CS G120 Artificial Intelligence

1
CS G120 Artificial Intelligence

• Prof. C. Hafner
• Class Notes Feb 26, 2009

2
AI planning

• Generating plans
• Given
• A way to describe the world
• 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

3
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,

4
Plannning 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 (have
  we been here before?)
• 3 approaches
• Situation calculus
• Strips approach
• With total order planning (state space search)
• Partial order planning (POP)

5
Classical Planning Assumptions

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

6
The situation calculus revisited(McCarthy 63)

• 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.

7
S1
S1 result(go(store), S0)
holds(at(home), S1)
S0
go(store)
holds(at(store), S1)
mow_lawn()
S2
holds(at(home), S0) holds(color(door, red), S0)
8
Monkeys and Bananas ProblemAssignment 5a
Planning by Logical Inference

• 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.

9
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)

10
Monkey/Bananas axioms

• ? x, s Reachable(x, BANANAS, s) ? Has(x,
  BANANAS, Result(Reach(x, BANANAS), s))
• ? s At(BOX, UNDER-BANANAS, s)
  On(MONKEY, BOX, s) ? Reachable(x, BANANAS, s)
• ? x, loc, s Can-move(x) ? At(x, loc,
  Result(Move(x, loc), s))
• ? x, y, s ? loc At(x, loc, s) At(y, loc, s)
  Climbable(y) ? On(x, y, Result(Climb-on(x, y),
  s))
• ? x, y, loc, s ? loc2 At(x, loc2, s) At(y,
  loc2, s) Moveable(y) ? At(y, loc,
  Result(Push(x, y, loc), s))
• At(x, loc, Result(Push(x, y,
  loc), s))
  

11
Monkey/Bananas axioms (initial state S0)

• Moveable(BOX)
• Climbable(BOX)
• Can-move(MONKEY)
• At(BOX, CORNER, S0)
• At(MONKEY, UNDER-BANANAS, S0)
• You will need to
• 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.
• (dont forget to standardize!)

12
Assignment 5b Due March 9

• Implement the backward chaining algorithm from
  your text book, in one file with the following
  interface/API
• readKB(file) returns a data structure of your
  choice
• folBCAsk(KB, goals, THETA)
• Goals is a list of FOLexps
• Theta is a substitution
• Returns ANS, a LIST of substitutions (initially
  empty)
• displayANS(ANS) prints the result
• These must be 3 separate functions callable by
  the user. Also KB must not be changed by
  folBCAsk or displayANS, so these can be called
  repeatedly.

13
Assignment 5b input file format

• FACT
• LikesJohn, Pizza
• RULE
• Likesx, Pizza
• gt
• Likesx, Spaghetti
• RULE
• Richx
• Healthyx
• gt
• Happyx

14
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/John

15
Return to Planning
16
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
  hasnt changed?

17
Successor state axioms

• Normally, things stay true from one state to the
  next --
• unless an action changes them
• holds(at(X),result(A,S)) iff A go(X)
• or holds(at(X),S) and A ! go(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

18
Strips (Fikes and Nilsson 71)

• For efficiency, separates theorem-proving within
  a world state from searching the space of
  possible states
• Highly influential representation for actions
• 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)

19
Example problem

• Initial state at(home), have(beer),
  have(chips)
• Goal have(beer), have(chips), at(home)
• Actions
• Buy (X)
• Pre at(store)
• Add have(X)

Go (X, Y) Pre at(X) Del at(X) Add at(Y)
20
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
  hasnt changed?
• Strips provides a good solution for simple actions

21
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?

22
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 programmer to avoid mistakes

23
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?

24
Example blocks world (Sussman anomaly)
Initial
A
Goal
C
B
A
B
C

• State I (on-table A) (on C A) (on-table B)
  (clear B) (clear 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!!!!

25
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.

26
How to get dressed

• State
• Goal RightShoeOn, LeftShoeOn
• Actions
• 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
27
Total-Order vs Partial-Order Plans
28
Remember the Sussman Anomaly
Initial State (on-table A) (on C A) (on-table B)
(clear B) (clear
C) Goal (on A B) (on B C)
29
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)
30
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
31
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)
32
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)
J
S
clear(b)
puton(b, c)
S
J
clear(c)
33
clear(a)
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)
34
Final plan
puton(c, X)
clear(a)
puton(a, b)
puton(b, c)
J
S
clear(b)
35
Action RepresentationPropositional STRIPS (real
STRIPS uses variables)

• Move-C-from-A-to-Table
• preconditions (on C A) (clear C)
• effects
• add (on-table C)
• delete (on C A)
• add (clear A)
• Solution to frame problem explicit effects are
  the only changes to the state.

36
Total-Order Plan GenerationSearch space of
world states

• Planning as a search problem
• Nodes world states
• Arcs actions
• Solution path from the initial state to one
  state that satisfies the goal
• Initial state is fully specified
• There may be many goal states

37
Search Space Blocks World
38
ProgressionForward search (I -gt G)

• ProgWS(state, goals, actions, path)
• If state satisfies goals, then return path
• else a choose(actions), s.t.
• preconditions(a) satisfied in state
• if no such a, then return failure
• else return
• ProgWS(apply(a, state), goals, actions,
• concatenate(path, a))
• First call ProgWS(IS, G, Actions, ())

39
Progression Example Sussman Anomaly

• I (on-table A) (on C A) (on-table B) (clear B)
  (clear C)
• G (on A B) (on B C)
• P(I, G, BlocksWorldActions, ())
• P(S1, G, BWA, (move-C-from-A-to-table))
• P(S2, G, BWA, (move-C-from-A-to-table,
• move-B-from-table-to-C))
• P(S3, G, BWA, (move-C-from-A-to-table,
• move-B-from-table-to-C,
• move-A-from-table-to-B))

G ? S3 gt Success!
40
RegressionBackward Search (I lt- G)

• RegWS(init-state, current-goals, actions, path)
• If init-state satisfies current-goals, then
  return path
• else a choose (actions), s.t. some effect of
  a satisfies one of current-goals
• If no such a, then return failure
  unachievable
• If some effect of a contradicts some of
  current-goals, then return failure
  inconsistent state
• CG current-goals effects(a)
  preconditions(a)
• If current-goals ? CG, then return failure
  useless
• RegWS(init-state, CG, actions,
  concatenate(a,path))
• First call RegWS(IS, G, Actions, ())

41
Regression Example Sussman Anomaly

• I (on-table A) (on C A) (on-table B) (clear B)
  (clear C)
• G (on A B) (on B C)
• R(I, G, BlocksWorldActions, ())
• R(I, ((clear A) (on-table A) (clear B) (on B C)),
  BWA,
• 
  (move-A-from-table-to-B))
• R(I, ((clear A) (on-table A) (clear B) (clear C),
  (on-table B)),
• BWA, (move-B-from-table-to-C,
  move-A-from-table-to-B))
• R(I, ((on-table A) (clear B) (clear C) (on-table
  B) (on C A)),
• BWA, (move-C-from-A-to-table,
  move-B-from-table-to-C,
• move-A-from-table-to-B))

current-goals ? I gt Success!
42
Progression vs. Regression

• Both algorithms are
• Sound the result plan is valid
• Complete if valid plan exists, they find one
• Non-deterministic choice gt search!
• Brute force DFS, BFS, Iterative Deepening, ..,
• Heuristic A
• Regression often faster, focused by goals
• Progression full state to compute heuristics

43
Plan Generation Search space of plans

• Partial-Order Planning (POP)
• Nodes are partial plans
• Arcs/Transitions are plan refinements
• Solution is a node (not a path).
• Principle of Least commitment
• e.g. do not commit to an order of actions until
  it is required

44
Partial Plan Representation

• Plan (A, O, L), where
• A set of actions in the plan
• O temporal orderings between actions (a lt b)
• L causal links linking actions via a literal
• Causal Link
• Action Ac (consumer) has precondition Q that
  is established in the plan by Ap (producer).
• move-a-from-b-to-table
  move-c-from-d-to-b

(clear b)
45
Threats to causal links

• Step At threatens link (Ap, Q, Ac) if
• 1. At has (not Q) as an effect, and
• 2. At could come between Ap and Ac, i.e.
• O ? (Ap lt At lt Ac ) is consistent
• Whats an example of an action that threatens the
  link example from the last slide?

46
POP algorithm takes account of threats to
causal links

• POP((A, O, L), agenda, actions)
• If agenda () then return (A, O, L)
• Pick (Q, aneed) from agenda
• aadd choose(actions) s.t. Q ?effects(aadd)
• If no such action aadd exists, fail.
• L L ? (aadd, Q, aneed) O O ? (aadd
  lt aneed)
• agenda agenda - (Q, aneed)
• If aadd is new, then A A ? aadd and
• ?P ?preconditions(aadd), add (P, aadd) to
  agenda
• For every action at that threatens any causal
  link (ap, Q, ac) in L
• choose to add at lt ap or ac lt at to O.
• If neither choice is consistent, fail.
• POP((A, O, L), agenda, actions)

Termination
Goal Selection
Action Selection
Update goals

• Protect causal links
• Demotion at lt ap
• Promotion ac lt at