# Complexity Theoretic Aspects of Property Testing

## Oded Goldreich

Notes for the MFO meeting on Complexity Theory.
In the meeting I only managed to talk of the first work.

For the purpose of this meeting, I would say that "Property Testing is
a PCP of Proximity without the proof part".

Normally, I'd say that
property Testing is concerned with approximate decisions, where the
task is distinguishing between objects having a predetermined property
and objects that are ``far'' from having this property.
Specifically, objects are modeled by functions,
and distance between functions is measured as the fraction
of the domain on which the functions differ.
We consider (randomized) algorithms that may query the function
at arguments of their choice,
and seek algorithms of very low complexity
(i.e., much lower than the size of the function's domain).

#### Relation between adaptive and non-adaptive query complexity
of graph properties in the adjacency matrix model

For any fixed property $Pi$, let $q$ denote the query complexity
of (general, i.e., adaptive) testing of $Pi$, and $Q$ dente
the corresponding non-adaptive query complexity
(i.e., which refers to non-adaptove testers of $Pi$).
Following is a list of known and conjectured results,
where $tildeOmega$ and $tildeTheta$ denote bounds with a slackness
of a polylogarithmic factor.
- THM: For any graph property in the adjacency matrix model,
it holds that $Q=O(q^2)$. (See GT)
- THM [GR]: There exist graph properties in the adjacency matrix model
such that $Q=tildeTheta(q)$.

Actually, $Q=O(q)$ and even $Q=q$ are known too.
- THM [GR]: There exists a graph property in the adjacency matrix model
such that $Q=tildeTheta(q^{4/3})$.
- THM [GR]: There exists a graph property in the adjacency matrix model
such that $Q=tildeOmega(q^{3/2})$.
- CONJ [GR]: For every integer $t>2$,
there exists a graph property in the adjacency matrix model
such that $Q=tildeTheta(q^{2-(2/t)})$.

This conjecture is supported by a theorem that establish
the same relation relation for a promise problem.

All existential results are proved using natural graph properties.
#### Hierarchy Theorems for Property Testing

Such results are proved for three central models of property testing:
the general model of generic function, the model of bounded-degree
graph properties, and the model of dense graph properties
(in the adjacency matrix model).
From a technical perspective,
the treatment of the latter is most interesting,
since it raises and resolves various natural questions
regarding graph blow-up. For details, see
my paper with M. Krivelevich, I. Newman and E. Rozenberg.
See also
text
for the official report.

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