PSPACE  IP Proshanto Mukherji CSC 486 April 23, 2001

Overview Definitions Proof Arithmetization The protocol Soundness and Completeness Related results Summary

Definitions(1): IP Two components: Verifier: polynomial time-bounded probabilistic oracle TM Prover: deterministic TM with unlimited computational power Interactive Proof Systems VERIFIER PROVER QUERY TAPE question answer

Definitions(1): IP Soundness and Completeness

Definitions(2): PSPACE But we still don’t know whether

Overview Definitions Proof Arithmetization The Protocol Soundness and Completeness Related results Summary

Proof Let L be an arbitrary language in PSPACE Let D be the corresponding PSPACE machine Assume that: D has M states, D’s alphabet has N symbols, D’s tape usage is bound by the polynomial p D has exactly one accepting configuration for any given length of input If D accepts x, it does so in exactly steps Setting it up

Arithmetization Transform a computational problem to one of evaluating a polynomial Let

Arithmetization Transform a computational problem to one of evaluating a polynomial Let

Arithmetization

Define: Configurations of D on x

Arithmetization What is a “legal” configuration?

Arithmetization What is a “legal” configuration? Define:

Arithmetization Let: Transitions of D on x

Arithmetization What is a “legal” transition? }

Arithmetization What is a “legal” transition? So, set

Arithmetization Reachability Now we define a polynomial that captures whether, if D is in configuration O, it is possible to reach configuration N in one step

Arithmetization Multi-step Reachability And recursively extend this to get a set of polynomials that capture whether it is possible to get from O to N in 2 k steps, for any

Arithmetization Multi-step Reachability Configuration B Configuration A If: Recall:

Arithmetization Multi-step Reachability Configuration B Configuration A Configuration C Then: Recall:

Arithmetization Multi-step Reachability Recall: N O

Arithmetization So, let C ini be the (unique) initial configuration, and C fin the (unique) final configuration of D on input x. Then

Arithmetization (recap) ANDNOTOR EQUNIQ exactly one trueequal LCONF legal configuration LTRANS legal transition R0R0 reachability (1 step) RkRk reachability (2 k steps)

Arithmetization Key Point All these polynomials have been discussed for cases where each variable is binary, but may be evaluated over any field Their values at points outside {0,1} may not preserve their “key properties”

Overview Definitions Proof Arithmetization The Protocol Soundness and Completeness Related results Summary

The Protocol Preliminaries Define:

The Protocol Preliminaries Therefore: (no constraint on  )

The Protocol

Overview Definitions Proof Arithmetization The Protocol Soundness and Completeness Related results Summary

Soundness and Completeness Proof Key

Soundness and Completeness Completeness Recall: Completeness means that, if x is in L, there is at least one prover that causes the protocol to accept with probability >.75

Soundness and Completeness Key Lemma

Soundness and Completeness Soundness Recall: Soundness means that, if x is not in L, there is no prover that causes the protocol to accept with probability .25

Overview Definitions Proof Arithmetization The Protocol Soundness and Completeness Related results Summary

Related Results IP  PSPACE MIP = NEXP

Overview Definitions Proof Arithmetization The Protocol Soundness and Completeness Related results Summary

Here’s how we proved it Choose an arbitrary language in PSPACE, let D be a PSPACE machine that decides it Get a polynomial that, on binary inputs, describes the “essential behavior” of D Evaluate that at numerous points randomly picked from a large finite field, and use that to bound the probability of erroneous acceptance

Finis (that’s all, folks)