Hierarchy Theorems for Property Testing

Oded Goldreich, Michael Krivelevich, Ilan Newman, Eyal Rozenberg

Research output: Contribution to journalArticlepeer-review


Referring to the query complexity of property testing, we prove the existence of a rich hierarchy of corresponding complexity classes. That is, for any relevant function q, we prove the existence of properties that have testing complexity Θ(q). Such results are proven in three standard domains often considered in property testing: generic functions, adjacency predicates describing (dense) graphs, and incidence functions describing bounded-degree graphs. While in two cases, the proofs are quite straightforward, and the techniques employed in the case of the dense graph model seem significantly more involved. Specifically, problems that arise and are treated in the latter case include (1) the preservation of distances between graph under a blow-up operation and (2) the construction of monotone graph properties that have local structure.

Original languageEnglish
Pages (from-to)129-192
Number of pages64
JournalComputational Complexity
Issue number1
StatePublished - Mar 2012

Bibliographical note

Funding Information:
An extended abstract of this work appeared in the proceedings of RANDOM’09. Oded Goldreich was partially supported by an Israel Science Foundation grant (No. 1041/08). Michael Krivele-vich was partially supported by an Israel Science Foundation grant No. 1063/08, an USA-Israel BSF grant (No. 2006322), and a Pazy Memorial Award. Ilan Newman was partially supported by an Israel Science Foundation grant (No. 1011/06).


  • Property testing
  • adaptivity versus non-adaptivity
  • graph blow-up
  • graph properties
  • hierarchy theorems
  • monotone graph properties
  • one-sided versus two-sided error
  • query complexity

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Mathematics
  • Computational Theory and Mathematics
  • Computational Mathematics


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