In its many guises and variations, propositional dynamic logic (PDL) plays an important role in various areas of computer science such as databases, artificial intelligence, and computer linguistics. One relevant and powerful variation is ICPDL, the extension of PDL with intersection and converse. Although ICPDL has several interesting applications, its computational properties have never been investigated. In this paper, we prove that ICPDL is decidable by developing a translation to the monadic second order logic of infinite trees. Our result has applications in information logic, description logic, and epistemic logic. In particular, we solve a long-standing open problem in information logic. Another virtue of our approach is that it provides a decidability proof that is more transparent than existing ones for PDL with intersection (but without converse).
The raw image sequences and reconstructions of original data supporting this research are downloadable from the Kings College London Research data server ( -720). Raw data from the Localisation Microscopy Challenge are also publicly available from Source Data is available as a source data file. Source data are provided with this paper.
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19. The root reason for human dignity lies in man's call to communion with God. From the very circumstance of his origin man is already invited to converse with God. For man would not exist were he not created by Gods love and constantly preserved by it; and he cannot live fully according to truth unless he freely acknowledges that love and devotes himself to His Creator. Still, many of our contemporaries have never recognized this intimate and vital link with God, or have explicitly rejected it. Thus atheism must be accounted among the most serious problems of this age, and is deserving of closer examination.
Christ, to be sure, gave His Church no proper mission in the political, economic or social order. The purpose which He set before her is a religious one.(11) But out of this religious mission itself come a function, a light and an energy which can serve to structure and consolidate the human community according to the divine law. As a matter of fact, when circumstances of time and place produce the need, she can and indeed should initiate activities on behalf of all men, especially those designed for the needy, such as the works of mercy and similar undertakings.
Thus the family, in which the various generations come together and help one another grow wiser and harmonize personal rights with the other requirements of social life, is the foundation of society. All those, therefore, who exercise influence over communities and social groups should work efficiently for the welfare of marriage and the family. Public authority should regard it as a sacred duty to recognize, protect and promote their authentic nature, to shield public morality and to favor the prosperity of home life. The right of parents to beget and educate their children in the bosom of the family must be safeguarded. Children too who unhappily lack the blessing of a family should be protected by prudent legislation and various undertakings and assisted by the help they need.
65. Economic development must remain under man's determination and must not be left to the judgment of a few men or groups possessing too much economic power or of the political community alone or of certain more powerful nations. It is necessary, on the contrary, that at every level the largest possible number of people and, when it is a question of international relations, all nations have an active share in directing that development. There is need as well of the coordination and fitting and harmonious combination of the spontaneous efforts of individuals and of free groups with the undertakings of public authorities. 1e1e36bf2d