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 Tr. Mat. Inst. Steklova, 2003, Volume 242, Pages 77–97 (Mi tm406)

Variants of Realizability for Propositional Formulas and the Logic of Weak Excluded Middle

N. K. Vereshchagina, D. P. Skvortsovb, E. Z. Skvortsovac, A. V. Chernovca

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b All-Russian Institute for Scientific and Technical Information of Russian Academy of Sciences
c M. V. Lomonosov Moscow State University

Abstract: It is unknown whether the logic of propositional formulas that are realizable in the sense of Kleene has a finite or recursive axiomatization. In this paper, another approach to the realizability of propositional formulas is studied. This approach is based on the following informal idea: a formula is realizable if it has a “simple” realization for each substitution. More precisely, logical connectives are interpreted as operations on the sets of natural numbers, and a formula is interpreted as a combined operation; if some sets are substituted for variables, then elements of the result are called realizations. A realization (a natural number) is simple if it has low Kolmogorov complexity, and a formula is called realizable if it has at least one simple realization whatever sets are substituted. Similar definitions can be formulated in arithmetic terms. A few “realizabilities” of this kind are considered, and it is proved that all of them give the same finitely axiomatizable logic, namely, the logic of weak excluded middle. The proof uses characterizations of superintuitionistic logics with an intuitionistic positive fragment that was obtained in 1960s by Medvedev and Yankov.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2003, 242, 67–85

Bibliographic databases:

UDC: 510.642+510.25+517.1

Citation: N. K. Vereshchagin, D. P. Skvortsov, E. Z. Skvortsova, A. V. Chernov, “Variants of Realizability for Propositional Formulas and the Logic of Weak Excluded Middle”, Mathematical logic and algebra, Collected papers. Dedicated to the 100th birthday of academician Petr Sergeevich Novikov, Tr. Mat. Inst. Steklova, 242, Nauka, MAIK «Nauka/Inteperiodika», M., 2003, 77–97; Proc. Steklov Inst. Math., 242 (2003), 67–85

Citation in format AMSBIB
\Bibitem{VerSkvSkv03} \by N.~K.~Vereshchagin, D.~P.~Skvortsov, E.~Z.~Skvortsova, A.~V.~Chernov \paper Variants of Realizability for Propositional Formulas and the Logic of Weak Excluded Middle \inbook Mathematical logic and algebra \bookinfo Collected papers. Dedicated to the 100th birthday of academician Petr Sergeevich Novikov \serial Tr. Mat. Inst. Steklova \yr 2003 \vol 242 \pages 77--97 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm406} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2054486} \zmath{https://zbmath.org/?q=an:1079.03016} \transl \jour Proc. Steklov Inst. Math. \yr 2003 \vol 242 \pages 67--85 

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This publication is cited in the following articles:
1. A. V. Chernov, “Complexity of Sets Obtained as Values of Propositional Formulas”, Math. Notes, 75:1 (2004), 131–139
2. A. V. Chernov, “Finite problems and the logic of the weak law of excluded middle”, Math. Notes, 77:2 (2005), 263–272
3. Plisko V., “A Survey of Propositional Realizability Logic”, Bulletin of Symbolic Logic, 15:1 (2009), 1–42
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