Relational Semantics of Non-Deterministic Dataflow

Hildebrandt, Thomas; Panangaden, Prakash; Winskel, Glynn

doi:10.7146/brics.v4i36.18962

Cited by 5 publications

(6 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The trace operation is, however, not definable in the present affine-linear language. It can be shown that if the trace operation of [9] were definable in the presheaf semantics, we could obtain (replacing Set by 2) a compositional relational semantics of nondeterministic dataflow with feedback, shown impossible by Brock&Ackerman [3].…”

Section: Nondeterministic Dataflowmentioning

confidence: 99%

See 1 more Smart Citation

Linearity in Process Languages

Nygaard¹,

Winskel²

2002

BRICS

Self Cite

View full text Add to dashboard Cite

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...

show abstract

Section: Nondeterministic Dataflowmentioning

confidence: 99%

“…A subcategory of Aff supports a "trace operation" to represent processes with feedback loops (see [9]). The trace operation is, however, not definable in the present affine-linear language.…”

Section: Nondeterministic Dataflowmentioning

confidence: 99%

Linearity in Process Languages

Nygaard¹,

Winskel²

2002

BRICS

Self Cite

View full text Add to dashboard Cite

show abstract

“…Thus, supposing that the trace operation of [13] were definable in the presheaf semantics, we would obtain a compositional relational semantics of nondeterministic dataflow with feedback, shown impossible by the Brock-Ackerman anomaly [3].…”

Section: Nondeterministic Dataflowmentioning

confidence: 99%

“…A subcategory of Aff Set supports a "trace operation" to represent processes with feedback loops (see [13]). The trace operation is, however, not definable in the present affine-linear language.…”

Section: Nondeterministic Dataflowmentioning

confidence: 99%

Linearity and Nonlinearity in Distributed Computation

Winskel

2004

Linear Logic in Computer Science

Self Cite

View full text Add to dashboard Cite

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...

show abstract

“…A subcategory of « supports a "trace operation" to represent processes with feedback loops (see [9]). The trace operation is, however, not definable in the present affinelinear language.…”

Section: Nondeterministic Dataflowmentioning

confidence: 99%