Extending fairness expressibility of ECTL+: a tree-style one-pass tableau approach

Temporal logic has become essential for various areas in computer science, most notably for the specification and verification of hardware and software systems. For the specification purposes rich temporal languages are required that, in particular, can express fairness constraints. For linear-time logics which deal with fairness in the linear-time setting, one-pass and two-pass tableau methods have been developed. In the repository of the CTL-type branching-time setting, the well-known logics ECTL and ECTL^+ were developed to explicitly deal with fairness. However, due to the syntactical restrictions, these logics can only express restricted versions of fairness. The logic CTL^*, often considered as "the full branching-time logic" overcomes these restrictions on expressing fairness. However, this logic itself, is extremely challenging for the application of verification techniques, and the tableau technique, in particular. For example, there is no one-pass tableau construction for this logic, while it is known that one-pass tableau has an additional benefit enabling the formulation of dual sequent calculi that are often treated as more "natural" being more friendly for human understanding. Based on these two considerations, the following problem arises - are there logics that have richer expressiveness than ECTL^+ yet "simpler" than CTL^* for which a one-pass tableau can be developed? In this paper we give a solution to this problem. We present a tree-style one-pass tableau for a sub-logic of CTL^* that we call ECTL^#, which is more expressive than ECTL^+ allowing the formulation of a new range of fairness constraints with "until" operator. The presentation of the tableau construction is accompanied by an algorithm for constructing a systematic tableau, for any given input of admissible branching-time formulae. We prove the termination, soundness and completeness of the method. As tree-shaped one-pass tableaux are well suited for the automation and are amenable for the implementation and for the formulation of sequent calculi, our results also open a prospect of relevant developments of the automation and implementation of the tableau method for ECTL^#, and of a dual sequent calculi

Branching-time logic ECTL# and its tree-style one-pass tableau: Extending fairness expressibility of ECTL+

Temporal logic has become essential for various areas in computer science, most notably for the specification and verification of hardware and software systems. For the specification purposes rich temporal languages are required that, in particular, can express fairness constraints. For linear-time logics which deal with fairness in the linear-time setting, one-pass and two-pass tableau methods have been developed. In the repository of the CTL-type branching-time setting, the well-known logics ECTL and ECTL+ were developed to explicitly deal with fairness. However, due to the syntactical restrictions, these logics can only express restricted versions of fairness. The logic CTL⋆, often considered as ‘the full branching-time logic’ overcomes these restrictions on expressing fairness. However, CTL⋆ is extremely challenging for the application of verification techniques, and the tableau technique, in particular. For example, there is no one-pass tableau construction for CTL⋆, while one-pass tableau has an additional benefit enabling the formulation of dual sequent calculi that are often treated as more ‘natural’ being more friendly for human understanding. These two considerations lead to the following problem - are there logics that have richer expressiveness than ECTL+, allowing the formulation of a new range of fairness constraints with ‘until’ operator, yet ‘simpler’ than CTL⋆, and for which a one-pass tableau can be developed? Here we give a positive answer to this question, introducing a sub-logic of CTL⋆ called ECTL#, its tree-style one-pass tableau, and an algorithm for obtaining a systematic tableau, for any given admissible branching-time formulae. We prove the termination, soundness and completeness of the method. As tree-shaped one-pass tableaux are well suited for the automation and are amenable for the implementation and for the formulation of sequent calculi. Our results also open a prospect of relevant developments of the automation and implementation of the tableau method for ECTL#, and of a dual sequent calculi

One-pass Context-based Tableaux Systems for CTL and ECTL

When building tableau for temporal logic formulae, applying a two-pass construction, we first check the validity of the given tableaux input by creating a tableau graph, and then, in the second `pass', we check if all the eventualities are satisfied. In one-pass tableaux checking the validity of the input does not require these auxiliary constructions. This paper continues the development of one-pass tableau method for temporal logics introducing tree-style one-pass tableau systems for Computation Tree Logic (CTL) and shows how this can be extended to capture Extended CTL (ECTL). A distinctive feature here is the utilisation, for the core tableau construction, of the concept of a context of an eventuality which forces its earliest fulfilment. Relevant algorithms for obtaining a systematic tableau for these branching-time logics are also defined. We prove the soundness and completeness of the method. With these developments of a tree-shaped one-pass tableau for CTL and ECTL, we have formalisms which are well suited for the automation and are amenable for the implementation, and for the formulation of dual sequent calculi. This brings us one step closer to the application of one pass context based tableaux in certified model checking for a variety of CTL-type branching-time logics

Forming factors and properties of soils developed over limestone in Galicia

[Abstract] We describe the forming factors, properties, typology and distribution of soils developed over limestone in Galicia. According to the degree of development, three main tipes are distinguished. AR soils in high-erosion localities are decarbonated. AC soils formed by colluvial deposits of limestone material on, hillslopes have high carbonate levels, partly due to the greater solubility of unconsolidated material. ABR soils are found in relatively stable hillslope positions, in flat areas and cracks of rocks. Some are moderately well developed, with Bw horizons that have high carbonate content. Others are highly developed, with thick, totally decarbonated Bt horizons sometimes with low base saturation. The properties of all these soils depend on the degree of decarbonation

Caracterización de suelos con horizonte mólico formado sobre calizas en clima templado húmedo (Galicia, NW España)

[Abstract] Soils with a mollic horizon lying directly over limestone were characterized physicochemically, mineralogically and micromorphologically. The development of these soils involves dissolution of limestone and leaching of the released calcium carbonate under favourable climatic and topographic conditions, the incorporation of insoluble residues (chiefly by inheritance rather than transformation), and the action of efficient humification mechanism. They are classified as Lithic Haprendolls by Soil Taxonomy (1998) and as Rendzic Leptosols by the FAO classification (1998).

Alteración de los monumentos graníticos en la provincia de La Coruña: datos preliminares

[Resumen] Se estudian una serie de edificios monumentales de la provincia de la Coruña construidos con materiales graniticos distribuidos por todo su territorio, tanto en el interior como en la zona costera. El objeto del trabajo es el de evaluar el estado de alteración de dichos edificios y conocer las principales causas y mecanismos que la producen Se ha encontrado que presentan diferentes formas de alteración según la parte del edificio que se considere y el tamaño de grano del material.[Résumé] On á etudié une serie de batiments monumentales á la province de la Coruña construits avec des materiels granitiques distribués dans tout le territoire tant al'interieur cornme á la zone de la cote. L'objet du travail c'est l'evaluation de l'etat d'altération de ces batiments et connaitre les principaux motifs et mechanismes qui la produissent. On á trouvé qu' ils pr~sent differents formes d'alteration selon la part du batiment qu' on considere et la grandeur des grains du materiel

Distribution and Origin of Iron Oxides in Soils over Limestone

[Resumen] Se pretende conocer la distribuci6n y el origen de los compuestos de hierro en suelos formados sobre calizas en Galicia. Predominan las formas minerales de hierro, encontrando en la fracción arena magnetita, goethita y/o hematites y excepcionalmente ilmenita, que se consideran heredadas del material de partida. En lámina delgada se observaron goethita y hematites como granos aislados, constituyendo nódulos e integrados en fragmentos de caliza. En la arcilla, trazas de goethita están presentes en casi todos los perfiles; s610 en uno de ellos la hematites es el único 6xido de hierro. La goethita puede haberse formado tanto por neoformaci6n como por microdivisi6n a partir de las fracciones gruesas, siendo este último fen6meno el que ha originado la hematites de la fracci6n arcilla. En dos perfiles la goethita va acompañada de magnetita y se discute la posible existencía y formaci6n de maghemita.[Abstract] Iron distribution in soils formed over limestone of Galicia is studied an the origin of the different iron forms is established. Organic iron is very scarce and mineral iron components dominate. Magnetite, goethite and / or hematite, less cornmonly ilmenite, are found in the sand fraction; they are considered as interited from the parent material. In polished sections goethite an hematite appear as single grains, nodules and il1corporated into limestone fragments. In the clay fraction, traces of goethite are present in aH except one of the profiles where only traces of hematite are indentified. Neoformation and microdivision from coarse fractions can act in the genesis of goethite; only microdivision is considered as the origin of hematite. Magnetite accompanies goethite in two profiles; the possible presence of maghemite and its origin is also discusse

Composición mineralógica de los suelos del área de Mera (Las Mariñas): relación con los factores edafonegéticos

[Resumen] Se estudia la composición mineralógica de la fracción arena y arcilla de los distintos tipos de suelos existentes en el área de Mera (Las Mariñas). Durante el Terciario Y Cuaternario el marco geológico, constituido por esquistos de la Formación de Ordenes-Bergondo, ha experimentado varios procesos de edafogénesis y morfogénesis que han onigtnado una serie de depósitos,saprolitas y suelos de diferente composición mineralogica, si bien se observa una clara tendencia a la formación de una facies constituida por caolinita, cuarzo y oxi-hidróxidos de Fe. - El estudio mineralógico permite establecer las secuencias de evolución de los materiales originales en las condiciones existentes[Abstract] A study of the minera1ogica1 composition of the sand and c1ay fractions of soils in the litle area shows that during the tertiary and quaternary periods the Ordenes-Bergondo Schists of the surrounding basin have undergone processes producing a series of sediments, sapro1ites and soils of varied composition in which a facies of cao1inite, quartz and oxy-hydroxides of iron predominates. The evo1utionary sequence of the parent material under the various existing conditions has been determine