Effectiveness in RPL, with applications to continuous logic |
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Authors: | Farzad Didehvar Massoud Pourmahdian |
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Institution: | a Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran b Department of Computer Sciences, University of Toronto, Toronto, ON, Canada c School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran |
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Abstract: | In this paper, we introduce a foundation for computable model theory of rational Pavelka logic (an extension of ?ukasiewicz logic) and continuous logic, and prove effective versions of some related theorems in model theory. We show how to reduce continuous logic to rational Pavelka logic. We also define notions of computability and decidability of a model for logics with computable, but uncountable, set of truth values; we show that provability degree of a formula with respect to a linear theory is computable, and use this to carry out an effective Henkin construction. Therefore, for any effectively given consistent linear theory in continuous logic, we effectively produce its decidable model. This is the best possible, since we show that the computable model theory of continuous logic is an extension of computable model theory of classical logic. We conclude with noting that the unique separable model of a separably categorical and computably axiomatizable theory (such as that of a probability space or an Lp Banach lattice) is decidable. |
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Keywords: | primary 03D45 secondary 03B50 |
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