An Ultimate Tradeoff in
Propositional Proof Complexity
Alexander A. Razborov
University of Chicago
Steklov Mathematical Institute
Toyota Technological Institute@Chicago
Dagstuhl workshop “Theory and Practice of SAT Solvers”, April 23, 2015
Overview of the talk
1. Ultimate tradeoffs? Hmm… what the heck is
that?
2. Width vs. size in resolution, a brief survey and
the statement of our result.
3. A few words about the proof.
4. Conclusion and open problems.
Ordinary and ultimate tradeoffs
• History of tradeoff results goes back (at least) to
the 70s.
• Reflect our inherent inability to achieve two
conflicting tasks at once.
• Originally: time-space. Now: literally anything.
T a task. PT a set of protocols achieving this task.
μ and ν – two complexity measures on PT.
Examples:
1. Algorithms: μ = SPACE and ν = TIME.
2. Resolution: μ = WIDTH and ν = SIZE.
μ
μmin
μmax
T ϵ Tn , the class of all problems of size n. νcr(n) –
the “obvious” upper bound (2n in our examples).
Ultimate tradeoff
νcr(n)
μ
μmin
μmax
Resolution: width and size
Selected results on simulation and separation
between these measures.
Width vs. general (DAG-like) size
While in the opposite direction…
Width vs. tree-like size
The opposite direction: small
width refutations imply…
Translation: even if your (say) CDCL proof
produces only clauses of small width, it
does not guarantee they can be eliminated
without significantly increasing the size of
the proof.
Our contribution: ultimate tradeoff
between width and tree-like size
Translation: even if your (say) CDCL proof
produced only clauses of small width, it
does not guarantee they can be eliminated
without significantly increasing the size of
the proof.
Our addition: it is even much (exponentially)
worse idea to go for a myopic DPLL
algorithm.
Construction and proof method
Buzz words: convert separation into ultimate tradeoff
using hardness compression.
Conclusion and Open Problems
Ultimate tradeoffs (and friends) in
the literature
Information Complexity
CC(fn)
IC(fn)
Information Compression
Open Problems
More ultimate tradeoffs?
Thank you