1. Explorations in Artificial Intelligence
Prof. Carla P. Gomes
Module 3-1-2
Logic Based Reasoning
Proof Methods

2. Proofs Methods

3. Proof methods Next modules
Current
module
Proof methods divide into (roughly) two kinds:
Application of inference rules
Legitimate (sound) generation of new sentences from old
Proof = a sequence of inference rule applications Can use inference rules as operators in a standard search algorithm
Different types of proofs
Model checking
truth table enumeration (always exponential in n)
(including some inference rules)
heuristic search in model space (sound but incomplete)
e.g., min-conflicts-like hill-climbing algorithms
we’ve talked about this approach
Next
modules

4. Proof
The sequence of wffs (w1, w2, …, wn) is called a proof (or deduction) of wn from
a set of wffs Δ iff each wi in the sequence is either in Δ or can be inferred from a
wff (or wffs) earlier in the sequence by using a valid rule of inference.
If there is a proof of wn from Δ, we say that wn is a theorem of the set Δ.
Δ├ wn
(read: wn can be proved or inferred from Δ)
The concept of proof is relative to a particular set of inference rules used. If we
denote the set of inference rules used by R, we can write the fact that wn can be
derived from Δ using the set of inference rules in R:
Δ├ R wn
(read: wn can be proved from Δ using the inference rules in R)

5. Propositional logic: Rules of Inference or Methods of Proof
How to produce additional wffs (sentences) from other ones? What steps can we
perform to show that a conclusion follows logically from a set of hypotheses?
Example
Modus Ponens P
P  Q
______________
 Q
The hypotheses (premises) are written in a column and the conclusions below the bar
The symbol  denotes “therefore”. Given the hypotheses, the conclusion follows.
The basis for this rule of inference is the tautology (P  (P  Q))  Q)
[aside: check tautology with truth table to make sure]
In words: when P and P  Q are True, then Q must be True also. (meaning of
second implication)

6. Propositional logic: Rules of Inference or Methods of Proof
Example
Modus Ponens
If you study the CS 372 material  You will pass
You study the CS372 material
______________
 you will pass
Nothing “deep”, but again remember the formal reason is that
((P ^ (P  Q))  Q is a tautology.

7. Propositional logic: Rules of Inference or Method of Proof
Rule of Inference
Tautology (Deduction Theorem)
Name P
 P  Q
P  (P  Q)
Addition
P  Q
 P
(P  Q)  P
Simplification Q
 P  Q
[(P)  (Q)]  (P  Q)
Conjunction
PQ
 Q
[(P)  (P Q)]  (P  Q)
Modus Ponens
 Q
P  Q
 P
[(Q)  (P Q)]  P
Modus Tollens
Q  R
 P R
[(PQ)  (Q  R)]  (PR)
Hypothetical Syllogism
(“chaining”)
P  Q P
[(P  Q)  (P)]  Q
Disjunctive syllogism
P  R
 Q  R
[(P  Q)  (P  R)]  (Q  R)
Resolution

8. Valid Arguments Show that (p1  p2  …pn)  q is a tautology
An argument is a sequence of propositions. The final proposition is called the conclusion of the argument while the other proposition are called the premises or hypotheses of the argument.
An argument is valid whenever the truth of all its premises implies the truth of its conclusion.
How to show that q logically follows from the hypotheses (p1  p2  …pn)?
Show that
(p1  p2  …pn)  q is a tautology
One can use the rules of inference to show the validity of an argument.

9. Proof Tree
Proofs can also be based on partial orders – we can represent them using a tree structure:
Each node in the proof tree is labeled by a wff, corresponding to a wff in the original set of hypotheses or be inferable from its parents in the tree using one of the rules of inference;
The labeled tree is a proof of the label of the root node.
Example:
Given the set of wffs:
P, R, PQ
Give a proof of Q  R

10. What rules of inference
Tree Proof
P, P Q, Q, R, Q  R
P PQ R Q
Q  R MP
Conj.
What rules of inference
did we use?

11. Length of Proofs Consider premises:
Why bother with inference rules? We could always use a truth table
to check the validity of a conclusion from a set of premises.
But, resulting proof can be much shorter than truth table method.
Consider premises:
p_1, p_1  p_2, p_2  p_3 … p_(n-1)  p_n
To prove conclusion: p_n
Inference rules: Truth table:
n-1 MP steps 2n
Key open question: Is there always a short proof for any valid
conclusion? Probably not. The NP vs. co-NP question.

12. Beyond Propositional Logic: Predicates and Quantifiers

13. Predicates Predicate, i.e. a property of x
Propositional logic assumes the world contains facts that are true or false.
But let’s consider a statement containing a variable:
x > 3 since we don’t know the value of x we cannot say whether the expression is true or false
x > 3 which corresponds to “x is greater than 3”
Predicate, i.e. a property of x

14. “x is greater than 3” can be represented as P(x), where P denotes “greater than 3”
In general a statement involving n variables x1, x2, … xn can be denoted by
P(x1, x2, … xn )
P is called a predicate or the propositional function P at the n-tuple (x1, x2, … xn ).

15. When all the variables in a predicate are assigned values 
Proposition, with a certain truth value.
Predicate: On(x,y)
Propositions:
ON(A,B) is False (in figure)
ON(B,A) is True
Clear(B) is True

16. Variables and Quantification
How would we say that every block in the world has a property – say “clear”  We would have to say:
Clear(A); Clear(B); … for all the blocks… (it may be long or worse we may have an infinite number of blocks…)
What we need is:
Quantifiers:
 Universal quantifier
x P(x)
- P(x) is true for all the values x in the universe of discourse
 Existential quantifier
x P(x)
- there exists an element x in the universe of discourse
such that P(x) is true.

17. Universal quantification
Everyone at Cornell is smart:
x At(x,Cornell)  Smart(x)
Implicity equivalent to the conjunction of instantiations of P
At(Mary,Cornell)  Smart(Mary)
 At(Richard,Cornell)  Smart(Richard)
 At(John,Cornell)  Smart(John)
 …

18. A common mistake to avoid
Typically,  is the main connective with 
Common mistake: using  as the main connective with :
x At(x,Cornell)  Smart(x)
means “Everyone is at Cornell and everyone is smart”

19. Existential quantification
Someone at Cornell is smart:
x (At(x,Cornell)  Smart(x))
x P(x) “ There exists an element x in the universe of discourse such that P(x) is true”
Equivalent to the disjunction of instantiations of P
(At(John,Cornell)  Smart(John))
 (At(Mary,Cornell)  Smart(Mary))
 (At(Richard,Cornell)  Smart(Richard))
 ...

20. Another common mistake to avoid
Typically,  is the main connective with
Common mistake: using  as the main connective with :
x At(x,Cornell)  Smart(x)
when is this true?
is true if there is anyone who is not at Cornell!

21. Quantified formulas
If ω is a wff and x is a variable symbol, then both x ω and x ω are
wffs.
x is the variable quantified over
ω is said to be within the scope of the quantifier
if all the variables in ω are quantified over in ω, we say that we have a closed wff or closed sentence.
Examples:
x [P(x)  R(x)]
x [P(x)(y [R(x,y)  S(x))]]

22. Properties of quantifiers
x y is the same as y x
x y is the same as y x
x y is not the same as y x
x y Loves(x,y)
“Everyone in the world loves at least one person”
y x Loves(x,y)
Quantifier duality: each can be expressed using the other
x Likes(x,IceCream) x Likes(x,IceCream)
x Likes(x,Broccoli) x Likes(x,Broccoli)
“There is a person who is loved by everyone in the world” 

23. Statement
When True
When False
x y P(x,y)
y x P(x,y)
P(x,y) is true for every pair
There is a pair for which P(x.y) is false
x y P(x,y)
For every x there is a y for which P(x,y) is true
There is an x such that P(x,y) is false for every y.
x y P(x,y)
There is an x such that P(x,y) is true for every y.
For every x there is a y for which P(x,y) is false
x  y P(x,y)
y  x P(x,y)
There is a pair x, y for which P(x,y) is true
P(x,y) is false for every pair x,y.

24. Negation Negation Equivalent Statement When is the negation True
When is False
x P(x)
x P(x)
For every x, P(x) is false
There is an x for which P(x) is true.
 x P(x)
x P(x)
There is an x for which P(x) is false.
For every x, P(x) is true.

25. Love Affairs Loves(x,y) x loves y
Everybody loves Jerry
x Loves (x, Jerry)
Everybody loves somebody
x y Loves (x, y)
There is somebody whom somebody loves
y x Loves (x, y)
Nobody loves everybody
 x y Loves (x, y) ≡ x y Loves (x, y)
There is somebody whom Lydia doesn’t love
y Loves (Lydia, y)
Note: flipping quantifiers when ¬ moves in.

26. Love Affairs continued…
There is somebody whom no one loves
y x Loves (x, y)
There is exactly one person whom everybody loves (uniqueness)
y(x Loves(x,y)  z((w Loves (w ,z) z=y))
There are exactly two people whom Lynn Loves
x y ((xy)  Loves(Lynn,x) Loves(Lynn,y) 
z( Loves (Lynn ,z) (z=x  z=y)))
Everybody loves himself or herself
x Loves(x,x)
There is someone who loves no one besides herself or himself
x y Loves(x,y) (x=y) (note biconditional )

27. Let Q(x,y) denote “xy =0”; consider the domain of discourse the real
numbers
What is the truth value of
a) y x Q(x,y)?
b) x y Q(x,y)?
False
True (additive inverse)

28. The kinship domain: Brothers are siblings
x,y Brother(x,y)  Sibling(x,y)
One's mother is one's female parent
m,c Mother(c) = m  (Female(m)  Parent(m,c))
“Sibling” is symmetric
x,y Sibling(x,y)  Sibling(y,x)

29. The set domain:
Sets are empty sets or those made by adjoining something to a set
s Set(s)  (s = {} )  (x,s2 Set(s2)  s = {x|s2})
The empty set has no element adjoined to it
x,s {x|s} = {}
Adjoining an element already in the set has no effect
x,s x  s  s = {x|s}
Only elements have been adjoined to it
x,s x  s  [ y,s2 (s = {y|s2}  (x = y  x  s2))]
Subset
s1,s2 s1  s2  (x x  s1  x  s2)
Equality of sets
s1,s2 (s1 = s2)  (s1  s2  s2  s1)
Intersection
x,s1,s2 x  (s1  s2) (x  s1  x  s2)
Uniion
x,s1,s2 x  (s1  s2)  (x  s1  x  s2)

30. Rules of Inference for Quantified Statements
(x) P(x)
P(c)
Universal Instantiation
P(c) for an arbitrary c
(x) P(x)
Universal Generalization
 (x) P(x)
 P(c) for some element c
Existential Instantiation
P(c) for some element c
 (x) P(x)
Existential Generalization

31. Example:
Let CS372(x) denote: x is taking CS372 class
Let CS(x) denote: x has taken a course in CS
Consider the premises x (CS372(x)  CS(x))
CS372(Ron)
We can conclude CS(Ron)

32. Arguments Argument (formal): Step Reason
1 x (CS372(x)  CS(x)) premise
2 CS372(Ron)  CS(Ron) Universal Instantiation
3 CS372(Ron) Premise
4 CS(Ron) Modus Ponens (2 and 3)

33. Example Show that the premises:
1- A student in this class has not read the textbook;
2- Everyone in this class passed the first homework
Imply
Someone who has passed the first homework has not read the textbook

34. Example Solution: Let C(x) x is in this class;
T(x) x has read the textbook;
P(x) x passed the first homework
Premises:
x (Cx  T(x))
x (C(x)  P(x))
Conclusion: we want to show x (P(x)  T(x))

35. Step Reason
1 x (Cx T(x)) premise
2 C(a)  T(a) Existential Instantiation from 1
3 C(a) Simplification 2
4 x (C(x)P(x)) Premise
5 C(a)  P(a) Universal Instantiation from 4
6 P(a) Modus ponens from 3 and 5
7 T(a) Simplification from 2
8 P(a)  T(a) Conjunction from 6 and 7
9 x P(x) T(x) Existential generalization from 8

36. Resolution in Propositional Logic

37. Resolution (for CNF)
Very important inference rule – several other inference rules
can be seen as special cases of resolution.
P  Q
P  R
 Q  R
Soundness of rule (validity of rule):
[(P  Q)  (P  R)]  (Q  R)
Resolution for CNF – applied to a special type of wffs: conjunction of clauses.
Literal – either an atom (e.g., P) or its negation (P).
Clause – disjunction of of literals (e.g., (P  Q  R)).
Note: Sometimes we use the notation of a set for a clause: e.g. {P,Q,R} corresponds
to the clause (PQ R); the empty clause (sometimes written as Nil or {}) is equivalent
to False;

38. Conjunctive Normal Form (CNF)
A wff is in CNF format when it is a conjunction of disjunctions of literals.
(P  Q  R) (S  P  T R) (Q  S)
Resolution for CNF – applied to wffs in CNF format.
{λ}  Σ1
{ λ}  Σ2
 Σ1  Σ2
Σi- sets of literals i =1 ,2
λ – atom;
Resolution
Resolvent of the
two clauses
atom resolved upon

39. Resolution: Notes 1 – Rule of Inference: Chaining R  P R  P
P  Q
 R  Q
R  P
P  Q
 R Q
can be re-written
Rule of Inference Chaining
2 – Rule of Inference: Modus Ponens P
P  Q
 Q
P  Q
can be re-written
Rule of Inference: Modus Ponens

40. Resolution: Notes
3 – Unit Resolution P
P  Q
 Q P
P  Q
 Q

41. Resolution: Notes 4 – No duplications in the resolvent set
only one instance of Q
appears in the resolvent,
which is a set!
P  Q  R  S
P  Q  W
 Q  R  S  W
5 – Resolving one pair at a time
DO NOT Resolve
on Q and R
P  Q  R
P  W  Q  R
 P  R  R  W
P  Q  R
P  W  Q  R
 P  Q  Q  W
P  Q  R
P  W  Q  R
 P  W
Resolving
on Q
Resolving
on R
True

42. Resolution: Notes 6 – Same atom with opposite signs {P} {P}  {}
 {}
False – any set of wffs containing two contradictory clauses is unsatisfiable. However, a clause {P, P} is True.

43. Soundness of Resolution: Validity of the Resolution Inference Rule
P  Q
P  R
 Q  R
resolving on P
Validity (Tautology): (P  Q) (P  R)  (Q  R) ; P Q R
(PQ)
(PR)
(PQ)(PR)
(QR)
(P  Q) (P  R)  (Q  R) 1

44. Conversion to CNF P  (Q  R)
1.Eliminate , replacing α  β with (α  β)(β  α).
(P  (Q  R))  ((Q R)  P)
2. Eliminate , replacing α  β with α β.
(P  Q  R)  ((Q R)  P)
3. Move  inwards using de Morgan's rules and double-negation:
(P Q R)  ((Q R)  P)
4. Apply distributivity law ( over ) and flatten:
(P  Q  R)  (Q P)  (R  P)

45. (P  S  Q)  (P  R  Q) (P  P  Q) (P  S  S)
Converting DNF (Disjunctions of conjunctions) into CNF
1 – create a table – each row corresponds to the literals in each conjunct;
2 - Select a literal in each row and make a disjunction of these literals;
Example:
(PQ R ) (S R P) (Q S  P) P Q R S R P
(P  S  Q)  (P  R  Q) (P  P  Q) (P  S  S)
(P  R  S) (P  P  S) (P  P  Q)…
How many clauses?

46. Resolution: Wumpus World
P31  P2,2,
P2,2
 P31 P?

47. Resolution Refutation
Resolution is sound – but resolution is not complete – e.g., (P R) ╞ (P  R) but
we cannot infer (P  R) using resolution 
we cannot use resolution directly to decide all logical entailments.
Resolution is Refutation Complete:
We can show that a particular wff W is entailed from a given KB how?
Proof by contradiction:
Write the negation of what we are trying to prove (W) as a conjunction of clauses;
Add those clauses (W) to the KB (also a set of clauses), obtaining KB’; prove inconsistency for KB’, i.e.,
Apply resolution to the KB’ until:
No more resolvents can be added
Empty clause is obtained
To show that (P  R) ╞Res (P  R) do: (1) negate (P  R), i.e.: (P) (R) ; (2) prove that
(P  R)  (P) (R) is inconsistent ! !

48. Propositional Logic: Proof by refutation or contradiction:
Satisfiability is connected to inference via the following:
KB ╞ α if and only if (KB α) is unsatisfiable
One assumes α and shows that this leads to a contradiction with the facts in KB

49. Resolution: Robot Domain
Example:
BatIsOk
RobotMoves
BatIsOk  BlockLiftable RobotMoves
Show that KB ╞ BlockLiftable KB
BatIsOk  BlockLiftable  RobotMoves
BlockLiftable
BatIsOk
RobotMoves
BatIsOk  BlockLiftable  RobotMoves
BlockLiftable
RobotMoves
KB’
BatIsOk  RobotMoves
BatIsOk
BatIsOk
Nil

50. Resolution
Resolution is refutation complete (Completeness of resolution refutation):
If KB ╞ W, the resolution refutation procedure, i.e., applying resolution on KB’, will produce the empty clause.
Decidability of propositional calculus by resolution refutation:
If KB is a set of finite clauses and if KB ╞ W, then the resolution refutation procedure will terminate without producing the empty clause.
Ground Resolution Theorem
If a set of clauses is not satisfiable, then resolution closure of those clauses contains the empty clause.
In general, resolution for propositional logic
is exponential !
The resolution closure of a set of clauses W in CNF, RC(W), is the set of all clauses derivable by repeated application
of the resolution rule to clauses in W or their derivatives.

51. Resolution algorithm
Proof by contradiction, i.e., show KBα unsatisfiable
Any complete search algorithm applying only the resolution rule, can derive any conclusion entailed by any knowledge base in propositional logic – resolution can always be used to either confirm or refute a sentence – refutation completeness (Given A, it’s true we cannot use resolution to derive A OR B; but
we can use resolution to answer the question of whether A OR B is true.)

52. Resolution example: Wumpus World
KB = (B1,1  (P1,2 P2,1))  B1, α = P1,2

53. Resolution example: Wumpus World
KB = (B1,1  (P1,2 P2,1))  B1, α = P1,2
KB = (B11  (P1,2 P2,1)) ^ ((P1,2 P2,1)  B11)  B1,1
=(B11  P1,2 P2,1) ^ ((P1,2 P2,1)  B11)  B1,1
=(B11  P1,2 P2,1) ^(( P1,2 ^  P2,1)  B11))  B1,1
=(B11  P1,2 P2,1) ^( P1,2  B11) ^ ( P2,1  B11)  B1,1

54. Resolution Refutation – Ordering Search Strategies
Original clauses – 0th level resolvents
Depth first strategy 
Produce a 1st level resolvent;
Resolve the 1st level resolvent with a 0th level resolvent to produce a 2nd level resolvent, etc.
With a depth bound, we can use a backtrack search strategy;
Breadth first strategy 
Generate all 1st level resolvents, then all 2nd level resolvents, etc.
BlockLiftable
BatIsOk  BlockLiftable  RobotMoves
BatIsOk  RobotMoves
RobotMoves
BatIsOk
BatIsOk
Nil
BatIsOk
RobotMoves
BatIsOk  BlockLiftable  RobotMoves
BlockLiftable
0th level resolvents
Depth first strategy

55. Refinement Resolution Strategies
Set-of-support Resolution Strategy
Definitions:
A clause γ2 is a descendant of a clause γ1 iif:
Is a resolvent of γ1 with some other clause
Or is a resolvent of a descendant of γ1 with some other clause;
If γ2 is a descendant of γ1, γ1 is an ancestor of γ2;
Set-of-support – set of clauses that are either clauses coming from the negation of the theorem to be proved or descendants of those clauses.
Set-of-support Strategy – it allows only refutations in which one of the clauses being resolved is in the set of support.
Set-of-support Strategy is refutation complete.
BlockLiftable
BatIsOk  BlockLiftable  RobotMoves
BatIsOk  RobotMoves
RobotMoves
BatIsOk
BatIsOk
Nil
Set-of-support Strategy

56. Refinement Strategies
Ancestry-filtered strategy – allows only resolutions in which at least one member of the clauses being resolved either is a member of the original set of clauses or is an ancestor of the other clause being resolved;
The ancestry-filtered strategy is refutation complete.

57. Refinement Strategies
Linear Input Resolution Strategy – at least one of the clauses being resolved is a member of the original set of clauses (including the theorem being proved).
Linear Input Resolution Strategy is not refutation complete.
Example:
(P Q) (P Q) (P  Q) (P  Q)
This set of clauses is inconsistent; but there is no linear-input
refutation strategy; but there is a resolution refutation strategy;
(P Q)
(P Q)
(P  Q) (P  Q) Q Q
This is NOT
Linear Input
Resolution Strategy
Nil

58. Horn Clauses

59. Horn Clauses Definition:
A Horn clause is a clause that has at most one positive literal.
Examples:
P; P  Q;  P  Q;  P  Q  R;
Types of Horn Clauses:
Fact – single atom – e.g., P;
Rule – implication, whose antecendent is a conjunction
of positive literals and whose consequent consists of a single
positive literal – e.g., PQ  R; Head is R; Tail is (PQ )
Set of negative literals - in implication form, the antecedent is
a conjunction of positive literals and the consequent is empty.
e.g., PQ  ; equivalent to  P  Q.
Inference with propositional Horn clauses can be done in linear time !

60. Forward chaining HORN (Expert Systems and Logic Programming)
Horn Form (restricted)
KB = conjunction of Horn clauses
Horn clause =
proposition symbol; or
(conjunction of symbols)  symbol
E.g., C  (B  A)  (C  D  B)
Modus Ponens (for Horn Form): complete for Horn KBs
α1, … ,αn, α1  …  αn  β β
Can be used with forward chaining 
Deciding entailment with Horn clauses can be done in linear time, in the size of the KB !

61. Forward Chaining: Diagnosis systems
Example: diagnostic system
IF the engine is getting gas and the engine turns over THEN the problem is spark plugs
IF the engine does not turn over and the lights do not come on THEN the problem is battery or cables
IF the engine does not turn over and the lights come on THEN the problem is starter motor
IF there is gas in the fuel tank and there is gas in the carburator
THEN the engine is getting gas

62. Forward chaining (Data driven reasoning)
Idea: fire any rule whose premises are satisfied in the KB,
add its conclusion to the KB, until query is found
AND-OR graph

63. Forward chaining algorithm
Forward chaining is sound and complete for Horn KB

64. Count
Inferred
Agenda A B
P => Q 1
L and M => P 2
B and L => M 2
A and P => L 2
A and B => L 2
P F
L F
M F
B F
A F

65. Forward chaining example

66. Forward chaining example

67. Forward chaining example

68. Forward chaining example

69. Forward chaining example

70. Forward chaining example

71. Forward chaining example

72. Forward chaining example

73. Backward chaining 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

74. Backward chaining example

75. Backward chaining example

76. Backward chaining example

77. Backward chaining example

78. Backward chaining example

79. Backward chaining example

80. Backward chaining example

81. Backward chaining example

82. Backward chaining example

83. Backward chaining example

84. Forward vs. backward chaining
FC is data-driven, automatic, unconscious processing,
e.g., object recognition, routine decisions
May do lots of work that is irrelevant to the goal
BC is goal-driven, appropriate for problem-solving,
e.g., Where are my keys? How do I get into a PhD program?
Complexity of BC can be much less than linear in size of KB