Automated Planning

1
Automated Planning

• Source
• Ch. 1
• Appendix B.3
• Dana Naus slides
• My own

• Dr. Héctor Muñoz-Avila

2
What is Planning? Classical Definition

• Planning finding a sequence of actions to
  achieve a goal

• Domain Independent symbolic descriptions of the
  problems and the domain. The plan generation
  algorithm remains the same
• Domain Specific The plan generation algorithm
  depends on the particular domain

Advantage - opportunity to have clear
semantics Disadvantage - symbolic description
requirement
Advantage - can be very efficient Disadvantag
e - lack of clear semantics
- knowledge-engineering for adaptation
3
Example of Planning Tasks Military Planning
4
Example of Planning Tasks Playing a Game
5
Example of Planning Tasks Route Planning
6
Classical Planning

• Classical planning makes a number of assumptions
• Symbolic information (i.e., non numerical)
• Actions always succeed
• The Strips assumption only changes that takes
  place are those indicated by the operators
• Next slide enumerates all assumptions

• Despite these (admittedly unrealistic)
  assumptions some work-around can be made (and
  have been made!) to apply the principles of
  classical planning to games
• Hot research topic to removes some of these
  assumptions

7
State Goals
A

• Initial state (on A Table) (on C A) (on B Table)
  (clear B) (clear C)
• Goals (on C Table) (on B C) (on A B) (clear A)

Initial state
Goals
C
B
A
B
C
(Ke Xu)
8
General-Purpose Planning Operators
?x
?y
?y
?x

• Operator (Unstack ?x)
• Preconditions (on ?x ?y) (clear ?x)
• Effects
• Add (on ?x table) (clear ?y)
• Delete (on ?x ?y)

9
Classical Planning can be Hard
C
A
B
C
A
B
C
B
A
B
A
C
A
B
C
B
A
B
C
B
C
A
B
A
C
C
A
A
A
B
C
B
C
C
B
A
A
B
C
(Michael Moll)
10
Conceptual Model1. Environment
System ?
State transition system ? (S,A,E,?)
Dana Nau Lecture slides for Automated
PlanningLicensed under the Creative Commons
Attribution-NonCommercial-ShareAlike License
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11
State Transition System

• ? (S,A,E,?)
• S states
• A actions
• E exogenous events
• State-transition function? S x (A ? E) ? 2S
• S s0, , s5
• A move1, move2, put, take, load, unload
• E
• ? see the arrows

The Dock Worker Robots (DWR) domain
Dana Nau Lecture slides for Automated
PlanningLicensed under the Creative Commons
Attribution-NonCommercial-ShareAlike License
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12
Conceptual Model2. Controller
Given observation o in O, produces action a in A
Controller
Observation function h S ? O
State transition system ? (S,A,E,?)
Dana Nau Lecture slides for Automated
PlanningLicensed under the Creative Commons
Attribution-NonCommercial-ShareAlike License
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13
Conceptual Model2. Controller
Complete observability h(s) s
Given observation o in O, produces action a in A
Controller
Observation function h S ? O
Given state s, produces action a
State transition system ? (S,A,E,?)
Dana Nau Lecture slides for Automated
PlanningLicensed under the Creative Commons
Attribution-NonCommercial-ShareAlike License
http//creativecommons.org/licenses/by-nc-sa/2.0/
14
Conceptual Model3. Planners Input
Planner
Depends on whether planning is online or offline
Given observation o in O, produces action a in A
Observation function h S ? O
State transition system ? (S,A,E,?)
Dana Nau Lecture slides for Automated
PlanningLicensed under the Creative Commons
Attribution-NonCommercial-ShareAlike License
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15
PlanningProblem
Description of ? Initial state or set of
states Initial state s0 Objective Goal state,
set of goal states, set of tasks, trajectory of
states, objective function, Goal state s5
The Dock Worker Robots (DWR) domain
Dana Nau Lecture slides for Automated
PlanningLicensed under the Creative Commons
Attribution-NonCommercial-ShareAlike License
http//creativecommons.org/licenses/by-nc-sa/2.0/
16
Conceptual Model4. Planners Output
Planner
Instructions tothe controller
Depends on whether planning is online or offline
Given observation o in O, produces action a in A
Observation function h(s) s
State transition system ? (S,A,E,?)
Dana Nau Lecture slides for Automated
PlanningLicensed under the Creative Commons
Attribution-NonCommercial-ShareAlike License
http//creativecommons.org/licenses/by-nc-sa/2.0/
17
Classical Assumptions (I)

• A0 Finite system
• finitely many states,actions, and events
• A1 Fully observable
• the controller alwaysknows what state ? is in
• A2 Deterministic
• each action or event hasonly one possible
  outcome
• A3 Static
• No exogenous events no changes except those
  performed by the controller

Dana Nau Lecture slides for Automated
PlanningLicensed under the Creative Commons
Attribution-NonCommercial-ShareAlike License
http//creativecommons.org/licenses/by-nc-sa/2.0/
18
Classical Assumptions (II)

• A4 Attainment goals
• a set of goal states Sg
• A5 Sequential plans
• a plan is a linearlyordered sequence of actions
  (a1, a2, an)
• A6 Implicit time
• no time durations
• linear sequence of instantaneous states
• A7 Off-line planning
• planner doesnt know the execution status

Dana Nau Lecture slides for Automated
PlanningLicensed under the Creative Commons
Attribution-NonCommercial-ShareAlike License
http//creativecommons.org/licenses/by-nc-sa/2.0/
19
This is Nice but How About Actual Deployed
Applications?

• We briefly discuss three deployed applications
• Fear application of a classical planner
• Bridge application of a new-classical planner
• MRB planning execution
• We will discuss these again in detail later in
  the semester

20
Detailed Discussion of Topics

• See web page

21
Math Background Logic
Source Appendix B.3
22
Introduction to Logic

• A logic is a formal system of representing
  knowledge
• A logic has
• Syntax indicates the valid expressions
• Semantics provides meaning to the expressions
• Inference mechanism draw conclusions from a set
  of statements

23
Example propositional Logic

• Definition. A propositonal formula is defined
  recursively as follows
• A symbol form a predefined list P is a
  proposition
• If ?1 and ?2, are propositions then
• (?1 ? ?2)
• (?1 ? ?2)
• (?1 ? ?2)
• are also propositions
• If ? is a proposition then (?) is a proposition

Example.
(a) ? (a ? b ? c ? d) ? (c ? d) ? (d)
Semantics. Truth tables Inference mechanism.
Modus ponens
24
Predicate Logic

• Definition. A term is defined as follows
• A constant is a term
• A variable is a term
• If t1, , tn are terms and f is a function
  symbols then f(t1,,tn) is a term
• Definition. If t1, , tn are terms and p is a
  symbol for an n-ary predicate then p(t1, , tn )
  are predicates

25
Predicate Logic Formulas

• Definition. An atomic formula is defined
  recursively as follows
• An atom is an atomic formula
• If ?1 and ?2, are atomic formulas then
• (?1 ? ?2)
• (?1 ? ?2)
• (?1 ? ?2)
• are also atomic formulas
• If ? is a atomic formula then (?) is an atomic
  formula
• If ? is a atomic formula and x is a variable
  then
• ?x(?) is an atomic formula
• ?x(?) is an atomic formula

Example ?x (likes(Mephistus,x) ?
evilThing(x)) How do we say
that Mephistus likes only evil things?
26
Predicate Logic Semantics

• (?1 ? ?2)
• (?1 ? ?2)
• (?1 ? ?2)
• (?)
• ?x(?)
• ?x(?)

27
Predicate Logic Literals and Clauses

• Definition. A literal is an atomic formula
  consisting of a single atom and no quantifiers
• likes(Mephistus,x)
• evilThing(x)
• Definition. A clause is a disjunction of literals
• likes(Mephistus,x) ? evilThing(x)

28
Resolution Inference Mechanism for Predicate
Logic

• Substitution, ?
• Unification
• Most general unifier
• Resolution Given two clauses
• L l1 ? l2 ? ? ln
• M m1 ? m2 ? ? mn
• If there is and li and mk such that
• li a and mk a and
• There is a most general unifier ? for a and a
• Then (?L li) ? (?M mk) is a resolvent of L
  and M
• Idea behind the resolution procedure