2. Hierarchy theorems Section 9.1 Giorgi Japaridze Theory of Computability

3. 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 1 n ) 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).

4. 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 10* 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 2 f(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?).

5. Corollaries of the space hierarchy theorem 9.1.c Giorgi Japaridze Theory of Computability Corollary 9.4 For any two functions f 1,f 2 : N  N, where f 1 is o(f 2 (n)) and f 2 is space constructible, SPACE(f 1 (n))  SPACE(f 2 (n)). Below and elsewhere “A  B” means “A  B and A≠B”. Corollary 9.5 For any two real numbers 0 ≤  1 <  2, SPACE(n  1 )  SPACE(n  2 ). Corollary 9.6 NL  PSPACE. Proof. This is so because NL  SPACE(log 2 n) (by Savitch’s theorem), and SPACE(log 2 n)  SPACE(n) (by the space hierarchy theorem). Corollary 9.7 PSPACE  EXPSPACE. EXPSPACE is defined as SPACE(2 n 1 )  SPACE(2 n 2 )  SPACE(2 n 3 )  …

6. Time hierarchy theorem 9.1.d Giorgi Japaridze Theory of Computability 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 10* for some TM M, reject. 4. Simulate M on w. 5. If M accepts, reject. If M rejects, accept.” 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)).

7. Corollaries of the time hierarchy theorem 9.1.e Giorgi Japaridze Theory of Computability Corollary 9.11 For any two functions t 1,t 2 : N  N, where t 1 is o ( t 2 (n)/log t 2 (n) ) and t 2 is time constructible, TIME(t 1 (n))  TIME(t 2 (n)). Corollary 9.12 For any two real numbers 0 ≤  1 <  2, TIME(n  1 )  TIME(n  2 ). Corollary 9.13 P  EXPTIME.

8. 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 E k is also a regular expression with exponentiation. The meaning of E k is E concatenated with itself k times. Examples: 0 24 000 000 (1 7  1*); 0 18 000 000 0 6 000 000 1* EQ REX  = { | Q and R are equivalent regular expressions with exponentiation}. Theorem 9.15 EQ REX  is EXPSPACE-complete. Proof omitted.