Kolmogorov complexity and formula size lower bounds
Troy Lee
Part I: Kolmogorov Complexity
Randomness is a concept familiar to all of us in our daily lives. From
changing weather, to the flip of a coin, to the shuffle function on a
music player---random is frequently used to describe the behavior of things
around us. Despite this familiarity, or perhaps because of it, randomness
eluded a precise mathematical definition until the second half of the
twentieth century.
In the early 1960's, independently and within a few years of each
other, Solomonoff, Kolmogorov, and Chaitin all developed an elegant
way to capture when a sequence of events (viewed as a string with
letters 0 and 1) is random. They proposed the following definition:
the complexity of a string is the length of a shortest description of
that string. Here a description can be thought of as a computer
program. We will refer to this measure as the Kolmogorov complexity
of the string, and write $C(x | y)$ for the length of a shortest
program for $x$ which is allowed to make use of the advice string $y$.
While the string $01010101010101010101010101010101010101010101010101$
has 50 symbols, it has a much shorter description saying ``write 25
times the string 01''. On the other hand, a random string will have
no structure which would allow a description of length shorter than
the length of the string itself. Thus we call a string random if its
Kolmogorov complexity is at least as large as its length.
Far beyond its initial purpose, Kolmogorov complexity has become an
important tool in theoretical computer science, witnessing many
diverse applications. Almost all of the applications use at least one
of the following four fundamental theorems, which we call the `four
pillars' of Kolmogorov complexity:
o Incompressibility: There is a random string of length $n$, for
each $n$.
o Language compression: Any element from a computable set $A$ can be
given a description of size about $\log |A|$.
o Source compression: Any element $x$ in the support of a computable
probability distribution $P$ can be given a description of size about
$-\log P(x)$.}
o Symmetry of information: The information which a string $x$ contains
about a string $y$ is about the same as that which $y$ contains about
$x$. In
symbols: $C(x) - C(x | y) = C(y)- C(y | x)$.
A drawback to Kolmogorov complexity is that it is uncomputable, and
this sometimes limits its range of applicability in computational
complexity. One way to scale down the theory into the feasible domain
is to require that the program which prints the string $x$ does so in
time which is polynomial in the length of $x$. The main goal of the
first part of this thesis is to see what analogues, if any, of the
four pillars hold in this resource bounded setting.
The first pillar holds unchanged in the resource bounded setting, as
adding restrictions to a program can only make it harder to describe a
string.
Things get much more interesting beginning with the second pillar.
The most natural resource bounded analog of the second pillar is the
statement: every element $x$ in a set $A$ decidable in polynomial time
has a description of length $\log |A|$ from which $x$ can be generated
in polynomial time. This statement is unlikely to hold, however, as
it implies the polynomial hierarchy collapses. We show that going up
one step higher in the polynomial hierarchy, however, we can find an
analog of the second pillar. That is, we are able to show every
element $x$ in a set $A$ which can be decided in nondeterministic
polynomial time can be given a description of length about $\log |A|$
from which $x$ can be generated in nondeterministic polynomial time.
The most natural analogue of the third pillar would be: any element in
the support of a polynomial time computable probability distribution
$P$ can be given a description of length about $-\log P(x)$. We show
that such a statement implies that randomized algorithms (those which
can make decisions based on the outcome of a coin flip) can be
simulated by deterministic algorithms (which cannot flip coins).
Precisely, we show this implies $\BPP \ne \EXP$. Recently, Antunes
and Fortnow were able to prove a weak converse to this statement:
under a derandomization assumption the polynomial time version of the
third pillar holds.
With these results we have a fairly complete picture of the third pillar in
the resource bounded setting.
Finally, we turn to symmetry of information which is perhaps the most
tricky of the four pillars in the resource bounded domain. To prove
symmetry of information seems to require the ability to do both
language compression and source compression. We are only able to
prove a weaker version of symmetry of information which says that the
polynomial time complexity of the pair $x,y$ is larger than the
randomized nondeterministic complexity of $x$ plus the randomized
nondeterministic complexity of $y$ given $x$. This result is tight
with respect to relativizing techniques---we give an oracle where even
the nondeterministic complexity of $x,y$ is nearly twice as large as
the nondeterministic complexity of $x$ plus the nondeterministic
complexity of $y$ given $x$.
Part II: Formula Size Lower Bounds
One of the most famous, important, and yes, most difficult problems in
complexity theory is the question of whether or not P is equal to NP.
P is the set of all problems that can be solved in time polynomial in
the length of the problem description; NP is the set of problems which
can be efficiently verified, meaning that if someone gives you the
solution to the problem you can indeed check that it is a proper
solution in polynomial time. Currently, it is widely believed that P
is not equal to NP. To prove this one must show a lower bound, that
is give an example of a problem in NP that requires more than
polynomial time to solve. A well-studied approach to doing this is
through circuit complexity, where a circuit consists of AND, OR, and
NOT gates. The current best lower bound on the size of a circuit
required to compute a function in NP is 5n. As the task of proving
larger circuit lower bounds seems extremely difficult, we look instead
at the weaker model of formula size, where a formula is simply a
circuit where every gate has exactly one outgoing wire. The current
best lower bound for formula size is $n^3$.
We give a new algebraic approach to proving formula size lower bounds.
We are not able to improve on the best $n^3$ lower bound; we are,
however, able to simulaneously generalize several techniques from the
literature and show them as part of an overarching theory. We also
give some concrete examples of functions for which our new method is
able to provably do better than previous methods. Perhaps the most
interesting consequence of our results, however, is that it gives
evidence for a connection between the formula size of a function and
its complexity in a very different model, namely its quantum query
complexity. In fact, our results can be taken as evidence for the
provocative conjecture that the square of the quantum query complexity
of a function lower bounds its formula size.