Geodesic
A survey of predicate realizability logic
Informatics and Automation, Algorithmic aspects of algebra and logic, Tome 274 (2011), pp. 222-251

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The paper contains a survey of the results and methods of studying interpretations of predicate formulas based on constructive semantics of the first-order language of arithmetic and its extensions. 
@article{TRSPY_2011_274_a12,
     author = {Valerii E. Plisko},
     title = {A survey of predicate realizability logic},
     journal = {Informatics and Automation},
     pages = {222--251},
     publisher = {mathdoc},
     volume = {274},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2011_274_a12/}
}
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Valerii E. Plisko. A survey of predicate realizability logic. Informatics and Automation, Algorithmic aspects of algebra and logic, Tome 274 (2011), pp. 222-251. http://geodesic.mathdoc.fr/item/TRSPY_2011_274_a12/