Multi-prover Interactive Proof Systems Research Articles

Provably Safe Artificial General Intelligence via Interactive Proofs

Methods are currently lacking to prove artificial general intelligence (AGI) safety. An AGI ‘hard takeoff’ is possible, in which first generation AGI1 rapidly triggers a succession of more powerful AGIn that differ dramatically in their computational capabilities (AGIn << AGIn+1). No proof exists that AGI will benefit humans or of a sound value-alignment method. Numerous paths toward human extinction or subjugation have been identified. We suggest that probabilistic proof methods are the fundamental paradigm for proving safety and value-alignment between disparately powerful autonomous agents. Interactive proof systems (IPS) describe mathematical communication protocols wherein a Verifier queries a computationally more powerful Prover and reduces the probability of the Prover deceiving the Verifier to any specified low probability (e.g., 2−100). IPS procedures can test AGI behavior control systems that incorporate hard-coded ethics or value-learning methods. Mapping the axioms and transformation rules of a behavior control system to a finite set of prime numbers allows validation of ‘safe’ behavior via IPS number-theoretic methods. Many other representations are needed for proving various AGI properties. Multi-prover IPS, program-checking IPS, and probabilistically checkable proofs further extend the paradigm. In toto, IPS provides a way to reduce AGIn ↔ AGIn+1 interaction hazards to an acceptably low level.

Parallelization of entanglement-resistant multi-prover interactive proofs

Multi-prover interactive proof systems are said to be entanglement-resistant if the soundness holds even when provers are allowed to share an arbitrary quantum state before the interaction starts. This letter proves that every entanglement-resistant multi-prover interactive proof system can be parallelized to two rounds without ruining its entanglement resistance at the expense of adding one prover.

Using Entanglement in Quantum Multi-Prover Interactive Proofs

The central question in quantum multi-prover interactive proof systems is whether or not entanglement shared among provers affects the verification power of the proof system. We study for the first time positive aspects of prior entanglement and show how it can be used to parallelize any multi-prover quantum interactive proof system to a one-round system with perfect completeness, soundness bounded away from one by an inverse-polynomial in the input size, and one extra prover. Alternatively, we can also parallelize to a three-turn system with the same number of provers, where the verifier only broadcasts the outcome of a coin flip. This “public-coin” property is somewhat surprising, since in the classical case public-coin multi-prover interactive proofs are equivalent to single-prover ones.

Quantum multi-prover interactive proof systems with limited prior entanglement

This paper gives the first formal treatment of a quantum analogue of multi-prover interactive proof systems. It is proved that the class of languages having quantum multi-prover interactive proof systems is necessarily contained in NEXP, under the assumption that provers are allowed to share at most polynomially many prior-entangled qubits. This implies that, in particular, if provers do not share any prior entanglement with each other, the class of languages having quantum multi-prover interactive proof systems is equal to NEXP. Related to these, it is shown that, in the case a prover does not have his private qubits, the class of languages having quantum single-prover interactive proof systems is also equal to NEXP.

P-Selective Sets and Reducing Search to Decision vs Self-Reducibility

We distinguish self-reducibility of a languageLwith the question of whether search reduces to decision forL. Results include: (i) If NE≠E, then there exists a setLin NP−P such that search reduces to decision forL, search doesnotnonadaptively reduce to decision forLandLis not self-reducible. (ii) If UE≠E, then there exists a languageL∈UP−P such that search nonadaptively reduces to decision for L, but L is not self-reducible. (iii) If UE∩co-UE≠E, then there is a disjunctive self-reducible languageL∈UP−P for which search doesnotnonadaptively reduce to decision. We prove that if NE⊈BPE, then there is a languageL∈NP−BPP such thatLis randomly self-reducible,notnonadaptively randomly self-reducible, andnotself-reducible. We obtain results concerning trade-offs in multiprover interactive proof systems and results that distinguish checkable languages from those that are nonadaptively checkable. Many of our results are proven by constructing p-selective sets. We obtain a p-selective set that isnot⩽Ptt-equivalent to any tally language, and we show that if P=PP, then every p-selective set is ⩽PT-equivalent to a tally language. Similarly, if P=NP, then every cheatable set is ⩽Pm-equivalent to a tally language. We construct a recursive p-selective tally set that isnotcheatable.

On games of incomplete information

We study two-person games of cooperation and multi-prover interactive proof systems. We first consider a two-person game G, which we call a free game, defined as follows. A Boolean function C& is given. Players I and II each pick a random number i and j in private, where 1 d i,j < s, and then each chooses a private numberf(i) and g(j), 1 <f(i), g(j) < s. If &( i,j,f( i), g(j)) = 1, then both players win; otherwise, they lose. The objective of both players is to win collectively. We ask whether, if such a game is played n times in parallel, the probability of winning all the games decays exponentially in n. This question was posed in a more general context by Fortnow [lo], which we will discuss soon. Formally, we define the nth product game G” as the following two-person game. Players I and II each pick a vector of independent random numbers i= ( iI,. . . , i,) and j=(j l,...,jn) in private, 1 <ik, jk<s, and then each chooses a private sequence of numbers fi(g, . . ..f.(fl and gl(j) , . . ., g&3. The goal for both players is to ensure Ai=1 &(&,j&.(i),g.Jj))= 1. We define th e winning probability of the game G to be maxS,, Pr[&( i, j,f( i), g(j)) = 11, where the probability is taken over all randomly and uniformly chosen i,j in the range 1 , . . . . s, and we denote it by w(G). The game G is called nontrivial if its winning probability is neither 0 nor 1. We shall consider only

Uses of randomness in algorithms and protocols

Uses of Randomness in Algorithms and Protocols makes fundamental contributions to two different fields of complexity theory: computational number theory and cryptography. The most famous result is Goldwasser and Kilian's invention of a new approach to distinguish prime numbers from composites, using methods from the theory of elliptic curves over finite fields.The Goldwasser-Kilian algorithm is the first to yield a polynomial size proof of its assertions, ensuring correctness while still provably running fast on most inputs. This new primality test implies for the first time and without any assumptions that large certified primes can be generated in expected polynomial time under a distribution that is close to uniform. It provides a provocative new link between algebraic geometry and primality testing, one of the most ancient algorithmic problems in number theory. Heuristic implementations of the algorithm are currently considered to be the fastest existing methods to certify primes.Kilian also provides two elegant and original contributions to theoretical cryptography. He shows how to base general two-party protocols on a simple protocol, known as oblivious transfer, proving the first completeness result of this kind. He also introduces a generalization of interactive proof known as multi-prover interactive proof systems, and shows that anything provable in this model is provable in zero knowledge.Joe Kilian is a National Science Foundation Postdoctoral Fellow at MIT and Harvard.Contents: Introduction. New Techniques in Primality Testing. Committing Bits Using Oblivious Transfer. Circuit Evaluation Using Oblivious Transfer: The NC1 Circuit Base. ObliviousEvaluation of Arbitrary Circuits. Interactive Proof Systems with Multiple Provers.