## On IP=PSPACE and theorems with narrow proofs

Venue: | EATCS Bulletin |

Citations: | 1 - 0 self |

### BibTeX

@ARTICLE{Hartmanis_onip=pspace,

author = {Juris Hartmanis and Richard Chang and Desh Ranjan and Pankaj Rohatgi},

title = {On IP=PSPACE and theorems with narrow proofs},

journal = {EATCS Bulletin},

year = {},

volume = {41},

pages = {166--174}

}

### OpenURL

### Abstract

It has been shown that the class of languages with interactive proofs, IP, is exactly the class PSPACE. This surprising result elegantly places IP in the standard classification of feasible computations. Furthermore, the IP = PSPACE result reveals some very interesting and unsuspected properties of mathematical proofs. In this column we define the width of a proof in a formal system F and show that it is an intuitively satisfying and robust definition. Then, using the IP = PSPACE result, it is seen that the width of a proof (as opposed to the length) determines how quickly one can give overwhelming evidence that a theorem is provable without showing the full proof. 1 On Proofs and Interactive Proofs A mathematician has the most confidence in the truth of a theorem when he/she is given a complete proof of the theorem in a trusted formal system. Let F be such a formal system in which the correctness of a proof can be checked by a verifier in polynomial time. The class NP clearly captures all the theorems which have polynomially long proofs. The NP =? P question is the question about the quantitative computational difference between finding a proof of a theorem and checking the correctness of a given proof. Some years ago, theoretical computer scientists asked whether it is possible to give convincing evidence that a theorem is provable in F without showing a complete

### Citations

405 | Non-Deterministic Exponential Time has TwoProver Interactive Protocols
- Babai, Fortnow, et al.
- 1990
(Show Context)
Citation Context ...Nisan [LFKN90] showed that IP actually contains the entire Polynomial Hierarchy. This result then led Shamir [Sha90] to completely characterize IP by showing that IP = PSPACE. Babai, Fortnow and Lund =-=[BFL90]-=- characterized the computational power of multiprover interactive protocols MIP = NEXP. In both cases, it is interesting to see that interactive proof systems provide alternative definitions of classi... |

309 | Algebraic Methods for Interactive Proof Systems
- Lund, Fortnow, et al.
- 1990
(Show Context)
Citation Context ...t contained in IP, because there are oracle worlds where co-NP �⊆ IP [FS88]. In fact, the computational power of interactive protocols was not fully appreciated until Lund, Fortnow, Karloff and Nisan =-=[LFKN90]-=- showed that IP actually contains the entire Polynomial Hierarchy. This result then led Shamir [Sha90] to completely characterize IP by showing that IP = PSPACE. Babai, Fortnow and Lund [BFL90] charac... |

164 |
Proofs that yield nothing but their validity and a methodology of cryptographic protocol design
- Goldreich, Micali, et al.
- 1986
(Show Context)
Citation Context ...guage not known to be in NP. Consider GNI, the set of pairs of graphs that are not isomorphic. GNI is known to be in co-NP and believed not to be in NP. However, GNI does have an interactive protocol =-=[GMW86]-=-. For small graphs, the Verifier can easily determine if the two graphs are not isomorphic. For sufficiently large graphs, the Verifier solicits help from the Prover to show that Gi and Gj are not iso... |

44 |
Ip = pspace
- Shamir
- 1992
(Show Context)
Citation Context ... power of interactive protocols was not fully appreciated until Lund, Fortnow, Karloff and Nisan [LFKN90] showed that IP actually contains the entire Polynomial Hierarchy. This result then led Shamir =-=[Sha90]-=- to completely characterize IP by showing that IP = PSPACE. Babai, Fortnow and Lund [BFL90] characterized the computational power of multiprover interactive protocols MIP = NEXP. In both cases, it is ... |

38 | Are there interactive protocols for coNP languages
- Fortnow, Sipser
- 1988
(Show Context)
Citation Context ... be incomplete for co-NP. So, the preceding discussion does not show that co-NP ⊆ IP. For a while, it was believed that co-NP is not contained in IP, because there are oracle worlds where co-NP �⊆ IP =-=[FS88]-=-. In fact, the computational power of interactive protocols was not fully appreciated until Lund, Fortnow, Karloff and Nisan [LFKN90] showed that IP actually contains the entire Polynomial Hierarchy. ... |

12 |
The knowledge-complexity of interactive proof systems
- Goldwasser, Micali, et al.
- 1989
(Show Context)
Citation Context ...obability that a given theorem is provable without seeing the whole proof? and how rapidly can this be done? This problem has been formulated and extensively studied in terms of interactive protocols =-=[Gol89]-=-. Informally, an interactive protocol consists of a Prover and a Verifier. The Prover is an all powerful Turing Machine (TM) and the Verifier is a TM which operates in time polynomial in the length of... |

12 |
Structural complexity theory: Recent surprises
- Hartmanis, Chang, et al.
- 1990
(Show Context)
Citation Context ...t problems with contradictory relativizations are beyond our proof techniques. The IP = PSPACE result also provides a very dramatic counterexample to the already battered Random Oracle Hypothesis. In =-=[HCRR90]-=-, we showed that ProbA[ IP A �= PSPACE A ] = 1. 2 On Theorems with Polynomially Wide Proofs In this section, we define the notion of the width of a proof of a theorem in a formal system. The notion of... |

3 |
Two characterizations of the context sensitive languages
- Fischer
- 1969
(Show Context)
Citation Context ...of the theorem is equivalent to restricting the size of the instantaneous descriptions of a Turing machine computation to be polynomial in the size of the input string. But, this is just PSPACE. (See =-=[Fis69]-=- for related results.) Theorem 5 PWT = PSPACE. Proof: Let L ∈ PWT. Then, using our first definition of “formal system”, there exist a constant k and a formal system with a DFA proof checker, DF, such ... |