CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-27

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IP is closed under complement

Statement

If $L \in I P$ then its complement $\overline{L}$ also lies in $I P$ . Consequently $I P$ is closed under complement.

Facts & Assumptions

Given: A language $L \in I P$ .

[A1]

$I P = P S P A C E$ : every language in IP lies in PSPACE and every language in PSPACE lies in IP (IP equals PSPACE).

[A2]

If $L \in P S P A C E$ then its complement lies in PSPACE; this is closure of deterministic polynomial space under complement (PSPACE equals NPSPACE and is closed under complement).

Proof technique: direct.

Proof

1.1

Since $L \in I P$ , the containment $I P \subseteq P S P A C E$ of [A1] gives $L \in P S P A C E$ .

A1given

2.1

In particular there is a deterministic polynomial-space machine $M$ deciding $L$ : it halts on every input with the correct yes or no answer and uses at most $p (n)$ cells for some polynomial $p$ .

step 1.1given

3.1

By [A2] the complement of $L$ lies in PSPACE; concretely, flipping the accept and reject states of the machine $M$ of step 2.1 yields a deterministic polynomial-space machine deciding $\overline{L}$ , since $M$ is total on all inputs.

step 2.1A2construct

4.1

Applying the containment $P S P A C E \subseteq I P$ of [A1] to the language $\overline{L}$ gives $\overline{L} \in I P$ , which is the claim. This argument uses complementation of a deterministic space-bounded machine and does not claim that complementing an arbitrary interactive protocol preserves completeness or soundness.

step 3.1A1given∎

Depends on

Used by

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Sources

Arora and Barak, Computational Complexity: A Modern Approach, §8.5 and the remarks after Theorem 8.17, author-hosted draft (standard reference, not scraped)
A. Shen, IP = PSPACE: Simplified Proof, JACM 39(4) 1992, pp. 878–880 (standard reference, not scraped)