1. L9. Planning Agents L7_exAnswer and explanation Review
Search vs. planning
Situation calculus
STRIPS operators

2. L7_ex 解答例

3. Forward chaining example
Let us add facts r1, r2, r3, f1, f2, f3 in turn into KB.
r1. Buffalo(x)  Pig(y)  Faster(x,y)
r2. Pig(y)  Slug(z)  Faster(y,z)
r3. Faster(x,y) Faster(y,z)  Faster(x,z)
f1. Buffalo(Bob) [r1-c1, Bob/x, yes]
f2. Pig(Pat) [r1-c2, Pat/y, yes]  f4. Faster(Bob, Pat)
f3. Slug(Steve) [r2-c2, Steve/z, yes]
[r2, f2, f3, Pat/y, Steve/z, yes]  f5. Faster(Pat, Steve)
[r3, f4, f5, Bob/x, Pat/y, Steve/z, yes]  f6. Faster(Bob, Steve)
 x, y, z

4. Backward chaining example
Bob is a buffalo | 1. Buffalo(Bob) f1
Pat is a pig | 2. Pig(Pat) f2
Buffaloes run faster than pigs | 3.  x, y Buffalo(x)  Pig(y)  Faster(x,y) --r1
Goal: to prove
Faster(Bob, Pat)
Faster(x, y) r1
Buffalo(x)  Pig(y)
R(2) – And Elimination
Buffalo(x)
Pig(y)
R(8) – Universal Elimination
R(8) – Universal Elimination
{x/Bob}
{y/Pat}
Buffalo(Bob)
Pig(Pat) {} {}

5. Search in problem solving
Problem solution:
A path through the state space tree
State space search:
Search is a traversal of the tree
until the goals is reached.
State transitions is performed
by operators
　　a b c

6. Search in problem solving Difficulty using standard search algorithms
Standard search algorithms seems to fail miserably since the goal test is inadequate.
What is “finish”?

7. Search in problem solving Planning problem in situation calculus
Consider the same task get milk, bananas, and a cordless drill
A planning problem is represented in situation calculus by logical sentences that describe the three main parts of a problem.
Initial state:
At(home, S0)  Have(Milk, S0)  Have(Banana, S0)  Have(Drill, S0)
Goal state:
 s At(home, s)  Have(Milk, s)  Have(Bananas, s)  Have(Drill, s)
Operators:
a, s Have(Milk, Result (a,s)) [ (a=Buy(Milk)) At(Supermarket, s)
 (Have(Milk, s) a Drop(Milk) )]
Plan:
p = [Go(Supermarket), Buy(Milk), Buy(Bananas), Go(HWS), …]

8. Search in problem solving Problem solving vs. planning
Representation of states
PS: Direct assignment of a symbol to each state
PL: Logic sentences
Representation of goals
PS: A goal state symbol
PL: Sentences that describe objective
Representation of actions
PS: Operators that transform one state symbol into another
PL: Addition/deletion of logic sentences describing world state
Representation of plans
PS: Path through state space
PL: Ordered or partially-orderer sequence of actions.

9. Search in problem solving Advantages of Planning Systems
Uniform language for describing states, goals, actions,
and their effects.
Ability to add actions to a plan whenever they are needed,
not just in an incremental sequence from some initial state.
Ability to capture the fact that most parts of the world are
independent of most other parts.
It performs better for complex worlds over standard search
algorithm since searching space becomes huge when there
are many initial states and operators in standard search
algorithms.

10. The Situation Calculus
Wff - well formed formula
A goal can be described by a wff:  x On(x, B)
if we want to have a block on B
Planning: finding a set of actions to achieve a goal wff.
Situation Calculus (McCarthy, Hayes, 1969, Green 1969)
A Predicate Calculus formalization of states, actions, and their effects.
So state in figure can be described by:
On(B, A)  On(A, C)  On(A, Fl)  Clear(B)
we reify the state and include them as arguments
The atoms denotes relations over states. On(B, A, S0)  On(A, C, S0)  On(C, Fl, S0)  Clear(B, S0)
We can also have. x, y, s On(x, y, s)  (y = Fl)   Clear(y, s)
s Clear(Fl, s)

11. Representing actions Reify the actions: denote an action by a symbol
actions are functions
move(B,A,Fl): move block B from block A to Fl
move (x,y,z) - action schema
do: A function constant, do denotes a function that maps actions and states into states
Express the effects of actions.
Example: (on, move) (expresses the effect of move on On)
positive effect axiom:
action
state

12. Effect axioms for (clear, move)
move(x, y, z)
matching?
precondition are satisfied with
B/x, A/y, S0/s, F1/z
what was true in S0 remains true
figure 21

13. Frame axioms
Not everything true can be inferred Such as, On(C,Fl) remains true but cannot be inferred
Actions have local effect
We need frame axioms for each action and each fluent that does not change as a result of the action
example: frame axioms for (move, on)
If a block is on another block and move is not relevant, it will stay the same.
Positive
Negative
Frame axioms for (move, clear)
A fluent is something that flows and changes across situations

14. STRIPS: describing goals and state
STRIPS: STanford Research Institute Planning System
Basic approach in GPS (general Problem Solver):
Find a “difference” (Something in G that is not provable in S0)
Find a relevant operator f for reducing the difference
Achieve precondition of f; apply f; from resultant state, achieve G.

15. STRIPS planning STRIPS uses logical formulas to represent the states
S0, G, P, etc:
Description of operator f:

16. A STRIPS planning example
On(B,A)
On(A,C)
On(C,F1)
Clear(B)
Clear(Fl)
The formula describes a set of world states
Planning search for a formula satisfying a goal description
On(A, C)
On(C, Fl)
On(B, Fl)
Clear(A)

17. STRIPS Description of Operators
A STRIPS operator has 3 parts:
A set, PC - preconditions
A set D - the delete list
A set A - the add list
Usually described by Schema: Move(x,y,z)
PC: On(x,y) and On(Clear(x) and Clear(z)
D: Clear(z) , On(x,y)
A: On(x,z), Clear(y), Clear(F1)
A state S1 is created applying operator O by adding A and deleting D from S1.

18. Example: The move operator

19. Example1: The move operatorG S0 
f(P)->G
S0->P   
x/B, y/A, z/Fl
x/B, y/A, z/Fl  
On(x,y)
Clear(x)
Clear(z)
f: move(x,y, z) add: On(x,z), Clear(y)
del: On(x,y), Clear(z)
On(x,z)
Clear(x)
Clear(y) P:
f(P):

20. Example1: The move operator
ABC +
Clear(F2) G S0 
f(P)->G
S0->P   
x/B, y/A, z/Fl
x/B, y/A, z/Fl  
On(x,y)
Clear(x)
Clear(z)
f: move(x,y, z) add: On(x,z), Clear(y)
del: On(x,y), Clear(z)
On(x,z)
Clear(x)
Clear(y) P:
f(P):

21. STRIPS algorithms are to search for operators!!!
We will leave the discussion of STRIPS algorithms to next week.

22. ?
Quiz：
What differences are between standard search algorithms for problem solving and planning systems?