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Assignment 5

- Why you must re-standardize variables every time a rule is retrieved.
- Test data in assignments/A5sampleInputs
- easy
- average
- hard
- veryhard

New Topic: AI planning

- Generating plans
- Given:
- A way to describe the world (“ontology”)
- An initial state of the world
- A goal description
- A set of possible actions to change the world
- Find:
- A sequence of actions to change the initial state into one that satisfies the goal
- Note similarity to state space search (e.g., 8 puzzle)
- Planning extends to more complex worlds and actions

States of the world have partial descriptions

(assertions of agent’s beliefs about its current situation)

S1

[ ┐holds(at(home)) ]

S0

go(store)

holds(at(store))

holds(color(door, red))

paint(door, green)

S2

holds(at(home))

holds(color(door, red))

Applications

- Mobile robots
- An initial motivator, and still being developed
- Simulated environments
- Goal-directed agents for training or games
- Web and grid environments
- Intelligent Web “bots”
- Workflows on a computational grid
- Managing crisis situations
- E.g. oil-spill, forest fires, urban evacuation, in factories, …
- And many more
- Factory automation, flying autonomous spacecraft, playing bridge, military planning, …

Plannning challenge: Representing change

- As actions change the world OR we consider possible actions, we need to:
- Know how an action will alter the world
- Keep track of the history of world states (avoid loops)
- 3 approaches:
- Strips approach with total order planning (state space search)
- Strips approach with partial order planning (POP)
- Situation calculus

“Classical Planning” Assumptions

- Discrete Time
- Instantaneous actions
- Deterministic Effects
- Omniscience
- Sole agent of change
- Goals of attainment (not avoidance)

Strips

- Highly influential representation for actions:
- Instead of F: state X action next state, uses a set of planning operators for achieving goals and subgoals
- Preconditions (list of propositions to be true)
- Delete list (list of propositions that will become false)
- Add list (list of propositions that will become true)
- [Implementation]
- More efficient to capture known strategies instead of searching the space of possible primitive actions and resulting states.

Example problem:

Initial state: at(home), ┐have(beer), ┐have(chips)

Goal: have(beer), have(chips), at(home)

Operators:

Buy (X):

Pre: at(store)

Add: have(X)

Go (X, Y):

Pre: at(X)

Del: at(X)

Add: at(Y)

States of the world have partial descriptions

(assertions of agent’s beliefs

S1

[ ┐holds(at(home)) ]

S0

go(store)

holds(at(store))

holds(color(door, red))

mow_lawn()

S2

holds(at(home))

holds(color(door, red))

Frame problem (again)

- I go from home to the store, creating a new situation S’. In S’:
- The store still sells chips
- My age is still the same
- Los Angeles is still the largest city in California…
- How can we efficiently represent everything that hasn’t changed?
- Strips provides a good solution for simple actions

Another problem: Ramification problem

- I go from home to the store, creating a new situation S’. In S’:
- I am now in Marina del Rey
- The number of people in the store went up by 1
- The contents of my pockets are now in the store..
- Do we want to say all that in the action definition?

Solutions to the frame and ramification problems

- In Strips, some facts are inferred within a world state,
- e.g. the number of people in the store
- All other facts, e.g. at(home) persist between states unless changed (remain unless on delete list)
- A challenge for knowledge engineer to avoid mistakes

Questions about Strips

- What would happen if the order of goals was

at(home), have(beer), have(chips) ?

- When Strips returns a plan, is it always correct? efficient?
- Can Strips always find a plan if there is one?

Strips operators for blocks world

- Move-to (x, b):
- Preconditions: Isa(b,Block), On(x, y), Cleartop(x), Cleartop(b)
- Add: On(x, b), Cleartop(y)
- Delete: On(x, y), Cleartop(b)
- Implementation: Puton(x, Topof(b))
- Move-to(x, Table):
- Preconditions:
- Add: On(x, Table)
- Delete:
- Implementation: Findspace(x, Table), Puton(x, Table)

Example: blocks world (Sussman anomaly)

Initial:

Goal:

A

State I: (On A Table) (On C A) (On B Table) (Cleartop B) (Cleartop C)

Goal: (On A B) (On B C)

“Naïve” planning algorithm output: Put C on table, put A on B [goal 1 accomplished], put A on table, put B on C [both goals accomplished] DONE!!!!

C

B

A

B

C

Partial Order Planning (POP)

- Explicitly views plans as a partial order of steps. Add ordering into the plan as needed to guarantee it will succeed.
- Avoids the problem in Strips, that focussing on one subgoal forces the actions that resolve that goal to be contiguous.

How to get dressed

- State: {}
- Goal {RightShoeOn, LeftShoeOn}
- Plan Operators:
- PutRshoe, Precond: RightSockOn, Effect: RightShoeOn
- PutLshoe, Precond: LeftSockOn, Effect: LeftShoeOn
- PutRsock, Effect: RightSockOn
- PutLsock, Effect: LeftSockOn
- Create a POP graph of solutions with causal links: “A achieves P for B” A B (also called protection links). This prevents another goal from causing a sock to be removed before the shoe goes on.

p

Goal:

A

C

B

A

B

C

Remember the “Sussman Anomaly”State I: (On A Table) (On C A) (On B Table) (Cleartop B) (Cleartop C)

Goal: (On A B) (On B C)

POP using Nets Of Action Hierarchies

on(a, b)

S

J

on(b, c)

clear(a)

puton(a, b)

S

J

clear(b)

S

J

clear(b)

puton(b, c)

S

J

clear(c)

Nets Of Action Hierarchies

on(a, b)

S

J

on(b, c)

clear(a)

puton(a, b)

S

J

clear(b)

S

J

clear(b)

puton(b, c)

S

J

clear(c)

Add a “threat” link to the network of plan actions

Resolve threat with an “order” link

clear(a)

puton(a, b)

S

J

clear(b)

S

J

clear(b)

puton(b, c)

S

J

clear(c)

clear(a)

puton(a, b)

S

J

clear(b)

S

J

clear(b)

puton(b, c)

S

J

clear(c)

puton(a, b)

S

J

clear(b)

S

J

clear(b)

puton(b, c)

S

J

clear(c)

clear(a)

puton(a, b)

J

S

clear(b)

puton(b, c)

S

J

clear(c)

puton(a, b)

J

S

clear(b)

puton(b, c)

S

J

clear(c)

puton(c, X)

clear(a)

puton(a, b)

J

S

clear(b)

puton(b, c)

S

J

clear(c)

Planning using logic and resolution: The situation calculus

- Key idea: represent a snapshot of the world, called a ‘situation’ explicitly.
- ‘Fluents’ are statements that are true or false in any given situation, e.g. ‘I am at home’
- Actions map situations to situations.

Blocks world example

- A move action: Move(x, loc)
- Use of the Result function:

Result(Move(x, loc), state) the state resulting from doing the Move action

- An axiom about moving:

x loc s [ At(x, loc, Result(Move(x, loc), s)) ]

“If you move some object to a location, then in the resulting state that object is at that location

- At(B1, Table, S0)
- At(B1, Top(B2), Result(Move(B1, Top(B2)), S0))

using the axiom

Monkeys and Bananas Problem

- The monkey-and-bananas problem is faced by a monkey standing under some bananas which are hanging out of reach from the ceiling. There is a box in the corner of the room that will enable the monkey to reach the bananas if he climbs on it.
- Use situation calculus to represent this problem and solve it using resolution theorem proving.

Representation of Monkey/Banana problem

- Fluents: Constants:
- At(x, loc, s) - BANANAS
- On(x, y, s) - MONKEY
- Reachable(x, Bananas, s) - BOX
- Has(x, y, s) - S0
- Other predicates - CORNER
- Moveable(x), Climbable(x) - UNDER-BANANAS
- Can-move(x)
- Actions
- Climb-on(x, y) -- Move(x, loc)
- Reach(x, y) -- Push(x, y, loc)

Monkey/Bananas axioms

1. ∀ x1, s1 [ Reachable(x1, BANANAS, s1) Has(x1, BANANAS, Result(Reach(x1, BANANAS), s1)) ]

If a person can reach the bananas then the result of reaching them is to have them.

2. ∀ s2 [At(BOX, UNDER-BANANAS, s2) ^ On(MONKEY, BOX, s2) Reachable(MONKEY, BANANAS, s2)

If a box is under the bananas and the monkey is on the box then the monkey can reach the bananas.

Monkey/Bananas axioms

3. ∀ x3, loc3, s3 [ Can-move(x3) At(x3, loc, Result(Move(x3, loc3), s3)) ]

The result of moving to a location is to be at that location

4. ∀ x4, y4, s4 [∃ loc4 [At(x4, loc4, s4) ^ At(y4, loc4, s4)] ^ Climbable(y4) On(x4, y4, Result(Climb-on(x4, y4), s4))]

The result of climbing on an object is to be on the object

5. ∀ x5, y5, loc5, s5 [∃ loc [At(x, loc0, s) ^ At(y5, loc0, s5) ] ^ Moveable(y5) At(y5, loc5, Result(Push(x5, y5, loc5), s5))

6. <same> At(x6, loc6, Result(Push(x6, y6, loc6), s6)) ]

The result of x pushing y to a location is both x and y are at that location.

Monkey/Bananas axioms (initial state S0)

F1. Moveable(BOX)

F2. Climbable(BOX)

F3. Can-move(MONKEY)

F4. At(BOX, CORNER, S0)

F5. At(MONKEY, UNDER-BANANAS, S0)

- To solve this for the goal Has(MONKEY, BANANAS, s):
- Convert to clause form
- Apply resolution to prove something like this:

Has(MONKEY, BANANAS, Result(Reach( . . . ), Result(. .) . .), S0)

which gives you the plan in reverse order.

(don’t forget to standardize!)We need 2 additional “frame axioms” for the proof

Frame Axioms for Monkey/Bananas world

7. ∀ x, y, loc, s [ At(x, loc, s) At(x, loc, Result(Move(y, loc), s)) ]

The location of an object does not change as a result of someone moving to the same location.

8. ∀ x, y, loc, s [ At(x, loc, s) At(x, loc, Result(Climb-on(y, x), s)) ]

The location of an object does not change as a result of someone climbing on it.

Refutation Resolution as the theoretical basis of BC

A query is conceptualized with existential variables:

? Likes(John, x) means ? ∃x Likes(John, x)

to answer the question, assert its NEGATION, and attempt to derive a contradiction by RESOLVING TO THE EMPTY CLAUSE!

~ ∃x Likes(John, x) is equivalent to ∀ x ~ Likes(John, x), so add that to the KB and try to derive the empty clause

Resolution rule:

[A1 V A2 V . . . An] ^

[B1 V B2 V . . Bm V ~A1’] where A1 and A1’ unify

---------------------------------------------------------------

[ A2’ V . . . An’ ] ^ [B1’ V B2’ V . . . Bm’]

Likes(John, Pizza) ^ ~ Likes(John, x) resolves to { }, given {x/Pizza}

A Limitation of Situation Calculus: The Frame problem

- I go from home (S) to the store, creating a new situation S’. In S’:
- My friend is still at home
- The store still sells chips
- My age is still the same
- Los Angeles is still the largest city in California…
- How can we efficiently represent everything that hasn’t changed?

Successor state axioms

- Normally, things stay true from one state to the next --

unless an action changes them:

At(p, loc, Result(a, s)) iff a = Go(p, x)

or [At(p, loc, s) and a != Go(p, y)]

- We need one or more of these for every fluent
- Now we can use theorem proving (or possibly backward chaining) to deduce a plan: not very practical

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