Polynomial Equivalence of Extended Chemical Reaction Models
🤖 AI Summary
The computational impact of “zero-detection”—explicitly sensing when a species count reaches zero—remains open in chemical reaction networks (CRNs) and their equivalent models (Petri nets, vector addition systems), particularly regarding whether common extensions (inhibition, priority, synchronization) genuinely enhance computational power.
Method: We introduce void-generation CRNs, a novel CRN variant that directly and explicitly models zero-count detection, enabling formal analysis of this capability.
Contribution/Results: We prove polynomial-time mutual simulation between void-generation CRNs and several strongly extended CRN models (e.g., with inhibition or priority). This equivalence reveals that zero-detection is not an incidental feature but the shared computational core underlying these extensions—unifying their expressive power beyond standard Turing completeness. Consequently, zero-detection constitutes the essential computational primitive enabling these extensions, rather than conferring additional advantage.
Application Category
📝 Abstract
The ability to detect whether a species (or dimension) is zero in Chemical Reaction Networks (CRN), Vector Addition Systems, or Petri Nets is known to increase the power of these models -- making them capable of universal computation. While this ability may appear in many forms, such as extending the models to allow transitions to be inhibited, prioritized, or synchronized, we present an extension that directly performs this zero checking. We introduce a new void genesis CRN variant with a simple design that merely increments the count of a specific species when any other species' count goes to zero. As with previous extensions, we show that the model is Turing Universal. We then analyze several other studied CRN variants and show that they are all equivalent through a polynomial simulation with the void genesis model, which does not merely follow from Turing-universality. Thus, inhibitor species, reactions that occur at different rates, being allowed to run reactions in parallel, or even being allowed to continually add more volume to the CRN, does not add additional simulation power beyond simply detecting if a species count becomes zero.
Problem
Research questions and friction points this paper is trying to address.
Extending chemical reaction networks to detect zero species counts
Comparing computational power of various CRN extensions through equivalence
Demonstrating Turing universality via void genesis model simulation
Innovation
Methods, ideas, or system contributions that make the work stand out.
Void genesis CRN for zero detection
Polynomial equivalence across CRN variants
Turing universal through species monitoring
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