Typing weak MSOL properties

International audienceWe consider non-interpreted functional programs: the result of the execution of a program is its normal form, that can be seen as the tree of calls to built-in operations. Weak monadic second-order logic (wMSO) is well suited to express properties of such trees. This is an extension of first order logic with quantification over finite sets. Many behavioral properties of programs can be expressed in wMSO. We use the simply typed lambda calculus with the fixpoint operator, $\lambda Y$-calculus, as an abstraction of functional programs that faithfully represents the higher-order control flow. We give a type system for ensuring that the result of the execution of a $\lambda Y$-program satisfies a given wMSO property. The type system is an extension of a standard intersection type system with both: the least-fixpoint rule, and a restricted version of the greatest-fixpoint rule. In order to prove soundness and completeness of the system we construct a denotational semantics of $\lambda Y$-calculus that is capable of computing properties expressed in wMSO. The model presents many symmetries reflecting dualities in the logic and has also other applications on its own. The type system is obtained from the model following the domain in logical form approach

Lambda-calculus and formal language theory

Formal and symbolic approaches have offered computer science many application fields. The rich and fruitful connection between logic, automata and algebra is one such approach. It has been used to model natural languages as well as in program verification. In the mathematics of language it is able to model phenomena ranging from syntax to phonology while in verification it gives model checking algorithms to a wide family of programs. This thesis extends this approach to simply typed lambda-calculus by providing a natural extension of recognizability to programs that are representable by simply typed terms. This notion is then applied to both the mathematics of language and program verification. In the case of the mathematics of language, it is used to generalize parsing algorithms and to propose high-level methods to describe languages. Concerning program verification, it is used to describe methods for verifying the behavioral properties of higher-order programs. In both cases, the link that is drawn between finite state methods and denotational semantics provide the means to mix powerful tools coming from the two worlds

Search for the sub-stellar lithium depletion boundary in the open star cluster Coma Berenices

We mainly aim to search for the lithium depletion boundary (LDB) among the sub-stellar population of the open star cluster Coma Berenices. We carried out a search for brown dwarf (BD) candidates using colour-magnitude diagrams combining optical and infrared photometry from the latest public releases of the following large-scale surveys: UKIRT/UKIDSS, Pan-STARRS, SDSS, and AllWISE. We checked astrometric consistency with cluster membership using $Gaia$ DR2. A couple dozen new candidate BDs located inside the tidal radius of Coma Ber are reported, but none of these are significantly fainter and cooler than previously known members. A search for Li in three new and five previously known BD candidate cluster members was performed via spectroscopic observations using the OSIRIS instrument at the 10.4-m GTC. No LiI resonance doublet at 6707.8 A was detected in any of eight Coma Ber targets in the magnitude range J=15--19 and G=20--23 observed with the GTC. Spectral types and radial velocities were derived from the GTC spectra. These values confirm the cluster membership of four L2--L2.5 dwarfs, two of which are new in the literature. The large Li depletion factors found among the four bona fide BD members in Coma Ber implies that the LDB must be located at spectral type later than L2.5 in this cluster. Using the latest evolutionary models for BDs, a lower limit of 550 Myr on the cluster age is set. This constraint has been combined with other dating methods to obtain an updated age estimate of 780$\pm$230 Myr for the Coma Ber open cluster. Identification of significantly cooler sub-stellar cluster members in Coma Ber awaits the advent of the Euclid wide survey, which should reach a depth of about J=23; this superb sensitivity will make it possible to determine the precise location of the sub-stellar LDB in this cluster and to carry out a complete census of its sub-stellar population.Comment: 10 pages, 7 figures, accepted for publication in A&

Matrix Graph Grammars

This book objective is to develop an algebraization of graph grammars. Equivalently, we study graph dynamics. From the point of view of a computer scientist, graph grammars are a natural generalization of Chomsky grammars for which a purely algebraic approach does not exist up to now. A Chomsky (or string) grammar is, roughly speaking, a precise description of a formal language (which in essence is a set of strings). On a more discrete mathematical style, it can be said that graph grammars -- Matrix Graph Grammars in particular -- study dynamics of graphs. Ideally, this algebraization would enforce our understanding of grammars in general, providing new analysis techniques and generalizations of concepts, problems and results known so far.Comment: 321 pages, 75 figures. This book has is publisehd by VDM verlag, ISBN 978-363921255

On Word and Frontier Languages of Unsafe Higher-Order Grammars

Higher-order grammars are an extension of regular and context-free grammars, where nonterminals may take parameters. They have been extensively studied in 1980\u27s, and restudied recently in the context of model checking and program verification. We show that the class of unsafe order-(n+1) word languages coincides with the class of frontier languages of unsafe order-n tree languages. We use intersection types for transforming an order-(n+1) word grammar to a corresponding order-n tree grammar. The result has been proved for safe languages by Damm in 1982, but it has been open for unsafe languages, to our knowledge. Various known results on higher-order grammars can be obtained as almost immediate corollaries of our result

Sizing Up the Stars

For the main part of this dissertation, I have executed a survey of nearby, main sequence A, F, and G-type stars with the CHARA Array, successfully measuring the angular diameters of forty-four stars to better than 4% accuracy. The results of these observations also yield empirical determinations of stellar linear radii and effective temperatures for the stars observed. In addition, these CHARA-determined temperatures, radii, and luminosities are fit to Yonsei-Yale isochrones to constrain the masses and ages of the stars. These quantities are compared to the results found in Allende Prieto & Lambert (1999), Holmberg et al. (2007), and Takeda (2007), who indirectly determine these same properties by fitting models to observed photometry. I find that for most cases, the models underestimate the radius of the star by ~12%, while in turn they overestimate the effective temperature by ~ 1.5 - 4%, when compared to my directly measured values, with no apparent correlation to the star\u27s metallicity or color index. These overestimated temperatures and underestimated radii in these works appear to cause an additional offset in the star\u27s surface gravity measurements, which consequently yield higher masses and younger ages, in particular for stars with masses greater than ~ 1.3 M_sol. Alternatively, these quantities I measure are also compared to direct measurements from a large sample of eclipsing binary stars in Andersen (1991), and excellent agreement is seen within both data sets. Finally, a multi-parameter solution is found to fit color-temperature-metallicity values of the stars in this sample to provide a new calibration of the effective temperature scale for these types of stars. Published work in the field of stellar interferometry and optical spectroscopy of early-type stars are presented in Appendix D and E, respectively

A model for divergence insensitive properties of lambdaY-terms

A term of a simply typed λ-calculus with fixpoints can be considered as an abstraction of a higher-order functional program. The result of the computation of a term is its Böhm tree. Given a tree automaton describing a property of Böhm trees, we are interested in constructing a model recognizing the property, in a sense that the value of a term determines if its Böhm tree satisfies the property. We show how to construct models recognizing properties expressed by parity automata that cannot detect divergence. We call them Ω-blind parity automata, as the symbol Ω is used in Böhm trees to represent divergence; an automaton is Ω-blind when it has to accept Ω from every state. The models we construct resemble standard Scott models of latices of monotone functions, but application needs to be modified and the the fixpoint operator should be interpreted as a particular non-extremal fixpoint in a lattice