1. Russell and Norvig Chapter 7
Propositional Logic
Russell and Norvig
Chapter 7

2. Knowledge-Based Agent
sensors
actuators
environment
agent
Knowledge
base ?

3. A simple knowledge-based agent
The agent must be able to:
Represent states, actions, etc.
Incorporate new percepts
Update internal representations of the world
Deduce hidden properties of the world
Deduce appropriate actions

4. Types of Knowledge
Procedural, e.g.: functions Such knowledge can only be used in one way -- by executing it
Declarative, e.g.: constraints It can be used to perform many different sorts of inferences

5. Logic Logic is a declarative language to:
Assert sentences representing facts that hold in a world W (these sentences are given the value true)
Deduce the true/false values to sentences representing other aspects of W

6. Wumpus World PEAS description
Performance measure
gold +1000, death -1000
-1 per step, -10 for using the arrow
Environment
Squares adjacent to wumpus are smelly
Squares adjacent to pit are breezy
Glitter iff gold is in the same square
Shooting kills wumpus if you are facing it
Shooting uses up the only arrow
Grabbing picks up gold if in same square
Releasing drops the gold in same square
Sensors: Stench, Breeze, Glitter, Bump, Scream
Actuators: Left turn, Right turn, Forward, Grab, Release, Shoot

7. Wumpus world characterization
Fully Observable No – only local perception
Deterministic Yes – outcomes exactly specified
Episodic No – sequential at the level of actions
Static Yes – Wumpus and Pits do not move
Discrete Yes
Single-agent? Yes – Wumpus is essentially a natural feature

8. Exploring a wumpus world

9. Exploring a wumpus world

10. Exploring a wumpus world

11. Exploring a wumpus world

12. Exploring a wumpus world

13. Exploring a wumpus world

14. Exploring a wumpus world

15. Exploring a wumpus world

16. Logic in general
Logics are formal languages for representing information such that conclusions can be drawn
Syntax defines the sentences in the language
Semantics define the "meaning" of sentences;
i.e., define truth of a sentence in a world

17. Connection World-Representation
Sentences
entail
Sentences
represent
Facts about W
represent
World W
Conceptualization
hold
Facts about W
hold

18. Examples of Logics Propositional calculus A  B  C
First-order predicate calculus ( x)( y) Mother(y,x)
Logic of Belief B(John,Father(Zeus,Cronus))

19. Model
A model of a sentence is an assignment of a truth value – true or false – to every atomic sentence such that the sentence evaluates to true.

20. Model of a KB Let KB be a set of sentences
A model m is a model of KB iff it is a model of all sentences in KB, that is, all sentences in KB are true in m.

21. Satisfiability of a KB
A KB is satisfiable iff it admits at least one model; otherwise it is unsatisfiable
KB1 = {P, QR} is satisfiable KB2 = {PP} is satisfiable
KB3 = {P, P} is unsatisfiable
valid sentence or tautology

22. Logical Entailment KB : set of sentences  : arbitrary sentence
KB entails  – written KB  – iff every model of KB is also a model of 
Alternatively, KB  iff
{KB,} is unsatisfiable
KB   is valid

23. Inference Rule
An inference rule {, }  consists of 2 sentence patterns  and  called the conditions and one sentence pattern  called the conclusion
If  and  match two sentences of KB then the corresponding  can be inferred according to the rule 

24. Inference I: Set of inference rules KB: Set of sentences
Inference is the process of applying successive inference rules from I to KB, each rule adding its conclusion to KB

25. Example: Modus Ponens {  ,  }   {, }  From
{  ,  } 
{, }  
From
Battery-OK  Bulbs-OK  Headlights-Work
Battery-OK  Bulbs-OK
Infer
Headlights-Work

26. Inference   Connective symbol (implication) Logical entailment
KB  iff KB   is valid 

27. Soundness
An inference rule is sound if it generates only entailed sentences
All inference rules previously given are sound, e.g.: modus ponens: {   , } 
The following rule: {   , }  is unsound, which does not mean it is useless (an inference rule for abduction, outside scope of this course)  

28. Is each of the following a sound inference rule? {   , } 
{   , } 
{   , }   

29. Completeness
A set of inference rules is complete if every entailed sentences can be obtained by applying some finite succession of these rules
Modus ponens alone is not complete, e.g.: from A  B and B, we cannot get A

30. Proof
The proof of a sentence  from a set of sentences KB is the derivation of  by applying a series of sound inference rules

31. Proof
The proof of a sentence  from a set of sentences KB is the derivation of  by applying a series of sound inference rules
Battery-OK  Bulbs-OK  Headlights-Work
Battery-OK  Starter-OK  Empty-Gas-Tank  Engine-Starts
Engine-Starts  Flat-Tire  Car-OK
Headlights-Work
Battery-OK
Starter-OK
Empty-Gas-Tank
Car-OK
Battery-OK  Starter-OK by 5,6
Battery-OK  Starter-OK  Empty-Gas-Tank by 9,7
Engine-Starts by 2,10
Engine-Starts  Flat-Tire by 3,8
Flat-Tire by 11,12

32. Inference Problem Given: Answer: KB: a set of sentence : a sentence

33. Deduction vs. Satisfiability Test
KB  iff {KB,} is unsatisfiable
Hence:
Deciding whether a set of sentences entails another sentence, or not
Testing whether a set of sentences is satisfiable, or not
are closely related problems

34. Complementary Literals
A literal is a either an atomic sentence or the negated atomic sentence, e.g.: P, P
Two literals are complementary if one is the negation of the other, e.g.: P and P

35. Unit Resolution Rule
Given two sentences: L1  …  Lp and M where Li,…, Lp and M are all literals, and M and Li are complementary literals
Infer: L1  …  Li-1  Li+1  …  Lp

36. Examples From: Engine-Starts  Car-OK Engine-Starts Infer: Car-OK
Modus ponens
Engine-Starts  Car-OK
From: Engine-Starts  Car-OK
Car-OK
Infer: Engine-Starts
Modus tollens

37. Shortcoming of Unit Resolution
From:
Engine-Starts  Flat-Tire  Car-OK
Engine-Starts  Empty-Gas-Tank
we can infer nothing!

38. Full Resolution Rule
Given two clauses: L1  …  Lp and M1  …  Mq where Li and Mj are complementrary
Infer the clause: L1 … Li-1Li+1…LkM1 … Mj-1Mj+1…Mk

39. Example From: Engine-Starts  Flat-Tire  Car-OK
Engine-Starts  Empty-Gas-Tank
Infer:
Empty-Gas-Tank  Flat-Tire  Car-OK

40. Example From: P  Q ( P  Q) Q  R ( Q  R) Infer:
P  R ( P  R)

41. Not All Inferences are Useful!
From:
Engine-Starts  Flat-Tire  Car-OK
Engine-Starts  Flat-Tire
Infer:
Flat-Tire  Flat-Tire  Car-OK

42. Not All Inferences are Useful!
From:
Engine-Starts  Flat-Tire  Car-OK
Engine-Starts  Flat-Tire
Infer:
Flat-Tire  Flat-Tire  Car-OK
tautology

43. Not All Inferences are Useful!
From:
Engine-Starts  Flat-Tire  Car-OK
Engine-Starts  Flat-Tire
Infer:
Flat-Tire  Flat-Tire  Car-OK  True
tautology

44. Example Battery-OK  Bulbs-OK  Headlights-Work
Battery-OK  Starter-OK  Empty-Gas-Tank  Engine-Starts
Engine-Starts  Flat-Tire  Car-OK
Headlights-Work
Battery-OK
Starter-OK
Empty-Gas-Tank
Car-OK
Flat-Tire
We want to show Flat-Tire, given clauses 1-8. Using resolution, we can show
that clauses 1-8 along with clause 9 deduce an empty clause.
Can you trace the resolution steps?

45. Sentence  Clause Form Example: (A  B)  (C  D)
1. Eliminate  (A  B)  (C  D) 2. Reduce scope of  (A  B)  (C  D) 3. Distribute  over  (A  (C  D))  (B  (C  D))
(A  C)  (A  D)  (B  C)  (B  D)
Set of clauses:
{A  C , A  D , B  C , B  D}

46. Resolution Refutation Algorithm
RESOLUTION-REFUTATION(KB,a)
clauses  set of clauses obtained from KB and a
new  {}
Repeat:
For each C, C’ in clauses do res  RESOLVE(C,C’) If res contains the empty clause then return yes
new  new U res If new  clauses then return no
clauses  clauses U new

47. Efficient Propositional Inference
Two families of efficient algorithms for propositional inference:
Complete backtracking search algorithms
DPLL algorithm (Davis, Putnam, Logemann, Loveland)
Incomplete local search algorithms
WalkSAT algorithm

48. The DPLL algorithm
Determine if an input propositional logic sentence (in CNF) is satisfiable.
Improvements over truth table enumeration:
Early termination
A clause is true if any literal is true.
A sentence is false if any clause is false.
Pure symbol heuristic
Pure symbol: always appears with the same "sign" in all clauses.
e.g., In the three clauses (A  B), (B  C), (C  A), A and B are pure, C is impure.
Make a pure symbol literal true.
Unit clause heuristic
Unit clause: only one literal in the clause
The only literal in a unit clause must be true.

49. Horn Clauses Horn Clause A clause with at most one positive literal.
KB: A Horn clause with one positive literal which can be written as
α1  …  αn  β
Query: A Horn clause without positive literal
α1  …  αn
I.e.
( α1  …  αn )
Horn clause logic is the basis for Logic Programming

50. Forward chaining for Horn Clauses
Idea: fire any rule whose premises are satisfied in the KB,
add its conclusion to the KB, until query is found

51. Backward chaining for Horn Clasues
Idea: work backwards from the query q:
to prove q by BC,
check if q is known already, or
prove by BC all premises of some rule concluding q
Avoid loops: check if new subgoal is already on the goal stack
Avoid repeated work: check if new subgoal
has already been proved true, or
has already failed

52. Summary Propositional Logic Model of a KB Logical entailment
Inference rules
Resolution rule
Clause form of a set of sentences
Resolution refutation algorithm
DPLL algorithm
Horn clauses