Hierarchy theorems

1
Hierarchy theorems
Giorgi Japaridze Theory of
Computability
Section 9.1
2
Space constructibility
9.1.a
Giorgi Japaridze Theory of Computability
Definition 9.1 A function f N?N, where f(n) is
at least O(log n), is called space constructible
if the function that maps any string w of length
n (equivalently, the string 1n) to the binary
representation of the number f(n) is computable
in space O(f(n)).
Intuition Assume a machine M runs in f(n)
space, and f(n) is space constructible. Then,
for any input w of size n, using only O(f(n))
space, we can not only tell whether M accepts w,
but can also compute the value of f(n) itself,
i.e. the amount of space used by M on input w.
Most natural functions are space
constructible. Those that are not are
pathological cases --- cases where computing
the space bound is so expensive that its space
cost exceeds the bound itself. This is like if
counting money was more expensive than the
amount of money itself. Motivation When f(n)
is space constructible, we can easily construct
machines for whatever purposes that control their
own space consumption and make sure that it does
not exceed f(n).

3
Space hierarchy theorem
9.1.b
Giorgi Japaridze Theory of Computability
Theorem 9.3 For any space constructible function
f N?N, a language A exists that is decidable in
O(f(n)) space but not in o(f(n)) space.
Proof. Such a language A is the one decided by
the following algorithm (TM) D On input w
1. Let n be the length of w. 2.
Compute f(n) using space constructibility, and
mark off this much tape. If later
stages ever attempt to use more, reject.
3. If w is not of the form ltMgt10 for some TM
M, reject. 4. Simulate M on w while
counting the number of steps used in the
simulation. If the count ever
exceeds 2f(n), reject. 5. If M accepts,
reject. If M rejects, accept.
Step 4 guarantees that D is indeed a decider
(why?).
Step 2 guarantees that D runs in space O(f(n))
(why?).
Step 5 guarantees that A is different from the
language decided by any o(f(n)) space
machine (why?).

4
Corollaries of the space hierarchy theorem
9.1.c
Giorgi Japaridze Theory of Computability
Below and elsewhere A?B means A?B and A?B.
EXPSPACE is defined as SPACE(2n1) ? SPACE(2n2) ?
SPACE(2n3) ?
Corollary 9.4 For any two functions f1,f2 N ?N,
where f1 is o(f2(n)) and f2 is space
constructible, SPACE(f1(n)) ? SPACE(f2(n)).
Corollary 9.5 For any two real numbers 0 ?1lt
?2, SPACE(n?1) ? SPACE(n?2).
Corollary 9.6 NL ? PSPACE.
Proof. This is so because NL?SPACE(log2n) (by
Savitchs theorem), and SPACE(log2n)?SPACE(n) (by
the space hierarchy theorem).
Corollary 9.7 PSPACE ? EXPSPACE.
5
Time hierarchy theorem
9.1.d
Giorgi Japaridze Theory of Computability
Definition 9.8 A function t N?N, where t(n) is
at least O(n log n), is called time
constructible if the function that maps any
string w of length n to the binary
representation of the number t(n) is
computable in time O(t(n)).
Theorem 9.10 For any time constructible function
t N?N, a language A exists that is decidable in
O(t(n)) time but not in time o(t(n)/log t(n)).
Proof. Such an A is the one decided by the
following O(t(n)) time algorithm (why?) D
On input w 1. Let n be the length of w.
2. Compute t(n) using time
constructibility, and store the value t(n)/log
t(n) (roun- ded up to the nearest
integer) in a binary counter. Decrement this
counter before each step used to
carry out stages 3, 4 and 5. If the counter ever
hits 0, reject. 3. If w is not of the
form ltMgt10 for some TM M, reject. 4.
Simulate M on w. 5. If M accepts, reject.
If M rejects, accept.

6
Corollaries of the time hierarchy theorem
9.1.e
Giorgi Japaridze Theory of Computability
Corollary 9.11 For any two functions t1,t2 N
?N, where t1 is o(t2(n)/log t2(n)) and t2 is
time constructible, TIME(t1(n)) ? TIME(t2(n)).
Corollary 9.12 For any two real numbers 0 ?1lt
?2, TIME(n?1) ? TIME(n?2).
Corollary 9.13 P ? EXPTIME.
7
EXPSPACE-completeness
9.1.f
Giorgi Japaridze Theory of Computability
Definition 9.14 A language B is
EXPSPACE-complete if 1. B is in EXPSPACE,
and 2. every language A in EXPSPACE is
polynomial time reducible to B.

• Regular expressions with exponentiation are
  defined in the same was as (ordinary)
• regular expressions, but with the additional
  formation rule
• If E is a regular expression with
  exponentiation and k is a decimal number, then
• Ek is also a regular expression with
  exponentiation.
• The meaning of Ek is E concatenated with itself k
  times.
• Examples 024 000 000(17?1) 018 000 00006
  000 0001

EQREX? ltQ,Rgt Q and R are equivalent regular
expressions with exponentiation.
Theorem 9.15 EQREX? is EXPSPACE-complete.
Proof omitted.