20 Jan 2017 at 14:00 in DRoom D2, Lindstedtsvägen 5
On Complexity Measures in Polynomial Calculus
(MMladen Mikša, KTH- TCS group)
Proof complexity is the study of different resources that a proof needs
in different proof systems for propositional logic. This line of inquiry
relates to the fundamental questions in theoretical computer science, as
lower bounds on proof size for an arbitrary proof system would separate P
from NP.
We study two simple proof systems: resolution and polynomial calculus. In
resolution we reason using clauses, while in polynomial calculus we use
polynomials. We study three measures of complexity of proofs: size, space,
and width/degree. Size is the number of clauses or monomials that appear
in a resolution or polynomial calculus proof, respectively. Space is the
maximum number of clauses/monomials we need to keep at each time step of
the proof. Width/degree is the size of the largest clause/monomial in a
proof.
Width is a lower bound for space in resolution. The original proof of this
claim used finite model theory. In this thesis we give a different, more
direct proof of the space-width relation. We can ask whether a similar
relation holds between space and degree in polynomial calculus. We make
some progress on this front by showing that when a formula F requires
resolution width w then the XORified version of F requires polynomial
calculus space Ω(w). We also show that space lower bounds do not imply
degree lower bounds in polynomial calculus.
Width/degree and size are also related, as strong lower bounds for
width/degree imply strong lower bounds for size. Currently, proving width
lower bounds has a well-developed machinery behind it. However, the degree
measure is much less well-understood. We provide a unified framework for
almost all previous degree lower bounds. We also prove some new degree and
size lower bounds. In addition, we explore the relation between theory
and practice by running experiments on some current state-of-the-art SAT
solvers.