Edmonds, Cobham circa 1960: Programs that use a polynomially
bounded resources (time, space) are generally reasonable
for moderately sized inputs. Programs that use exponential
resources are generally not reasonable for moderately
size inputs.

Definition 1: TIME(T(n)) = { language L : There exists a Turing machine M such that
(1) M accepts x iff x is in L and (2) for all but finitely many x M(x) halts in T(|x|) steps  }

Definition 2: TIME(T(n)) = { language L : There exists a Turing machine M and a constant b such that
(1) M accepts x iff x is in L and (2) for all x M(x) halts in b*T(|x|) steps  }

Homework: Prove that the two definitions are equivalaent. 

Linear Speedup Theorem: TIME(2n^2) subset TIME(n^2)


Time Hierarchy Theorem: 
TIME(f(n)) is a strict subset of TIME(g(n)) if f(n) \log n = O(g(n))
Proof for time and f(n)=n^2 and g(n)= n^3:

Let L = {x |  machine x doesn't accept x in |x|^2.5 steps}

*L in TIME(n^3) by simulation 
*L not in TIME(n^2) by diagonalization

Note that as there are infinitely many copies of any n^2 time machine in the
rows of A, and thus anymachine that accepts L must run longer than n^2.5 steps infinitely often.
End Proof

Defintion of standard complexity classes: 
SPACE(T(n)), P = Union_k TIME(n^k), 
PSPACE= Union_k SPACE(n^k), EXPTIME = Union_k TIME(2^n^k)
LogSpace.

Example: 0^n1^n in LogSpace


Theorem: TIME(f(n)) subset SPACE(f(n)) subset TIME(2^f(n))
proof: First inclusion is obvious.

A program that uses space S(n) only has O(Sigma^S(n) possible
configurations. Here Sigma is the tape alphabet size. 
Once a program repeats configurations then its is an infinite loop.
End proof

Corollary:
Polynomial space is contained in exponential time
Log space is contained in polynomial time

Definition: A language L is complete for a complexity class C 
(with respect to a type of reduction Q) iff 
* L is in C, and
* For all X in C, X is reducible to L via a reduction of type Q.
So intuitively L is the hardest problem in C, where the degree of
refinement in this hardness measure is given by the courseness of Q.

Example of Completeness use:
Say you want to prove a theorem of the form: If exponential time
problem X has a poly space algorithm, then all exponential 
time algorithms have poly space algorithms. 
Question: What sort of completeness would you need for X?
Answer: Polynomial space completeness since a poly space algorithm
that calls another poly space algorithm runs in poly space

Question: Does PSPACE have a complete problem L ? 
Wrong Answer: Yes,
L = {(M, k, I)  : M accepts I in |I|^k space}
This is wrong because it is not clear that there is a unique l such that this language
can be accepted in n^l space


Answer:
Yes,
L = {(M, I, 1^|I|^k)  : M accepts I in |I|^k space}
Proof:
In PSPACE 
Reduction from PSPACE problem accepted by polynomnial space time machine M that uses space n^k
Program 
	Read I
	Call program for L with input (M, I, 1^|I|^k)
Question: What is the unique l such that this language can be accepted in n^l space?



Theorem: Circuit Value Problem is log space complete for P
Proof: Consider a particular polynomial Turing machine and its computation history.
Show how to simulate the computation with a circuit




Theorem: Quantifiable Boolean Formula (QBF) is PSPACE-complete
proof: 
QBF in PSPACE
Show L is reducible to QBF

Try 1: THEREEXISTS x such that x is computation history showing the 
M accepts I using less than |I|^k space
Problem: x is expoentially large

Try 2:
Phi_i(A,B) = configuration B of M is reachable from configuration
A of M in 2^i steps

Question: How to recursively define Phi_i

There is a formula for Phi_0(A,B), which is reachability in one step.

Question: What is wrong with
Phi_{i+1}(A, B) = THEREEXISTS C   Phi_i(A,C) and Phi_i(C,B)
Answer: Grows exponentially. May be as large as the time used
by M, which may be exponential in |I|

Try 3:
Phi_{i+1}(A, B) = THEREEXISTS C   FORALL X, Y
(X=A and Y=C) or (X=C and Y=B) implies Phi_i(X, Y)


Fact: Determining whether one player has a winning strategy in obvious generalizations
of common two person games is often PSPACE-complete.