3 research outputs found
Resolution and the binary encoding of combinatorial principles.
Res(s) is an extension of Resolution working on s-DNFs. We prove tight n (k) lower bounds for the size
of refutations of the binary version of the k-Clique Principle in Res(o(log log n)). Our result improves that of
Lauria, Pudlák et al. [27] who proved the lower bound for Res(1), i.e. Resolution. The exact complexity of
the (unary) k-Clique Principle in Resolution is unknown. To prove the lower bound we do not use any form of
the Switching Lemma [35], instead we apply a recursive argument specific for binary encodings. Since for the
k-Clique and other principles lower bounds in Resolution for the unary version follow from lower bounds in
Res(log n) for their binary version we start a systematic study of the complexity of proofs in Resolution-based
systems for families of contradictions given in the binary encoding.
We go on to consider the binary version of the weak Pigeonhole Principle Bin-PHPmn
for m > n. Using
the the same recursive approach we prove the new result that for any > 0, Bin-PHPmn
requires proofs of size
2n1− in Res(s) for s = o(log1/2 n). Our lower bound is almost optimal since for m 2
p
n log n there are
quasipolynomial size proofs of Bin-PHPmn
in Res(log n).
Finally we propose a general theory in which to compare the complexity of refuting the binary and unary
versions of large classes of combinatorial principles, namely those expressible as first order formulae in 2-form
and with no finite model
Resolution and the binary encoding of combinatorial principles
Res(s) is an extension of Resolution working on s-DNFs. We prove tight n (k) lower bounds for the size of refutations of the binary version of the k-Clique Principle in Res(o(log log n)). Our result improves that of Lauria, Pudlák et al. [27] who proved the lower bound for Res(1), i.e. Resolution. The exact complexity of the (unary) k-Clique Principle in Resolution is unknown. To prove the lower bound we do not use any form of the Switching Lemma [35], instead we apply a recursive argument specific for binary encodings. Since for the k-Clique and other principles lower bounds in Resolution for the unary version follow from lower bounds in Res(log n) for their binary version we start a systematic study of the complexity of proofs in Resolution-based systems for families of contradictions given in the binary encoding. We go on to consider the binary version of the weak Pigeonhole Principle Bin-PHPmn for m > n. Using the the same recursive approach we prove the new result that for any > 0, Bin-PHPmn requires proofs of size 2n1− in Res(s) for s = o(log1/2 n). Our lower bound is almost optimal since for m 2 p n log n there are quasipolynomial size proofs of Bin-PHPmn in Res(log n). Finally we propose a general theory in which to compare the complexity of refuting the binary and unary versions of large classes of combinatorial principles, namely those expressible as first order formulae in 2-form and with no finite model