Multiport Interaction Nets and Concurrency Damiano Mazza Institut - - PowerPoint PPT Presentation

Multiport Interaction Nets and Concurrency

Damiano Mazza

Institut de Math´ ematiques de Luminy Marseilles, France

CONCUR 2005 San Francisco, 23 August 2005

Interaction Net Systems: cells, wires, nets

Interaction Net Systems were introduced by Yves Lafont (1990) as a generalization

• f MLL proof-nets. The basic elements of an Interaction Net System (INS for

short) are cells and wires:

α . . .

auxiliary ports principal port wire

Wires can be used to connect occurrences of cells to form nets:

δ δ γ γ γ γ δ ε ε ε ε

A set Σ of cells is called an alphabet. Slide 1/19

Interaction Net Systems: interaction

When two cells are connected through their principal ports, they form an active pair. Active pairs can be reduced by means of interaction rules of the following form:

α . . . . . . β → . . . . . . α ⊲ ⊳ β

◮ α ⊲ ⊳ β is a net whose free ports are in bijection (implicitly given by the graphical

representation) with the auxiliary ports of α, β.

◮ Reduction step: replace an active pair α, β of a net µ with α ⊲ ⊳ β (this is why

a bijection is needed), yielding a new net µ′. We then write µ → µ′.

◮ An INS is defined by an alphabet (maybe infinite, but denumerable) and a set of

interaction rules (at most one per active pair).

◮ We can use a type discipline to limit the possible connections/active pairs.

Slide 2/19

Interaction Net Systems: properties

◮ Strong confluence → no reduction strategies → strong determinism. ◮ Turing-completeness: given a Turing machine, there exists an INS simulating it. ◮ The “right” syntax for the Geometry of Interaction (Girard, 1988, Gonthier-Abadi-

L´ evy, 1992, Interaction Combinators, Yves Lafont, 1995).

◮ All of these properties are a consequence of the locality of interaction. ◮ Limitation: INS’s model distributed, but sequential computation, i.e., there is no

parallel-or, gustave, etc. → truly concurrent behavior is excluded. Slide 3/19

Multiport Interaction Net Systems

Cells are allowed to have more than one principal port (multicells, opposed to mono- cells). Active pairs are still cells having one of their principal ports wired together:

α σi β σj . . . . . . . . . . . . . . . . . . . . . . . . − → . . . . . . . . . . . . αi ⊲ ⊳ βj

We have used the two wirings σi and σj just for graphical convenience; they are not part of the interaction. Notice that a multicell can be involved in several active pairs; the choice of which

• ne is reduced is non-deterministic. Thus, in general, we have lost confluence, but

we have gained in expressive power (parallel-or is easily encodable). Slide 4/19

The “core” π-calculus

The “core” π-calculus (Cπ for short) can be seen as a synchronous and extended version of the Pict language (Pierce and Turner, 1997): Prefixes:

π ::= xy

• x(z)

Extended prefixes:

κ ::= x(z)

• π

Processes:

P, Q ::= 0

• π.P
• P | Q
• ν(z)P
• κ ⊲ P

Reduction:

xy.P | x(z).Q − → P | Q{y/z} xy.P | x(z) ⊲ Q − → P | Q{y/z} | x(z) ⊲ Q x(z) ⊲ P | x(z).Q − → ν(z)(P | Q) | x(z) ⊲ P

Structural congruence: as in the finite π-calculus, i.e., no such thing as

!P ≡ P | !P

Slide 5/19

Names as multicells

We shall now define an encoding of · of Cπ processes into Multiport Interaction Nets. The key point is encoding names, i.e., concurrent communication channels. Multicells are going to be fundamental here:

Xm m principal ports . . .

i/o requests input queue

• utput queue

synchronization

Slide 6/19

Synchronization

Prefixes are encoded by means of two binary monocells π−, π+:

Xm+2 . . . . . . . . . π+ π− − + N C N C

xy.P | x(z).Q | R − → P | Q{y/z} | R

Slide 7/19

Synchronization

Upon interaction, prefixes become queue constructors:

. . . π+ ρ− . . . Xm+1 − + N C N C

xy.P | x(z).Q | R − → P | Q{y/z} | R

Slide 7/19

Synchronization

The i/o queues are built according to the (non-deterministic) reduction order:

. . . Xm − + ρ+ ρ− N C N C

xy.P | x(z).Q | R − → P | Q{y/z} | R

Slide 7/19

Synchronization

The top elements of the two queues eventually meet, implementing synchronization:

. . . Xm ρ+ ρ− N C C N

xy.P | x(z).Q | R − → P | Q{y/z} | R

Slide 7/19

Synchronization

The name sent is connected to the receiver, and the two continuations communicate:

. . . Xm N C C N

xy.P | x(z).Q | R − → P | Q{y/z} | R

Slide 7/19

Name passing

We use a family of monocells Bn (of arity n) to collect the occurrences of a name bound by an input prefix. Name passing is thus implemented by the following rule:

Bn − → . . . . . . . . . . . . . . . ωi Xm+1 Xm+n . . . . . .

Slide 8/19

Guards

To encode guards, we use two families of monocells Λ+

n , Λ− n , of arity 2n:

Λ+

k

Λ−

n

→ . . . . . . . . . . . .

Suppose P is already defined; then we build the net .P as follows:

. . . µ . . . . . . x1 xk c active pairs

contained in P

P = . . . x1 xk µ . . . . . . . . . Λ

k+c

.P = = ⇒

In practice, active pairs (both present and future) are “inhibited”, and restored upon synchronization. Slide 9/19

Replication

We can use two standard duplicator cells δ+, δ−, with the following rules:

− → δn δ1 α α α δ . . . . . . . . . . . . δ+ δ− − → α is any cell except Xm and δ− ωi Xm+1 . . . . . . − → Xm+2 . . . . . . δ+

Problem: only reduced nets containing no Xm or δ cells can be duplicated. Slide 10/19

Replicated guards

Suppose that .P is a net containing n X-cells and (among others) k free ports labelled with the name z:

. . . z z . . . Xm1 . . . Xmn .P = µ . . . . . .

Then, we define ⊲P and ⊲P z as follows:

. . . z z µ . . . . . . . . . . . . !Xm1 !Xk+1 µ . . . σ . . . Nn . . . . . . !Xmn !Xm1 Nn+1 . . . . . . !Xmn . . . ⊲P ⊲P z

Slide 11/19

Replicated guards

We also add a zeroary monocell ϕ with the following interaction rules:

→ ϕ Nn . . . . . . ϕ ϕ . . . Xm . . . ϕ → !Xm

The following property holds:

σ ⊲P . . . ϕ δ− δ+ .P . . . σ ⊲P . . . δ1 δn →∗ . . .

Slide 12/19

Replicated prefixes

Additional monocells !π−, !π+ (arity 4) and !ρ−, !ρ+ (arity 5) are used to encode replicated prefixes and their respective queue constructors. Synchronization of replicated prefixes involves dispatching duplicators and ϕ cells appropriately:

!ρ+ !ρ− − → ϕ δ+ δ+ δ− δ+ ϕ δ+ δ+ δ+ δ− !π+ !π−

There is of course also the “asymmetric” case in which a !ρ cell interacts with a

ρ− cell, in which duplicators are dispatched only on one side.

Slide 13/19

Inductive definition of the encoding

◮ 0 is the empty net. ◮ P | Q is the net obtained by juxtaposing P and Q. ◮ π.P is the following net, depending on the nature of π:

x π+ .P +

xy.P =

. . . y . . . . . . Bn .P − π− x

x(z).P =

. . .

In the encoding of the input prefix, the n free ports of .P − labelled by z are connected to the Bn cell. Slide 14/19

Inductive definition of the encoding

◮ κ ⊲ P is the following net, depending on the nature of κ:

!π+ x x y ⊲P + . . . Bn . . . . . . . . . ⊲P − x !π− x . . . x ⊲P z x !π+ x(z) ⊲ P x(z) ⊲ P xy ⊲ P

◮ If P has m free ports labelled by z, then ν(z)P is the net obtained from P by connecting all such free ports to the free ports of the following net:

. . . Xm

Remark: this is the only case in which active pairs may be introduced. Slide 15/19

The strong encoding

So far the encoding is only able to implement synchronizations which are due on restricted channels, so we force our processes to be “closed”: Definition 1 Let x range over fn(P); if in P there are m free ports labelled by

x, we define [P] as the net obtained from P by connecting all such ports to a Xm+1 cell, which will be left with one free port labelled by x:

Xm+1 . . . x

[P] is virtually identical to ν( x)P, where x are the free names of P , the only

difference being that we want to remember the names we artificially bound. Slide 16/19

Readback

The encoding commutes with reduction only modulo a readback procedure:

. . . Q−

n

. . . Xm Q+

p

. . .

• Xm+n+p

π+ π− !π− !π+ . . . . . . . . . Xm+1 . . . x

• m

m x x

The readback procedure “undoes” the choices made in queuing up prefixes and removes artificial name restrictions. We denote by

ν the readback of a net ν.

Only the reduction of active pairs involving ρ and/or !ρ cells affects the readback. We write µ ։ ν iff the reduction from µ to ν contains exactly one such reduction. Slide 17/19

Properties of the encoding

Proposition 1 If P ≡ Q, then P = Q and [P] = [Q]. This would be impossible if we had admitted !P ≡ P | !P . Notice also that the converse is false; in fact, whenever z does not appear in the prefix π,

ν(z)π.P = π.ν(z)P.

Theorem 2 Let P be a Cπ process.

• 1. The observables of P are in bijection with the free principal ports of P.
• 2. If P

τ

→ Q, then [P] ։ ν and ν = Q.

• 3. If [P] ։ ν, then there exists Q such that P

τ

→ Q and Q = ν.

The need for a “closure” reminds of what happens when interpreting linear logic into process calculi. This might find a justification if one thinks that (multiport) interaction net reduction is basically generalized MLL cut-elimination. Slide 18/19

Conclusions

◮ Expressivity: mINS’s are able to represent every feature of the π-calculus. With

some (rather ad hoc and technical) additions, the encoding can be extended to guarded choice and match prefixes.

◮ Locality: unlike virtually any other approach to model concurrency, the dynamics

in mINS’s is completely local; in particular, no “boxes” are needed to cope with replication (just like in sharing graphs for the λ-calculus). Moreover, we don’t need an infinite alphabet (it is only convenient for conciseness).

◮ Semantical interest: the Geometry of Interaction might be extended to mINS’s,

possibly yielding a novel semantical approach to concurrency.

◮ mINS’s are not “ad hoc”: unlike any other graphical encoding of process calculi,

mINS’s were not built around the π-calculus. In fact, we see our work more as a “benchmark” for mINS’s than as “yet-another-graphical-encoding-of-π”. Mul- tiport Interaction Nets should be seen as an independent, alternative model of concurrency. Slide 19/19