1 Mapping Fusion and Synchronized Hyperedge Replacement into Logic Programming joint work with Ugo Montanari Ivan Lanese Dipartimento di Informatica Università

1

Mapping Fusion and Synchronized Hyperedge Replacement into Logic Programming

joint work withUgo Montanari

Ivan LaneseDipartimento di Informatica Università di Pisa

Dagstuhl Seminar #05081, 20-25 February 2005

To be published on a special issue of Theory and Practice of Logic Programming

2

Roadmap

Lot of background– Fusion Calculus– Synchronized Hyperedge Replacement– Logic programming

From Fusion Calculus to Hoare SHR From Hoare SHR to logic programming Conclusions

3

Motivations

Many models proposed for global computing systems

Each model has its strengths and its weaknesses

Comparing different models– To understand the relationships among them– To devise new (hybrid) models

Cannot analyze all the models, naturally…

4

Roadmap



5

Fusion Calculus

Process calculus that is an evolution of -calculus

Simpler and more symmetric but also more expressive

Introduces fusions of names

6

Syntax for Fusion Calculus

Agents:

S::=i i.Pi

P::=0 | S | P1|P2 | (x)P | rec X. P | X

Processes are agents up to a standard structural

congruence

nameson relation eequivalenc:

||::

:actions Free

xuxu

7

Reduction semantics

'

'',',

QQ

QPPPQP

)||)(()).(|).(|)(( QPRzQyuPxuRz

)|)(()).(|)(( PRzPRz

yxyx

of mgu and where ||||

of mgu where

8

Synchronized Hyperedge Replacement

Follows the approach of graph transformation (Hyper)edges are systems connected through common

nodes Productions describe the evolution of single edges

– Local effect, easy to implement

Productions are synchronized via constraints on nodes– Global constraint solving algorithm to find allowed transitions– Productions applied indipendently– Allows to define complex transformations

9

Hyperedge Replacement Systems

A production describes how the hyperedge L is transformed into the graph R

R

1

2 3 4

L

1

2 3 4 H H

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Hyperedge Replacement Systems

A production describes how the hyperedge L is transformed into the graph R

R

R’

1

2 3 4

1

2

3

Many concurrent rewritings are allowed

L

L’

1

2 3 4

1

2

3

H H

11

Synchronizing productions

Productions associate actions to nodes A transition is allowed iff the synchronization

constraints imposed on actions are satisfied Many synchronization models are possible

(Hoare, Milner, ...)

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An example: Hoare SHR

Hoare synchronization: all the edges must perform the same action

Milner synchronization: pairs of edges do complementary actions

aB1 A1

B2 A2

3 3

aa a

13

SHR with mobility

– Actions carry tuples of references to nodes (new or already

existent)

– References associated to synchronized actions are matched and

corresponding nodes are merged

We use name mobility

a<x>B1 A1

B2 A2

a<y>a<x>a<y>

(x) (y)x=y

14

Very quickly… Syntax Semantics

Logic programming

nBBA ,...,1 nBB ,...,1

goal atomic

),...,(

)mgu(,...,

1

1

k

k

BBA

HAPBBH

goal econjunctiv

',', GFGG

FG

15

Roadmap



16

From Fusion to HSHR

We separate the topological structure (graph) from the behaviour (productions)

We give a visual representation to processes– Processes translated into graphs– Sequential processes become hyperedges– Names become structures called amoeboids

Our approach deals only with closed processes

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Translation by example

We take agents in standard form– restrictions with inside parallel composition of

sequential agents We transform it into a linear agent + substitution

)),(.|),(.|),,()(( wzSuzwxuRxyuzyxQuxyzw

)),(.|),(.|),,()(( 2312332221111 wzSwzuxuRyxuzyxQuxyzw

18


We translate the process into a graph– Each linear agent becomes an edge labelled with a

copy of the agent with standard names» Q(x1,y1,z1) becomes an edge labelled by Q(x1,x2,x3)

attached to x1,y1,z1

– σ transformed into amoeboids» Each amoeboid connects a group of names merged by σ

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)),(.|),(.|),,()(( 2312332221111 wzSwzuxuRyxuzyxQuxyzw

Q(x1,x2,x3)

x1x2x3.R(x4,x5)

x1x2x3.S(x4,x5)

x

uw

zy

x1

y1

z1

u1

x2

y2

u2

x3

u3

z2 z3

w2

w1

20

Dynamics

We have actions for input and output prefixes Productions for process edges

– Correspond to executions of prefixes of normalized linear agents

– Prefixes modeled by corresponding SHR actions– RHS contains the translation of the resulting

sequential process– Fusions implemented by connecting nodes via

amoeboids

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A sample production

),(),(. 3232221221 xuRxuRyxu yxu

x1x2x3.R(x4,x5)

u1

x2

y2

u2

x3

R(x1,x2)

x2

y2

u2

x3

u1

out2x2y2

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What is an amoeboid?

Amoeboids must allow two complementary actions on the interface and create new amoeboids connecting corresponding names

Connected amoeboids are merged

Px

Qy

in x out yP Q

=

23

Implementing amoeboids in HSHR

Amoeboids implemented as networks of edges with a particular structure– composed essentially by edges that act as routers– each internal node is shared by two edges– some technical conditions

Satisfy the desired properties but…– interleaving must be imposed from the outside– produce some garbage (disconnected from the

system)

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Correspondence theorem

Reductions of fusion processes correspond to (interleaving) HSHR transitions– up to garbage– up to equivalence of amoeboids that does not

change the behaviour The correspondence can be extended to

computations

25


Q(x1,x2,x3)

x1x2x3.R(x4,x5)

x1x2x3.S(x4,x5)

x

uw

zy

out x2 y2

in z2 w1

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Q(x1,x2,x3)

R(x1,x2)

S(x1,x2)

x

uw

zy

)),(|),(|),,()(( yxSxuRxyxQuxy

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Summary : Fusion Calculus vs HSHR

Fusion Hoare SHR

Closed process Graph

Sequential process Hyperedge

Name Amoeboid

Prefix Action

Prefix execution Production

Reduction Transition

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Roadmap



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From Hoare SHR to logic programming

Useful for implementation purposes

Logic programming as goal rewriting engine

Very similar syntax (with the textual representation for HSHR)

Logic programming allows for many execution strategies and

data structures we need some restrictions– limited function nesting

– synchronized execution

We define Synchronized Logic Programming (SLP)

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Synchronized Logic Programming

A transactional version of logic programming (in the zero-safe

nets style)

Safe states are goals without function symbols (goal-graphs)

Transactions are sequences of SLD steps

During a transaction each atom can be rewritten at most once

Transactions begin and end in safe states

Transactions are called big-steps

A computation is a sequence of big-steps

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Synchronized clauses

Clauses with syntactic restrictions

– bodies are goal-graphs

– heads are A(t1,…,tn) where ti is either a variable or

a single function symbol applied to variables

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HSHR vs logic programming

Graphs translated to goal-graphs– edges modeled by predicates applied to the attachment nodes

Productions are synchronized clauses

Transitions are matched by big-steps

Actions are implemented by function symbols– the constraint that all function symbols have to be removed corresponds

to the condition for Hoare synchronization

– names are the arguments of the function symbol

– we choose the first one to represent the new name for the node where

the interaction is performed (needed since substitutions are idempotent)

– fusions performed by unification

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Correspondence theorem

Correspondence between HSHR transitions

and big-steps

An injective (at each step) substitution keeps

track of the correspondence between HSHR

nodes and logic programming variables

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An example (simpler than Fusion…)

y

C

x

y

C

z

C

x

y

C

x

y

S

xr <w>

r <w>

(w)

C(x,y)←C(x,z),C(z,y)

C(r(x,w),r(y,w))←S(y,w)

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Dynamics

x C

C

C

C

C

C

C

CC C

S

S

SS

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Dynamics

x C

C

C

C

C

C

C

CC C

S

S

SS

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Summary : HSHR vs SLP

Hoare SHR SLP

Graph Goal

Hyperedge Atom

Node Variable

Parallel comp. AND comp.

Action Function sym.

Production Clause

Transition Big-step

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Roadmap



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Conclusions

Many relations among the three models– similar underlying structure (e.g. parallel composition)

– name generations ability

– fusions

Distinctive features– Fusion: same structure for system and elementary actions,

interleaving semantics, Milner synchronization, restriction

– HSHR: distributed parallel computations, Hoare synchronization, synchronous execution

– SLP: Hoare synchronization, asynchronous execution engine

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Future work

Analyzing different name-handling mechanisms– In π calculus bound names are guarenteed distinct

– Useful for analyzing protocols (nonces, key generation)

Hybrid models– Different synchronizations for Fusion or logic programming

– Process calculi with unification (of terms)

– Logic programming with restriction

Logic programming for implementation purposes– For HSHR systems

– For Fusion Calculus

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End of talk

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x,y,w,z C(x,w) | C(w,y) | C (y,z) | C(z,x)

A textual notation for graphs

Ring Example

w z

43

: (A x N* ) (x, a , y) if (x) = (a , y)

Transitions as judgements

Transitions

G1 ,, G2

is the set of new names that are used in synchronization

= {z | x. (x) = (a , y), z , z set(y)}

I contains new internal names

44

Transitions as judgements

Computations

0 G0 1 G1 … n Gn

1 2 n

x1,…,xn L(x1,…,xn) x1,…,xn , , I G

Productions

Names can be merged (and new names can be added)Identity productions are always available

Transitions are generated from the productions by applying the transition rules for the chosen synchronization model

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Textual representation for productions

y

C

x

y

C

z

C

x

x, y C(x,y)(x,ε,<>)(y,ε,<>)

x, y, z .C(x,z) | C(z,y)

y

C

x

y

S

xr <w>

r <w>

(w)

x, y C(x,y)(x,r,<w>)(y,r,<w>),

x, y S(w,y)

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From SHR to SLP

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Structural congruence

Process: agent up to the following laws:– | and + are associative, commutative and with 0 as

unit -conversion– (x)0 = 0, (x)(y)P=(y)(x) P– P|(x)Q=(x)(P|Q) if x not free in P– rec X.P=P[rec X.P/X]

Documents

1 Mapping Fusion and Synchronized Hyperedge Replacement into Logic Programming joint work with Ugo Montanari Ivan Lanese Dipartimento di Informatica Università