1. Warning
I estimate 3 weeks of intensive reading for you to understand your assigned topic
These are not self contained and you will need to understand other topics first before we cover them inb class.
I estimate 4-5 weeks before you have successfully installed the planning system and be able to run it in your domain

2. Plan Representation in Classical Planning
Sources:
Ch. 2
Appendix B.2
Slides from Dana Nau’s lecture
Dr. Héctor Muñoz-Avila

3. Classical Assumptions (I)
A0: Finite system
finitely many states, actions, and events
A1: Fully observable
the controller always knows what state  is in
A2: Deterministic
each action or event has only one possible outcome
A3: Static
No exogenous events: no changes except those performed by the controller
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4. Classical Assumptions (II)
A4: Attainment goals
a set of goal states Sg
A5: Sequential plans
a plan is a linearly ordered sequence of actions (a1, a2, … an)
A6 :Implicit time
no time durations
linear sequence of instantaneous states
A7: Off-line planning
planner doesn’t know the execution status
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5. Ok So Why Even Study Classical Planning?
By restricting the complex processes we can study properties and limitations (e.g., complexity)
Some classical planning techniques can be expanded to deal with more realistic situations
We will study these later
Surprisingly some classical planners have made it into actual deployed applications

6. Classical Planning: Conceptual Model
State transition system
 = (S,A,E,)
S; States: s1, s2, …, sn
A: Actions
: valid transitions: General definition of transitions says S x A  2S but here we restrict to S x A  S or
Why?
E: events. E is assumed to be empty. Why? aw si sk
Because actions are deterministic
System 
Because world is static

7. Three Representations
Set-Theoretic:
uses propositional logic
Classical representation:
uses predicate logic
State-variable representations
Represent states in terms of variables used

8. Propositional Logic
Definition. A proposition over a set P is defined as follows:
any element in a set 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: tired  ¬hungry
Semantics defined in the usual way
Difference with predicate logic?
No variables
No quantifiers: ,  (Hai!)

9. Set-Theoretic Representation: State
If L= {p1, .., pn} then S 2L
Lets define possible propositions for the Dock Worker Robots (DWR) domain
Lets define the state above
Suppose that there are 8 containers how many states would be needed?

10. Set-Theoretic Representation: Actions and state-transition functions
An action a is defined as a triple: (precs(a), eff+(a), eff(a)) such that:
precs(a), eff+(a), eff(a)  2L
An action a is applicable to an state s if precs(a)  s
State-transition function (s,a): If
an action a is applicable in an state s, and
eff(a)  s
Then the transition (s,a) is defined as
(s  eff(a)) eff+(a)
Otherwise (s,a) is undefined

11. Set-Theoretic Representation: Example
take
put
move1
move2
load
unload
location 1
location 2 s0 s1 s4 s5 s3 s2
Set-Theoretic Representation: Example
Lets define a few actions
Note that if more containers are added then:
Existing actions may need to change
New actions are needed
Drawback?
A large number of actions must be defined

12. Classical Representation: State
States: sets of ground (i.e., without variables) positive literals
Lets define possible predicates for the Dock Worker Robots (DWR) domain
Lets define the state above

13. Classical Representation: State (2)
Lets compare with the one in the book…

14. Classical Representation: Operators
Operator: a triple o=(name(o), precond(o), effects(o))
name(o) is a syntactic expression of the form n(x1,…,xk)
n: operator symbol - must be unique for each operator
x1,…,xk: variable symbols (parameters)
must include every variable symbol in o
precond(o): preconditions
literals that must be true in order to use the operator
effects(o): effects
literals the operator will make true
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15. Classical Representation: Actions
Action: ground instance (via substitution) of an operator
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16. Notation Let S be a set of literals. Then
S+ = {atoms that appear positively in S}
S– = {atoms that appear negatively in S}
More specifically, let a be an operator or action. Then
precond+(a) = {atoms that appear positively in a’s preconditions}
precond–(a) = {atoms that appear negatively in a’s preconditions}
effects+(a) = {atoms that appear positively in a’s effects}
effects–(a) = {atoms that appear negatively in a’s effects}
effects+(take(k,l,c,d,p)) = {holding(k,c), top(d,p)}
effects–(take(k,l,c,d,p)) = {empty(k), in(c,p), top(c,p), on(c,d)}
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17. Applicability
An action a is applicable to a state s if s satisfies precond(a),
i.e., if precond+(a)  s and precond–(a)  s = 
Here are an action and a state that it’s applicable to:
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18. Result of Performing an Action
If a is applicable to s, the result of performing it is
(s,a) = (s – effects–(a))  effects+(a)
Delete the negative effects, and add the positive ones

19. Planning domain: language plus operators
Corresponds to a set of state-transition systems
Example: operators for the DWR domain

20. Planning Problems Given a planning domain (language L, operators O)
Statement of a planning problem: a triple P=(O,s0,g)
O is the collection of operators
s0 is a state (the initial state)
g is a set of literals (the goal formula)
The actual planning problem: P = (,s0,Sg)
s0 and Sg are as above
 = (S,A,) is a state-transition system
S = {all sets of ground atoms in L}
A = {all ground instances of operators in O}
 = the state-transition function determined by the operators
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21. Plans and Solutions
Plan: any sequence of actions  = a1, a2, …, an such that each ai is a ground instance of an operator in O
The plan is a solution for P=(O,s0,g) if it is executable and achieves g
i.e., if there are states s0, s1, …, sn such that
 (s0,a1) = s1
 (s1,a2) = s2 …
 (sn–1,an) = sn
sn satisfies g
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22. Example g1={loaded(r1,c3), at(r1,loc2)} Let P1 = (O, s1, g1), where
O is the set of operators given earlier
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23. Example (continued) Here are three solutions for P1:
take(crane1,loc1,c3,c1,p1), move(r1,loc2,loc1), move(r1,loc1,loc2), move(r1,loc2,loc1), load(crane1,loc1,c3,r1), move(r1,loc1,loc2)
take(crane1,loc1,c3,c1,p1), move(r1,loc2,loc1), load(crane1,loc1,c3,r1), move(r1,loc1,loc2)
move(r1,loc2,loc1), take(crane1,loc1,c3,c1,p1), load(crane1,loc1,c3,r1), move(r1,loc1,loc2)
Each of them produces the state shown here:
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24. Comparison: Classical versus Set-Theoretical Representations
A set-theoretic representation is equivalent to a classical representation in which all of the atoms are ground
From classical to set-theoretical: Exponential blowup?
If a classical operator contains n atoms and each atom has arity k, then it corresponds to cnk actions where c = |{constant symbols}|
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25. Extensions to Classical Representation
Conditional effects
Quantified expressions
Disjunctive preconditions
Axioms
Extended goals

26. State-Variable Representation
Use ground atoms for properties that do not change, e.g., adjacent(loc1,loc2)
For properties that can change, assign values to state variables
Like fields in a record structure
Classical and state-variable representations take similar amounts of space
Each can be translated into the other in low-order polynomial time
{top(p1)=c3, cpos(c3)=c1, cpos(c1)=pallet, holding(crane1)=nil, rloc(r1)=loc2, loaded(r1)=nil, …}
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27. Expressive Power
Any problem that can be represented in one representation can also be represented in the other two
Can convert in linear time and space, except when converting to set-theoretic (where we get an exponential blowup)
P(x1,…,xn)
becomes
fP(x1,…,xn)=1
trivial
Set-theoretic
representation
Classical
representation
State-variable
representation
write all of
the ground instances
f(x1,…,xn)=y
becomes
Pf(x1,…,xn,y)
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28. Comparison Classical representation
The most popular for classical planning, partly for historical reasons
Set-theoretic representation
Can take much more space than classical representation
Useful in algorithms that manipulate ground atoms directly
e.g., planning graphs (Chapter 6), satisfiability (Chapters 7)
Useful for certain kinds of theoretical studies
State-variable representation
Equivalent to classical representation
Less natural for logicians, more natural for engineers
Useful in non-classical planning problems as a way to handle numbers, functions, time
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29. “Hidden” Issues Closed World Assumption
Ground literals not in the state are assumed to be false
Unification:
When unifying two atoms there is a unique Most General Unifier (sans variable names)
But: when matching sets of atoms (e.g., operator preconditions versus state) multiple different Most General Unifiers may exist
This increases branching factor of search space