Linearity in Process Languages

Abstract

The meaning and mathematical consequences of linearity (managing without a presumed ability to copy) are studied for a path-based model of processes which is also a model of affine-linear logic. This connection yields an affine-linear language for processes, automatically respecting open-map bisimulation, in which a range of process operations can be expressed. An operational semantics is provided for the tensor fragment of the language. Different ways to make assemblies of processes lead to different choices of exponential, some of which respect bisimulation. Path-based models of processesIn distributed computation it can be hard or impossible for a process to copy a process, while it is generally easy for a process to ignore another process. For this reason an operation on processes associated with distributed computation often has the following property: a computation path of the process arising from the application of the operation to an input process has resulted from at most one computation path of the input process. As we will see, this property expresses that the operation is an affine linear map within a model of linear logic, a fact which carries many consequences.This article presents a model of processes based on computation paths and so can make precise the sense in which many process operations are associated with linear maps, investigates the consequences of linearity for the important equivalence of bisimulation, and delineates the boundaries of linearity with respect to one, fairly broad, mathematical model, in which processes are represented as presheaves. This data, what paths are present and how they restrict to smaller paths, is precisely that caught in a presheaf over a category in which the objects are path shapes and the maps express how one path shape can extend to another. In the category of all such presheaves we can view the tree as a colimit of its paths-another kind of "collection" of its paths.To illustrate the idea, suppose that actions are drawn from some alphabet L, and consider processes whose computation paths have the shape of strings of actions, so members of L * . The substring ordering ≤ makes L * a partial order, and so a category with an arrow from s to t precisely when s ≤ t. A presheaf over L * is a functor from the opposite category (L * ) op , where all the arrows are reversed, to the category of sets and functions Set. When thinking of a presheaf X as representing a process, for a string s, the set X(s) is the set of computation paths of shape s that the process can perform, and, when s ≤ t, the function X(s, t) : X(t) → X(s) tells how paths of shape t restrict to subpaths of shape s.Suppose that we replace the category of sets used in the definition of presheaves by the simple subcategory 2, consisting of the empty set, ∅, and the singleton set, 1, with the only non-identity map being ∅ ⊆ 1. A functor X from (L * ) op to 2 is the same as a monotonic function from the reverse order (L * ) op to the order 2, so that if s ≤ t then X(t) ≤ X(s). When thinking of...

Profunctors, Open Maps and Bisimulation

“…The mathematics has a life of its own, which is only patchily covered and understood in terms of existing process languages and their operational semantics. There have been successes in applying the mathematics, in connecting with process languages and operational semantics [39,40], the semantics of nondeterministic dataflow [24], independence/causal models [23,38], fairness [22], pi-Calculus and name generation for higher order processes [13,57], and weak bisimulation [17]. These are all examples of how we can bring categorical reasoning to bear on issues of concurrent computation.…”

“…Such functors do not have to send the empty presheaf to the empty presheaf, but will still preserve open map bisimulation. This relaxation makes the category of connected colimit preserving functors between presheaf categories a suitable framework in which to give semantics to a wide range of process languages [52,11,39]. • On objects: P ⊥ is the category P to which it has been added a new strict initial object, often referred to as ⊥.…”

“…Affine HOPLA is such a linear process passing language, introduced in [39,40]; its operations, definable within Conn preserve open map bisimulation leading automatically to congruence results [52,11]. The category Conn also supports a trace operation associated with a feedback loop in nondeterministic dataflow [24].…”

“…(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.…”

Full Abstraction for HOPLA

“…It has a straightforward operational semantics supporting a standard bisimulation congruence, and can directly encode calculi like CCS, higher-order CCS and mobile ambients with public names. The language came out of work on a linear domain theory for concurrency, based on a categorical model of linear logic and associated comonads [4,18], the comonad used for HOPLA being an exponential ! of linear logic.…”