The Split Semantics

This section gives abstract characterizations of the [[]]^MUSTsplit- and [[]]^TESTsplit- semantics on WT Nets, shows that they form algebraic cpo's, and proves that all the corresponding process operations from Chapter 3 are continuous functions on these cpo's.

We dene^hDMUST-split-

Act

0

;

^vMUST-split-

Act

0 ias a sub-partial-order of^hDActMUST

;

^{vMUSTAct i}corresponding to [[]]^MUST meanings of

+Mdupl-split nets. In order to ensure that every compact element of

D

MUST-split-

Act

0 is denable as the [[]]^MUSTsplit- meaning of some WT Net, we require that ^DMUST-split-

Act

satisfy some additional closure conditions. 0

First, we must ensure that Act is a \dupl-split alphabet," and that

PF

and

PD

are closed under \0-splitting" any

a

0-labeled events. Dually, any minimal failure or pomset-divergence must be the result of \0-splitting" some 1-2-respecting pomset. We note that the denition of 1-2-respecting ensures that no

a

1-labeled event must be a maximal cause of any divergence. Furthermore, any maximal

a

1-labeled events corresponds to \half-red"

a

0-events and hence can be relabeled with

a

0. Also, ring any

a

1-labeled event additionally enables only a

a

2-labeled event. The special role of ^p and

is also reected in the closure conditions. In particular, (1e) reects the presence of initial

-moves.

Denition 4.3.1

Let Act be a nite alphabet containing ^p and let Act⁰ = ^f

a

0

;a

1

;a

2:

a

² Act^,f

;

^pgg[fp

;

^g. A triple ^h

PF;PD;

Act⁰ⁱ is said to be must-split-respecting i it is a must-respecting triple and satises the following properties:

1. Additional closure properties of

PF

: (a) 0-split(

PF

)

PF

.

(b) ^h

p;F

^{i 2} 1-2-respect(

PF

) and

p

^{0 2}

(

p

) implies that ^h

p

⁰

;

^{;i 2}

PF

, where

is the sequence of choice renements ^hchoice(a¹;a¹;a⁰):

a

²Act^,f

;

^pgi.

(c) ^h

p;F

^{i 2}

PF

,

c

² Act^,f

;

^pg, and ^h

p;F

^[f

c

1^gi62

PF

implies that there is some

p

⁰²

p

with

c

1 such that ^h

p

⁰

;F

^,f

c

2^gi2

PF

.

(d) ^h

;

Actⁱ²

PF

.

(e) ^h;

;F

^[f

a

^gi2

PF

and ^h

a;

^;i2

PF

implies that^h;

;F

^[f

a;

^gi2

PF

. 2. Additional closure properties of

PD

:

(a) ^h

p;D

ⁱ²min^v(

PD

) implies that^h

p;D

ⁱ²augment(0-split(1-2-respect(

PD

))).

(b) 0-split(

PD

)

PD

.

(c) ^h

p;D

^{i 2} 1-2-respect(

PD

) and

p

^{0 2}

(

p

) implies that ^h

p

⁰

;D

^{i 2}

PD

, where

is the sequence of choice renements ^hchoice(a¹;a¹;a⁰):

a

²Act^,f

;

^pgi.

(d) ^h

p;D

ⁱ²min^v(

PD

) implies that

p

contains no

-labeled events.

3. Additional mixed properties:

(a) ^h

p;F

ⁱ²

PF

and ^h

p;

^fEventsp^gi62

PD

implies that

h

p;F

ⁱ²augment(0-split(1-2-respect(

PF

)))

(b) ^h

p;F

ⁱ²

PF

and ^h

p;

^fEventsp^gi62

PD

and

p

contains a^p-labeled event implies that this is the sole event in

p

.

(c) ^h

p;F

ⁱ²

PF

, ^h

p;

^fEventsp^gi62

PD

, and

F

^\f

a

0

;a

1^{g 6}=^; for some

a

²Act^,f

;

^pg implies that

F

^f

a

0

;a

1^g.

(d) ^h

p;F

ⁱ²1-2-respect(

PF

),^h

p;

^fEventsp^gi62

PD

, and

a

2²

F

for some

a

²Act^,f

;

^pg implies that no event in

p

is

a

1-labeled.

Denition 4.3.2

Let Act be a nite alphabet containing ^p and let Act⁰ = ^f

a

0

;a

1

;a

2:

a

² Act^,f

;

^pgg[fp

;

^g. A pair^hh

PT;

Actⁱ

;

^h

PF;PD;

Act⁰ⁱⁱsaid to be test-split-respecting i

h

PT;

Actⁱ ismay-respecting ,^h

PF;PD;

Act⁰ⁱis must-split-respecting , and

1.

p

²

PT

and

p

^{0 2}

(

(

p

)) implies that ^h

p

⁰

;

^{;i 2}

PF

, where

is the sequence of choice

0 is the restriction of ^DMUSTAct0 tomust-split-respecting triples and ^vMUST-split-

Act

0 is the restriction of ^vMUSTAct0 to^DMUST-split-

Act

Act;^Act0 is dened to be the set of all test-split-respecting pairs^hh

PT;

Actⁱ

;

^h

PF;PD;

Act⁰ⁱⁱ. Furthermore,^vTEST-split-

Act;^Act0 is the binary relation on^DTEST-split-

Act;^Act0

Proof.

We rst prove the case for [[]]^MUSTsplit-. Since WT Nets are closed under +M and dupl-split, the denition of [[]]^MUSTsplit- and Theorem 4.2.8 together imply that [[^h

N;

Actⁱ]]^{MUSTsplit- 2}

D MUST

Act

0 . The additional closure conditions of Denition 4.3.3 follow directly from the properties of

+M and dupl-split nets, and are easy to verify. The details are left to the reader.

For [[]]^TESTsplit-, it is straightforwardto see from the denitions of pomset-traces, pomset-failures, and pomset-divergences of WT Nets, the denition of [[]]^TESTsplit-, the denition of dupl-split and

+M, and Proposition 3.2.15 that the additional closure conditions hold. The theorem then follows easily from Theorem 4.2.8 and the above case.

Theorem 4.3.6

Let Act be a nite alphabet containing ^p and let Act⁰ = ^f

a

0

;a

1

;a

2:

a

²

The proof of the theorem is an easy combination and adaptation of the proof of Theo-rem 4.2.9. The details are left to the reader.

We now give a nite characterization of the compact elements of^DMUST-split-

Act

As an immediate consequence of Denitions 4.3.3 and 4.3.4 and Lemma 4.2.11, we have:

Lemma 4.3.8

Let Act be a nite alphabet containing ^p and let Act⁰ = ^f

a

0

;a

1

;a

2:

a

²

Proof.

The proof for^DMUST-split-

Act

0 is a minor modication of that of Theorem 4.2.12, needed in order to ensure that closure condition (2a) of Denition 4.3.3 holds for every approximation.

For every

n

0, we dene

n

^th approximation, ^h

PF

n

;PD

n

;

Act⁰ⁱ to^h

PF;PD;

Act⁰ⁱ as follows:

PD

n-n=^fh

p;D

^[

D

⁰ⁱ:^h

p;

^;i2PF0n-_n

; D

^[

D

⁰⁶=^;

;

either^h

p;D

ⁱ²

PD

or

D

=^;

;

and for all

d

²

D

⁰

;

either

d

= downp(

x

)^[f

x

^gfor some

x

such that depth_p(

x

) =

n;

and

l

p(

x

)⁶=

a

1 for any

a

²Act^,f

;

^pg

;

or

d

= downp(

y

)^[f

y

^gfor some

y

such that

y

is a maximal cause of some

x

with depth_p(

x

) =

n;

and

l

p(

x

) =

b

1 for some

b

²Act^,f

;

^pgg

PD

n= augment(extend^Act0(^PDn-n))

PF

n-n= augment(^PF0n-_n)

PF

n=^PFn-n^[implied-failures^Act0(

PD

n)

The proof of this case is then a straightforward adaptation of that of Theorem 4.2.12; the details are left to the reader.

The proof of ^DTEST-split-

Act;^Act0 is a straightforward adaptation of the proof of Theorems 4.2.12 and the above case; the details are left to the reader.

We now have:

Theorem 4.3.10

Let Act be a nite alphabet containing ^p and let Act⁰ = ^f

a

0

;a

1

;a

2:

a

² Act^,f

;

^pgg[fp

;

^g. Then ^DMUST-split-

Act

0 and ^DTEST-split-

Act;^Act0 are algebraic cpo's.

The theorem is a simple consequence of Lemma 4.3.8 and Lemma 4.3.9 (cf. [18]).

We now show that all compact elements are denable as the meanings of WT Nets:

Theorem 4.3.11

Let Act be a nite alphabet containing ^p and let Act⁰ = ^f

a

0

;a

1

;a

2:

a

² Act^,f

;

^pgg[fp

;

^g. For every compact element ^h

PF;PD;

Act^{0i 2 D}MUST-split-

Act

0 , there is some WT Net ^h

N

1

;

Actⁱ with [[^h

N

1

;

Actⁱ]]^MUSTsplit- = ^h

PF;PD;

Act⁰ⁱ. For every compact element

hh

PT;

Actⁱ

;

^h

PF;PD;

Act^0ii2DTEST-split-

Act;^Act0 , there is some WT Net^h

N

2

;

Actⁱwith [[^h

N

2

;

Actⁱ]]^TESTsplit-

=^hh

PT;

Actⁱ

;

^h

PF;PD;

Act⁰ⁱⁱ.

Proof.

For the rst case, let^h

PF;PD;

Act⁰ⁱ be a compact element of^DMUSTAct . As a simple consequence of Lemma 4.3.9 and the denition of compactness (cf. [18]), ^h

PF;PD;

Act⁰ⁱ is a nite candidate of^DMUST-split-

Act

0 .

The construction of the tree is analogous to the proof of the proof of Theorem 4.2.15, except that nodes are labeled with pomset-failures only from 0-split(1-2-respect(^PFn)) (rather than

PF

n) and with pomset-divergences only from 0-split(1-2-respect(^PDn)) (rather than ^PDn).

Furthermore, the root has

-labeled arcs to nodes labeled with valid triples of the form^h;

;

^f;g

;

ⁱ or^h;

;F

^[f

^g

;

ⁱ. Finally, an additional restriction is that a node labeled ^h

p;F

p

;x

pⁱ has an

a

1 -labeled arc to a node -labeled with ^h

p

⁰

;F

p⁰

;x

p⁰ⁱ, then

F

p^,f

a

2^g

F

p⁰. The remainder of the proof is a straightforward modication of the proof of Theorem 4.2.15; the details are left to

the reader. In particular, using closure condition (3a), it is easy to rewire the net to simulate duplicate-splitting.

We remark that the \patching-up" process is done rst on prexes whose maximal nodes are all

0-labeled or

2-labeled. The failure sets for the remaining prexes are then chosen appropriately.

The resulting net is then isomorphic to

+M(dupl-split(^h

N

1

;

Actⁱ) grow^f

^g) for some WT Net^h

N

2

;

Actⁱ, proving this case.

The proof for ^DTEST-split-

Act;^Act0 is then a straightforward combination of the proofs of Theo-rems 4.2.16 and the previous case; the details are left to the reader.

The operations on ^DMUST-split-

Act

0 are given in Denition 3.2.45. We do not restate the def-initions here. Let the operations on ^DTEST-split-

Act;^Act0 be the natural pairwise combination of the operations on^DMAYAct and ^DMUST-split-

Act

Act;^Act0 are closed under prexing, restric-tion, renaming, hiding, sequencing, internal choice, CCS choice, non-communicating parallel composition, CSP-style parallel composition, CCS-style parallel composition, split renements, and choice renements. Furthermore, all of these operations are continuous functions on the respective domains.

Let Act1 be a nite alphabet containing ^pand let Act1⁰=^f

a

0

;a

1

;a

2:

a

²Act1^,f

;

^pgg[

f

p

;

^g. Then grow Act1and shrink Act1are continuous functions from^hDMUST-split-

Act

Proof.

It is straightforward but tedious to show that ^DMUST-split-

Act

0 is closed under all of the operations except alphabet growing and shrinking, and that the domain and range of grow Act1 and shrink Act1 are as specied. It is easy to show that +M, augment, extend 1-2-respect, 0-split, 0-1-choice, and 0-1-split are continuous functions. The theorem then fol-lows easily from Theorem 4.2.17.

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