Reversing a (forward) computation history means undoing the history. In concurrent systems, undoing the history is not performed in a deterministic way but in a causally consistent fashion, where states that are reached during a backward computation are states that could have been reached during the computation history by just performing independent actions in a different order.
is therefore implemented by an artifact that binds the two inputs one after the other and then releases the output out of its own structure. Of course, if the second input never comes it must release the first input, because the first input may be legitimately used by some other operator. This means that the binding of the first input must be reversible, and the natural reversibility of reversible structures is exploited to achieve the correctness.
In order to bridge the gap between reversible process calculi and massive concurrent systems, we consider reversible structures where multiplicities are dropped (terms have multiplicity one) -the coherence constraint. Coherence in this strong sense is not realizable in well-mixed chemical solutions, but may become realizable in the future if we learn how to control individual molecules. We demonstrate that coherent reversible structures implement the asynchronous fragment of RCCS.
The exact distance between coherent and uncoherent reversible structures (that is, between reversible process calculi and massive systems) is manifested by the computational complexity of the reachability problem (verifying whether a configuration is reachable from an initial one). We demonstrate that reachability in coherent reversible structures has a computational complexity that is quadratic with respect to the size of the structures, a problem that is EXPSPACE-complete in generic structures.
Our study prompts a thorough analysis of reversible calculi where processes have multiplicities and the causal dependencies between copies may be exchanged. Open questions are (i) What synchronization schemas can be programmed in massive concurrent systems? (ii) Are there other constraints, different than coherence, such that relevant bio-chemical properties retains better algorithms than in standard structures? (iii) What is the theory of massive (reversible) systems with irreversible operators and what is the relationship with standard programming languages?
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