# Best current space lower bound for SAT?

Following on from a previous question,

what are the best current space lower bounds for SAT?

With a space lower bound I here mean the number of worktape cells used by a Turing machine which uses a binary worktape alphabet. A constant additive term is unavoidable since a TM can use internal states to simulate any fixed number of worktape cells. However, I am interested in controlling the multiplicative constant that is often left implicit: the usual setup allows arbitrary constant compression via larger alphabets so the multiplicative constant is not relevant there, but with a fixed alphabet it should be possible to take it into account.

For instance, SAT requires more than $\log\log n + c$ space; if not then this space upper bound would lead to a time upper bound of $n^{1+o(1)}$ by simulation, and thereby the combined $n^{1.801+o(1)}$ space-time lower bound for SAT would be violated (see the linked question). It also seems possible to improve this argument to argue that SAT requires at least $\delta\log n + c$ space for some small positive $\delta$ that is something like $0.801/C$, where $C$ is the constant exponent in simulation of a space-bounded TM by a time-bounded TM.

Unfortunately $C$ is usually quite large (and certainly at least 2 in the usual simulation, where the tapes of a TM are first encoded on a single tape via a larger alphabet). Such bounds with $\delta \ll 1$ are rather weak, and I would be especially interested in a space lower bound of $\log n + c$. An unconditional time lower bound of $\Omega(n^d)$ steps, for some large enough constant $d > 1$, would imply such a space lower bound via simulation. However, time lower bounds of $\Omega(n^d)$ for $d>1$ are not currently known, let alone for large $d$.

Put differently, I'm looking for something that would be a consequence of superlinear time lower bounds for SAT, but which might be possible to obtain more directly.

• as in that other answer (eg by RW), focusing on time or space lower bounds separately seems to be out of reach and have only weak/ generic known bounds, and the leading research in the area appears to give rise to a relatively newer concept of combined time-space complexity. – vzn Nov 16 '14 at 15:49

Looks like the best bound known (for multitape Turing machines) is logarithmic.

Suppose $\delta\log n$ bits of binary worktape is enough to decide whether any $n$-bit CNF formula is satisfiable, for all large enough $n$. By the standard simulation, a TM with $q$ states that uses at most $s$ bits of space can be simulated by a TM that has at most $qns2^s = 2^{s+\log n + \log s + \log q}$ different configurations. Whenever the machine accepts, there is a sequence of (nondeterministic) moves reaching an accepting state that is at most as long as this number of configurations. When $s = \Omega(\log n)$, this is at most $2^{s(2+o(1))}$ (note that $q$ remains the same for all input lengths $n$). On a separate counter tape, $M$ can first write this quantity in unary, then at each step of the simulation erase one of the symbols of the counter, and terminate the computation if it ever runs out of counter symbols. This creates a constant factor of overhead (something like 3), which is absorbed by the $o(1)$ term in the exponent. Hence $2^{s(2+o(1))}$ steps are sufficient.

By assumption $s \le \delta\log n$, so the time-space product is at most $\delta\log n2^{\delta\log n(2+o(1))} = n^{\delta(2+o(1))}$.

Rahul Santhanam showed in 2001 (see doi:10.1016/S0020-0190(00)00227-1) that the space-time product for a Turing machine deciding SAT must be at least $\Omega(n^{2-o(1)})$; his argument applies also to nondeterministic machines. Hence $\delta \ge 1$, and at least $\log n$ bits of binary worktape are needed.

More generally, additional worktapes and a larger worktape alphabet change the exponent by a constant factor. This ultimately reduces the factor $\delta$, but the space lower bound is still $\Omega(\log n)$.

Perhaps we can prove a $\log n$ space lower bound for SAT in this way (but I'm not confident with limit/asymptotic analysis, so my answer can be totally wrong).

On a Turing machine model with one read-only input tape and one work tape, both over binary alphabet $\Sigma = \{0,1\}$, for every decider with $c$ states on an input of size $n$ we have that :

$$T(n) \leq c \, 2^{S(n)} \, n \, S(n) \quad (1)$$

otherwise the Turing machine will loop forever (the $2^{S(n)}$ component represents all possible tape configurations, the $n$ component represents the input tape head positions, while the $S(n)$ component represents the work tape head positions). On a one tape single head TM over binary alphabet (1) becomes $T(n) \leq c 2^{S(n)} S(n)$.

Multiplying both terms by $S(n)$ and applying the general space-time tradeoff for SAT, we get:

$$n^{1.801+o(1)} \leq S(n)\cdot T(n) \leq c \, S(n)^2 \, 2^{S(n)} \, n$$

So picking a space upper bound like $S(n) \leq (\log n)^{1 - \epsilon}$ for SAT would lead to a contraddiction, indeed

$$\lim_{n \to \infty} \frac{n^{1.801}}{ c \, ((\log n)^{1-\epsilon})^2 2^{(\log n)^{1-\epsilon}} n } =$$

$$\lim_{n \to \infty} \Big( 0.801 \cdot \log n - \log c - 2 (1- \epsilon) \log (\log n) - (\log n)^{1-\epsilon}\Big) = \infty$$

• There seem to be at least two general ways to show that an $o(\log n)$ upper bound leads to a contradiction. Primarily I had in mind using the (essentially identical, but slightly easier to work with) inequality $T(n) \le 2^{\log n + C.S(n)}$ for some constant $C$. The last step you provide can also be made stronger, as a contradiction follows even from $S(n) \le \delta\log n$ for $\delta < 0.801/C$. – András Salamon Nov 20 '14 at 21:36
• @AndrásSalamon: on the $S\cdot T$ bound side, you cannot expect easy improvements: from S. Buss and R. Williams. Limits on Alternation-Trading Proofs for Time-Space Lower Bounds, 2012: "We show that new techniques are provably necessary in order to prove any better time-space lower bounds for the satisfiability problem. That is, the method of "alternation-trading proofs" used to establish that SAT cannot be solved in $n^{2\cos(\pi /7)}$ time and $n^{o(1)}$ space cannot prove an $n^{2\cos(\pi /7)}+\epsilon$ time lower bound, for every $\epsilon >0$". Do you have any idea :-)? – Marzio De Biasi Nov 22 '14 at 9:24
• I think this is about as far as one can go using the space-time bounds, precisely because Ryan's approach is as far as these bounds go. – András Salamon Nov 22 '14 at 15:49
• To even store a SAT instance you need $\Omega(n)$ and read it you need $\Omega(n)$ time. Doesn't this prove $\Omega(n^2)$ ST lower bound? – 1.. Aug 1 '15 at 13:34
• @Turbo, it is not clear that every algorithm to decide SAT has to store the instance: proving an $\Omega(n)$ bit deterministic space lower bound would show $\mathsf{L} \subsetneq \mathsf{NP}$. – András Salamon Mar 27 '17 at 13:54