Book ChapterDOI
Some Results on Interactive Proofs for Real Computations
Martijn Baartse,Klaus Meer +1 more
- pp 107-116
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This work introduces interactive proofs in the full BSS model and proves an upper bound for the class IP, giving rise to the conjecture that a characterization of IP will not be given via one of the real complexity classes, PAT or PAR.Abstract:
We study interactive proofs in the framework of real number complexity theory as introduced by Blum, Shub, and Smale. Shamir’s famous result characterizes the class IP as PSPACE or, equivalently, as PAT and PAR in the Turing model. Since space resources alone are known not to make much sense in real number computations the question arises whether IP can be similarly characterized by one of the latter classes. Ivanov and de Rougemont [9] started this line of research showing that an analogue of Shamir’s result holds in the additive Blum-Shub-Smale model of computation when only Boolean messages can be exchanged. Here, we introduce interactive proofs in the full BSS model. As main result we prove an upper bound for the class \(\mathrm{IP}_{{\mathbb R}}\). It gives rise to the conjecture that a characterization of \(\mathrm{IP}_{{\mathbb R}}\) will not be given via one of the real complexity classes \(\mathrm{PAR}_{{\mathbb R}}\) or \(\mathrm{PAT}_{{\mathbb R}}\). We report on ongoing approaches to prove as well interesting lower bounds for \(\mathrm{IP}_{{\mathbb R}}\).read more
Citations
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Proceedings ArticleDOI
Real Interactive Proofs for VPSPACE.
Martijn Baartse,Klaus Meer +1 more
TL;DR: Interactive real protocols for a large class of functions introduced by Koiran and Perifel and denoted by UniformVSPACE^0 are designed and it is shown PAR_R is a subset of IP_R, which in particular implies co-NP_R has been identified, and P_R^{Res} is a subsets of IP-R, where Res denotes certain multivariate Resultant polynomials.
Journal ArticleDOI
Interactive proofs and a Shamir-like result for real number computations
Martijn Baartse,Klaus Meer +1 more
TL;DR: In this article, interactive proofs in the framework of real number computations were introduced and studied in the full BSS model in which also multiplications can be performed and reals can be exchanged.
References
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MonographDOI
Computational Complexity: A Modern Approach
Sanjeev Arora,Boaz Barak +1 more
TL;DR: This beginning graduate textbook describes both recent achievements and classical results of computational complexity theory and can be used as a reference for self-study for anyone interested in complexity.
Book
Complexity and Real Computation
TL;DR: This chapter discusses decision problems and Complexity over a Ring and the Fundamental Theorem of Algebra: Complexity Aspects.
Journal ArticleDOI
Ip = pspace
TL;DR: It is proven that when both randomization and interaction are allowed, the proofs that can be verified in polynomial time are exactly those proofs that could be generated with polynometric space.
Journal ArticleDOI
Algebraic methods for interactive proof systems
TL;DR: This technique is used to prove that every language in the polynomial-time hierarchy has an interactive proof system and played a pivotal role in the recent proofs that IP = PSPACE and MIP = NEXP.