Document Open Access Logo

Optimal Information Encoding in Chemical Reaction Networks

Authors Austin Luchsinger , David Doty , David Soloveichik

PDF
Thumbnail PDF

File

LIPIcs.DNA.29.9.pdf
  • Filesize: 0.85 MB
  • 16 pages

Document Identifiers

Author Details

Austin Luchsinger
  • Electrical and Computer Engineering, University of Texas at Austin, TX, USA
David Doty
  • Computer Science, University of California-Davis, CA, USA
David Soloveichik
  • Electrical and Computer Engineering, University of Texas at Austin, TX, USA

Funding

  • Luchsinger, Austin: Carroll H. Dunn Endowed Graduate Fellowship in Engineering.
  • Doty, David: NSF grants 2211793, 1900931, and CAREER-1844976.
  • Soloveichik, David: NSF grant 1901025, Sloan Foundation Research Fellowship.

Acknowledgements

The authors thank Eric Severson for the invaluable discussions on chemical reaction network computation that helped lead to the results presented here.

Cite AsGet BibTex

Austin Luchsinger, David Doty, and David Soloveichik. Optimal Information Encoding in Chemical Reaction Networks. In 29th International Conference on DNA Computing and Molecular Programming (DNA 29). Leibniz International Proceedings in Informatics (LIPIcs), Volume 276, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.DNA.29.9

Abstract

Discrete chemical reaction networks formalize the interactions of molecular species in a well-mixed solution as stochastic events. Given their basic mathematical and physical role, the computational power of chemical reaction networks has been widely studied in the molecular programming and distributed computing communities. While for Turing-universal systems there is a universal measure of optimal information encoding based on Kolmogorov complexity, chemical reaction networks are not Turing universal unless error and unbounded molecular counts are permitted. Nonetheless, here we show that the optimal number of reactions to generate a specific count x ∈ ℕ with probability 1 is asymptotically equal to a "space-aware" version of the Kolmogorov complexity of x, defined as K̃s(x) = min_p {|p|/log|p| + log(space(𝒰(p))) : 𝒰(p) = x}, where p is a program for universal Turing machine 𝒰. This version of Kolmogorov complexity incorporates not just the length of the shortest program for generating x, but also the space usage of that program. Probability 1 computation is captured by the standard notion of stable computation from distributed computing, but we limit our consideration to chemical reaction networks obeying a stronger constraint: they "know when they are done" in the sense that they produce a special species to indicate completion. As part of our results, we develop a module for encoding and unpacking any b bits of information via O(b/log{b}) reactions, which is information-theoretically optimal for incompressible information. Our work provides one answer to the question of how succinctly chemical self-organization can be encoded - in the sense of generating precise molecular counts of species as the desired state.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
Keywords
  • chemical reaction networks
  • Kolmogorov complexity
  • stable computation

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Leonard Adleman, Qi Cheng, Ashish Goel, and Ming-Deh Huang. Running time and program size for self-assembled squares. In Proceedings of the thirty-third annual ACM symposium on Theory of computing, pages 740-748, 2001. Google Scholar
  2. Dan Alistarh, James Aspnes, and Rati Gelashvili. Space-optimal majority in population protocols. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2221-2239. SIAM, 2018. Google Scholar
  3. Eric Allender, Michal Kouckỳ, Detlef Ronneburger, and Sambuddha Roy. The pervasive reach of resource-bounded Kolmogorov complexity in computational complexity theory. Journal of Computer and System Sciences, 77(1):14-40, 2011. Google Scholar
  4. Dana Angluin, James Aspnes, Zoë Diamadi, Michael J Fischer, and René Peralta. Computation in networks of passively mobile finite-state sensors. In Proceedings of the twenty-third annual ACM symposium on Principles of distributed computing, pages 290-299, 2004. Google Scholar
  5. Dana Angluin, James Aspnes, and David Eisenstat. Stably computable predicates are semilinear. In Proceedings of the twenty-fifth annual ACM symposium on Principles of distributed computing, pages 292-299, 2006. Google Scholar
  6. Petra Berenbrink, George Giakkoupis, and Peter Kling. Optimal time and space leader election in population protocols. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pages 119-129, 2020. Google Scholar
  7. Michael Blondin, Javier Esparza, Blaise Genest, Martin Helfrich, and Stefan Jaax. Succinct population protocols for presburger arithmetic. arXiv preprint arXiv:1910.04600, 2019. Google Scholar
  8. Michael Blondin, Javier Esparza, and Stefan Jaax. Large flocks of small birds: on the minimal size of population protocols. arXiv preprint arXiv:1801.00742, 2018. Google Scholar
  9. Allan Borodin. On relating time and space to size and depth. SIAM journal on computing, 6(4):733-744, 1977. Google Scholar
  10. Daniele Cappelletti, Andrés Ortiz-Muñoz, David F Anderson, and Erik Winfree. Stochastic chemical reaction networks for robustly approximating arbitrary probability distributions. Theoretical Computer Science, 801:64-95, 2020. Google Scholar
  11. E Cardoza, R Lipton, and Albert R Meyer. Exponential space complete problems for Petri nets and commutative semigroups (preliminary report). In Proceedings of the eighth annual ACM symposium on Theory of computing, pages 50-54, 1976. Google Scholar
  12. Cameron Chalk, Niels Kornerup, Wyatt Reeves, and David Soloveichik. Composable rate-independent computation in continuous chemical reaction networks. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 18(1):250-260, 2019. Google Scholar
  13. Ho-Lin Chen, David Doty, and David Soloveichik. Deterministic function computation with chemical reaction networks. Natural computing, 13:517-534, 2014. Google Scholar
  14. Ben Chugg, Hooman Hashemi, and Anne Condon. Output-oblivious stochastic chemical reaction networks. In 22nd International Conference on Principles of Distributed Systems (OPODIS 2018). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2018. Google Scholar
  15. Rachel Cummings, David Doty, and David Soloveichik. Probability 1 computation with chemical reaction networks. Natural Computing, 15(2):245-261, 2016. Special issue of invited papers from DNA 2014. URL: https://doi.org/10.1007/s11047-015-9501-x.
  16. Philipp Czerner. Leaderless population protocols decide double-exponential thresholds. arXiv preprint arXiv:2204.02115, 2022. Google Scholar
  17. Philipp Czerner and Javier Esparza. Lower bounds on the state complexity of population protocols. In Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing, pages 45-54, 2021. Google Scholar
  18. Philipp Czerner, Roland Guttenberg, Martin Helfrich, and Javier Esparza. Fast and succinct population protocols for presburger arithmetic. arXiv preprint arXiv:2202.11601, 2022. Google Scholar
  19. David Doty, Mahsa Eftekhari, Leszek Gąsieniec, Eric Severson, Przemyslaw Uznański, and Grzegorz Stachowiak. A time and space optimal stable population protocol solving exact majority. In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS), pages 1044-1055. IEEE, 2022. Google Scholar
  20. Javier Esparza and Mogens Nielsen. Decidability issues for Petri nets - a survey. Journal of Information Processes and Cybernetics, 3:143-160, 1994. Google Scholar
  21. Daniel T Gillespie. Stochastic simulation of chemical kinetics. Annu. Rev. Phys. Chem., 58:35-55, 2007. Google Scholar
  22. Richard M Karp and Raymond E Miller. Parallel program schemata. Journal of Computer and system Sciences, 3(2):147-195, 1969. Google Scholar
  23. Ulla Koppenhagen and Ernst W Mayr. Optimal algorithms for the coverability, the subword, the containment, and the equivalence problems for commutative semigroups. Information and Computation, 158(2):98-124, 2000. Google Scholar
  24. Marvin Künnemann, Filip Mazowiecki, Lia Schütze, Henry Sinclair-Banks, and Karol Węgrzycki. Coverability in VASS revisited: Improving Rackoff’s bound to obtain conditional optimality. arXiv preprint arXiv:2305.01581, 2023. Google Scholar
  25. Derrick H Lehmer. Teaching combinatorial tricks to a computer. In Proc. Sympos. Appl. Math. Combinatorial Analysis, volume 10, pages 179-193, 1960. Google Scholar
  26. Jérôme Leroux. State complexity of protocols with leaders. In Proceedings of the 2022 ACM Symposium on Principles of Distributed Computing, pages 257-264, 2022. Google Scholar
  27. Ming Li, Paul Vitányi, et al. An introduction to Kolmogorov complexity and its applications, volume 3. Springer, 2008. Google Scholar
  28. Richard Lipton. The reachability problem requires exponential space. Research Report 62. Department of Computer Science, Yale University, 1976. Google Scholar
  29. Ernst W Mayr and Albert R Meyer. The complexity of the word problems for commutative semigroups and polynomial ideals. Advances in mathematics, 46(3):305-329, 1982. Google Scholar
  30. Marvin Lee Minsky. Computation. Prentice-Hall Englewood Cliffs, 1967. Google Scholar
  31. Charles Rackoff. The covering and boundedness problems for vector addition systems. Theoretical Computer Science, 6(2):223-231, 1978. Google Scholar
  32. Walter J Savitch. Relationships between nondeterministic and deterministic tape complexities. Journal of computer and system sciences, 4(2):177-192, 1970. Google Scholar
  33. Rich Schroeppel. A two counter machine cannot calculate 2^N. Technical report, Massachusetts Institute Of Technology, Artificial Intelligence Lab, 1972. Google Scholar
  34. Robert Sedgewick. Permutation generation methods. ACM Computing Surveys (CSUR), 9(2):137-164, 1977. Google Scholar
  35. Eric E Severson, David Haley, and David Doty. Composable computation in discrete chemical reaction networks. In Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing, pages 14-23, 2019. Google Scholar
  36. David Soloveichik, Matthew Cook, Erik Winfree, and Jehoshua Bruck. Computation with finite stochastic chemical reaction networks. natural computing, 7:615-633, 2008. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact