We study query-efficient Probabilistically Checkable
Proofs (PCPs) and linearity tests. We focus on the number
of amortized query bits. A testing algorithm uses $q$ amortized
query bits if, for some constant $k$, it reads $qk$ bits and has
error probability at most $2^{-k}$. The best known PCP construction
for NP in this respect uses 3 amortized query bits (Hastad, STOC97);
at least one amortized query bit is necessary, unless P=NP (Bellare,
Goldreich and Sudan, FOCS95). This parameter is an extremely natural
one and has applications to proving non-approximability for
constraint satisfaction problems. Furthermore, a PCP characterization
of NP with less than 2 amortized query bits would imply a separation
of the PCP model from the 2-Prover 1-Round model.

Our approach is to take an atomic verification procedure and
then iterate it several times, saving queries by recycling them
between different iterations of the atomic test.

We first apply this idea in order to develop query-efficient
linearity tests. Linearity testing is a problem closely related
to testing the Long Code and making PCP constructions. It is also a
significant combinatorial problem still lacking tight characterizations,
except for the case of three queries (Bellare et al. FOCS95).
The best known linearity test uses 3 amortized query bits (Bellare
et al., FOCS95); a different one achieves 1 amortized free bit (a
different parameter related to the Max Clique problem) but uses an
unbounded number of amortized query bits (Bellare, Goldreich and
Sudan, FOCS95). We develop a general analysis technique and a linearity
test achieving simultaneously amortized query complexity 1.5 and
amortized free bit complexity 1/2. This test answers an open question
raised by Bellare, Goldreich and Sudan.

We then show how to adapt a weaker result to the PCP setting, and we
obtain a PCP for NP that makes 5 queries and has error probability
1/4, so that its amortized query complexity is 2.5.