# IP = PSPACE

@article{Shamir1992IPP, title={IP = PSPACE}, author={Adi Shamir}, journal={J. ACM}, year={1992}, volume={39}, pages={869-877} }

In this paper, it is proven that when both randomization and interaction are allowed, the proofs that can be verified in polynomial time are exactly those proofs that can be generated with polynomial space.

## 786 Citations

Proofs for Quantified Boolean Formulas

- Mathematics
- 2013

In [4], it was claimed that the amount of communication in an interactive protocol for QBFormulaSAT can be bounded by a polynomial in the number of variables in the input formula. However, the proof…

Two-Message Quantum Interactive Proofs Are in PSPACE

- Computer Science2009 50th Annual IEEE Symposium on Foundations of Computer Science
- 2009

It is proved that QIP(2), the class of problems having two-message quantum interactive proof systems, is a subset of PSPACE by means of an efficient parallel algorithm, based on the matrix multiplicative weights update method, for approximately solving a certain class of semidefinite programs.

IP = SPACE: simplified proof

- EducationJACM
- 1992

A slightly simplified version of Shamir's proof is presented, using degree reductions instead of simple QBFs, to prove that PH is contained in IP.

Computational Complexity and Mathematical Proofs

- Mathematics, Computer ScienceInformatics
- 2001

How the major computational complexity classes, P, NP and PSPACE, capture different computational properties of mathematical proofs and reveal new quantitative aspects of mathematics are discussed.

Interactive Proof Systems and Alternating Time-Space Complexity

- Computer Science, MathematicsTheor. Comput. Sci.
- 1993

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

- Computer Science, MathematicsLecture Notes in Computer Science
- 2012

We show that given a satisfiable instance of the 2-to-1 Label Cover problem, it is NP-hard to find a ( 23 24 + )-satisfying assignment.

Probabilistically checkable proofs and their consequences for approximation algorithms

- Mathematics, Computer ScienceDiscret. Math.
- 1994

More on BPP and the Polynomial-Time Hierarchy

- Mathematics, Computer ScienceInf. Process. Lett.
- 1996

Progress on Polynomial Identity Testing - II

- MathematicsElectron. Colloquium Comput. Complex.
- 2013

The area of algebraic complexity theory is surveyed, with the focus being on the problem of polynomial identity testing (PIT), and the key ideas are discussed.

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This technique is used to prove that every language in the polynomial-time hierarchy has an interactive proof system and played a pivotal role in the recent proofs that IP = PSPACE and MIP = NEXP.

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Permission to copy without fee all or part of this material is granted provided that the copies arc not made or distributed for direct commercial advantage. rhe ACM copyright notice and the title of…

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A model in which a computationally bounded verifier consults with an oracle in the presence of malicious faults on the communication lines, and it is shown that a deterministic polynomial time verifier can test membership in any language in P-space, but cannottest membership in languages not in P -space, even if he is allowed to toss random coins in private.