Cyclic Proofs for First-Order Logic with Inductive Definitions

  • James Brotherston
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3702)


We consider a cyclic approach to inductive reasoning in the setting of first-order logic with inductive definitions. We present a proof system for this language in which proofs are represented as finite, locally sound derivation trees with a “repeat function” identifying cyclic proof sections. Soundness is guaranteed by a well-foundedness condition formulated globally in terms of traces over the proof tree, following an idea due to Sprenger and Dam. However, in contrast to their work, our proof system does not require an extension of logical syntax by ordinal variables.

A fundamental question in our setting is the strength of the cyclic proof system compared to the more familiar use of a non-cyclic proof system using explicit induction rules. We show that the cyclic proof system subsumes the use of explicit induction rules. In addition, we provide machinery for manipulating and analysing the structure of cyclic proofs, based primarily on viewing them as generating regular infinite trees, and also formulate a finitary trace condition sufficient (but not necessary) for soundness, that is computationally and combinatorially simpler than the general trace condition.


Proof System Predicate Symbol Trace Condition Sequent Calculus Induction Rule 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • James Brotherston
    • 1
  1. 1.Laboratory for Foundations of Computer Science, Division of InformaticsUniversity of EdinburghEdinburgh, ScotlandUK

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