# Lecture 23

## Space Complexity

Definition The space complexity of an input/work tape Turing machine is a function $$s:\mathbb{N}\rightarrow\mathbb{N}$$ such that on any input $$x\in\{0,1\}^n$$, $$M$$ writes on at most $$O(s(n))$$ cells in the work tape.

Definition $$SPACE(s(n))$$ is the class of languages that can be decided by an input/work tape Turing machine with space complexity $$s(n)$$.

E.g. If $$L$$ is regular, the $$L\in SPACE(1)$$ $$L=\{0^k1^k:k\geq 1\}$$ is in $$SPACE(\log n)$$

Definition $$L=SPACE(\log n)$$ $$PSPACE=\cup_{c\geq 1}SPACE(n^c)$$

Theorem $$L\subseteq NL\subseteq P\subseteq NP\subseteq PSPACE\subseteq EXP\subseteq NEXP$$

Proof $$NP\subseteq PSPACE$$

With polynomial space, I can enumerate and check all paths of a non-deterministic Turing machine and accept if any path accepts.

Proof $$L\subseteq P$$

If $$M$$ uses $$s$$ cells in the work tape on input $$x$$, it can be in at most $$|\Gamma|^s\times |Q|\times |x|\times s$$ distinct configuration.

If $$M$$ is halting, then it can never be in the same configuration twice. So if its space complexity is $$O(\log n)$$, then its time complexity is $$O(|\Gamma|^{O(\log n)\times |Q|\times\log n\times |n|})=O(n^c)$$

### Time Hierarchy Theorem

We know: $$P\neq EXP$$, $$NP\neq NEXP$$, $$L\neq PSPACE$$, rest is open.

## State of the Art on Interactive Proof

Recall: $$IP=$$ class of languages decidable by polynomial-time verifier that interacts with a prover. E.g. $$NP\subseteq IP$$, $$GNI\in IP$$.

Theorem $$IP=PSPACE$$

Proof $$IP\subseteq PSPACE$$

Brute force, simulate all possible interactions (results of coin flips + prover response) to compute the probability that verifier accepts on the best response strategy (reuse space).

Proof $$PSPACE\subseteq IP$$

Key idea: look at $$TQBF=\{\langle \Phi\rangle: \Phi$$ is a true quantified Boolean formula$$\}$$ where, e.g. $$\Phi=\exists x\forall y (x\lor y)\land(\lnot x\lor\lnot y)$$

Idea 1: $$TQBF$$ is $$PSPACE-complete$$

Idea 2: $$TQBF\in IP$$ (Arithmetization)

## Multiple-prover Interactive Proof

Definition $$MIP$$ is the class of languages deciplable by polynomial-time probabilistic verifier that con interact with $$2$$ provers.

Theorem $$IP\subseteq MIP$$

Proof Verifier can ignore one of the provers.

Theorem $$MIP=NEXP$$

Proof $$MIP\subseteq NEXP$$ Brute force

Proof $$NEXP\subseteq MIP$$ harder.

## Probabilistically Checkable Proof

Theorem $$MIP$$ = class of languages decidable by a probabilistic polynomial-time verifier in the RAM model that receives a proof of exponential length.

Proof Exercise