CS 4100 Artificial Intelligence

CS 4100 Artificial Intelligence Prof. C. Hafner Class Notes Feb 14, 2012

Assignment 5 • Why you must re-standardize variables every time a rule is retrieved. • Test data in assignments/A5sampleInputs • easy • average • hard • veryhard

State space search examples 4 4 3 S E A B F C G D 5 C B 5 4 6.7 10.4 4.0 A 3 S 2 4 11.0 G 8.9 D 3.0 E 6.9 F

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

Total-Order vs Partial-Order Plans

Initial State: 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)

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)

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)

Final plan puton(c, X) clear(a) puton(b, c) puton(a, b) J S clear(b)

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

CS 4100 Artificial Intelligence