Action%20Planning%20for%20Graph%20Transition%20Systems

1
Action Planning for Graph Transition Systems

• Stefan Edelkamp
• Shahid Jabbar
• Computer Science Department
• University of Dortmund, Dortmund, Germany
• Alberto Lluch-Lafuente
• Department of Informatics, University of Pisa,
  Pisa, Italy

2
Graph Transition Systems
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Graph Transition Systems

• A graph transition system (GTS) is a pair ltM,ggt,
  where
• M is a weighted transition system and
• g is a partial graph morphism.
• The weights of a transition system is modeled by
  a generalize cost-algebra based on monoids
  structure.
• See Edelkamp, Jabbar, and Lafuente,
  Cost-Algebraic Heuristic Search, in Proc. of 20th
  National Conference on Artificial Intelligence
  (AAAI05), July 2005, Pittsburgh, PA. (to
  appear). for more details.
• Applications Biology Changing molecular
  structure, Networks Clients appearing and
  disappearing.

4
Objectives

• How to model and check GTS?
• How to encode the application of partial graph
  morphism?
• How to deal with flexible goals?
• How to guide the search process?
• Solution Model the problem of checking graph
  transition systems in PDDL and use planning
  heuristics to guide the search process.

5
Outline

• Arrow Distributed Directory Protocol An example
  of GTS.
• PDDL Encoding of Arrow Distributed Directory
  Protocol.
• Experimental Results
• Performance Planner vs. Model Checker
• Scaling behavior of the model
• Conclusions

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Directory Service Protocol

• Assume a distributed environment.
• Clients The nodes in the distributed network
  e.g., different computers.
• Mobile Objects
• Could be a file, a process or any other data
  structure.
• It can be transmitted over a network from one
  node to another.
• It lives only on one node at a time.
• Purpose of a Directory Service
• Navigation To provide the ability to locate a
  mobile object.
• Synchronization To ensure mutual exclusion in
  the presence of concurrent requests.

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Usual Approach

• home-based structure.
• Each object has its own home.
• home keeps track of the objects location.
• All requests are send to the home.
• home sends a message to the client currently
  holding the object.
• That client forwards the object to the requesting
  client.
• Bottleneck Communication costs between home
  and clients.

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The Arrow Distributed Directory Protocol (Demmer
and Herlihy)

• Based on the idea of a trail of pointers
• Distributed Network G (V,E,w)

v
z
u4
u1
u2
u3
u
w
o
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Properties of the Protocol

• If link(v) v (self-loop) gt The object either
  resides at v, or will soon reside at v.
• Else, the object resides some where in the region
  of the directory containing link(v).

link(v) w
o
v
w
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Messages and Constructs

• link(u,v) Defines the spanning tree.
• find(v) Request for the object issued by the
  node v.
• move(v) The object is free to be moved to v. It
  travels with the object, following the links in
  the original graph.
• pending(u,v) Every link(u,v) has a buffer that
  keeps the request. Not a FIFO, but reliable.
• queue(u) v, NULL A predicate attached with
  every node. Tells that u has to transfer the
  object to v when it is finished with the object.

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Working of the Protocol

• v issues a request find(v) for the object.

find(v)
o
move(v)
u issues move(v) when it is finished with the
object
The object is moved to v following the shortest
path in G (blue edges)
A queue predicate is declared for v queue(u) v
find(v) inserted in the pending buffer
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Concurrent Requests

• find(v) stuck in the communication channel.
• w also issues a request in the meanwhile.

queue(v) w
queue(z) v
find(z)
ws request would be diverted to v instead
find(v) stuck in the com. channel
find(v)
z also issues a request
All future requests will be forwarded to w
o
find(v) released
queue(u) z
find(w)
Object Path u z v - w
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Advantages

• A distributed queue structure.
• Object request messages travel the shortest path
  in the spanning tree and not in the original
  graph.
• The queue structure ensures locality all
  requests will go directly to the object or to
  another terminal. Do not have to pass through a
  home.

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Properties to Verify / Types of Goals

• Can a particular node u be a terminal? (Subgraph
  matching)
• Can a particular node u be a terminal and all
  arrow paths end at u? (Graph Matching)
• Can an arbitrary node ui be a terminal? (Subgraph
  isomorphism)
• Can an arbitrary node ui be a terminal and all
  arrow paths end at ui? (Graph isomorphism)

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Part IIPDDL Encoding
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PDDL Encoding Morphism as PDDL Actions

• A morphism operation that inverses an edge can
  easily be defined as a very simple action.
• (action morphism-inverse
• parameters(?u ?v - node)
• precondition
• (link ?u ?v)
• effect
• (and
• (not (link ?u ?v))
• (link ?v ?u)))

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PDDL Encoding of Goals Graph and Subgraph
Matching.

• Subgraph and graph matching are easy to encode.
• Encode the goal graph with (link u v)
• and owner with (owner w) predicates.

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PDDL Encoding of Goals Subgraph Isomorphism

• Goals are strictly more expressive.
• Need an existential quantification over all the
  nodes to be described.
• ADL (Pednault 1989) ?
• (goal ltexistential-expressiongt
• ltgoal-conditiongt)
• Using ADL, subgraph isomorphism can be encoded as
  
• (goal (exists (?n - node) (owner ?n)))

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PDDL Encoding of Goals Graph
Isomorphism

• Existential quantifier can again be used ..
• (goal (exists ?v0 ?v1 ?v2 ?v3 ?v4
• ?v5 - node)
• (and (link ?v0 ?v0) (link ?v1 ?v0)
• (link ?v2 ?v0) (link ?v3 ?v1)
• (link ?v4 ?v0) (link ?v5 ?v4)
• (owner ?v3)))

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Performance Model Checker(HSF-Spin) vs. Planner
(FF) Subgraph Matching
Star DFS BFS hf EHC RPH
Stored nodes 6,253 30 6
Sol. length 134 58 5
Chain DFS BFS hf EHC RPH
Stored nodes 78,112 38 6
Sol. length 118 74 5
Tree DFS BFS hf EHC RPH
Stored nodes 24,875 34 6
Sol. length 126 66 5
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Scaling Behavior of the Model
Topology Nodes Stored Plan Length
Star 10 6 5
Star 25 7 6
Star 50 7 6
Star 70 7 6
Chain 10 6 5
Chain 25 33 28
Chain 50 100 73
Chain 70 138 101
Tree 10 6 5
Tree 25 22 16
Tree 50 47 25
Tree 70 61 31
1.9 GB
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Conclusions

• First such efforts to model GTS with PDDL.
• Advantages Planning heuristics can be employed
  directly.
• Successfully modeled Arrow Distributed Directory
  Protocol.
• Still some limitations in modeling the
  full-fledged specification of GTS.
• Dynamic insertions and deletions of nodes and
  edges.
• Can be circumvented to some extent by using a
  pool of available objects.