A Finite Model Property for Gödel Modal Logics

A new semantics with the finite model property is provided and used to establish decidability for Gödel modal logics based on (crisp or fuzzy) Kripke frames combined locally with Gödel logic. A similar methodology is also used to establish decidability, and indeed co-NP-completeness for a Gödel S5 logic that coincides with the one-variable fragment of first-order Gödel logic.

Experimental Validation of Finite Element Analysis of Human Vertebral Collapse Under Large Compressive Strains

Osteoporosis-related vertebral fractures represent a major health problem in elderly populations. Such fractures can often only be diagnosed after a substantial deformation history of the vertebral body. Therefore, it remains a challenge for clinicians to distinguish between stable and progressive potentially harmful fractures. Accordingly, novel criteria for selection of the appropriate conservative or surgical treatment are urgently needed. Computer tomography-based finite element analysis is an increasingly accepted method to predict the quasi-static vertebral strength and to follow up this small strain property longitudinally in time. A recent development in constitutive modeling allows us to simulate strain localization and densification in trabecular bone under large compressive strains without mesh dependence. The aim of this work was to validate this recently developed constitutive model of trabecular bone for the prediction of strain localization and densification in the human vertebral body subjected to large compressive deformation. A custom-made stepwise loading device mounted in a high resolution peripheral computer tomography system was used to describe the progressive collapse of 13 human vertebrae under axial compression. Continuum finite element analyses of the 13 compression tests were realized and the zones of high volumetric strain were compared with the experiments. A fair qualitative correspondence of the strain localization zone between the experiment and finite element analysis was achieved in 9 out of 13 tests and significant correlations of the volumetric strains were obtained throughout the range of applied axial compression. Interestingly, the stepwise propagating localization zones in trabecular bone converged to the buckling locations in the cortical shell. While the adopted continuum finite element approach still suffers from several limitations, these encouraging preliminary results towardsthe prediction of extended vertebral collapse may help in assessing fracture stability in future work.

A Hennessy-Milner Property for Many-Valued Modal Logics

A Hennessy-Milner property, relating modal equivalence and bisimulations, is defined for many-valued modal logics that combine a local semantics based on a complete MTL-chain (a linearly ordered commutative integral residuated lattice) with crisp Kripke frames. A necessary and sufficient algebraic condition is then provided for the class of image-finite models of these logics to admit the Hennessy-Milner property. Complete characterizations are obtained in the case of many-valued modal logics based on BL-chains (divisible MTL-chains) that are finite or have universe [0,1], including crisp Lukasiewicz, Gödel, and product modal logics.