NC = P consequences?

Question

The Complexity Zoo points out in the entry on EXP that if L = P then PSPACE = EXP. Since NPSPACE = PSPACE by Savitch, as far as I can tell the underlying padding argument extends to show that $$(\text{NL} = \text{P}) \Rightarrow (\text{PSPACE} = \text{EXP}).$$ We also know that L $\subseteq$ NL $\subseteq$ NC $\subseteq$ P via Ruzzo's resource-bounded alternating hierarchy.

If NC = P, does it follow that PSPACE = EXP?

A different interpretation of the question, in the spirit of Richard Lipton: is it more likely that some problems in P cannot be parallelized, than that no exponential-time procedure requires more than polynomial space?

I would also be interested in other "surprising" consequences of NC = P (the more unlikely the better).

Edit: Ryan's answer leads to a further question: what is the weakest hypothesis that is known to guarantee PSPACE = EXP?

• W. Savitch. Relationships between nondeterministic and deterministic tape complexities, Journal of Computer and System Sciences 4(2):177-192, 1970.
• W. L. Ruzzo. On uniform circuit complexity, Journal of Computer and System Sciences 22(3):365-383, 1971.

Edit (2014): updated old Zoo link and added links for all other classes.

Answer 1

Yes. $NC$ can be seen as the class of languages recognized by alternating Turing machines that use $O(\log n)$ space and $(\log n)^{O(1)}$ time. (This was first proved by Ruzzo.) $P$ is the class where alternating Turing machines use $O(\log n)$ space but can take up to $n^{O(1)}$ time. For brevity let's call these classes $ATISP[(\log n)^{O(1)},\log n] = NC$ and $ASPACE[O(\log n)] = P$.

Suppose the two classes are equal. Replacing the $n$ with $2^n$ in the above (i.e., applying standard translation lemmas), one obtains

$TIME[2^{O(n)}] = ASPACE[O(n)] = ATISP[n^{O(1)}, n] \subseteq ATIME[n^{O(1)}] = PSPACE$.

If $TIME[2^{O(n)}] \subseteq PSPACE$ then $EXP = PSPACE$ as well, since there are $EXP$-complete languages in $TIME[2^{O(n)}]$.

Edit: Although the above answer is perhaps more educational, here's a simpler argument: $EXP = PSPACE$ already follows from "$P$ is contained in polylog space" and standard translation. Note "$P$ is contained in polylog space" is a much weaker hypothesis than $NC = P$.

More details: Since $NC$ circuit families have depth $(\log n)^c$ for some constant, every such circuit family can be evaluated in $O((\log n)^c)$ space. Hence $NC \subseteq \bigcup_{c > 0} SPACE[(\log n)^c]$. So $P = NC$ implies $P \subseteq \bigcup_{c > 0} SPACE[(\log n)^c]$. Applying translation (replacing $n$ with $2^n$) implies $TIME[2^{O(n)}] \subseteq PSPACE$. The existence of an $EXP$-complete language in $TIME[2^{O(n)}]$ finishes the argument.

Update: Addressing Andreas' additional question, I believe it should be possible to prove something like: $EXP=PSPACE$ iff for all $c$, every polynomially sparse language in $n^{O(\log^c n)}$ time is solvable in polylog space. (Being polynomially sparse means that there are at most $poly(n)$ strings of length $n$ in the language, for all $n$.) If true, the proof would probably go along the lines of Hartmanis, Immerman, and Sewelson's proof that $NE = E$ iff every polynomially sparse language in $NP$ is contained in $P$. (Note, $n^{O(\log^c n)}$ time in polylog space is still enough to imply $PSPACE=EXP$.)

Answer 2

(I've seen Ryan's answer, but I just wanted to provide another perspective, which was too long to fit into a comment.)

In the $L = P \Rightarrow PSPACE = EXP$ proof, all that you need to know about L, informally, is that when blown up by an exponential, L becomes PSPACE. The same proof goes through for NL, because NL blown up by an exponential also becomes PSPACE.

Similarly, when NC is blown up by an exponential, you do get PSPACE. I like to see this in terms of circuits: NC is the class of polynomial size circuits with polylog depth. When blown up, this becomes exponential size circuits with polynomial depth. One can show that this is exactly PSPACE, once the appropriate uniformity conditions are added in. I guess if NC is defined with L-uniformity, then this will get PSPACE-uniformity.

The proof should be easy. In one direction, take a PSPACE-complete problem like TQBF and express the quantifiers using AND and OR gates of exponential size. In the other direction, try traversing the polynomial depth circuit recursively. The stack size will be polynomial, so this can be done in PSPACE.

Finally, I came up with this argument when I saw the question (and before reading Ryan's answer), so there might be bugs. Please point them out.

Answer 3

Here's a little more detail from the perspective of simulating time-space bounded Alternating Turing machine.

Suppose that $P = NC$.

Since $NC = ATISP((\log(n))^{O(1)}, O(\log(n)))$, we get $$P = ATISP((\log(n))^{O(1)}, O(\log(n))).$$

Now, consider the linear time universal simulation problem $LinU$ where we are given an encoding on a Turing machine $M$ and an input string $x$ of length $n$ and we want to know if $M$ accepts $x$ in at most $n$ steps.

We know that $LinU \in P$. Therefore, there exists a constant $c$ (sufficiently large) such that $$(*) \; LinU \in ATISP(\log^c(n), c\log(n)).$$

As a result of a padding argument (a little tricky see comments), we have $$(1) \; DTIME(n) \subseteq ATISP(\log^c(n), c\log(n)).$$

Extending the padding argument, we get $$(2) \; DTIME(n^k) \subseteq ATISP(k^c\log^c(n), kc\log(n)).$$ $$(3) \; DTIME(2^{n^k}) \subseteq ATISP(k^cn^{kc}, kcn^{k}).$$

Further, there are known results about the simulation of Alternating time-space bounded Turing machines. In particular, we know that $$ATISP(\log^c(n), c\log(n)) \subseteq DSPACE(O(\log^{c+1}(n))).$$

Therefore, we (essentially) have the following for all natural numbers $k$:

$$(2^{*}) \; DTIME(n^k) \subseteq DSPACE(k^{c+1}\log^{c+1}(n))$$ $$(3^{*}) \; DTIME(2^{n^k}) \subseteq DSPACE(n^{k(c+1)}).$$

From $(3^{*})$, we would get that $EXP = PSPACE$.

====================After Thought===================

It is important to notice that $P = NC$ implies $$ATISP((\log(n))^{O(1)}, O(\log(n))) = ATISP(\log^c(n), O(\log(n)))$$ for some constant $c$.

Any comments or corrections are welcomed. :)