### Abstract

The interpolation method has been one of the main tools for proving lower bounds for propositional proof systems. Loosely speaking, if one can prove that a particular proof system has the feasible interpolation property, then a generic reduction can (usually) be applied to prove lower bounds for the proof system, sometimes assuming a (usually modest) complexity-theoretic assumption. In this paper, we show that this method cannot be used to obtain lower bounds for Frege systems, or even for TC^{0}-Frege systems. More specifically, we show that unless factoring is feasible, neither Frege nor TC^{0}-Frege has the feasible interpolation property. In order to carry out our argument, we show how to carry out proofs of many elementary axioms/theorems of arithmetic in polynomial-size TC^{0}-Frege. In particular, we show how to carry out the proof for the Chinese Remainder Theorem, which may be of independent interest. As a corollary, we obtain that TC^{0}-Frege as well as any proof system that polynomially simulates it, is not automatizable (under a hardness assumption).

Original language | English (US) |
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Pages (from-to) | 254-263 |

Number of pages | 10 |

Journal | Annual Symposium on Foundations of Computer Science - Proceedings |

State | Published - Dec 1 1997 |

Externally published | Yes |

Event | Proceedings of the 1997 38th IEEE Annual Symposium on Foundations of Computer Science - Miami Beach, FL, USA Duration: Oct 20 1997 → Oct 22 1997 |

### All Science Journal Classification (ASJC) codes

- Hardware and Architecture

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## Cite this

^{0}-Frege proofs.

*Annual Symposium on Foundations of Computer Science - Proceedings*, 254-263.