Free Heyting algebra endomorphisms: Ruitenburg’s Theorem and beyond

Ruitenburg\u2019s Theorem says that every endomorphism f of a finitely generated free Heyting algebra is ulti- mately periodic if f fixes all the generators but one. More precisely, there is N 65 0 such that f^N+2 = f^N , thus the period equals 2. We give a semantic proof of this theorem, using duality techniques and bounded bisimulation ranks. By the same techniques, we tackle investigation of arbitrary endomorphisms of free algebras. We show that they are not, in general, ultimately periodic. Yet, when they are (e.g. in the case of locally finite subvarieties), the period can be explicitly bounded as function of the cardinality of the set of generators

Quantifier-Free Interpolation of a Theory of Arrays

The use of interpolants in model checking is becoming an enabling technology to allow fast and robust verification of hardware and software. The application of encodings based on the theory of arrays, however, is limited by the impossibility of deriving quantifier- free interpolants in general. In this paper, we show that it is possible to obtain quantifier-free interpolants for a Skolemized version of the extensional theory of arrays. We prove this in two ways: (1) non-constructively, by using the model theoretic notion of amalgamation, which is known to be equivalent to admit quantifier-free interpolation for universal theories; and (2) constructively, by designing an interpolating procedure, based on solving equations between array updates. (Interestingly, rewriting techniques are used in the key steps of the solver and its proof of correctness.) To the best of our knowledge, this is the first successful attempt of computing quantifier- free interpolants for a variant of the theory of arrays with extensionality

Fixed-point elimination in the intuitionistic propositional calculus

It is a consequence of existing literature that least and greatest fixed-points of monotone polynomials on Heyting algebras-that is, the algebraic models of the Intuitionistic Propositional Calculus-always exist, even when these algebras are not complete as lattices. The reason is that these extremal fixed-points are definable by formulas of the IPC. Consequently, the $\mu$-calculus based on intuitionistic logic is trivial, every $\mu$-formula being equivalent to a fixed-point free formula. We give in this paper an axiomatization of least and greatest fixed-points of formulas, and an algorithm to compute a fixed-point free formula equivalent to a given $\mu$-formula. The axiomatization of the greatest fixed-point is simple. The axiomatization of the least fixed-point is more complex, in particular every monotone formula converges to its least fixed-point by Kleene's iteration in a finite number of steps, but there is no uniform upper bound on the number of iterations. We extract, out of the algorithm, upper bounds for such n, depending on the size of the formula. For some formulas, we show that these upper bounds are polynomial and optimal

Admissibility of Π₂-inference rules: Interpolation, model completion, and contact algebras

We devise three strategies for recognizing admissibility of non-standard inference rules via interpolation, uniform interpolation, and model completions. We apply our machinery to the case of symmetric implication calculus S2IC, where we also supply a finite axiomatization of the model completion of its algebraic counterpart, via the equivalent theory of contact algebras. Using this result we obtain a finite basis for admissible Π2-rules

Zinc increases the effects of essential amino acids-whey protein supplements in frail elderly

Abstract: Protein undernutrition is frequent in the elderly. It contributes to the development of osteoporosis, possibly via lower IGF-I. Dietary zinc can influence IGF-I production. Objectives: To determine the influence of dietary zinc addition on IGF-I and bone turnover responses to essential amino acids-whey (EAA-W) protein supplements in frail elderly. Design and setting: A daily oral protein supplement was given to hospitalized patients for 4 weeks. On a randomized, double-blind basis, patients received either an additional 30 mg/day of zinc or control. Participants: Sixty-one hospitalized elderly aged 66.7 to 105.8, with a mini-nutritional assessment score between 17 and 24 were enrolled. Measurements: Activities of daily living; dietary intakes; serum IGF-I, IGF-BP3, CrossLaps™, osteocalcin and zinc were measured before and after 1, 2 and 4 weeks of protein supplementation. Results: Serum IGF-I rapidly increased in both groups. Zinc accelerated this increase with changes of +48.2±14.3 and +22.4±4.7% (p<.05) by 1 week, in the zinc-supplemented and control groups, respectively. Zinc significantly decreased the serum bone resorption marker CrossLaps™ by already 1 week. Activities of daily living improved by +27.0±3.1 and +18.3±4.5% in zinc-supplemented and control groups, respectively. Conclusion: In the elderly, zinc supplementation accelerated the serum IGF-I response to EAA-W protein by 1 week and decreased a biochemical marker of bone resorptio

Finitely generated free Heyting algebras via Birkhoff duality and coalgebra

Algebras axiomatized entirely by rank 1 axioms are algebras for a functor and thus the free algebras can be obtained by a direct limit process. Dually, the final coalgebras can be obtained by an inverse limit process. In order to explore the limits of this method we look at Heyting algebras which have mixed rank 0-1 axiomatizations. We will see that Heyting algebras are special in that they are almost rank 1 axiomatized and can be handled by a slight variant of the rank 1 coalgebraic methods