Reversibility in Massive Concurrent Systems

Proceedings of 5th W orkshop on Membrane Computing and Biologically Inspired Process Calculi (MeCBIC 2011) Pages 3–4, 2011. c  Luca Cardelli and Cosimo Lane ve All the rights to the paper remain with the authors. Reversibility in Massive Co ncurrent Systems Luca Cardelli Microsoft Research, Cambridge luca@micro soft.com Cosimo Lane ve Univ ersit ` a di Bologna laneve@cs. unibo.it Rev ersi ng a (forwa rd) computation history m eans undoi ng the history . In con current systems, undo- ing the histo ry is not performed in a deterministi c way b ut in a caus ally consi stent fashio n, w here states that are reac hed during a backwa rd computation are states that cou ld hav e been reache d during the com- putati on history by j ust pe rforming inde pendent actions in a dif feren t order . In R C C S , Danos and Kri vine achie ve this b y atta ching a memory m t o each process P , i n t he monit ored proces s construct m : P . Mem- ories in R C C S are stacks of info rmation ne eded for proce sses to backtrack . Alternati vely , Phillips and Ulido wski propose a techni que for rev ersin g process calculi without using memories . In this technique, the structure of proces ses is not destro yed and the progress is noted by underlinin g the action s that hav e been pe rformed. In o rder to tag th e communi cating proce sses, they generate uni que iden tifiers on-th e-fly during the communica tions. These found ational studies of rev ersible and concurre nt comp utation s ha ve been lar gely stimulate d by areas such as chemica l and biolog ical systems – called massive concurr ent systems in the f ollo wing – where oper ations are rev ersibl e, and only an appropriate injection of ener gy and/or a change of entrop y can mov e the computat ional system in a desired directi on. Ho wev er there is a mismatch between chemical and biolo gical systems and the abo ve conc urrent formalisms . In the latter ones, re ve rsibility means desynchr oniz ing pr ocesses that actually inter acte d in the pa st while, in massi ve concu rrent s ystems, rev ersib ility means r eversi bility of configu ratio ns . In order to make massi ve con current systems re ve rsible with the process calcu lus meaning, one has to remember the position a nd momentum o f each mole cule, whic h i s precisely contrary to the w ell-mixin g assumpti on of biochemical soups, namely that the probabilit y of collision between two m olecul es is independen t of their position ( cf. G illespi e’ s algo rithm). T o comply with the well-mixin g assumptio n, notions of causality and independ ence of e ven ts need to be ad apted to reflec t the fun damental fact tha t diffe rent proces ses of the same specie s are indistin- guisha ble. Their intera ctions can cause ef fects, bu t not to the poin t of being able to identify the precise molecule that caused an effe ct. W e introduce an algebra for m assi v e concurre nt sys tems, c alled re ver sible struct ur es , and, follo wing L ´ evy , we define an equi valen ce on computa tions that abstracts aw ay from the order of causally indepen dent red uctions – the permutation equivale nce . Because of multiplicit ies th is abstra ction does not alway s e xchang e ind epende nt reductio ns. F or exampl e, two reduct ions tha t use a same signal c annot be e xchan ged beca use one canno t grasp whether the two reduction s are competing on a same sign al or are using two d if ferent occurren ces of it. Notwithstandin g this inadequ acy , permutat ion equi valenc e in reve rsible structur es yields a standardi zation theorem that allows one to remov e con verse reduct ions from computa tions. Rev ersi ble struct ures may implement significant C C S -style inter action patter ns (Cardelli already no- ticed this by studying a class of re ve rsible systems – the D NA chemical syste ms). Conside r for exa mple a binary operat or that takes two input molecu les and produces one unrelated output molecule when (and only when ) both i nputs are present . It is too difficu lt to en gineer the input machin ery in o rder to any p os- sible pattern of interaction, and to produce the output m olecule out of their o wn struct ure. This operator is there fore implemented by an artif act that binds the two inputs one after the oth er and then releas es the outpu t out of its own st ructure. Of course, if the second input ne ver comes it must relea se the fi rst input , becaus e the first input may be le gitimate ly used by some oth er operat or . This means that the binding of the first input must be rev ersible , and the natural rev ersib ility of rev ersib le structures is exploi ted to achie ve the corre ctness. In o rder to br idge the gap betwee n re ver sible process calc uli and m assi v e concur rent systems, we consid er rev ersib le structure s where multiplicitie s are dropped (terms hav e multip licity one) – the c oher - ence constra int. Coherence in this strong sense is not realiza ble in well-mixed chemical solut ions, b ut may become realizab le in the futur e if we learn how to control in di vidual molecules. W e demonstrate that cohere nt re ver sible structures implement the asynchr onous fragment of R C C S . The exact distanc e between coherent an d unc oherent re versible st ructure s (that i s, be tween re versible proces s calcul i and massi v e systems) is manifested by the computational complexi ty of the reachab ility proble m (verify ing whether a configuratio n is r eachabl e from an in itial one ). W e demonstrate that r each- ability in coherent rev ersible struc tures has a computation al complex ity that is quadratic with respect to the size of the struct ures, a probl em that is E X P S P A C E -complete in generic struc tures. Our study prompts a thorough analys is of rev ersib le calculi where proces ses ha ve m ultipli cities and the causal d epende ncies bet ween copies may be exch anged. Open questio ns ar e (i) What synchron ization schemas can be programmed in massi ve concurren t sys tems? (ii ) Are there other const raints, dif ferent than coherence , such that rele v ant bio-chemical propert ies retains better algorithms than in standa rd structu res? (iii) What is the theory of massi ve (re versi ble) systems with irr ever sible operator s and what is the relatio nship with standar d prog ramming langu ages? 4