Natural Computing

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

Probability 1 computation with chemical reaction networks

Article

Abstract

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.

Keywords

Arithmetical hierarchy Chemical reaction network Deterministic computation Probabilistic computation 

References

  1. Angluin D, Aspnes J, Eisenstat D (2006) Stably computable predicates are semilinear. In: PODC 2006: Proceedings of the twenty-fifth annual ACM symposium on Principles of distributed computing. ACM Press, New York, NY, USA, pp 292–299Google Scholar
  2. Aspnes J, Ruppert E (2007) An introduction to population protocols. Bull Eur Assoc Theor Comput Sci 93:98–117MathSciNetMATHGoogle Scholar
  3. Brijder R (2014) Output stability and semilinear sets in chemical reaction networks and deciders. In: DNA 2014: Proceedings of the 20th international meeting on DNA computing and molecular programming, lecture notes in computer science. Springer, BerlinGoogle Scholar
  4. Cardelli L (2011) Strand algebras for DNA computing. Nat Comput 10(1):407–428MathSciNetCrossRefMATHGoogle Scholar
  5. Chen Y-J, Dalchau N, Srinivas N, Phillips A, Cardelli L, Soloveichik D, Seelig G (2013) Programmable chemical controllers made from DNA. Nat Nanotechnol 8(10):755–762CrossRefGoogle Scholar
  6. Chen H-L, Doty D, Soloveichik D (2014) Deterministic function computation with chemical reaction networks. Nat Comput 13(4):517–534 Preliminary version appeared in DNA 2012MathSciNetCrossRefMATHGoogle Scholar
  7. Cook M, Soloveichik D, Winfree E, Bruck J (2009) Programmability of chemical reaction networks. In: Condon A, Harel D, Kok JN, Salomaa A, Winfree E (eds) Algorithmic Bioprocess. Springer, Berlin, pp 543–584CrossRefGoogle Scholar
  8. Feller W (1968) An introduction to probability theory and its applications, vol 1. Wiley, New YorkMATHGoogle Scholar
  9. Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81(25):2340–2361CrossRefGoogle Scholar
  10. Karp RM, Miller RE(1969) Parallel program schemata. J Comput Syst Sci 3(2):147–195Google Scholar
  11. Mayr EW (1981) An algorithm for the general Petri net reachability problem. In: Proceedings of the thirteenth annual ACM symposium on theory of computing (STOC ’81). ACM, New York, NY, USA, pp 238–246Google Scholar
  12. Petri CA (1966) Communication with automata. Technical report, DTIC DocumentGoogle Scholar
  13. Rogers H Jr (1967) Theory of recursive functions and effective computability. McGraw-Hill series in higher mathematics. McGraw-Hill, New YorkGoogle Scholar
  14. Shoenfield JR (1959) On degrees of unsolvability. Ann Math 69(3):644–653MathSciNetCrossRefMATHGoogle Scholar
  15. Soare RI (2013) Interactive computing and relativized computability. In: Copeland BJ, Posy CJ, Shagrir O (eds) Computability: Turing, Gödel, Church, and beyond. MIT Press, Cambridge, pp 203–260Google Scholar
  16. Soloveichik D, Cook M, Winfree E, Bruck J (2008) Computation with finite stochastic chemical reaction networks. Nat Comput 7(4):615–633MathSciNetCrossRefMATHGoogle Scholar
  17. Soloveichik D, Seelig G, Winfree E (2010) DNA as a universal substrate for chemical kinetics. Proc Natl Acad Sci 107(12):5393 Preliminary version appeared in DNA 2008CrossRefGoogle Scholar
  18. Volterra V (1926) Variazioni e fluttuazioni del numero dindividui in specie animali conviventi. Mem Acad Lincei Roma 2:31–113MATHGoogle Scholar
  19. Zavattaro G, Cardelli L (2008) Termination problems in chemical kinetics. In: CONCUR 2008-concurrency theory, pp 477–491Google Scholar

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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