no. 1
Deciding whether a relation defined in Presburger logic can be defined in weaker logics
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 1, pp. 121-135.

We consider logics on ℤ and ℕ which are weaker than Presburger arithmetic and we settle the following decision problem: given a k-ary relation on ℤ and ℕ which are first order definable in Presburger arithmetic, are they definable in these weaker logics? These logics, intuitively, are obtained by considering modulo and threshold counting predicates for differences of two variables.

DOI : 10.1051/ita:2007047
Classification : 03B10, 68Q70
Mots clés : Presburger arithmetic, first order logic, decidability 
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     author = {Choffrut, Christian},
     title = {Deciding whether a relation defined in {Presburger} logic can be defined in weaker logics},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {121--135},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {1},
     year = {2008},
     doi = {10.1051/ita:2007047},
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Choffrut, Christian. Deciding whether a relation defined in Presburger logic can be defined in weaker logics. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 42 (2008) no. 1, pp. 121-135. doi : 10.1051/ita:2007047. http://archive.numdam.org/articles/10.1051/ita:2007047/

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