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We show that the state reduction problem for fuzzy automata is related to the problem of finding a solution to a particular system of fuzzy relation equations in the set of all fuzzy equivalences on its set of states. This system may consist of infinitely many equations, and finding its non-trivial solutions may be a very difficult task. For that reason we aim our attention to some instances of this system which consist of finitely many equations and are easier to solve. First, we study right invariant fuzzy equivalences, and their duals, the left invariant ones. We prove that each fuzzy automaton possesses the greatest right (resp. left) invariant fuzzy equivalence, which provides the best reduction by means of fuzzy equivalences of this type, and we give an effective procedure for computing this fuzzy equivalence, which works if the underlying structure of truth values is a locally finite residuated lattice. Moreover, we show that even better reductions can be achieved alternating reductions by means of right and left invariant fuzzy equivalences. We also study strongly right and left invariant fuzzy equivalences, which give worse reductions than right and left invariant ones, but whose computing is much easier. We give an effective procedure for computing the greatest strongly right (resp. left) invariant fuzzy equivalence, which is applicable to fuzzy automata over an arbitrary complete residuated lattice.

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Background: Several registries of idiopathic pulmonary fibrosis (IPF) have been established to better understand its natural history, though their size and duration of follow-up are limited. Here, we describe the large European MultiPartner IPF Registry (EMPIRE) and validate predictors of long-term survival in IPF. Methods: The multinational prospective EMPIRE registry enrolled IPF patients from 48 sites in 10 Central and Eastern European countries since 2014. Survival from IPF diagnosis until death was estimated, accounting for lefttruncation. The Cox proportional hazards regression model was used to estimate adjusted hazard ratios (HR) of death for prognostic factors, using restricted cubic splines to fit continuous factors. Results: The cohort included 1620 patients (mean age at diagnosis 67.6 years, 71% male, 63% smoking history), including 75% enrolled within 6 months of diagnosis. Median survival was 4.5 years, with 45% surviving 5 years post-diagnosis. Compared with GAP stage I, mortality was higher with GAP stages II (HR 2.9; 95% CI: 2.3-3.7) and III (HR 4.0; 95% CI: 2.8-5.7) while, with redefined cutoffs , the corresponding HRs were 2.7 (95% CI: 1.8-4.0) and 5.8 (95% CI: 4.0-8.3) respectively. Mortality was higher with concurrent pulmonary hypertension (HR 2.0; 95% CI: 1.5-2.9) and lung cancer (HR 2.6; 95% CI: 1.3-4.9). Conclusions: EMPIRE, one of the largest long-term registries of patients with IPF, provides a more accurate confirmation of prognostic factors and co-morbidities on longer term five-year mortality. It also suggests that some fine-tuning of the indices for mortality may provide a more accurate long-term prognostic profile for these patients.