Circuit Complexity, Kolmogorov Complexity, and Prospects for Lower Bounds. DCFS 2008. Today’s Goal:. To raise awareness of the tight connection between circuit complexity and Kolmogorov complexity. And to show that this is useful.
Circuit Complexity, Kolmogorov Complexity, and Prospects for Lower Bounds
We’re interested in NC1 (for instance) not because we want to build formulae for these functions…
… but because we want to know if the blocks of this partition are distinct.
These classes are real.
We’ll focus on questions such as:
Is BFE in TC0?
Is BFE in AC0?
This is known [IPS’97]
This implies TC0≠ NC1 [A, Koucky]
A subformula near the root
Subformulae near inputs
d levels of oracle gates
d2 levels of oracle gates
After log log rounds,
the depth is logO(1)n
d3 levels of oracle gates
Connections between Kolmogorov Complexity and Circuit Complexity might be relevant to the question of whether NEXP is contained in (non-uniform) TC0 (depth 3).
Similar techniques show:
What else happens in such a collapse?