Natural Computing

, Volume 15, Issue 2, pp 245–261 | Cite as

Probability 1 computation with chemical reaction networks



The computational power of stochastic chemical reaction networks (CRNs) varies significantly with the output convention and whether or not error is permitted. Focusing on probability 1 computation, we demonstrate a striking difference between stable computation that converges to a state where the output cannot change, and the notion of limit-stable computation where the output eventually stops changing with probability 1. While stable computation is known to be restricted to semilinear predicates (essentially piecewise linear), we show that limit-stable computation encompasses the set of predicates \(\phi :{\mathbb {N}}\rightarrow \{0,1\}\) in \(\Delta ^0_2\) in the arithmetical hierarchy (a superset of Turing-computable). In finite time, our construction achieves an error-correction scheme for Turing universal computation. We show an analogous characterization of the functions \(f:{\mathbb {N}}\rightarrow {\mathbb {N}}\) computable by CRNs with probability 1, which encode their output into the count of a certain species. This work refines our understanding of the tradeoffs between error and computational power in CRNs.


Arithmetical hierarchy Chemical reaction network Deterministic computation Probabilistic computation 



We thank Shinnosuke Seki, Chris Thachuk, and Luca Cardelli for many useful and insightful discussions. The first author was supported by NSF Grants CCF-1049899 and CCF-1217770, the second author was supported by NSF Grants CCF-1219274 and CCF-1162589 and the Molecular Programming Project under NSF Grant 1317694, and the third author was supported by NIGMS Systems Biology Center Grant P50 GM081879.


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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Rachel Cummings
    • 1
  • David Doty
    • 2
  • David Soloveichik
    • 3
  1. 1.Northwestern UniversityEvanstonUSA
  2. 2.California Institute of TechnologyPasadenaUSA
  3. 3.University of California, San FranciscoSan FranciscoUSA

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