 
Summary: Com S 631: Lower bounds and Separation Results
Lecture 2 Scribe: Ruchi Chaudary
1. Introduction
While we believe that SAT / P, we can not even prove that SAT can not be solved in time
n1+
. Similarly we believe that SAT requires linear space, we do not know how to show that
SAT can not be solved with logarithmic space. How about an algorithm that runs in time
n1+
and simultaneously uses logarithmic space? Can we show that SAT does not admit
such algorithms?
Definition 1. Let t(n) and s(n) be two functions. A language L is in class TISP(t(n), s(n))
if there is a deterministic Turing machine M that is both t(n)time bounded and s(n)space
bounded and M decides L.
We will show that SAT does not belong to the class TISP(nc
, log n) for some small constant
c > 0. Before we proceed with the proof, we will introduce alternating Turing machines and
the polynomialtime hierarchy.
2. Quantifier Characterizations
Recall that a language L is in NP if there exists a polynomial p and polynomialtime
computable relation R such that
