1. Automated Reasoning in Propositional Logic
Russell and Norvig:
Chapters 6 and 9 Chapter 7, Sections 7.5—7.6
CS121 – Winter 2003

2. Problem
Given:
KB: a set of sentence
: a sentence
Answer:
KB  ?

3. 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

4. Computational Approaches
Enumeration of models
Construction of a proof
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  (5+6)
Battery-OK  Starter-OK  Empty-Gas-Tank  (9+7)
Engine-Starts  (2+10)
Engine-Starts  Flat-Tire  (3+8)
Flat-Tire  (11+12)

5. Enumeration of Models P: Set of propositional symbols in {KB,}
n: Size of P
ENTAILS?(KB,)
For each of the 2n models on P do
If it is a model of {KB,} then return no Return yes

6. Satisfiability Test as CSP
Each propositional symbol is a variable
The domain of each variable is {True, False}
Each sentence in {KB,} is a constraint on the value(s) taken by one or several variables
Recursive backtracking CSP techniques and heuristics are applicable

7. Construction of a Proof
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  (5+6)
Battery-OK  Starter-OK  Empty-Gas-Tank  (9+7)
Engine-Starts  (2+10)
Engine-Starts  Flat-Tire  (3+8)
Flat-Tire  (11+12)

8. Construction of a Proof
Battery-OK  Bulbs-OK  Headlights-Work
Battery-OK  Starter-OK  Empty-Gas-Tank  Engine-Starts
Engine-Starts  Flat-Tire  Car-OK
Headlight-Work
Battery-OK
Starter-OK
Empty-Gas-Tank
Car-OK
Battery-OK  Starter-OK  (5+6)
Battery-OK  Starter-OK  Empty-Gas-Tank  (9+7)
Engine-Starts  (2+10)
Engine-Starts  Flat-Tire  (3+8)
Flat-Tire  (11+12)
What do we need?
A complete set of sound inference rules
A complete search algorithm to decide which rule to apply next and to which sentences

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

10. 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

11. 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 tolens

12. Another Example Engine-Starts  Flat-Tire  Car-OK Engine-Starts

13. Detection of Unsatisfiability
Car-OK
Car-OK
False

14. Soundness of Unit Resolution
Let m be a model of: L1  …  Lp and M where M and Li are complementary literals
Li must be False in m, hence L1  …  Li-1  Li+1  …  Lp must be True

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

16. Full Resolution Rule
Given two sentences: L1  …  Lp and M1  …  Mq where L1,…, Lp, M1,…, Mq are all literals, and Li and Mj are complementary literals
Infer: L1 … Li-1Li+1…LkM1 … Mj-1Mj+1…Mk in which only one copy of each literal is retained (factoring)

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

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

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

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

21. 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

22. Soundness of Full Resolution
Left as an exercise

23. Full Resolution Rule clauses
Given two sentences: L1  …  Lp and M1  …  Mq
Infer: L1 … Li-1Li+1…LkM1 … Mj-1Mj+1…Mk
clauses

24. Sentence  Clause Form
Example: (A  B)  (C  D)

25. Sentence  Clause Form Example: (A  B)  (C  D)
1. Eliminate  (A  B)  (C  D)

26. Sentence  Clause Form Example: (A  B)  (C  D)
1. Eliminate  (A  B)  (C  D) 2. Reduce scope of  (A  B)  (C  D)

27. 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)

28. 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}

29. 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

30. Completeness of Resolution Refutation
Left as an exercise

31. 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
Starter-OK  Empty-Gas-Tank  Engine-Starts
Battery-OK  Empty-Gas-Tank  Engine-Starts
Battery-OK  Starter-OK  Engine-Starts
Engine-Starts  Flat-Tire
Engine-Starts  Car-OK

32. Example Battery-OK  Bulbs-OK  Headlights-Work
Battery-OK  Starter-OK  Empty-Gas-Tank  Engine-Starts
Engine-Starts  Flat-Tire  Car-OK
Headlight-Work
Battery-OK
Starter-OK
Empty-Gas-Tank
Car-OK
Flat-Tire
Starter-OK  Empty-Gas-Tank  Engine-Starts
Battery-OK  Empty-Gas-Tank  Engine-Starts
Battery-OK  Starter-OK  Engine-Starts
Engine-Starts  Flat-Tire
Engine-Starts  Car-OK

33. Example … Battery-OK Starter-OK Empty-Gas-Tank Car-OK Flat-Tire
Battery-OK  Starter-OK  Engine-Starts
Engine-Starts  Flat-Tire

34. Example … Battery-OK Starter-OK Empty-Gas-Tank Car-OK Flat-Tire
Battery-OK  Starter-OK  Engine-Starts
Engine-Starts  Flat-Tire
Battery-OK  Starter-OK  Flat-Tire
Starter-OK  Flat-Tire
Flat-Tire
False (empty clause)

35. Resolution Heuristics
Set-of-support heuristics: At least one ancestor of every inferred clause comes from a
Shortest-clause heuristics: Generate a clause with the fewest literals first
Simplifications heuristics:
Remove any clause containing two complementary literals (tautology)
Remove any clause C that contains all the literals of another clause C’
If a symbol always appears with the same “sign”, remove all the clauses that contain it (pure symbol)

36. Example (Set-of-Support)
Battery-OK  Bulbs-OK  Headlights-Work
Battery-OK  Starter-OK  Empty-Gas-Tank  Engine-Starts
Engine-Starts  Flat-Tire  Car-OK
Headlight-Work
Battery-OK
Starter-OK
Empty-Gas-Tank
Car-OK
Flat-Tire

37. Example (Set-of-Support)
Battery-OK  Bulbs-OK  Headlights-Work
Battery-OK  Starter-OK  Empty-Gas-Tank  Engine-Starts
Engine-Starts  Flat-Tire  Car-OK
Headlight-Work
Battery-OK
Starter-OK
Empty-Gas-Tank
Car-OK
Flat-Tire
Engine-Starts  Car-OK
Engine-Starts
Battery-OK  Starter-OK  Empty-Gas-Tank
Starter-OK  Empty-Gas-Tank
Empty-Gas-Tank
False
Note the goal-directed
flavor

38. Resolution Heuristics
Set-of-support heuristics: At least one ancestor of every inferred clause comes from a
Shortest-clause heuristics: Generate a clause with the fewest literals first
Simplifications heuristics:
Remove any clause containing two complementary literals (tautology)
Remove any clause C that contains all the literals of another clause C’
If a symbol always appears with the same “sign”, remove all the clauses that contain it (pure symbol)

39. Example (Shortest-Clause)
Battery-OK  Bulbs-OK  Headlights-Work
Battery-OK  Starter-OK  Empty-Gas-Tank  Engine-Starts
Engine-Starts  Flat-Tire  Car-OK
Headlight-Work
Battery-OK
Starter-OK
Empty-Gas-Tank
Car-OK
Flat-Tire

40. Example (Shortest-Clause)
Battery-OK  Bulbs-OK  Headlights-Work
Battery-OK  Starter-OK  Empty-Gas-Tank  Engine-Starts
Engine-Starts  Flat-Tire  Car-OK
Headlight-Work
Battery-OK
Starter-OK
Empty-Gas-Tank
Car-OK
Flat-Tire
Engine-Starts  Car-OK
Engine-Starts
Bulbs-OK  Headlights-Work
Battery-OK  Starter-OK  Empty-Gas-Tank
Starter-OK  Empty-Gas-Tank
Empty-Gas-Tank
False

41. Resolution Heuristics
Set-of-support heuristics: At least one ancestor of every inferred clause comes from a
Shortest-clause heuristics: Generate a clause with the fewest literals first
Simplifications heuristics:
Remove any clause containing two complementary literals (tautology)
Remove any clause C that contains all the literals of another clause C’
If a symbol always appears with the same “sign”, remove all the clauses that contain it (pure symbol)

42. Example (Pure Literal)
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

43. When to Use Logic?
When the knowledge base is large and can be made explicit
When formulating a problem as a CSP problem would yield complex constraints
When the agent’s environment is conveniently described by true or false sentences

44. Summary Entailment problem Resolution rule
Clause form of a set of sentences
Resolution refutation algorithm
Resolution heuristics