Mapping Fusion and Synchronized Hyperedge Replacement into Logic Programming

1
Mapping Fusion and Synchronized Hyperedge
Replacement into Logic Programming
Dagstuhl Seminar 05081, 20-25 February 2005
Ivan Lanese Dipartimento di Informatica
Università di Pisa
joint work with Ugo Montanari
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

• Lot of background
• Fusion Calculus
• Synchronized Hyperedge Replacement
• Logic programming
• From Fusion Calculus to Hoare SHR
• From Hoare SHR to logic programming
• Conclusions

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
• P0 S P1P2 (x)P rec X. P X
• Processes are agents up to a standard structural
• congruence

7
Reduction semantics
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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

L
R
H
H
3
3
4
4
2
2
1
1
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Hyperedge Replacement Systems

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

H
Many concurrent rewritings are allowed
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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, ...)

12
An example Hoare SHR

• Hoare synchronization all the edges must
  perform the same action

• Milner synchronization pairs of edges do
  complementary actions

13
SHR with mobility
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Logic programming

• Very quickly
• Syntax
• Semantics

15
Roadmap

• Lot of background
• Fusion Calculus
• Synchronized Hyperedge Replacement
• Logic programming
• From Fusion Calculus to Hoare SHR
• From Hoare SHR to logic programming
• Conclusions

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

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Translation by example

• 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
• s transformed into amoeboids
• Each amoeboid connects a group of names merged by
  s

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Translation by example
x
x1
z1
Q(x1,x2,x3)
y1
z
x2
x3
y
y2
z2
z3
u2
u1
w1
u3
x1x2x3.S(x4,x5)
u
w2
w
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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

21
A sample production
x2
x3
?
y2
u2
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

P
Q

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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
Translation by example
x
Q(x1,x2,x3)
z
y
out x2 y2
x1x2x3.S(x4,x5)
u
in z2 w1
w
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Translation by example
x
Q(x1,x2,x3)
z
y
R(x1,x2)
S(x1,x2)
u
w
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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

• Lot of background
• Fusion Calculus
• Synchronized Hyperedge Replacement
• Logic programming
• From Fusion Calculus to Hoare SHR
• From Hoare SHR to logic programming
• Conclusions

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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)
C(x,y)?C(x,z),C(z,y)
r ltwgt
y
y
C(r(x,w),r(y,w))?S(y,w)
C
S
(w)
x
x
r ltwgt
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Dynamics
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Dynamics
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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

• Lot of background
• Fusion Calculus
• Synchronized Hyperedge Replacement
• Logic programming
• From Fusion Calculus to Hoare SHR
• From Hoare SHR to logic programming
• Conclusions

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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 p 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
Thanks
Questions?
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A textual notation for graphs

• Ring Example

x,y,w,z C(x,w) C(w,y) C (y,z) C(z,x)
?
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Transitions as judgements

• Transitions

?
?
? G1 ?? ?,??,?I G2
?
? ? ? (A x N ) (x, a , y) ? ? if ? (x)
(a , y)
?? 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
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Transitions as judgements

• Productions

?
x1,,xn L(x1,,xn) ?? x1,,xn , ??, ?I G
?
?
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
(x,e,ltgt)(y,e,ltgt)
?
x, y
?
C(x,y)
x, y, z
.C(x,z) C(z,y)
r ltwgt
y
y
C
S
(w)
x
x
r ltwgt
,
(x,r,ltwgt)(y,r,ltwgt)
?
?
x, y
x, y
S(w,y)
C(x,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)(PQ) if x not free in P
• rec X.PPrec X.P/X