Learning Reductions to Sparse Sets

  • Harry Buhrman
  • Lance Fortnow
  • John M. Hitchcock
  • Bruno Loff
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8087)


We study the consequences of NP having non-uniform polynomial size circuits of various types. We continue the work of Agrawal and Arvind [1] who study the consequences of Sat being many-one reducible to functions computable by non-uniform circuits consisting of a single weighted threshold gate. (Sat \(\leq_m^p \mathrm{LT}_1\)). They claim that P= NP follows as a consequence, but unfortunately their proof was incorrect.

We take up this question and use results from computational learning theory to show that if Sat \(\leq_m^p \mathrm{LT}_1\) then PH = PNP.

We furthermore show that if Sat disjunctive truth-table (or majority truth-table) reduces to a sparse set then Sat \(\leq_m^p\) LT1 and hence a collapse of PH to PNP also follows. Lastly we show several interesting consequences of Sat \(\leq_{dtt}^p\) SPARSE.


Binary String Satisfying Assignment Computational Learning Theory Polynomial Time Hierarchy Polynomial Size Circuit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Harry Buhrman
    • 1
  • Lance Fortnow
    • 2
  • John M. Hitchcock
    • 3
  • Bruno Loff
    • 4
  1. 1.CWI and University of AmsterdamThe Netherlands
  2. 2.Northwestern UniversityUSA
  3. 3.University of WyomingUSA
  4. 4.CWIThe Netherlands

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