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Composable Computation in Discrete Chemical Reaction Networks

This paper explores the concept of obliviously-computable functions in discrete chemical reaction networks (CRNs) and their composition. It presents a classification theorem and a general construction method for obliviously-computable functions. The paper also discusses the limitations of output-oblivious CRNs and presents an open question regarding leaderless, output-oblivious CRNs.

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Composable Computation in Discrete Chemical Reaction Networks

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  1. Composable Computation in Discrete Chemical Reaction Networks min 2x Eric E. Severson (joint work with David Haley and David Doty) PODC: Principles of Distributed Computing Toronto, ON, July 20, 2019

  2. Acknowledgements Co-authors David Haley David Doty Special Thanks Anne Condon, Cameron Chalk, Niels Kornerup, Wyatt Reeves, David Soloveichik

  3. Discrete Chemical Reaction Network (CRN) Model • Finite set ofspeciesand finite set of reactions • Configuration: integer counts of species, changes by successive asynchronous reactions • Similar Models: Population Protocols, Petri Nets, Vector Addition Systems

  4. CRN Function Computation Input: Output: Stably computes Input: Output: Stably computes initial configuration to compute initial configuration to compute Stable Computation: Intuitively, correct # output with probability 1

  5. Goal: Function Composition • Rename intermediate species • Combine reactions min min 2x 2xmin min 2x 2x 2xmin

  6. When Composition Works min 2x Stably computes Input: Output: Output-oblivious Stably computes Input: Output: Since max 2x NOT Output-oblivious • Combine CRNs to compose functions • Works correctly upstream CRN is output-oblivious(output isn’t a reactant) reactions compete

  7. The Main Question Classify the obliviously-computable that can be stably computed by some output-obliviousCRN (doesn’t consume its output) obliviously-computable Modular CRN Design output-oblivious CRN to compute

  8. What functions can be stably computed? Theorem: is stably computable is semilinear Example semilinear function: Piecewise Affine Domains are semilinear sets (threshold / mod) [Ho-Lin Chen, David Doty, and David Soloveichik. Deterministic function computation with chemical reaction networks.  Natural Computing, 2014.] [Dana Angluin, James Aspnes, and David Eisenstat. Stably computable predicates are semilinear. PODC, 2006.]

  9. New Constraint for Oblivious Computation Obliviously-computable must be nondecreasing: ie. if , must consume to stably compute Other species To stably compute Other species Same reactions overproduce when computing

  10. 1D Case: Exact Characterization Theorem: is obliviously-computable is semilinear and nondecreasing • Representative example semilinear, nondecreasing • General construction which crucially uses initial leader (initial configuration contains )

  11. CRN Construction Input species: Output species: Leader: 1 Other species: Invariant: exactly 1 copy of red species in every reachable configuration Correct starting value Correct finite differences has seen copies of input Correct periodic differences has seen copies of , where 1 1 2 0 2 0 1 1

  12. Dimension Simple classification fails: is semilinear and nondecreasing, but NOT obliviously-computable No output-oblivious CRN (even with a leader) can compute max NOT Output-oblivious

  13. Max is not Obliviously-Computable Idea: Any CRN will overproduce General Impossibility Lemma: Lemma: Let , with sequence . If for allthere exists such that . Then is not obliviously-computable.

  14. Generalized Characterization Obliviously-computable Obliviously-computable 16

  15. Quilt-Affine Functions Linear function • Affine function: linear with constant offset • Quilt-affine function: linear with periodic offset • is obliviously-computable with a leader Periodic offset Linear function Periodic offset 17

  16. Main Result min of quilt-affine functions quilt- affine functions Theorem: is obliviously-computable is nondecreasing. quilt-affine and s.t. for all . [recursive condition] every fixed-input restriction fixing some input to a constant value is obliviously-computable (so is also eventually-min of quilt affine functions). quilt- affine arbitrary finite behavior • Obliviously-computable . 18

  17. Continuous Limit Agrees with the continuous output-oblivious CRN result [1]: is positive-continuous, piecewise-linear, superadditive min of linear functions when min of quilt-affine functions quilt- affine functions linear at linear at quilt- affine arbitrary finite behavior • The scaling limit is obliviously-computable by a continuous CRN. • Obliviously-computable . [1] Cameron Chalk, Niels Kornerup, Wyatt Reeves, and David Soloveichik. Composable rate-independent computation in continuous chemical reaction networks. In Computational Methods in Systems Biology, 2018. 19

  18. Open Question: Leaderless Output-Oblivious CRNs • Without a leader, obliviously-computable must be superadditive: for all inputs • Dimension is classified: • Dimension is open: Theorem: is leaderlessly obliviously-computable is semilinear, nondecreasing, and superadditive Conjecture: is leaderlessly obliviously-computable is obliviously-computable and superadditive

  19. Thank you for your attention. Questions?

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