In the context of formal verification in general and model checking in particular, parity games serve as a mighty vehicle: many problems are encoded as parity games, which are then solved by the seminal algorithm by Jurdzinski. In this paper we identify the essence of this workflow to be the notion of progress measure, and formalize it in general, possibly infinitary, lattice-theoretic terms. Our view on progress measures is that they are to nested/alternating fixed points what invariants are to safety/greatest fixed points, and what ranking functions are to liveness/least fixed points. That is, progress measures are combination of the latter two notions (invariant and ranking function) that have been extensively studied in the context of (program) verification.We then apply our theory of progress measures to a general model-checking framework, where systems are categorically presented as coalgebras. The framework's theoretical robustness is witnessed by a smooth transfer from the branching-time setting to the linear-time one. Although the framework can be used to derive some decision procedures for finite settings, we also expect the proposed framework to form a basis for sound proof methods for some undecidable/infinitary problems.

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We classified unsuccessful 60-year-old forests, planted mainly with hinoki cypress (Chamaecyparis obtusa), into types based on the species composition of invading trees, and determined the change of each forest type. Thematic maps of topographic factors were derived from a digital elevation model generated from LiDAR data. The study area was classified into four forest types, based on tree regression analysis of species composition and the topographic factors. Then the dominance of C. obtusa and invading trees was determined from the distribution maps of their top heights created based on tree regression analyses of them and the topographic factors. The relationship between the dominance of them and forest types was shown. Finally, changes in the stand volume of each forest type were determined. Based on the results, we suggest the following three future management policies for each forest type, taking into consideration the timber price of invading hardwood trees, the growth of C. obtusa, and the increase in stand volume: no operations, reinforcement planting, or switching to mixed conifers and hardwood forests.