" and "always in the future" require the use of induction tools. In this paper, we present a classification of inductive schemes for propositional linear temporal logic that allows the detection of loops in decision procedures. In the design of automatic theorem provers, these schemes are responsible for the searching of efficient solutions for the detection and management of loops. We study which of these schemes have a good behavior in order to give a set of reduction rules that allow us to compute these schemes efficiently and, therefore, be able to eliminate these loops. These reduction laws can be applied previously and during the execution of any automatic theorem prover. All the reductions introduced in this paper can be considered a part of the process for obtaining a normal form of a given formula.

We continue the study of (isotone) Galois connections, also called adjunctions, in the framework of fuzzy preordered structures, which generalize fuzzy preposets by considering underlying fuzzy equivalence relations. Specifically, we present necessary and sufficient conditions so that, given a mapping f : A → B from a fuzzy preordered structure A = 〈A, ≈A,ρA〉 into a fuzzy structure 〈B, ≈B〉, it is possible to construct a fuzzy relation ρBthat induces a suitable fuzzy preorder structure on B and such that there exists a mapping g : B → A such that the pair (f, g) constitutes an Galois connection.

The problem of studying the existence of a right adjoint for a mapping defined between sets with different fuzzy structure naturally leads to the search of new notions of adjunction which fit better with the underlying structure of domain and codomain. In this work, we introduce a version of relational fuzzy adjunction between fuzzy preposets which generalizes previous approaches in that its components are fuzzy relations. We also prove that the construction behaves properly with respect to the formation of quotient with respect to the symmetric kernel relation.

In this paper, we define a sound and complete inference system for triadic implications generated from a formal triadic context , where G, M, and B are object, attribute, and condition sets, respectively, and I is a ternary relation . The inference system is expressed as a set of axioms “à la Armstrong.” The type of triadic implications we are considering in this paper is called conditional attribute implication (CAI) and has the following form: , where X and Y are subsets of M, and is a subset of B. Such implication states that Ximplies Y under all conditions in and any subset of it. Moreover, we propose a method to compute CAIs from Biedermann's implications. We also introduce an algorithm to compute the closure of an attribute set X w.r.t. a set Σ of CAIs given a set of conditions.

Landscapes exhibiting multiple secondary structures arise in natural RNA molecules that modulate gene expression, protein synthesis, and viral infection [corrected]. We report herein that high-throughput chemical experiments can isolate an RNA's multiple alternative secondary structures as they are stabilized by systematic mutagenesis (mutate-and-map, M2) and that a computational algorithm, REEFFIT, enables unbiased reconstruction of these states' structures and populations. In an in silico benchmark on non-coding RNAs with complex landscapes, M2-REEFFIT recovers 95% of RNA helices present with at least 25% population while maintaining a low false discovery rate (10%) and conservative error estimates. In experimental benchmarks, M2-REEFFIT recovers the structure landscapes of a 35-nt MedLoop hairpin, a 110-nt 16S rRNA four-way junction with an excited state, a 25-nt bistable hairpin, and a 112-nt three-state adenine riboswitch with its expression platform, molecules whose characterization previously required expert mutational analysis and specialized NMR or chemical mapping experiments. With this validation, M2-REEFFIT enabled tests of whether artificial RNA sequences might exhibit complex landscapes in the absence of explicit design. An artificial flavin mononucleotide riboswitch and a randomly generated RNA sequence are found to interconvert between three or more states, including structures for which there was no design, but that could be stabilized through mutations. These results highlight the likely pervasiveness of rich landscapes with multiple secondary structures in both natural and artificial RNAs and demonstrate an automated chemical/computational route for their empirical characterization.

SUMMARY: We have established an RNA mapping database (RMDB) to enable structural, thermodynamic and kinetic comparisons across single-nucleotide-resolution RNA structure mapping experiments. The volume of structure mapping data has greatly increased since the development of high-throughput sequencing techniques, accelerated software pipelines and large-scale mutagenesis. For scientists wishing to infer relationships between RNA sequence/structure and these mapping data, there is a need for a database that is curated, tagged with error estimates and interfaced with tools for sharing, visualization, search and meta-analysis. Through its on-line front-end, the RMDB allows users to explore single-nucleotide-resolution mapping data in heat-map, bar-graph and colored secondary structure graphics; to leverage these data to generate secondary structure hypotheses; and to download the data in standardized and computer-friendly files, including the RDAT and community-consensus SNRNASM formats. At the time of writing, the database houses 53 entries, describing more than 2848 experiments of 1098 RNA constructs in several solution conditions and is growing rapidly. AVAILABILITY: Freely available on the web at http://rmdb.stanford.edu. CONTACT: [email protected]. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics Online.