Logical Agents Reasoning [Ch 6] Propositional Logic [Ch 6] - PDF document

Logical Agents • Reasoning [Ch 6] • Propositional Logic [Ch 6] • Predicate Calculus – Representation [Ch 7] • Predicate Calculus – Inference [Ch 9] • Implemented Systems [Ch 10] • Applications [Ch 8] • Planning [Ch 11] − Using “Situation Calculus” − Simple − GraphPlan [Notes] PC-Plan 1

Typical Wumpus World Breeze Stench 4 PIT Breeze Breeze 3 PIT Stench Gold Breeze Stench 2 Breeze Breeze 1 PIT START 1 2 3 4 PC-Plan 2

Simple Reflex Agent Rep’n: At time = t , specify Percept([s,b,g,u,c], t ) where s ∈ { Stench , −} , b ∈ { Breeze, −} , . . . Eg: Tell(KB, Percept([Stench,-,Glitter,-,-], 3) ) • Connect percepts directly to actions: ∀ s, b, u, c, t Percept ([ s, b, Glitter , u, c ] , t ) ⇒ Action ( Grab , t ) • Or, more indirectly: ∀ s, b, u, c, t Percept ([ s, b, Glitter , u, c ] , t ) ⇒ AtGold ( t ) ∀ t AtGold ( t ) ⇒ Action ( Grab , t ) Q1: Which is more flexible? Q2: Limitations of reflex approach? PC-Plan 3

Problem with Reflex Agents Q: When to Climb ? A: @ [ 1 , 1 ], and have Gold . . . but . . . � being @ [ 1 , 1 ] agent cannot sense having Gold Also. . . May be @ [ 2 , 3 ] when hunting for gold Then reach same [ 2 , 3 ] when returning As SAME percepts both times & reflex agent can ONLY use percept to decide on action agent must take SAME actions both times. . . ∞ -loop! ⇒ ⇒ Agent needs INTERNAL MODEL of state/world PC-Plan 4

Tracking a Changing World • Consider FINDING KEYS . . . when “Keys are in pocket” . . . agent could keep percept history, & replay it Better: just store this info! Any decision based on past/present percepts FACT: can be based on current “world state” (. . . which is updated each action) So... perhaps KB should keep ONLY info about current situation PC-Plan 5

Single-Time Knowledge Base Time 0: Initial configuration At(Agent, [ 1 , 1 ] ) Facing(Agent, North) Smell(No) ... Time 1: Then take a step Action = Forward NOW in new situation: Remove false statements remove At(Agent, [ 1 , 1 ] ) , . . . Add in statement that are now true: add At(Agent, [ 1 , 2 ] ) , . . . ⇒ use revised KB: At(Agent, [ 1 , 2 ] ) Facing(Agent, North) Smell(Yes) ... Time 2: Turn to the right Actions = TurnRight ⇒ use revised KB: At(Agent, [ 1 , 2 ] ) Facing(Agent, East) Smell(Yes) ... PC-Plan 6

Problems with Single-Time KBs but . . . may need to reason about MANY times Eg: “Was there a stench in [ 1 , 2 ] and [ 2 , 3 ]?” • Need to Maintain info from previous states . . . labeled with state • Kinda like “time stamp” . . . but “time” is not relevant better to record SEQUENCE of ACTIONS! Compare: Having GOLD at time 4 with Having GOLD after Going forward , then Turning right , then Going forward , then Grabbing ⇒ Sequence of actions ≈ “plan”! PC-Plan 7

Situation Calculus • Tag each “changable predicate” with “time”: Eg: At( Agent, [ 1 , 1 ] , S 0 ) At( Agent, [ 1 , 2 ] , Result( Forward, S 0 ) ) At( Agent, [ 1 , 3 ] , Result(Forward, Result( Forward, S 0 ))) . . . Notice: all stay around! • Only “label” predicates that can change. As Pit doesn’t move, At(Pit, � 3 , 2 � ) sufficient Similarly, just 2+2=4 , . . . • “Time” only wrt actions Snapshot of SITUATION. . . World represented as SERIES OF SITUATIONS connected by actions PC-Plan 8

Updating State, Based on Action PIT PIT G o l d P I T PIT P I T Gold S 3 P I T P I T P I T Gold Forward S 2 P I T P I T P I Gold T TurnRight S 1 P I T Forward S 0 S 1 = Result( Forward, S 0 ) S 2 = Result( TurnRight, S 1 ) = Result( TurnRight, Result(Forward, S 0 ) ) S 3 Result( Forward, S 2 ) = = Result( Forward, Result( TurnRight, Result( Forward, S 0 ) ) ) Result : Action × State �→ State PC-Plan 9

Computing Location • Can compute location: ∀ i, j, s At ( Ag , [ i, j ] , s ) ∧ Facing ( Ag , North , s ) ⇒ At( Ag , [ i, j + 1 ] , Result ( Forward , s )) ∀ i, j, s At ( Ag , [ i, j ] , s ) ∧ Facing ( Ag , East , s ) ⇒ At( Ag , [ i − 1 , j ] , Result ( Forward , s )) . . . • Can compute orientation: ∀ i, j, s Facing(Ag, North, s) ⇒ Facing(Ag, East, Result( TurnRight, s) ) . . . PC-Plan 10

Interpreting Percepts • Extract Individual Percepts ∀ b, g, u, c, t Percept([Stench, b, g, u, c], t) ⇒ Stench(t) ∀ b, u, c, s, t Percept([s, Breeze, g, u, c], t) ⇒ Breeze(t) • Combine with State to make Infer- ences about Locations ∀ ℓ, s At ( Agent, ℓ, s ) ∧ Stench ( s ) ⇒ Smelly ( ℓ ) ∀ ℓ, s At ( Agent, ℓ, s ) ∧ Breeze ( s ) ⇒ Breezy ( ℓ ) • Combine with Causal Rules to Infer Locations of Wumpus, Pits ∀ ℓ Breezy ( ℓ ) ⇔ [ ∃ x PitAt ( x ) ∧ Adj ( ℓ, x )] ∀ ℓ Stench ( ℓ ) ⇔ [ ∃ x WumpusAt ( x ) ∧ Adj ( ℓ, x )] ∀ ℓ OK ( ℓ ) ⇔ ( ¬ WumpusAt ( ℓ ) ∧ ¬ PitAt ( ℓ )) PC-Plan 11

Deducing Hidden Properties • Squares are breezy near a pit: – Diagnostic rule — infer cause from effect ∀ ℓ Breezy ( ℓ ) ⇒ ∃ x Pit ( x ) ∧ Adj ( ℓ, x ) – Causal rule — infer effect from cause ∀ ℓ, x Pit ( x ) ∧ Adj ( ℓ, x ) ⇒ Breezy ( ℓ ) – Neither is complete. . . Eg, the causal rule doesn’t say whether squares far away from pits can be breezy ⇒ Definition for Breezy predicate: ∀ ℓ Breezy ( ℓ ) ⇔ [ ∃ x PitAt ( x ) ∧ Adj ( ℓ, x )] • Squares are stenchy near the wumpus: ∀ ℓ Stench ( ℓ ) ⇔ [ ∃ x WumpusAt ( x ) ∧ Adj ( ℓ, x )] PC-Plan 12

Using Information • After concluding ¬ Stench ([ 1 , 1 ]) ¬ Stench ([ 1 , 2 ]) Stench ([ 2 , 1 ]) ∀ ℓ Adj ([ 1 , 1 ] , ℓ ) ( ℓ = [ 2 , 1 ] ∨ ℓ = [ 1 , 2 ]) ⇔ ∀ ℓ Adj ([ 2 , 1 ] , ℓ ) ( ℓ = [ 1 , 1 ] ∨ ℓ = [ 2 , 2 ] ∨ ℓ = [ 3 , 1 ]) ⇔ ∀ ℓ Adj ([ 1 , 2 ] , ℓ ) ( ℓ = [ 1 , 1 ] ∨ ℓ = [ 2 , 2 ] ∨ ℓ = [ 1 , 3 ]) ⇔ • Can derive ¬ [ ∃ x WumpusAt ( x ) ∧ Adj ([ 1 , 1 ] , x )] ∀ x Adj ([ 1 , 1 ] , x ) ⇒ ¬ WumpusAt ( x ) ¬ WumpusAt ([ 1 , 2 ]), ¬ WumpusAt ([ 2 , 1 ]) ¬ [ ∃ x WumpusAt ( x ) ∧ Adj ([ 1 , 2 ] , x )] ¬ WumpusAt ([ 1 , 1 ]), ¬ WumpusAt ([ 2 , 2 ]), ¬ WumpusAt ([ 3 , 1 ]) [ ∃ x WumpusAt ( x ) ∧ Adj ([ 2 , 1 ] , x )] WumpusAt ([ 1 , 1 ]) ∨ WumpusAt ([ 2 , 2 ]) ∨ WumpusAt ([ 1 , 3 ]) . . . ⇒ WumpusAt ([ 1 , 3 ]) PC-Plan 13

Connecting Inferences to Actions • Rate Each Action ∀ ℓ 1 , s WumpusAt( ℓ 1 ) ∧ LocationAhead(Agent, s) = ℓ 1 ⇒ Deadly( Forward, s ) ∀ ℓ 1 , s OK( ℓ 1 ,s) ∧ LocationAhead(Agent,s) = ℓ 1 ∧ ¬ Visited( ℓ 1 ,s) ⇒ Good(Forward, s ) ∀ ℓ 1 , s Gold( ℓ 1 ,s) ⇒ Great(Grab, s ) • Choose Best Action ∀ a, s Great ( a, s ) ⇒ Action ( a, s ) ∀ a, s Good ( a, s ) ∧ ( ¬∃ b Great ( b, s )) ⇒ Action ( a, s ) • Now, for each situation S , Ask( KB , ∃ a Action( a , S) ) . . . find a s.t. KB | = Action ( a, S ) PC-Plan 14

Propagating Information Effect Actions: If agent Grabs, when in room with gold, he will be holding gold. ∀ ℓ, s Glitters ( ℓ ) ∧ At ( Agent , ℓ, s ) ⇒ AtGold ( s ) ∀ ℓ, s AtGold ( s ) ⇒ Holding ( Gold , Result ( Grab , s ) ) So, if Glitters( [ 3 , 2 ] ) , At ( Agent , [ 3,2 ] , S 6 ), then Holding ( Gold , S 7 ) where S 7 = Result ( Grab , S 6 ) What about NEXT situation? eg, S 8 = Result ( Turn Right , S 7 )? Want to derive Holding ( Gold , Result ( Turn Right , S 7 ) ), This requires . . . PC-Plan 15

Frame Axioms • ∀ a, x, s Holding ( x, s ) ∧ ( a � = Release ) Holding ( x, Result ( a, s ) ) ⇒ ∀ a, s ¬ Holding ( Gold , s ) ∧ ( a � = Grab ∨ ¬ AtGold ( s ) ⇒ ¬ Holding ( Gold , Result ( a, s ) ) true afterwards ⇔ [an action made it true ∨ Gen’l: (true already & no action made it false)] Here: ∀ a, s Holding ( Gold , Result ( a, s )) ⇔ [ ( a = Grab ∧ AtGold ( s ) ) ∨ ( Holding ( Gold , s ) ∧ a � = Release ) ] Similarly: ∀ a, d, p, s At ( p, ℓ, Result ( a, s ) ) ⇔ [ ( a = Forward ∧ ℓ = LocAhead ( p, s ) ∧ ¬ Wall ( ℓ )) ∨ ( At ( p, ℓ, s ) ∧ a � = Forward ) ] • “Successor State Axiom” Lists all ways predicate can become true/false PC-Plan 16

Frame, and Related, Problems • Representational Frame Problem Encoding what doesn’t change, as ac- tions take place (Solved via “success-state axioms”) • Inferential Frame Problem . . . deal with long sequences of actions . . . • Qualification Problem dealing with all qualifications . . . gold brick is not slippery, not screwed to table, . . . • Ramification When picking up the gold brick, also pick up the associated dust . . . PC-Plan 17

Goal-Based Agent • These rules sufficient to FIND gold Then what? • Need to change strategies: – Was “ Find gold ” – Now: “ Get out! ” ∀ s Holding ( Gold , s ) ⇒ GoalLocation ( [ 1 , 1 ] , s ) Need to incorporate. . . How? PC-Plan 18

How to Plan? Planning agents seek plan ≡ sequence of actions that achieve agent’s goals. Inference: Let logical reasoning system per- form search: Ask(KB, ∃ a 1 , a 2 , a 3 , a 4 , a 5 , t t = Result ( a 5 , Result ( a 4 , Result ( a 3 , Result ( a 2 , Result ( a 1 , S 0 ))))) ∧ Holding ( Agent, Gold, t ) ∧ At ( Agent, Outside, t )) Problematic, as not easy to heuristically guide reasoning system. . . . . . what if > 5 actions required? . . . Search: View actions as operations on KB , Goal = “ KB includes Holding ( Agent , Gold , t ) At ( Agent , Outside , t ))” Planning: Special purpose reasoning systems. . . PC-Plan 19