The Surprising Power Of Interaction: How Ip = Pspace
The Power of Interaction
Interactive proofs are a model of computation where a verifier has a conversation with a prover to check the validity of a statement. This interaction between two parties, rather than just a one-way verification, adds considerable power. The class IP captures the complexity of problems that have interactive proofs. In this article, we will explore how interaction enables the proof of more complex theorems, making IP equivalent in power to PSPACE – the class of problems solvable with polynomial space.
Defining Interactive Proofs
An interactive proof system has a prover P trying to convince a probabilistic polynomial-time verifier V that some statement is true. The verifier can ask the prover questions and evaluate the responses. At the end, V outputs either “accept” or “reject” for the statement based on the interaction. If the statement is true, V must accept with high probability. If false, V must reject with high probability. The power comes from this back-and-forth conversation.
Complete and Sound Proofs
For a proof system to be useful, it should satisfy completeness – true statements can be proved successfully, and soundness – false statements get rejected. The interaction allows more power in constructing complete proofs compared to non-interactive proofs, while the randomized verifier still ensures soundness.
Proof Systems Characteristics
Some key characteristics of these systems include:
- Number of rounds of interaction between P and V
- Resource usage limits for P and V
- Soundness and completeness parameters
These characterize the proof system’s efficiency and effectiveness.
The Classes P and IP
The class P contains decision problems efficiently solvable by a deterministic Turing machine. IP contains problems with interactive proofs. Surprisingly, it turns out P is strictly contained in IP – interaction adds power.
IP Systems for NP Problems
Many known IP systems can prove statements involving NP search problems. For example, consider the NP-complete problem of determining if a Boolean formula has satisfiable assignments. An IP system can allow P to send a satisfying assignment, which V can easily check. This shows SAT has interactive proofs, enabling it to be in IP.
IP Systems Beyond NP
However, IP also contains problems believed to be harder than those in NP. For example, some games like Go are in IP but not known to be in NP. So the interaction enables proof systems even for complex problems intractable for noninteractive verification.
IP Systems
There are various interactive proof systems using different techniques. We survey some important classes here.
Arthur-Merlin Games
Arthur-Merlin games involve a randomized Arthur who can toss coins during the protocol, interacting with all powerful Merlin. This randomness allows soundness while Merlin’s power provides completeness.
Multi Prover Systems
In multi-prover systems, V can interact with multiple provers P1 and P2 who cannot communicate during protocol. This prevents them from coordinating false claims.
Zero Knowledge Proofs
These IP systems have an extra condition – the verifier learns nothing beyond the bare fact something is true. This provides secrecy along with proof validity.
Arthur-Merlin Games
Arthur-Merlin (AM) proof systems are elegant and important types of IP systems introduced by Babai. Here Arthur is a randomized polynomial-time machine representing the verifier. Merlin is computationally unbounded, representing the prover.
Arthur’s Randomness
Arthur’s random coin tosses during interaction with Merlin leads to soundness. Though Merlin has infinite power, he does not know Arthur’s random choices so cannot always fake proofs.
Merlin Ensures Completeness
As Merlin can do unlimited computation, he can convince Arthur efficiently if asked good questions. This leads to polytime completeness while retaining soundness.
The symmetry between Arthur and Merlin makes AM games both intuitive and theoretically rich.
Randomness is Powerful
The power of randomness in IP systems merits deeper study. Randomized verifiers enable statistical checking of answers from untrusted provers.
Coin Flipping Protocols
Design of cryptographic protocols for secure coin flipping allowing generation of unbiased, mutually random bits is important.
Derandomization Questions
Derandomization research tries to efficiently simulate randomized computation with few random bits. Advances here could improve IP system efficiency.
Random Oracle Model
Modeling hash functions as random oracles is common technique in cryptography protocols. More rigorous analysis would boost confidence.
PSPACE and IP are Equivalent
A remarkable theorem called the IP = PSPACE theorem states that interactive proofs can solve exactly the problems solvable with polynomial space computations.
IP ⊆ PSPACE
This containment is reasonably clear – the verifier’s computation in any IP system protocol can be done with polynomial space resources.
PSPACE ⊆ IP
The breakthrough was constructing an IP system for any problem in PSPACE. The prover guides the verification through a graph representing configurations of the space-bounded computation.
Proof Techniques
Key proof ideas include graph encodings, randomness for checking history independence, and composition of IP systems.
Implications for Computational Complexity
The IP = PSPACE theorem has many implications for our understanding of efficient computation and proof systems.
IP Class Power
Showing IP can solve all problems efficiently solvable with space resources alone was surprising. IP may have further hidden powers.
Proof Techniques Generalizations
Can proof techniques used for showing IP=PSPACE be extended to characterize other complexity classes like NC? Open area worth exploring.
Philosophy of Mathematics
Humans can have subjective certainty about mathematical truths. But constructing formal proofs requires external tokens to avoid bias. IP systems provide a compelling analogue.
Open Problems in Interactive Proofs
Some open questions around interaction and randomness in proofs systems:
- Is AM = IP? Currently open.
- Are there problems in BQP outside PH? Quantum proofs may yield progress.
- Can PSPACE have non-interactive proofs? May need stronger assumptions.
Overall, IP systems keep revealing deep connections between interaction, randomness, knowledge, proofs and computation. More exciting discoveries likely lie ahead at this fascinating frontier.
