RevCRN: Reversible Analog Computation using Chemical Reaction Networks

The computability of real numbers and functions using Turing Machines has been a central area of theoretical computer science since the mid-20th century. In the late 20th century, it was shown that chemical reactions can serve as a basis for computation using the Chemical Reaction Network (CRN) model. Recent advances in computing real numbers using Deterministic Chemical Reaction Networks (DCRNs) have identified numerous classes of DCRN-computable real numbers. In parallel, the works of R. Landauer and C. H. Bennett, spanning the 1960s to the early 2000s, showed that reversible computing offers significant advantages over irreversible methods, particularly in energy efficiency, motivating extensive research on reversible computation. In this work, we investigate the computability of real numbers using Reversible Chemical Reaction Networks (RevCRNs). The paper has two primary contributions: (1) establishing relationships among CRN-computable real number classes including Lyapunov CRN (ℝLCRN), Real-Time CRN (ℝRTCRN), rational numbers (mathbbQ), and RevCRNs (ℝRevCRN), with key results: (i) mathbbQ is a strict subset of ℝRevCRN; (ii) the set of positive algebraic numbers (ALG), ℝLCRN, and real numbers computable by 1-species RevCRN (ℝRevCRN1s) are equal; (iii) ℝRTCRN and ℝRevCRN exhibit non-empty overlap; and (iv) the set of real numbers computable by detailed-balanced RevCRNs (ℝDetBalRevCRN) is a subset of ALG; and (2) exploring the existence of a hierarchy within ℝRevCRN. Finally, we leave open the exact relationship between ℝRevCRN and ℝRTCRN while conjecturing a general hierarchy of RevCRN-computable reals.