PLANNING

1
PLANNING

• Partial order regression planning

Temporal representation 1 Deductive planning in
Logic Temporal representation 2
2
Temporal representation 1

• From STRIPS to a simple situation calculus

3
Towards planning as Automated Reasoning

• Formulate in a logic theory T

• The initial situation
• For each operator
• its preconditions
• the relation between the previous
  situation and the next one.

• Prove the logical consequence F

There exists a sequence of operations, such
that if they are applied to the initial
situation, the resulting situation satisfies some
given properties.

• Extract the plan from the proof.

4
What the STRIPS represen-tation lacks to do this
do NOT describe WHEN (at which time point or in
which situation) these properties hold.

• Remember logic is monotonic no
  consequences get removed!

• 2) Add and Delete list are procedural.
• These do not declaratively describe the relation
  between the previous situation and the next.

5
Introducing temporal information

• Two dimensions of options

on(A,B,T)
on(A,B,S)
holds(on(A,B),T)
holds(on(A,B),S)
6
A tiny example problem
7
Situation calculus, extra argument.
8
The situations in this world
9
The plan ?
10
Completing situations

• In STRIPS (and in LP) these are implicitly
  represented.

• In FOL, we could instead add

?x,y,z,s on(x,y,s) ? yTable ? xz ?
on(z,y,s)
?x,y,z,s on(x,y,s) ? yz ? on(x,z,s)
?x,y,s on(x,y,s) ? yTable ? clear(y,s)
PS also completes all later situations
11
Relating different situations (1)

• For each operator we need to represent the
  relation between previous and next situation

?s on(B,A,s) ? clear(B,s) ? on(B,Table,moveBATable
(s)) ? clear(A,moveBATable(s))

• Note delete on(B,A) is implicitly represented
  because of add on(B,Table) and the completing
  formulae.

12
Relating different situations Frame axioms (1)

• STRIPS operators only express which things change
  !

13
Relating different situations Frame axioms (2)

• Anything that was on something else before,
  except for B on A, still is

?x,y,s on(B,A,s) ? clear(B,s) ? on(x,y,s) ?
(xB ? yA) ?
on(x,y,moveBATable(s))

• Anything that was clear before, still is

?x,s on(B,A,s) ? clear(B,s) ? clear(x,s)
? clear(x,moveBATable(s))
14
Discussion

• This is one of the most simple options.

• Generalizing to operators patterns in not easy.
• But this is sufficient to illustrate planning as
  deduction.
• We discuss more refined representations later.

15
Planning as deduction

• Automated reasoning applied to a simple situation
  calculus representation

16
The theory and goal
17
Normalization
18
A linear top-down proof
What is the plan?? The answer substitution
s1/moveBATable(S0)
19
The relevance of Frame axioms?

• Find a plan such that B gets on the Table AND A
  is still on the Table.

20
A linear top-down proofextended
No longer resolves with anything, except frame
axioms !
21
Normalization of Frame Axiom 1
?x,y,s on(B,A,s) ? clear(B,s) ? on(x,y,s) ?
(xB ? yA) ? on(x,y,moveBATable(s))
?x,y,s on(x,y,moveBATable(s)) ? on(B,A,s) ?
clear(B,s) ? on(x,y,s) ? (xB ? yA)
?x,y,s (on(x,y,moveBATable(s)) ? xB ? on(B,A,s)
? clear(B,s) ? on(x,y,s)) ? (on(x,y,moveBATable
(s)) ? yA ? on(B,A,s) ? clear(B,s) ?
on(x,y,s))
22
The continuation of the proof
23
Unique Names Axioms

• For each two non-unifiable syntactic objects o1
  and o2 o1o2.

• Here AB, ATable, BTable

24
Can deductive planning achieve the STRIPS
control?

• Definitely does regression (goal directed search)
  !

• Partial order planning?

• Reason ? s property(s)
• Resolution steps construct
• s move(move(move( . . . )))
• this is a total order !!

• Need more refined temporal representations to do
  these.

25
Temporal representation 2

• Representing operator patterns
• Meta-representation
• situation calculus
• Time-point representation
• event calculus

26
Representing operator patterns (1)

• S0 still represents the initial situation.

• Representation of a situation resulting from an
  operator pattern functor move/4

• WE get MUCH LESS different axioms!

27
Representing operator patterns (2)

• Better represent other operator patterns with
  other functors (to avoid confusion - and
  unification with wrong ones).

• Still 3 such axioms 6 frame axioms needed !

28
The full theory
Initial situation
Consistency axioms
New initiated properties for each action
29
The full theory (2)
Frame axioms for move x from y to z
Frame axioms for move x from y to table
Frame axioms for move x from Table to y
30
Difference with previous representation?
31
Difference with previousrepresentation (2)
Also for the frame axioms!
32
Meta-representation

• The Situation Calculus

33
Meta-representation

• Could not be done with extra-argument
  representation.

34
Meta-representationinitial situation
35
Situation Namesfinal version
1
2
Reason The name of the operator is now a term
that we can abstract with a variable
3
movetoT(B,A)
36
Formally situation names

• Situations are now represented as result(o,s) ,
  where o is an operator pattern and s a situation.

37
Improvement ?

• Use we can now define if list, add list and
  delete list of each operator pattern explicitly
  with new predicates !

38
If-list legal/2 predicate
39
Add-list initiates/2 predicate
40
Delete-list terminates/2 predicate
41
What is gained ?The Situation Calculus!
42
Problem independence

• Observe that these axioms defining holds are
  problem independent
• and can be applied to any planning problem.
• For each new planning problem only initially,
  legal, initiates and terminates need to be
  defined !
• These correspond to the initial situation and to
  the if- add- and delete-lists of operator
  patterns.

43
Planning

• should be added, where p1,,pn are the properties
  that should hold in the goal situation.

• Deduction proceeds similarly to what was
  presented in the extra-argument representation.

• In particular the final value for s in the
  unifier is the plan !

44
Completion and consistency

• Note that all formulae are essentially Horn
  clauses (possibly extended with negation in the
  bodies - case terminates).

• Disjunctions in bodies are readily transformed to
  a set of Horn clauses.

45
Completion and consistency (2)

• Assume that this description is complete
  (anything else than described is false !), then
  we can interpret these Horn clauses as a Logic
  program.

• Consequence the consistency axioms are no
  longer needed

• Requires that we understand holds(p,s) as
  holds(p,s),
• then information such as holds(on(B,Table)
  ,S0) follows from our specification.

46
Meta-representation (2)

• The Event Calculus

47
Using time points

• Alternative to situations

• Ontology something holds on a specific moment in
  time.

48
The main axioms
Where the concepts initially/1, initiates/2 and
terminates/2 are the same as before (and problem
dependent), and legal/2 is completely similar as
before,but defined in terms of holds(p,t) instead
of holds(p,s) .
49
Event calculus pictured

• When does something hold at time t ?

• If is was true from the beginning and not
  clipped

• If something happened to make it true and not
  clipped

50
The event calculus

• The new elements in this ontology
• Temporal identification through time points
  instead of situations.
• The relation lt/2 should be defined as a (strict)
  partial order (the temporal order on time points)
  .
• There is now only 1 definition of holds !!
• This includes the initial situation relating
  situations the frame axioms!
• In situations calculus situations are related to
  the previous situation In event
  calculus they are related to some previous moment
  in time that initiated something (which hasnt
  been clipped).
• Allows to make bigger steps than just 1 at a
  time.

51
Relation to planning?

• A plan in this representation is a set of atoms

event(O1,T1) event(O2,T2) event(O3,T3) ..
where each Oi is an operator, each Ti a time
point, Ti lt Ti1 and executing O1, O2, O3, in
sequence in the initial state gives the goal
state.

• We CAN NOT (deductively) !!!!

52
Relation to planning (2)

• Deduction from a goal
• needs to go through initiated/2.
• Unless the goal was already satisfied in the
  initial state, this requires event/2 facts to
  hold !
• But there is NO definition for event/2
• This predicate is the object of our search !!
• SO
• is not a logical consequence of the event
  calculus theory.
• It is not true in ALL models.

?t holds(on(A,Table) ,t)
53
Model generation

• SO the goal
• is not true in all models.
• BUT if there exists a model (with some event/2
  facts true) in which the goal is also true, then
  the true event/2 atoms in that model give us the
  plan.
• Model-generation techniques!
• Or alternatively abductive reasoning
  techniques.
• Meaning find a set of hypothesis ? of event/2
  atoms such that

?t holds(on(A,Table) ,t)
?t holds(on(A,Table) ,t)
? ? Theory entails