Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0908.2363

Polynomial-Space Approximation of No-Signaling Provers

Tsuyoshi Ito
View original source ↗Read source document ↗Export metadata ↓

Abstract

In two-prover one-round interactive proof systems, no-signaling provers are those who are allowed to use arbitrary strategies, not limited to local operations, as long as their strategies cannot be used for communication between them. Study of multi-prover interactive proof systems with no-signaling provers is motivated by study of those with provers sharing quantum states. The relation between them is that no-signaling strategies include all the strategies realizable by provers sharing arbitrary entangled quantum states, and more. This paper shows that two-prover one-round interactive proof systems with no-signaling provers only accept languages in PSPACE. Combined with the protocol for PSPACE by Ito, Kobayashi and Matsumoto (CCC 2009), this implies MIPns(2,1)=PSPACE, where MIPns(2,1) is the class of languages having a two-prover one-round interactive proof system with no-signaling provers. This is proved by constructing a fast parallel algorithm which approximates within an additive error the maximum value of a two-player one-round game achievable by cooperative no-signaling players. The algorithm uses the fast parallel algorithm for the mixed packing and covering problem by Young (FOCS 2001).

Explore related subjects

cs.CCquant-ph

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Tsuyoshi Ito. 2009-10-20. Polynomial-Space Approximation of No-Signaling Provers. https://arxiv.org/abs/0908.2363

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Improved Algorithms for the Remote Point Problem

The Remote Point Problem (RPP) is an algorithmic problem that asks, given a linear subspace $L \subseteq \mathbb{F}^n$ of dimension $k$, to deterministically find a vector $v \in \mathbb{F}^n$ far in Hamming distance from $L$. This problem was introduced by Alon, Panigrahy and Yekhanin [APY09], motivated in part by the matrix rigidity approach for proving circuit lower bounds. An algorithm is said to achieve remoteness $d$ if it finds a vector $v$ whose Hamming distance from $L$ is at least $d$. We observe that over the rational numbers, the problem admits a deterministic polynomial-time algorithm that achieves optimal remoteness $n-k$. Over finite fields, we obtain a (modest) improvement of a result of Alon, Panigrahy and Yekhanin [APY09], and give an algorithm that achieves remoteness $Ω\left(\frac{n}{\max\{k, \log n\}} \log n\right)$.

cs.CC↗
arXiv2026

Bit-counting complexity classes

We define bit-counting complexity classes whose membership depends on the binary profile of the number of accepting paths of non-deterministic polynomial time Turing machines. We study the relationship between this new family of complexity classes and the classical complexity classes. We prove that the complexity class ${\bf PP}$ is contained in our comparison based bit-counting complexity classes ${\bf B_{|0|=|1|}P}$, ${\bf B_{|0|<|1|}P}$ and ${\bf B_{|0|>|1|}P}$. We then show that the comparison based bit-counting complexity classes and the complexity class ${\bf PP}$ are Turing equivalent, that is ${\bf P}^{\bf PP} = {\bf P}^{{\bf B_{|0|=|1|}P}}={\bf P}^{{\bf B_{|0|>|1|}P}}={\bf P}^{{\bf B_{|0|<|1|}P}}$. We then prove that the complexity classes ${\bf NP}$ and ${\bf CoNP}$ are contained in both of our parity based bit-counting complexity classes ${\bf B_{|0| \oplus}P}$ and ${\bf B_{|1| \oplus}P}$. We also show that the Turing closures of the parity based bit-counting complexity classes coincide, that is ${\bf P}^{{\bf B_{|0|\oplus}P}}={\bf P}^{{\bf B_{|1|\oplus}P}}$. We do this by proving that when either parity based bit-counting complexity class is provided as an oracle for a polynomial time Turing machine, then it can simulate the other one, that is ${\bf B_{|1| \oplus}P}\subseteq {\bf P}^{{\bf B_{|0| \oplus}P}}$ and ${\bf B_{|0| \oplus}P}\subseteq {\bf P}^{{\bf B_{|1| \oplus}P}}$.

cs.CC↗
arXiv2026

Geometric Complexity Theory and Graph Isomorphism

We investigate ideas from the Geometric Complexity Theory approach to separating complexity classes (Mulmuley & Sohoni, SIAM J. Comput., 2001) in the setting of graph isomorphism. This provides us a playground of finite combinatorial objects on which to explore these techniques. We seek to separate non-isomorphic pairs of graphs using vector spaces of polynomials that are set-wise invariant under permutations (so-called separating modules). We characterize the power of this method for distinguishing graphs under several different complexity measures: - We show that separating modules of "support-degree" $k$ are equivalent in power to the counts of $O(k)$-vertex subgraphs. - We show that separating modules of symmetric algebraic circuit size $n^{Θ(k)}$ are equivalent to $Θ(k)$-dimensional Weisfeiler-Leman. This generalizes and strengthens the result of Dawar & Wilsenach (CSL '18; ICALP '20; ACM Trans. Comput. Log., 2022; Theory Comput., 2025). - When considering only the representation-theoretic multiplicities of separating modules, we show that two graphs are separated by multiplicities if and only if their automorphism groups have different multiplicity of cycle types (cycle index). The latter result is notable in the analogy with GCT, as it is the only result we are aware of in which the multiplicity approach to separating isomorphism types of objects has been given an "intrinsic" characterization in terms of the objects themselves. We show that for graphs, multiplicity obstructions are stronger than occurrence obstructions. We also connect support size (from the study of WL) to complexity measures on $S_n$ (Dafni, Filmus, Lifshitz, Lindzey, & Vinyals, ITCS '21); as well as connections between invariant polynomials, the Graph Reconstruction Conjectures, and Forman's "invariants of finite type" (Adv. Math., 2004).

cs.CC↗