Action Planning for Graph Transition Systems

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 Action Planning for GTS - Edelkamp, Jabbar, Lafuente

3. Graph Transition Systems • A graph transition system (GTS) is a pair <M,g>, 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 (AAAI’05), July 2005, Pittsburgh, PA. (to appear).” for more details. • Applications: Biology – Changing molecular structure, Networks – Clients appearing and disappearing. Action Planning for GTS - Edelkamp, Jabbar, Lafuente

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. Action Planning for GTS - Edelkamp, Jabbar, Lafuente

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 Action Planning for GTS - Edelkamp, Jabbar, Lafuente

6. 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. Action Planning for GTS - Edelkamp, Jabbar, Lafuente

7. Usual Approach • “home”-based structure. • Each object has its own “home”. • “home” keeps track of the object’s 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. Action Planning for GTS - Edelkamp, Jabbar, Lafuente

8. link(u3 ) = u link(u) = u Following green links will take you to the object. Spanning Tree – defined by the link predicates. Mobile Object 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 Action Planning for GTS - Edelkamp, Jabbar, Lafuente

9. Properties of the Protocol • Iflink(v) = v (self-loop) => 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 Action Planning for GTS - Edelkamp, Jabbar, Lafuente

10. 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. Action Planning for GTS - Edelkamp, Jabbar, Lafuente

11. v z u4 u1 u2 u3 u w 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 Action Planning for GTS - Edelkamp, Jabbar, Lafuente

12. v z u4 u1 u2 u3 u w 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) w’s 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 Action Planning for GTS - Edelkamp, Jabbar, Lafuente

13. 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”. Action Planning for GTS - Edelkamp, Jabbar, Lafuente

14. 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) Action Planning for GTS - Edelkamp, Jabbar, Lafuente

15. Part IIPDDL Encoding Action Planning for GTS - Edelkamp, Jabbar, Lafuente

16. 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))) Action Planning for GTS - Edelkamp, Jabbar, Lafuente

17. 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. Action Planning for GTS - Edelkamp, Jabbar, Lafuente

18. 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 <existential-expression> <goal-condition>) • Using ADL, subgraph isomorphism can be encoded as • (:goal (exists (?n - node) (owner ?n))) Action Planning for GTS - Edelkamp, Jabbar, Lafuente

19. 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))) Action Planning for GTS - Edelkamp, Jabbar, Lafuente

20. Performance: Model Checker(HSF-Spin) vs. Planner (FF) – Subgraph Matching Action Planning for GTS - Edelkamp, Jabbar, Lafuente

21. Scaling Behavior of the Model 1.9 GB Action Planning for GTS - Edelkamp, Jabbar, Lafuente

22. 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. Action Planning for GTS - Edelkamp, Jabbar, Lafuente