Automated Planning

Automated Planning

Dr. Héctor Muñoz-Avila

Source:• Ch. 1• Appendix B.3• Dana Nau’s slides• My own

What is Planning? Classical Definition

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 semanticsDisadvantage: - symbolic description requirement

Advantage: - can be very efficientDisadvantage: - lack of clear semantics - knowledge-engineering for adaptation

Planning: finding a sequence of actions to achieve a goal

Example of Planning Tasks: Military Planning

Example of Planning Tasks: Playing a Game

Example of Planning Tasks: Route Planning

• Classical planning makes a number of assumptions:Symbolic information (i.e., non numerical)Actions always succeedThe “Strips” assumption: only changes that takes place

are those indicated by the operatorsNext 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

Classical Planning

State & Goals

• 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)

A

C

B C

B

AInitial state Goals

(Ke Xu)

General-Purpose Planning: Operators

?y

?x

No block on top of ?x

transformation

?y

?x…

…

No block on top of ?y nor ?x

Operator: (Unstack ?x)• Preconditions: (on ?x ?y) (clear ?x) • Effects:

Add: (on ?x table) (clear ?y)Delete: (on ?x ?y)

On table

Classical Planning can be Hard

AC

B A B C A CB

CBA

BA

C

BAC

B CA

CAB

ACB

BCA

A BC

AB

C

ABC

(Michael Moll)

Conceptual Model1. Environment

State transition system = (S,A,E,)

System

Dana Nau: Lecture slides for Automated PlanningLicensed under the Creative Commons Attribution-NonCommercial-ShareAlike License: http://creativecommons.org/licenses/by-nc-sa/2.0/

= (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

State Transition System

take

put

move1

put

take

move1

move1move2

loadunload

move2

move2

location 1 location 2

s0

location 1 location 2

s1

s4

location 1 location 2

s5

location 1 location 2

location 1 location 2

s3

location 1 location 2

s2

The Dock Worker Robots (DWR) domainDana Nau: Lecture slides for Automated PlanningLicensed under the Creative Commons Attribution-NonCommercial-ShareAlike License: http://creativecommons.org/licenses/by-nc-sa/2.0/

Observation functionh: S O

State transition system = (S,A,E,)location 1 location 2

s3

Given observation o in O, produces action a in A

Conceptual Model2. Controller

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/

State transition system = (S,A,E,)location 1 location 2

s3

Complete observability:

h(s) = s

Observation functionh: S O

Given observation o in O, produces action a in A

Given state s, produces action a

Conceptual Model2. Controller

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/

State transition system = (S,A,E,)

Depends on whether planning is online or offline

Observation functionh: S O

Given observation o in O, produces action a in A

Conceptual Model3. Planner’s Input

Planner

Planning problemPlanning problemPlanning problem

Dana Nau: Lecture slides for Automated PlanningLicensed under the Creative Commons Attribution-NonCommercial-ShareAlike License: http://creativecommons.org/licenses/by-nc-sa/2.0/

PlanningProblem

take

put

move1

put

take

move1

move1move2

loadunload

move2

move2

location 1 location 2

s0

location 1 location 2

s1

s4

location 1 location 2

s5

location 1 location 2

location 1 location 2

s3

location 1 location 2

s2

Description of Initial state or set of states

Initial state = s0

ObjectiveGoal state, set of goal states, set of tasks, “trajectory” of states, objective function, …Goal state = s5

The Dock Worker Robots (DWR) domainDana Nau: Lecture slides for Automated PlanningLicensed under the Creative Commons Attribution-NonCommercial-ShareAlike License: http://creativecommons.org/licenses/by-nc-sa/2.0/

State transition system = (S,A,E,)

Given observation o in O, produces action a in A

Instructions tothe controller

Depends on whether planning is online or offline

Observation functionh(s) = s

Conceptual Model4. Planner’s Output

Planner

Planning problemPlanning problemPlanning problem

Dana Nau: Lecture slides for Automated PlanningLicensed under the Creative Commons Attribution-NonCommercial-ShareAlike License: http://creativecommons.org/licenses/by-nc-sa/2.0/

Classical Assumptions (I)

• A0: Finite system finitely many states,

actions, and events• A1: Fully observable

the controller alwaysknows what state is in

• A2: Deterministiceach action or event has

only 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/

Classical Assumptions (II)

A4: Attainment goalsa set of goal states Sg

A5: Sequential plansa plan is a linearly

ordered sequence of actions (a1, a2, … an)

A6 :Implicit timeno time durations linear sequence of instantaneous states

A7: Off-line planning planner doesn’t 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/

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

Detailed Discussion of Topics

• See web page

Math Background: Logic

Source: Appendix B.3

Introduction to Logic

• A logic is a formal system of representing knowledge

• A logic has:Syntax – indicates the valid expressionsSemantics – provides meaning to the expressions Inference mechanism – draw conclusions from a set

of statements

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 tablesInference mechanism. Modus ponens

Predicate Logic

• Definition. A term is defined as follows:A constant is a termA 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

Predicate Logic: FormulasDefinition. 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?

Predicate Logic: Semantics

(1 2)

(1 2)

(1 2) ¬()x()x()

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)

Resolution: Inference Mechanism for Predicate Logic

• Substitution,

• UnificationMost 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