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CS221: Computational Complexity Prof. Salil Vadhan
Lecture 30: Arthur-Merlin Games
12/6 Scribe: Noam Zeilberger
Contents
1 Remarks on P#P^ ⊆ IP and IP = PSPACE
- These constructions are based on specific complete problems and do not relativize.
- They provide interactive protocols with perfect completeness, hence a corollary of IP = PSPACE is that IP = IP with perfect completeness (a result known before IP = PSPACE).
- They rely on no hidden randomness for the verifier (i.e. they are “public coin” interactive proofs).
2 Arthur-Merlin Games
2.1 Introduction
The final remark leads to the notion of Arthur-Merlin games—interactive proofs between a verifier (Arthur) and a prover (Merlin) in which the verifier’s messages are simply its random coin tosses:
M
r 1 ∈R { 0 , 1 }p(n) A m 1
.. . rk∈R { 0 , 1 }p(n)
mk (^) A(x, r 1 , m 1 ,... , rk, mk) = accept or reject
A corollary to IP = PSPACE is that Arthur-Merlin games are as powerful as interactive proofs:
Corollary 1 (Goldwasser, Sipser) IP = {L | L has an Arthur-Merlin game }
However, Arthur-Merlin games were developed by L´aszl´o Babai independently of Goldwasser, Micali, and Rackoff’s interactive proofs, and his motivations originated from complexity theory, whereas theirs from cryptography.
Another interpretation of Arthur-Merlin games is as randomized alternation. Consider that if we assume perfect completeness, then
x ∈ L ⇒ ∀r 1 ∃m 1 ∀r 2 · · · A(x, r, m) = accept x /∈ L ⇒ ∃r 1 ∀m 1 ∃r 2 · · · A(x, r, m) = reject
But in fact the soundness condition is stronger, namely that
x /∈ L ⇒ ∃+r 1 ∀m 1 ∃+r 2 · · · A(x, r, m) = reject
where “∃+” means “there exist many”/“for almost all.” A similar way of thinking about Arthur- Merlin games is as “games versus Nature”—Arthur representing random choices made by Nature, like the random shuffling of a deck in Solataire (and Merlin, perhaps, the supernatural).
2.2 MA and AM
It is natural to treat the number of messages as a resources:
Definition 2 AM[k] is the class of languages that have Arthur-Merlin games in which k total messages are sent, beginning with Arthur. Likewise, MA[k] is the class of languages with k-message Arthur-Merlin games in which Merlin begins.
We write AM+[k] for AM[k] with perfect completeness. By the above casting of Arthur-Merlin games in terms of randomized alternation, it is immediate that AM+[k] and MA+[k] are contained in the polynomial hierarchy, for all k. But in fact, we will prove something much stronger.
Definition 3 We write AM def = AM[2] = AM+[2] and MA def = MA[2] = MA+[2] (where the equalities giving perfect completeness are by Problem Set 6). In general, a string of As and Ms denotes an Arthur-Merlin game where a message is sent by the player corresponding to the first symbol, then by the player corresponding to the second symbol, etc. For example, AMAM = AM[4]. Note that AA = A, MM = M, since separate messages sent on two successive rounds by the same player can be combined into one.
Theorem 4 ∀k ≥ 2 AM[k] = AM
Proof: We start by showing that MA ⊆ AM. The basic idea is to simply swap the order of the messages in the exchange, but then also use amplification in order to reduce the verifier’s error rate to a sufficient extent that soundness still holds for the AM game. Explicitly, suppose that m 1 is the message sent by Merlin, r 1 the random string sent by Arthur, and that the original MA verifier A has has two-sided error rate ≤ = ≤(n) on inputs of length n. Then the following holds:
x ∈ L ⇒ ∃m 1 Pr r 1 [A(x, m 1 , r 1 ) = accept] ≥ 1 − ≤ ⇒ Pr r 1 [∃m 1 A(x, m 1 , r 1 ) = accept] ≥ 1 − ≤
x /∈ L ⇒ ∀m 1 Pr r 1 [A(x, m 1 , r 1 ) = accept] ≤ ≤
⇒ Pr r 1 [∃m 1 A(x, m 1 , r 1 ) = accept] ≤ 2 |m^1 |^ · ≤
