Classical System of Martin-Löf’s Inductive Definitions Is Not Equivalent to Cyclic Proof System

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10203)


A cyclic proof system, called \( \mathtt{CLKID}^\omega \), gives us another way of representing inductive definitions and efficient proof search. The 2011 paper by Brotherston and Simpson showed that the provability of \( \mathtt{CLKID}^\omega \) includes the provability of Martin-Löf’s system of inductive definitions, called \( \mathtt{LKID} \), and conjectured the equivalence. Since then, the equivalence has been left an open question. This paper shows that \( \mathtt{CLKID}^\omega \) and \( \mathtt{LKID} \) are indeed not equivalent. This paper considers a statement called 2-Hydra in these two systems with the first-order language formed by 0, the successor, the natural number predicate, and a binary predicate symbol used to express 2-Hydra. This paper shows that the 2-Hydra statement is provable in \( \mathtt{CLKID}^\omega \), but the statement is not provable in \( \mathtt{LKID} \), by constructing some Henkin model where the statement is false.


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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Università di TorinoTurinItaly
  2. 2.National Institute of Informatics/SokendaiTokyoJapan

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