Simple Derivation Systems for Proving Sufficient Completeness of Non-Terminating Term Rewriting Systems

Authors Kentaro Kikuchi, Takahito Aoto

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Author Details

Kentaro Kikuchi
  • Tohoku University, Sendai, Japan
Takahito Aoto
  • Niigata University, Niigata, Japan


We are grateful to the anonymous reviewers for valuable comments.

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Kentaro Kikuchi and Takahito Aoto. Simple Derivation Systems for Proving Sufficient Completeness of Non-Terminating Term Rewriting Systems. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 49:1-49:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


A term rewriting system (TRS) is said to be sufficiently complete when each function yields some value for any input. Proof methods for sufficient completeness of terminating TRSs have been well studied. In this paper, we introduce a simple derivation system for proving sufficient completeness of possibly non-terminating TRSs. The derivation system consists of rules to manipulate a set of guarded terms, and sufficient completeness of a TRS holds if there exists a successful derivation for each function symbol. We also show that variations of the derivation system are useful for proving special cases of local sufficient completeness of TRSs, which is a generalised notion of sufficient completeness.

Subject Classification

ACM Subject Classification
  • Theory of computation → Rewrite systems
  • Theory of computation → Equational logic and rewriting
  • Term rewriting
  • Sufficient completeness
  • Local sufficient completeness
  • Non-termination
  • Derivation rule
  • Well-founded induction schema


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