What Are the Limits of Interactive Proofs: Why Does MIP Fall Within NEXP?

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SUMMARY

The discussion centers on the relationship between interactive proofs (IP) and multiple interactive proofs (MIP), establishing that IP is a subset of PSPACE, while MIP is contained within NEXP. The reasoning is based on the traversal of decision trees, where a single prover can be managed within polynomial space, but multiple provers complicate the scenario, necessitating exponential space. This distinction highlights the limitations of MIP compared to IP in computational complexity theory.

PREREQUISITES
  • Understanding of computational complexity classes, specifically IP, PSPACE, and NEXP.
  • Familiarity with decision tree traversal and its implications in proof systems.
  • Knowledge of interactive proof systems and their operational mechanics.
  • Basic concepts of multiple provers and their impact on computational resources.
NEXT STEPS
  • Research the implications of IP and PSPACE in computational theory.
  • Study the characteristics and limitations of MIP within NEXP.
  • Explore decision tree complexity and its relevance to proof systems.
  • Investigate the differences between single and multiple prover systems in interactive proofs.
USEFUL FOR

Theoretical computer scientists, researchers in computational complexity, and students studying interactive proof systems will benefit from this discussion.

Dragonfall
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You can prove that [tex]\mathbf{IP} \subset \mathbf{PSPACE}[/tex] by traversing the exponentially large decision tree of the prover, which takes only polynomial amount of space at a time.

Why can't you do the same in the case of multiple provers? You'd traverse polynomially many trees at a time instead of one, but that should only require poly space. Instead, [tex]\mathbf{MIP} \subset \mathbf{NEXP}[/tex].
 
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I'd have posted this in the CS subforums, but this is more of a theoretical question.
 

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