09 Jun 2017 at 14:00 in E2, Lindstedtsvägen 3, KTH Campus
Space in Proof Compexity
(Marc Vinyals, TCS, KTH)
Propositional proof complexity is the study of the resources that are
needed to prove formulas in propositional logic. In this thesis we are
concerned with the size and space of proofs, and in particular with the
latter.
Different approaches to reasoning are captured by corresponding proof
systems. The simplest and most well studied proof system is resolution, and
we try to get our understanding of other proof systems closer to that of
resolution.
In resolution we can prove a space lower bound just by showing that any
proof must have a large clause. We prove a similar relation between
resolution width and polynomial calculus space that lets us derive space
lower bounds, and we use it to separate degree and space.
For cutting planes we show length-space trade-offs. This is, there are
formulas that have a proof in small space and a proof in small length, but
there is no proof that can optimize both measures at the same time.
We introduce a new measure of space, cumulative space, that accounts for
the space used throughout a proof rather than only its maximum. This is
exploratory work, but we can also prove new results for the usual space
measure.
We define a new proof system that aims to capture the power of current SAT
solvers, and we show a landscape of length-space trade-offs comparable to
those in resolution.
To prove these results we build and use tools from other areas of
computational complexity. One area is pebble games, very simple
computational models that are useful for modelling space. In addition to
results with applications to proof complexity, we show that pebble game
cost is PSPACE-hard to approximate.
Another area is communication complexity, the study of the amount of
communication that is needed to solve a problem when its description is
shared by multiple parties. We prove a simulation theorem that relates
the query complexity of a function with the communication complexity of a
composed function.
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.