In computational complexity theory, the class IP (which stands for interactive proof) is the class of problems solvable by an interactive proof system. It is equal to the class PSPACE. The result was established in a series of papers: the first by Lund, Karloff, Fortnow, and Nisan showed that co-NP had multiple prover interactive proofs; and the second, by Shamir, employed their technique to establish that IP=PSPACE. The result is a famous example where the proof does not relativize.

The concept of an interactive proof system was first introduced by Shafi Goldwasser, Silvio Micali, and Charles Rackoff in 1985. An interactive proof system consists of two machines, a prover, P, which presents a proof that a given string n is a member of some language, and a verifier, V, that checks that the presented proof is correct. The prover is assumed to be infinite in computation and storage, while the verifier is a probabilistic polynomial-time machine with access to a random bit string whose length is polynomial on the size of n. These two machines exchange a polynomial number, p(n), of messages and once the interaction is completed, the verifier must decide whether or not n is in the language, with only a 1/3 chance of error. (So any language in BPP is in IP, since then the verifier could simply ignore the prover and make the decision on its own.)

A language L belongs to IP if there exist V, P such that for all Q, w:

The Arthur–Merlin protocol, introduced by László Babai, is similar in nature, except that the number of rounds of interaction is bounded by a constant rather than a polynomial.

Goldwasser et al. have shown that public-coin protocols, where the random numbers used by the verifier are provided to the prover along with the challenges, are no less powerful than private-coin protocols. At most two additional rounds of interaction are required to replicate the effect of a private-coin protocol. The opposite inclusion is straightforward, because the verifier can always send to the prover the results of their private coin tosses, which proves that the two types of protocols are equivalent.

In the following section we prove that IP = PSPACE, an important theorem in computational complexity, which demonstrates that an interactive proof system can be used to decide whether a string is a member of a language in polynomial time, even though the traditional PSPACE proof may be exponentially long.

The proof can be divided in two parts, we show that IP ⊆ PSPACE and PSPACE ⊆ IP.

In order to demonstrate that IP ⊆ PSPACE, we present a simulation of an interactive proof system by a polynomial space machine. Now, we can define: