tiling transactions in rewriting logic roberto bruni (pisa/illinois) josé meseguer (illinois) ugo...

Tiling Transactions in Rewriting

LogicRoberto Bruni (Pisa/Illinois)José Meseguer (Illinois)Ugo Montanari (Pisa)

WRLA 2002 Pisa, Italia, 19-21 Sept. 2002

Research supported by• IST-2001-32747 Project AGILE• Italian MIUR Project COMETA• ONR Grant N00014-02-1-0715• CNR Fellowship on Information Sciences and

Technologies

Outline

• Transactions• Zero-Safe Nets (shortly)• A Word on Adjunctions & Coreflections• Zero-Safe Nets as Tiles• Tiling Transactions• Example: Minimal Communication Tree• Conclusions & Future Work

“Mission”

Fully General Notion of Concurrent Transaction

– Graphical Representation– Formal Specification– Refined and Abstract view

• Refinement morphism

– (Distributed Commit)

Transactions: one, no one, one hundred

thousands• In databases:

– a transaction groups actions into a single logical unit which terminates either with a commit or with an abort

• the commit operation permanently applies all the changes to the database, making them visible

• the abort operation cancels all the changes, as they were never executed

– ACID: Atomic, Consistent, Isolated, Durable

• Several variants: – transactions with checkpoints, hierarchical,

timed, long-lived, nested, with compensation

Distributed Transactions

• In “global computing” applications:– data integration and process cooperation

is required across sites and organizations– different platforms, policies, data formats

• Multi-way transactions:– distributed processes start local

transactions, then– they interact by exchanging information

and joining their transactions– they must agree on commit/abort

Theory, Pragmatics & Ethics

• What are the abstract models?– Mathematical description– How do they relate to concrete model?– How to define formal abstraction/refinement

techniques?

• TM Servers can be overloaded– A dedicated machine is necessary (e.g. .NET)– Distribution is important

• When two or more organizations participate in the same transaction, which TM should take control?– Mediator trusted sites– Inter-corporation agreement

Zero-Safe Nets

stable places

zero places

transitions

tokens

copy

Example: Multicasting I

2

Interacting Processes

Administrator

buffer

new send

receive

Disconnected Processes

connect

Parallel composition

receive

receive

Horizontal Composition *

copyreceive

receive

Transaction

copyreceive

receive

send

Example: Multicasting II

Interacting Processes

Administrator

new

1-to-1

Disconnected Processes

connect

2

2

Abstract Net

1-to-2

3

3

1-to-n

n+1

n+1

… …

Main Features of ZS Nets• Graphical Presentation• Formal Theory

– Horizontal states in S

– Vertical states in Z (observations) – Horizontal composition as coordination

• parallel composition on (horizontal) states• sequential composition on observations

• Abstraction / Refinement– Refinement morphism– Abstraction via coreflection

• Concurrent Transactions & Distributed Interpreter• “Portability” under “Orthogonal” Flavors of Nets:

– coloured, reconfigurable, dynamic, probabilistic, timed, with read and inhibitor arcs, with priorities,…

Objects & Messages

• Object & Messages as Places in Coloured Nets

• Rewrites as Transitions• Partitioning stable/zero used to model

participation in transactions (for objects) and transactional messages

• Example: A Day at WRLA2002– Objects: participants, overhead projector,

beamer, microphone, …– Messages: words, gestures, …– Transitions: speaking, moving, changing slides, …– Transactions: presentations, intervals, …– (TM: chair-men, organizers, …)

A Constructive View of Adjunctions

A

B


A

B

F

F(A)


A

B

F

F(A)

b

What is the best approximation of b?

Adjunctions: Formally

b

BA

For any bB


F(Gb) b

B

Gb

A

We want to find a GbA with approximationb: F(Gb)b Bwhich is universal

b


F(Gb) b

B

Gb

A

i.e., any other approximation (via aA)g: F(a)b Bfactorizes uniquely through b

b

F(a)g

a


F(Gb) b

B

Gb

A

i.e., ! ĝ: aGb A such thatg = F (ĝ) ; gb

= {b}bB is called the counit

b

F(a)g

a

ĝ F(ĝ)

Duality

A

B

F

F(A)

Duality

A

B

F

F(A)

G

G(B)

Duality

A

B

F

F(A)

G

G(B)

F(Gb’) b’

F(Gb) b

Gb’

Gb

b’

b

Duality

A

B

F

F(A)

G

G(B)

F(Gb’) b’

F(Gb) b

Gb’

Gb

h

b’

b

Duality

A

B

F

F(A)

G

G(B)

F(Gb’) b’

F(Gb)

Gb’

Gb

b ; h

b’

b ; h

Duality

A

B

F

F(A)

G

G(B)

F(GF(a))

F(a)aidF(a)

F(a) F(a)F(a)F(a)

idF(a)

GF(a)

a = idF(a) is called the unit

Adjunctions: Properties

• Left Adjoints (e.g. F) preserve Colimits• Right Adjoints (e.g. G) preserve Limits• (Co)Limits are the categorical way of

expressing operations Adjunctions are Compositional

Interpretations• Typical Scenario:

– B has more structure than A– G is the obvious forgetful functor– F is the best way of adding the extra structure to A

Building Transactions I

ZPetri

HCatZPetri

objects: ZS netsarrows: graph homomorphisms, preserving place partitioning &disjoint image property for zero places

objects: ZS nets whose transitions possess ,*, idarrows: “disjoint” graph homomorphisms, preserving place partitioning and alloperations on transitions

forgets ,*, idZ

Building Transactions II

• Given NZPetri, take Z(N):– arrows from uS to vS are the

concurrent transactions of N– Prime arrows are the basic transactions

of N• from uS to vS

• non decomposable in parallel– e.g., in the multicast example, each basic

transaction contains at most one “sender”

Coreflection

A

B

F

F(A)

G

G(B)

when a: aG(F(a)) isa natural iso

Coreflection

B

A

when a: aG(F(a)) isa natural iso, thenA can be seen as asubcategory of B,with F the inclusion

Coreflection

B

A

when a: aG(F(a)) isa natural iso, thenA can be seen as asubcategory of B,with F the inclusion

• Typical Scenario:– B category of operational models– A category of abstract models– G(b) is the same as b when observed from

A

Abstract Net via Coreflection

ZSN

Petri

objects: ZS netsarrows: refinement morphisms

• A refinement morphism R: N1 N2 – is a zero-safe net morphism R’: N1 Z(N2)– that maps transitions of N1

• either to prime arrows of Z(N2)• or to transitions of N2

AZ

AZ maps ZS nets to their abstract counterpartsthe counit maps transitions to the transactions they represent

N

AZ(N)

N

From Nets to Computads

• Can we generalize the ZS approach?– Meta-theory for transactions

• formal models• fully general notion refinement / abstraction

– Extension of the distributed commit algorithm

• Petri Nets: based on multiset rewriting• Zero-Safe Nets: bidimensional multiset

rewriting– horizontal composition computes transactions– vertical composition builds computations on the

abstract net

• RL & TL seem to be the natural candidates

RL & 2-Computads• Main Ingredients of Rewriting Logic:

– Signature of system configurations– Structural axioms E– Rewrite rules over [t]E

• Categorically (in the initial model):– States form a cartesian category L,E –the Lawvere

Theory associated with (,E)• underlined natural number as objects• substitutions as arrows• composition = substitution application

– Proof terms form a cartesian 2-category

• 2-Computad– Monoidal category of configurations C– Rewrite rules over the arrows of C

Nets as 2-Computads

• S seen as monoidal category– unique object 0– uS seen as an arrow u:00– sequential composition = tensor

product• u;v = uv

• Transitions as rewrites– pre-set = source– post-set = target

RL & TL

• In (Unconditional) RL:– Rewrites : f(x)g(x) can be applied

• in any context C[.]: C[(x)]: C[f(x)] C[g(x)] • with any argument p(y): (p(y)): f(p(y)) g(p(y))

– Horizontal composition is total

• In TL:– Rewrites are coordinated via observations

– applicable in context if C accept the effect of

– applicable with argument p if it provides the trigger

– Horizontal composition is partial

pf

g

C

p f

g

C

TL & D-Computads• Main Ingredients of Tile Logic:

– (H,EH): system configurations

– (V,EV): Observations

– Tiles: : [p]EH [q]EH

• D-Computad– Monoidal category of configurations H– Monoidal category of observations V– Tiles over the arrows of H and C

[u]EV

[v]EV

ZS Nets as D-Computads• S , Z seen as monoidal categories

– unique object 0– uS seen as a configuration u:00– aZ seen as an observation a:00– (SZ) = S Z – sequential composition = tensor product

• u;v = uv• a;b = ab

• Transitions as tiles– stable pre-set = initial configuration (north)– stable post-set = final configuration (south)– zero pre-set = trigger (west)– zero post-set = effect (east)

Tiling Transactions I

ZComp

HCatZComp

objects: D-Computadsarrows: disjoint D-Computads morphisms (on observations: injective on objects, faithful on arrows)

objects: D-Computads whose tiles possess ,*, idarrows: disjoint D-Computad morphisms, preserving all operations on tiles

forgets ,*, idD

Tiling Transactions II

• Given DZComp, take D(D):– cells with indentities as trigger and effect

are the concurrent transactions of D– Elementary cells are the basic

transactions of D• trigger is an identity• effect is an identity• minimal

– non decomposable in parallel– non decomposable sequentially– without unnecessary idle configurations

Tiling Transactions III

ZComp

HCatZComp

forgets ,*, idD

ZPetri

HCatZPetri

D recovers Z

Abstraction via Coreflection I

Rcomp

2Comp

objects: D-Computadsarrows: refinement morphisms

• A refinement morphism M: D1 D2

– is a disjoint D-Computad morphism M’: D1 D(D2)

– that maps tiles of D1

• either to elementary elements of D(D2)

• or to tiles of D2

Ad

Ad maps D-Computads to their abstract 2-Computadsthe counit maps rewrite rules to the transactions they represent

D

Ad(D)

D

Abstraction via Coreflection II

Rcomp

2Comp

Ad

Petri

ZSN

Ad recovers AZ

Example: a Tile Transaction

0

act

a.0

aid

a.0

rpar

_|_

aida

_|_

_|_

P1

lpar

_|_

aaid

_|_

_|_

P2

com

_|_

idaa

_|__|_

0

act

a.0

aid

a.0

(X1|a.0)|(a.0|X2)

(X1|0)|(0|X2)

id

id

idid

id

id

id

idid

id

Abstract Communications

_|_

_|_ _|_

_|__|_

P3P2

_|_ _|_

P4 P5 P6 P7 P8P1


_|_

_|_ _|_

_|_X1

a.X2

X4 _|_

X3 a.X5 X6

Each time, we want toconsider just themost general “context”,which is essential to the interaction

(X1|(a.X2|X3))|(X4|(a.X5|X6))

(X1|(X2|X3))|(X4|(X5|X6))


_|_

_|_ _|_

_|_X1

a.X2

X4

X3

a.X5

(X1|(a.X2|X3))|(X4|a.X5)

(X1|(X2|X3))|(X4|X5)

OK


_|_

_|_ _|_

a.X2X1 _|_

(X1|a.X2)|(X3|(X4|a.X5))

(X1|X2)|(X3|(X4|X5))

OK

X4 a.X5

X3


_|_

_|_ _|_

a.X2X1 _|_ a.X5

(X1|a.X2)|((X3|X4)|a.X5)

(X1|X2)|((X3|X4)|X5)

NO!

X3 X4


_|_

_|_ _|_

a.X2X1 _|_ a.X4

Concurrent rewrites

b.Y1 b.Y2

Conclusions & Future Work

• We have fully generalized the ZS approach– Refinement morphisms between Tiles– Abstract Rewrite Theory (aka 2-

computad)– “Compositional” specification of

transactions• To investigate:

– Connection with conditional RL

Tiling Transactions in RL

a paper by Roberto Bruni José Meseguer Ugo Montanari

a WRLA presentation by Roberto Bruni

Research supported byIST-2001-32747 Project AGILEItalian MIUR Project COMETAONR Grant N00014-02-1-0715CNR Fellowship on Inf. Sci. and Techn.

Electronic watercolor by Roberto Bruni

tiling transactions in rewriting logic roberto bruni (pisa/illinois) josé meseguer (illinois) ugo...

Documents

technologies slide

compensation slide

commitabort slide

distributed transactions

transactions example

tiling transactions

outline transactions

local transactions