Planning with Incomplete, Unbounded Information

May 20, 2003

Tal Shaked

Types of Incomplete Information

• Finite set of states (objects, relations)– Uncertain of current state– Realistic?

• Infinite set of states– Many objects and relations (most unknown)– Too large to reason about directly

Example Domains

• Unix/Internet– Puccini and Rodney

• Web– Web services– DAML-S (DAML+OIL)

Topics

• Problems with existing planners– Puccini, PKS

• What is difficult

• LCW review

• Ideas to improve planners

• How PKS works

Two Recent Planners

• PKS (2002)• Petrick and Bacchus• Contingent plans

• Puccini (1998)• Etzioni, Golden, Weld• Internet softbot

Puccini

• Partial-order-planner

• Expressive – SADL

• Interleaves Planning with Execution– Not clear how

• Slow– Required domain-specific knowledge

PKS

• Contingent, forward-chaining planner

• Not Expressive

• Slow

– Constructs a complete, correct plan– Separates plan-time and execution-time effects

– No universal quantification or LCW

– No search control

Some Problems

• Slow– How do we solve these planning problems?– What heuristics can we add?

• Execution model– Are contingent plans practical?– When should actions be executed?

• How can we find structure?

Extending Planning Graph

• What is different?

• Mutexes?

• Scalability?

– LCW– Unlimited sensing, run-time objects/relations– Contingent plans– Interleaved planning and execution

LCW Review

• Why is LCW useful?

• How does it work?

– Avoid repeated sensing– Universal quantification

– Inference– Compression– Lazy evaluation

Querying Ground Conjunction

Querying LCW

How is inference done?

If we know all files in jokes/, then we know if the file giggy is in jokes/

If we know all files in jokes/, and know all dirty jokes, then we know if giggy is dirty and in jokes/

Updating LCW

• Information Gain: A formula that is originally U, becomes T or F– Generally cannot lose LCW

• Information Loss: A formula initially T or F, becomes U– Generally, all LCW “relevant” to that literal are lost

– Know the size of all files in root/. Execute compress root/passwords.txt

Sensing Actions

• Bounded sensing– Set of possible observations

• Unbounded sensing– Generic types and relations

• Consider potential bindings at next level

Finding a Plan

• Search for plans in the graph– Consider one branch at a time

• Heuristics– Reachability

• Amount of sensing to reach a literal

• Depth in planning graph

– Control execution• Agent-centered search?

Scalability

• Mutexes– Same as normal Graphplan– LCW?– Generic types and relations?

• Quick growth due to sensing– Limit to relevant actions– Learn relevance probabilities

Book Trading Example

Predicates:(own ?book)(book_subject ?book ?subject)(at_store ?book ?book_store)

action: trade(?book1 ?book2 ?book_store)precond: ((own ?book1) (at_store ?book2 ?book_store))effect: ((own ?book2) (not (own?book1)))

action: search(?book_store ?subject)precond: ()effect: (forall (!book)

(when (at_store !book ?book_store)(at_store !book ?book_store)(book_subject !book ?subject)))

(LCW((at_store #book ?book_store)(book_subject #book ?subject))

Start: ((own my_book) (book_subject my_book chess))

Goal: ((own ?book)(book_subject ?book go))

Sample Graph

(ow n my_book)

(book_subject my_book chess)

(search amazon go)

(LCW((at_store !b amazon)(book_subject !b go)))

(at_store *b amazon)

(book_subject *b go)

(trade my_book *b amazon)(ow n *b)

(book_subject *b go)

(ow n my_book)

(book_subject my_book chess)

(not (ow n my_book))

(book_subject my_book chess)

(LCW((at_store !b amazon)(book_subject !b go)))

Using the Graph

• Similar Graphplan search

• LPG-like search (local search on graph)

• Propagating sensing action links

• Executing to reach ‘better’ states

• Forward/backward chaining heuristics?

Other Problems

• Agent wants to find a *.pdf file– Try ls

• hope some file exists, possibly a *.pdf file

– latex(paper.tex), dvipdf(paper.dvi)• check for read/write permissions

• How can the agent learn?• Can this be represented in a planning graph?

Knowledge-Based Approach

• Only represent what agent knows

• Actions manipulate knowledge

• Advantages– Compact Representation– Introducing new objects

• Disadvantages?– Unable to distinguish between possible worlds

Modal Logic of Knowledge

• Fancy way of just adding K is true at a particular world w iff it is true

by standard rules

• K() is true at w iff is true at every possible world

can be true, yet the agent may not know

Knowledge Representation

• Databases store agent’s knowledge

• Can be converted to modal logic formulas

• Preconditions as knowledge

• Effects as database modifications

• Goals as knowledge

Databases

• Kf – stores facts like STRIPs

• Kw – agent either knows or negation

– know(this) Kw K(know(this)) v K(¬know(this))

– With variables, can model universal effects– At run time, generates LCW– Construct conditional branches

Databases (continued)

• Kv – function values agent will know

– Plan time just know value will exist– Execution time will know actual value

• Kx – “exclusive or” knowledge

– Exactly one proposition in a set is true

Knowledge State

• Databases are conjunctions of formulas– Limits what the agent can know

• Cannot represent some sets of worlds– w1: P(a), ¬P(b); w2: ¬P(a), P(b)

– {w1, w2} K(P(a) v P(b))

– If a directory contains the file a.out, then it also contains core

Querying Databases

• K() – is known to be true

• K(¬) – is known to be false

• Kw() – is known either true or false

• Kv(t) – is t known to have fixed value

• Negation of the above

• What about LCW?

Planning Problem

• {I, G, A, U}

• I = initial state

• G = goal conditions (primitive queries)

• A = set of actions

• U = domain specific update rules (optional)

Planning AlgorithmPlanPKS

if(goalsSatisfied) return plan

else choose some action, apply it, PlanPKS

or choose some ground instance in Kw

PlanPKS with added to Kf

PlanPKS with ¬ added to Kf

return merged, contingent plan

When does this search terminate?What are some problems and limitations?

Example

Initial State:Kf = {(= (pwd) root), (indir papers root), (indir planner root), (dir root), (dir papers), (dir planner), (file paper_tex)}Kx = {((indir paper_tex planner) | (indir paper_tex papers))}Goal:K(indir paper_tex (pwd))

root

papers planner

Directory Structure

Start: (pwd) = root

Goal: Know paper_tex is in the current directory (pwd)

Exclusive Or: paper_tex is in either papers or planner

Contingent Plan

start

cd-down(papers)

ls(paper_tex, papers)

cd-up(root)

add(Kf, indir(paper_tex, papers)) add(Kf, ¬indir(paper_tex, papers))

cd-down(papers) cd-down(planner)

Is this plan optimal?

What are problems with this representation of plans?

Expressiveness of Goals

• Conditions that hold in final state– No universal quantification

• What about SADL?– Initially?– Restore?– Hands-off?

Reasoning about Conditional Plans

• Conditional plan is a tree– Nodes are knowledge states– Edges are actions

• Each leave corresponds to one branch

• Each branch one linear sequences of worlds

• Reason about each linear sequence– How?

Example

Initial State: bottle of liquid, a healthy lawn

Goal: know whether liquid is poisonous

Inference Procedure

• Consider two consecutive states, s1 and s2, in a linearization and the related action, a newly known in s2 and a does not change , then s1

newly known in s1 and a does not change , then s2

newly known in s1 and a has conditional effect , then s2

– More inferences using similar ideas…

• When can we apply these inferences?– At what points in conditional plans?

• What about initially, restore, hands-off?

Flaws

• Experiments misleading

• Unclear about LCW

• Not clear what is new and important

• More discussion about incompleteness

Future Work

• Heuristic search– Dealing with scalability issues

• Contingent planning with universal quantification

• Further implementation and testing

• Parallel plans

• Probabilistic knowledge