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. ´Øµ´×µ. When thinking of as representing a process, ´×µ ½ means that the process can perform a path of shape × while ´×µ means that it can't. If ´Øµ ½ and × Ø, then ´×µ ½. The functor is a characteristic 1

Profunctors, open maps and bisimulation

“…Proof: In relation to Lemma 2.6, we have the following situation: Proof: From Proposition 8.7, we know that G ∼ = Lan j P ⊥ (F ) for some functor F : Via the reflection Cocont ⊥ Conn , the category Conn inherits a monoidal closed structure from Cocont, and is sufficiently rich in operations to give semantics to a broad spectrum of process languages, including those with a form of linear process passing. 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].…”

“…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.…”

“…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 (Winskel 1999;Cattani 1999;Nygaard and Winskel 2002).…”

“…the category Conn inherits a monoidal closed structure from Cocont, and is sufficiently rich in operations to give semantics to a broad spectrum of process languages, including those with a form of linear process passing. Affine HOPLA is such a linear process passing language, and was introduced in Nygaard and Winskel (2002) and Nygaard and Winskel (2004); its operations, which are definable within Conn, preserve open map bisimulation, which leads automatically to congruence results (Winskel 1999;Cattani 1999). The category Conn also supports a trace operation associated with a feedback loop in non-deterministic dataflow (Hildebrandt et al 1998).…”

Strategies as Concurrent Processes

“…However, the mathematical abstraction of profunctors comes at a price: it can be hard to give an operational reading to denotations as profunctors. There are examples of semantics of higher-order process languages and "strong correspondence" where elements of profunctor denotations correspond to derivations in an operational semantics [11,15]. But in general it is hard to extract operational semantics from the profunctor denotations alone because they have abstracted too far.…”