Open Problems

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The following are a list of open problems in self-assembly:

1) In his 1998 thesis [1], Winfree showed that the class of directed aTAM systems at temperature 2 is computationally universal, and in [2] Cook et al. showed that undirected temperature 1 aTAM systems could perform computations with a given amount of certainty. They also showed that in 2 planes directed aTAM temperature 1 systems are computationally universal. Is the class of directed aTAM systems at temperature 1 in the plane computationally universal? In [3] Doty, Patitz, and Summers provide deep insights into this question.

2) In [4], Doty et al. introduced a model known as the fuzzy temperature model and showed that in this model n by n squares could efficiently self-assemble. Can systems in this model also perform general computation?

3) In [5] Doty et al. showed that the aTAM is intrinsically universal for itself, but for directed systems the intrinsically universal tile set fundamentally relies on nondeterminism. Is the class of directed aTAM systems intrinsically universal for itself?

4) Is the STAM intrinsically universal for itself?

5) Is the class of aTAM systems at temperature 1 intrinsically universal for itself?

6) In his 1998 thesis [1], Winfree introduced both the aTAM and its more practical counterpart the kTAM. Currently, the 2HAM does not have a more realistic counterpart. Formulate a "k2HAM" model which takes into account the size of assemblies binding together (the bigger the assemblies the less likely it is they will bind together) and the lack of rigidity when assemblies come together.

7) In Self-Assembly of Discrete Self-Similar Fractals, Patitz and Summers proved several results about self-assembling discrete self-similar fractals, but it is still an open questions as to whether or not there exists a discrete self-similar fractal that can be self-assembled in the aTAM.


References

  1. 1.0 1.1 Erik Winfree - Algorithmic Self-Assembly of DNA
    Ph.D. Thesis, California Institute of Technology , June 1998
    Bibtex
    Author : Erik Winfree
    Title : Algorithmic Self-Assembly of DNA
    In : Ph.D. Thesis, California Institute of Technology -
    Address :
    Date : June 1998
  2. Matthew Cook, Yunhui Fu, Robert T. Schweller - Temperature 1 Self-Assembly: Deterministic Assembly in 3{D} and Probabilistic Assembly in 2{D}
    SODA 2011: Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms ,2011
    Bibtex
    Author : Matthew Cook, Yunhui Fu, Robert T. Schweller
    Title : Temperature 1 Self-Assembly: Deterministic Assembly in 3{D} and Probabilistic Assembly in 2{D}
    In : SODA 2011: Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms -
    Address :
    Date : 2011
  3. David Doty, Matthew J. Patitz, Scott M. Summers - Limitations of Self-Assembly at Temperature 1
    Theoretical Computer Science 412:145-158,2011
    Bibtex
    Author : David Doty, Matthew J. Patitz, Scott M. Summers
    Title : Limitations of Self-Assembly at Temperature 1
    In : Theoretical Computer Science -
    Address :
    Date : 2011
  4. David Doty, Matthew J. Patitz, Dustin Reishus, Robert T. Schweller, Scott M. Summers - Strong Fault-Tolerance for Self-Assembly with Fuzzy Temperature
    Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010) pp. 417--426,2010
    Bibtex
    Author : David Doty, Matthew J. Patitz, Dustin Reishus, Robert T. Schweller, Scott M. Summers
    Title : Strong Fault-Tolerance for Self-Assembly with Fuzzy Temperature
    In : Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010) -
    Address :
    Date : 2010
  5. David Doty, Jack H. Lutz, Matthew J. Patitz, Robert T. Schweller, Scott M. Summers, Damien Woods - The tile assembly model is intrinsically universal
    Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science ,2012
    Bibtex
    Author : David Doty, Jack H. Lutz, Matthew J. Patitz, Robert T. Schweller, Scott M. Summers, Damien Woods
    Title : The tile assembly model is intrinsically universal
    In : Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science -
    Address :
    Date : 2012