1. Propositional Logic

2. Recall: Wumpus World Environment: Actuators: 4x4 grid of rooms
Agent starts in [1,1]
Gold in a random room
Wumpus in a different random room
Bottomless pits in some rooms
Wumpus can eat agent if in same room
Agent can shoot Wumpus with arrow
Actuators:
Move left
Move right
Move up
Move down
Grab gold
Shoot

3. Wumpus World Continued
Sensors:
Stench (adjacent square contains Wumpus)
Breeze (adjacent square contains pit)
Glitter (this square contains gold)
Scream (Wumpus killed)
Performance measures:
Gold: +1000
Death: (falling into pit or eaten by Wumpus)
-1 per step
-10 for using the arrow

4. Propositional logic Syntax: Defines what makes a valid statement:
Statements are constructed from propositions
A proposition can be either true or false
Proposition made up of symbols and connectives
Semantics: Rules for determining the truth of a statement
<Later>

5. Propositional Logic - Syntax
Variables: represent a proposition that can be true or false
ex. Breeze in [2, 1]  Breeze2,1
Pit in [2, 2]  Pit2,2
ex. n is Odd  n_odd
Connectives: proposition operators
Negation: not, , ~
Conjunction: and, 
Disjunction: or, 
Implication: implies, =>
Biconditional: iff, <=>

6. Propositional Logic - Syntax
Sentence: statement composed of variables and operators
ex. Pit2,2  Pit1,3
Formally:
Sentence: True | False | Symbol |
Sentence| Sentence  Sentence |
Sentence  Sentence | Sentence => Sentence |
Sentence <=> Sentence

7. Propositional logic Syntax: Defines what makes a valid statement:
Statements are constructed from propositions
A proposition can be either true or false
Proposition made up of symbols and connectives
Semantics: Rules for determining the truth of a statement
Truth table
Rules of logic

8. Propositional Logic Semantics
Some Rules of Logic:
Modus Ponens: P => Q, P: can derive Q
deMorgan’s: (AB): can derive A  B
Contrapositive: : (P => Q) => (Q => P )

9. Inference with propositional logic
View it as a search problem:
starting state:
Initial Knowledge Base (KB)
actions:
all ways of deriving new propositions from the current KB
result:
add the new proposition to the KB
goal:
end up with the proposition we want to prove

10. Inference with propositional logic
View it as a search problem:
starting state:
Initial Knowledge Base (KB)
actions:
All the ways of deriving new propositions from the current KB
result:
Add the new proposition to the KB
goal:
End up with the proposition we want to prove

11. More Logical Equivalences

12. Propositional Logic for Wumpus World
Goal: Infer (prove) no Pit in [2, 1]
(On board)

13. Inference as Search Initial Knowledge Base Rule 4 Rule 1 Rule 2 Rule 3
Expanded Knowledge Base …
GOAL

14. Drawback of Propositional Logic
Need variable for each condition/rule, for each square
Breeze1,1 = false, Breeze1,2 = true, …
Breeze1,1 <=> Pit1,2 V Pit2,1, …
Breeze2,1 <=> Pit2,2 V Pit3,1, …
Need these rules for each square!

15. First order logic (FOL aka predicate calculus)
Syntax:
Objects – represent entities (~nouns)
Constants: e.g. Wumpus, Square1,2
Variables: x, y
Functions – describe relationships between objects
e.g. Adjacent(Square1,2, Square1,3),
AgentAt(x, t): (x is location, t is time)
Predicates – boolean function – describes a single object(~adjective)
e.g. Breezy(Square1,2), Pit(Square1,2)

16. First order logic (aka predicate calculus)
Quantifiers: describe “scope” of variables
“for all”:
“there exists”:
Connectives: operators (as with Propositional Logic)
Negation: not, , ~
Conjunction: and, 
Disjunction: or, 
Implication: implies, =>
Biconditional: iff, <=>

17. From text to FOL Given Predicates:
Purple(x), Mushroom(x), Poisonous(x)
“All purple mushrooms are poisonous”
“No purple mushroom is poisonous”
Given:
CSStudent(x), ProgLang(x), Knows(x,y)
“Every CS student knows a programming language”
“There is some programming language that all CS students know”
1) For all x purple(x) n mushroom(x) => poisonous(x)
2) For all x purple(x) n mushroom(x) => not poisonous(x) OR
not There exists x purple(x) n mushroom(x) n poisonous(x)
3) For all x there exists y CSS(x) => PL(y) n knows(x,y)
4) There exists x PL(x) n [for all y CSS(y) n knows(x,y)]

18. From FOL to text “Every rose has its thorn” “Everybody loves somebody”
“Everybody loves somebody, sometime”

19. FOL for Wumpus World
adjacentTo(), Pit(), Square(), Breezy() “All squares adjacent to Pits are Breezy” Smelly(), Contains() Wumpus “There is a smelly room adjacent to the room containing the Wumpus”

20. FOL for Wumpus World
adjacentTo(x, y), Pit(x), Square(x), Breezy(x) “All squares adjacent to Pits are Breezy” Smelly(), Contains() Wumpus “There is a smelly room adjacent to the room containing the Wumpus”

21. FOL for Wumpus World
adjacentTo(x, y), Pit(x), Square(x), Breezy(x) “All squares adjacent to Pits are Breezy” Smelly(), Contains() Wumpus “There is a smelly room adjacent to the room containing the Wumpus”

22. FOL for Wumpus World
adjacentTo(), Pit(), Square(), Breezy() “All squares adjacent to Pits are Breezy” Smelly(), Contains() Wumpus “There is a smelly room adjacent to the room containing the Wumpus”

23. FOL in Planning/Robotics

24. Wumpus World via Planning
Use Inference to formulate a Plan (for example):
Locate Smelly squares
Locate Breezy squares
Determine Wumpus location
Determine Pit locations
Locate Gold …

25. Wumpus World via Planning
Use Inference to formulate a Plan (for example):
Locate Smelly squares
Locate Breezy squares
Determine Wumpus location
Determine Pit locations
Locate Gold …
Planning – process of determining sequence of steps to solve a problem

26. Terminology State: conjunction (∧’s) of Predicates
ex. AgentAt(Square1,2) ∧ Breezy(Square1,2) ∧Smelly(Square1,2)
Action: function applied to a state that results in a new state
ex. MoveRight(Agent, Square1,2) -> AgentAt(Square2,2)

27. Terminology Precondition: state required prior to applying action
Postcondition/effect: state yielded after applying function

28. Example: Blocks World 3 blocks on a table
At most 1 block can fit on top of another
Robot can pick up one block and move it to table or on top of another block

29. Planning as Search START Action 4 Action 1 Action 2 Action 3 STATE …
GOAL

30. Shakey the Robot Stanford Research Institute (1966)
First general purpose mobile robot

34. Robotics’ Shakey Start
Living Room (L)
Kitchen (K)
Climable object
obj tv Sh
swi
Light switch
Bedroom (B)

35. Shakey collected video and sensor information and computer processed this information

36. Robotics's Shakey start
Go(from, to)
Preconditions: At(Sh, from)
Postconditions: At(Sh, to)
Push(obj, fr, to)
Preconditions: At(Sh, fr)  At(obj, fr)
Postconditions: At(Sh, to)  At(obj, to)
ACTIONS
At(Sh,L)  At(obj,K)  At(swi,B)  At(tv,L)
Go(L,B)
Push(tv,L,B)
Go(L,K)
Push(tv,L,K)
At(Sh,K)  At(tv,K)
STRIPS planning
explicit representation of high-level environmental states…
-- low-level engineering of actuation and control had to be improved
- -algorithms for reasoning about the robot's environment had to be improved
VIDEO!
At(Sh,L)  At(tv,K)  At(obj,L)
GOAL

37. Shakey video