Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

A FORM OF FEASIBLE INTERPOLATION FOR CONSTANT DEPTH FREGE SYSTEMS

JAN KRAJÍČEK
The Journal of Symbolic Logic
Vol. 75, No. 2 (JUNE 2010), pp. 774-784
Stable URL: http://www.jstor.org/stable/25676806
Page Count: 11
  • Read Online (Free)
  • Download ($10.00)
  • Subscribe ($19.50)
  • Cite this Item
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
A FORM OF FEASIBLE INTERPOLATION FOR CONSTANT DEPTH FREGE SYSTEMS
Preview not available

Abstract

Let L be a first-order language and Φ and ψ two $\Sigma _{1}^{1}$ L-sentences that cannot be satisfied simultaneously in any finite L-structure. Then obviously the following principle Chain L,Φ,ψ (n,m) holds: For any chain of finite L-structures C 1 ,...,C m with the universe [n] one of the following conditions must fail: 1. $C_{1}\vDash \Phi $ , 2. C i ≅ C i+1 , for i = 1,...,m — 1, 3. $C_{m}\vDash \Psi $ . For each fixed L and parameters n,m the principle Chain L,Φ,ψ (n,m) can be encoded into a propositional DNF formula of size polynomial in n,m. For any language L containing only constants and unary predicates we show that there is a constant c L such that the following holds: If a constant depth Frege system in DeMorgan language proves Chain L,Φ,ψ (n, c L · n) by a size s proof then the class of finite L-structures with universe [n] satisfying Φ can be separated from the class of those L-structures on [n] satisfying ψ by a depth 3 formula of size $2^{{\rm log}(s)^{O(1)}}$ and with bottom fan-in ${\rm log}(s)^{O(1)}$ .

Page Thumbnails

  • Thumbnail: Page 
774
    774
  • Thumbnail: Page 
775
    775
  • Thumbnail: Page 
776
    776
  • Thumbnail: Page 
777
    777
  • Thumbnail: Page 
778
    778
  • Thumbnail: Page 
779
    779
  • Thumbnail: Page 
780
    780
  • Thumbnail: Page 
781
    781
  • Thumbnail: Page 
782
    782
  • Thumbnail: Page 
783
    783
  • Thumbnail: Page 
784
    784