Linearity and Nonlinearity in Distributed Computation

Abstract

The copying of processes is limited in the context of distributed computation, either as a fact of life, often because remote networks are simply too complicated to have control over, or deliberately, as in the design of security protocols. Roughly, linearity is about how to manage without a presumed ability to copy. The meaning and mathematical consequences of linearity are studied for path-based models of processes which are also models of affine-linear logic. This connection yields an affine-linear language for processes in which processes are typed according to the kind of computation paths they can perform. One consequence is that the affine-linear language automatically respects open-map bisimulation. A range of process operations (from CCS, CCS with process-passing, mobile ambients, and dataflow) can be expressed within the affine-linear language showing the ubiquity of linearity. Of course, process code can be sent explicitly to be copied. Following the discipline of linear logic, suitable nonlinear maps are obtained as linear maps whose domain is under an exponential. Different ways to make assemblies of processes lead to different choices of exponential; the nonlinear maps of only some of which will respect bisimulation. IntroductionIn the film "Groundhog Day" the main character comes to relive and remember the same day repeatedly, until finally he gets it right (and the girl). Real life isn't like that. The world moves on and we cannot rehearse, repeat or reverse the effects of the more important decisions we take. 2 Glynn WinskelAs computation becomes increasingly distributed and interactive the more it resembles life in this respect, and the more difficult or impossible it is for a state of computation to be frozen and copied, so that interaction is most often conducted in a single irreversible run. Mathematically, this amounts to a form of linearity, as it is understood within models of linear logic.Although it can be hard for processes to copy processes, it is generally easy for processes to ignore other processes. For this reason distributed computation often involves affine-linear maps.To get an idea of the nature of affine-linear maps, imagine a network of interacting processes with a single hole, into which a process, the input process, may be plugged to form a complete process, the output process. The network with the hole represents an affine-linear map; given an input process, one obtains an output process.The input process cannot be copied, but can be ignored. Consequently, any computation path (or run) of the output process will have depended on at most one computation path (or run) of the input. This property of affine-linear maps rests on an understanding of the nature of the computation paths of a process. The property can be simplified provided we understand a computation path of the input processes to also admit the empty computation path-a computation path obtained even when the input process is ignored. Any output got while ignoring the input process, results from the empty...

Profunctors, open maps and bisimulation

“…Functors − : P → P ⊥ are a form of prefixing operation, as prevalent in process calculi. (Lifting constitutes the basic prefix operation in the presheaf semantics of affine HOPLA, the higher order affine language in [40], and underlies the semantics of many essentially affine process languages [51,13,52,53].) They also play a key role in harnessing open map preservation in Prof to connected colimit preserving functors.…”

“…But, in general, F Ω (P&Q) and F Ω (P) × F Ω (Q) are not isomorphic (the analogue of the Seely condition [47] is not met), so that (F Ω (Q)) op × R is not a function space for the polynomials with respect to −&−. (This example is dealt with in more detail in [39,53].) Example 9.7 Now consider the full subcategory of sets F consisting of all finite sets with functions as arrows.…”

“…Functors − : P → P ⊥ are a form of prefixing operation, which are prevalent in process calculi. (Lifting constitutes the basic prefix operation in the presheaf semantics of affine HOPLA, the higher order affine language in Nygaard and Winskel (2004), and underlies the semantics of many essentially affine process languages (Winskel 1996;Winskel 1999;Winskel 2004) Proof. In relation to Lemma 2.6, we have the following situation:…”

Profunctors, Open Maps and Bisimulation

“…(Lifting constitutes the basic prefix operation in the presheaf semantics of affine HOPLA, the higher order affine language in [40], and underlies the semantics of many essentially affine process languages [51,13,52,53].) They also play a key role in harnessing open map preservation in Prof to connected colimit preserving functors.…”

“…(This example is dealt with in more detail in [39,53].) Example 9.7 Now consider the full subcategory of sets F consisting of all finite sets with functions as arrows.…”