1. Based on Russell and Norvig slides
Propositional Logic
First Order Logic
Based on Russell and Norvig slides

2. Propositional Logic
Propositional logic is concerned with the truth or falsehood of statements (propositions) like:
the valve is closed
five plus four equals nine
Connectives: and 
or 
not 
implies 
equivalent 
(X  (Y  Z))  ((X  Y)  (X  Z))
“X implies Y and Z is the same as X implies Y and X implies Z”

3. Propositional Logic
Propositional Logic is declarative, pieces of syntax correspond to facts
Propositional Logic is compositional
Meaning of A  B derived from meaning of A and B
Meaning in Propositional Logic is context-independent
Propositional Logic has limited expressive power
Cannot say “Scottish men are careful with money”

4. Propositional logic Logical constants: true, false
Propositional symbols: P, Q, S, ...
Wrapping parentheses: ( … )
Sentences are combined by connectives:
 ...and
 ...or
...implies
..is equivalent
 ...not

5. Propositional logic (PL)
A simple language useful for showing key ideas and definitions
User defines a set of propositional symbols, like P and Q.
User defines the semantics of each of these symbols, e.g.:
P means "It is hot"
Q means "It is humid"
R means "It is raining"
A sentence (aka formula, well-formed formula, wff) defined as:
A symbol
If S is a sentence, then ~S is a sentence (e.g., "not”)
If S is a sentence, then so is (S)
If S and T are sentences, then (S v T), (S ^ T), (S => T), and (S <=> T) are sentences (e.g., "or," "and," "implies," and "if and only if”)
A finite number of applications of the above

6. Examples of PL sentences
(P ^ Q) => R
“If it is hot and humid, then it is raining”
Q => P
“If it is humid, then it is hot” Q
“It is humid.”

7. A BNF grammar of sentences in propositional logic
S := <Sentence> ;
<Sentence> := <AtomicSentence> | <ComplexSentence> ;
<AtomicSentence> := "TRUE" | "FALSE" |
"P" | "Q" | "S" ;
<ComplexSentence> := "(" <Sentence> ")" |
<Sentence> <Connective> <Sentence> |
"NOT" <Sentence> ;
<Connective> := "NOT" | "AND" | "OR" | "IMPLIES" | "EQUIVALENT" ;

8. Some terminology
The meaning or semantics of a sentence determines its interpretation.
Given the truth values of all of symbols in a sentence, it can be “evaluated” to determine its truth value (True or False).
A model for a Knowledge Base (a set of sentences) is a “possible world” in which each sentence in the KB is True.
We say that the KB is satisfiable
A valid sentence or tautology is a sentence that is True under all interpretations, no matter what the world is actually like or what the semantics is. Example?
An inconsistent sentence or contradiction is a sentence that is False under all interpretations. The world is never like what it describes. Example?
P entails Q, written P |= Q, means that whenever P is True, so is Q. In other words, all models of P are also models of Q.

9. Truth tables

10. Truth tables II
The five logical connectives:
A complex sentence:

11. Models of complex sentences

12. Inference rules
Logical inference is used to create new sentences that logically follow from a given set of sentences (KB).
An inference rule is sound if every sentence X produced by an inference rule operating on a KB logically follows from the KB. (That is, the inference rule does not create any contradictions)
An inference rule is complete if it is able to produce every expression that logically follows from (is entailed by) the KB.
Note the analogy to sound and complete search algorithms.

13. Sound rules of inference
Here are some examples of sound rules of inference.
Each can be shown to be sound using a truth table: A rule is sound if its conclusion is true whenever the premise is true.
RULE PREMISE CONCLUSION
Modus Ponens A, A => B B
And Introduction A, B A ^ B
And Elimination A ^ B A
Double Negation ~~A A
Unit Resolution A v B, ~B A
Resolution A v B, ~B v C A v C

14. Soundness of modus ponensA B
A → B
OK?
True 
False

15. Soundness of the resolution inference rule

16. Proving things 1 Hu Premise “It is humid”
A proof is a sequence of sentences, where each sentence is either a premise or a sentence derived from earlier sentences in the proof by one of the rules of inference.
The last sentence is the theorem (also called goal or query) that we want to prove.
Example for the “weather problem” given above.
1 Hu Premise “It is humid”
2 Hu=>Ho Premise “If it is humid, it is hot”
3 Ho Modus Ponens(1,2) “It is hot”
4 (Ho^Hu)=>R Premise “If it’s hot & humid, it’s raining”
5 Ho^Hu And Introduction(1,2) “It is hot and humid”
6 R Modus Ponens(4,5) “It is raining”

17. Entailment and derivation
Entailment: KB |= Q
Q is entailed by KB (a set of premises or assumptions) if and only if there is no logically possible world in which Q is false while all the premises in KB are true.
Or, stated positively, Q is entailed by KB if and only if the conclusion is true in every logically possible world in which all the premises in KB are true.
Derivation: KB |- Q
We can derive Q from KB if there is a proof consisting of a sequence of valid inference steps starting from the premises in KB and resulting in Q

18. Two important properties for inference
Soundness: If KB |- Q then KB |= Q
If Q is derived from a set of sentences KB using a given set of rules of inference, then Q is entailed by KB.
Hence, inference produces only real entailments, or any sentence that follows deductively from the premises is valid.
Completeness: If KB |= Q then KB |- Q
If Q is entailed by a set of sentences KB, then Q can be derived from KB using the rules of inference.
Hence, inference produces all entailments, or all valid sentences can be proved from the premises.

19. Complexity
Theorem: Determining if a set of sentences in PL is satisfiable (i.e. has a model) in NP-complete (Cook, 1971)
Determining if a sentence is valid is co-NP-complete
It is unlikely that we can find a polynomial algorithms for these problems

20. Complexity
A sentence in PL is called a Horn sentence if it is of the form
P1  P2  ...  Pn  Q
or equivalently ¬P1  ¬P2  ...  ¬Pn  Q
Theorem: If φ is a conjunction of Horn sentences then determining if φ is satisfiable can be done in polynomial time

21. Reasoning in PL
A proof is a sequence of applications of inference rules
Finding proofs is exactly like finding solutions in search problems
If we model the successor function to generate all possible applications of inference rules, we can use any search algorithm
BFS, DFS, A*, etc.
Forward or Backward Search

22. Resolution
Resolution yields a complete inference algorithm when coupled with any complete search algorithm
Unit Resolution
One resolvent is a unit clause
A v B v C, ~B -> A v C
Binary Resolution
One resolvent is a binary clause
A v B v C, D v ~B -> A v C v D
Full Resolution
Resolvents of any arity
A v B v C, A v D v ~B -> ?
factoring

23. Resolution It is easy to see that Resolution is sound
Consider two clauses l1 v … v lk and m1 v … mn, where li and mj are complementary (i.e. li = ~mj)
If li is true then mj is false and m1 v … v mj-1 v mj+1 v … v mn must be true
If li is false then l1 v … v lj-1 v lj+1 v … v ln must be true
This is what resolution gives: l1 v … v lj-1 v lj+1 v … v ln v m1 v … v mj-1 v mj+1 v … v mn

24. Resolution Resolution is also refutation complete
It can always be used to either confirm or refute a sentence
E.g. we can use it to answer the question of whether A v B is true
But it cannot be used to enumerate true sentences
E.g. if we know that A is true we cannot automatically generate the consequence A v B

25. A Resolution Algorithm
Based on proof by contradiction
Prove KB |= a by showing KB  ~a is unsatisfiable
First the query is converted into CNF (conjunctive normal form)
Then the resolution rule is applied to the resulting clauses
New clauses are added to the set of clauses (if not already there)
The process continues until either
No new clauses can be added
The empty clause is derived (i.e. l  ~l)
more on this later

26. Propositional logic is a weak language
Hard to identify “individuals.” E.g., Mary, 3
Can’t directly talk about properties of individuals or relations between individuals. E.g. “Bill is tall”
Generalizations, patterns, regularities can’t easily be represented. E.g., all triangles have 3 sides
First-Order Logic (abbreviated FOL or FOPC) is expressive enough to concisely represent this kind of situation.
FOL adds relations, variables, and quantifiers, e.g.,
“Every elephant is gray”:  x (elephant(x) → gray(x))
“There is a white alligator”:  x (alligator(X) ^ white(X))

27. Example
Consider the problem of representing the following information:
Every person is mortal.
Confucius is a person.
Confucius is mortal.
How can these sentences be represented so that we can infer the third sentence from the first two?

28. Example II
In PL we have to create propositional symbols to stand for all or part of each sentence. For example, we might do:
P = “person”; Q = “mortal”; R = “Confucius”
so the above 3 sentences are represented as:
P => Q; R => P; R => Q
Although the third sentence is entailed by the first two, we needed an explicit symbol, R, to represent an individual, Confucius, who is a member of the classes “person” and “mortal.”
To represent other individuals we must introduce separate symbols for each one, with means for representing the fact that all individuals who are “people” are also "mortal.”

29. Summary (so far)
The process of deriving new sentences from old one is called inference.
Sound inference processes derives true conclusions given true premises.
Complete inference processes derive all true conclusions from a set of premises.
A valid sentence is true in all worlds under all interpretations.
If an implication sentence can be shown to be valid, then - given its premise - its consequent can be derived.
Different logics make different commitments about what the world is made of and what kind of beliefs we can have regarding the facts.
Logics are useful for the commitments they do not make because lack of commitment gives the knowledge base write more freedom.

30. Summary (so far)
Propositional logic commits only to the existence of facts that may or may not be the case in the world being represented.
It has a simple syntax and a simple semantic. It suffices to illustrate the process of inference.
Propositional logic was considered impractical for real-world problems, even if they are small.
But recent (huge) advances in reasoning techniques have made people reconsider these
It can now be used to model and reason with several real problems (SAT)

31. First-order Logic
Whereas Propositional Logic assumes the world contains facts, First-order Logic assumes the world contains:
Objects; people, houses, games, wars, colours…
Relations; red, round, bogus, prime, brother, bigger_than, part_of…
Functions; father_of, best_friend, second_half_of, number_of_wheels…

32. Aside: Logics in General

33. Predicate Calculus
Two important extensions to Propositional Logic are:
predicates and quantifiers
Predicates are statements about objects by themselves or in relation to other objects. Some examples are:
less-than-zero
weighs-more-than
The quantifiers then operate over the predicates. There are two quantifiers:
 for all (the universal quantifier)
 there exists (the existential quantifier)
So with predicate calculus we can make statements like:
 XYZ: Smaller(X,Y)  Smaller(X,Z)  Smaller(X,Z)

34. Predicate Calculus (cont’d)
Predicate calculus is very general but not very powerful
Two useful additions are functions and the predicate equals
absolute-value
number-of-wheels
colour
Functions do not return true or false but return objects.

35. First Order Logic
Two individuals are “equal” if and only if they are indistinguishable under all predicates and functions.
Predicate calculus with these additions is a variety of first order logic.
A logic is of first order if it permits quantification over individuals but not over predicates and functions. For example a statement like “all predicates have only one argument” cannot be expressed in first order logic.
The advantage of first order logic as a representational formalism lies in its formal structure. It is relatively easy to check for consistency and redundancy.

36. First Order Logic
Advantages stem from rigid mathematical basis for first order logic
This rigidity gives rise to problems. The main problem with first order logic is that it is monotonic.
“If a sentence S is a logical consequence of a set of sentences A then S is still a logical consequence of any set of sentences that includes A. So if we think of A as embodying the set of beliefs we started with, the addition of new beliefs cannot lead to the logical repudiation of old consequences.”
So the set of theorems derivable from the premises is not reduced (increases monotonically) by the adding of new premises.

37. Syntax of FOL: Basic Elements
Constants: KingJohn, 2, TCD, …
Predicates: Brother, >, …
Functions: Sqrt, LeftLegOf,…
Variables: x, y, a, b,…
Connectives: , , , , 
Equality: =
Quantifiers: , 

38. FOL Example Sentences Brother(KingJohn, RichardTheLionHart)
> (Length (LeftLegOf(RichardThe…)
(Length (LeftLegOf(KingJohn)
Sibling(KingJohn, Richard…) 
Sibling(Richard…, KingJohn)

39. Quantifiers Universal Quantification
x At(x, Aegean)  Smart(x)
“Everyone at Aegean is Smart”
Being at Aegean implies you are smart
x P is true in a model m iff P is true with x being each possible object in the model
Roughly speaking, equivalent to the conjunction of instantiations of P
At(KingJohn,Aegean)  Smart(KingJohn)
 At(Richard, Aegean)  Smart(Richard)
 At(Robin, Aegean)  Smart(Robin)
 ...

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

41. Quantifiers Existential Quantification
x At(x, Aegean)  Smart(x)
“Someone at Aegean is Smart”
There exists someone who is both at Aegean and Smart
x P is true in a model m iff P is true with x being some possible object in the model
Roughly speaking, equivalent to the disjunction of instantiations of P
At(KingJohn,Aegean)  Smart(KingJohn)
 At(Richard, Aegean)  Smart(Richard)
 At(Robin, Aegean)  Smart(Aegean)
 ...

42. Another common mistake to avoid
Typically,  is the main connective with 
Common mistake: using  as the main connective with :
x At(x,Aegean)  Smart(x)
is true if there is anyone who is not at Aegean!

43. 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)
“There is a person who loves everyone in the world”
y x Loves(x,y)
“Everyone in the world is loved by at least one person”
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)

44. Equality
term1 = term2 is true under a given interpretation if and only if term1 and term2 refer to the same object
E.g., definition of Sibling in terms of Parent:
x,y Sibling(x,y)  [(x = y)  m,f  (m = f)  Parent(m,x)  Parent(f,x)  Parent(m,y)  Parent(f,y)]

45. Inference Rules in FOL In FOL there are three new inference rules
Universal Elimination:
For each sentence α, variable v, and ground term g
(v)α |= SUBST({v/g},α)
E.g. From (x) Likes(x,Icecream), by doing the substitution {x/John} we can infer Likes(John,Icecream)

46. Inference Rules in FOL
Existential Elimination: For each sentence α, variable v, and new constant k
(v)α |= SUBST({v/k},α)
E.g. From ( x) Likes(x,Icecream), we can infer Likes(Somebody,Icecream), as long as Somebody is a new constant which has not been used before

47. Inference Rules in FOL
Existential Introduction: For each sentence α, variable v not appearing in α, and ground term g appearing in α
α |= (v) SUBST({g/v},α)
E.g. from Likes(John,Icecream), we can infer ( x) Likes(John,x)

48. Using FOL 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)

49. Knowledge engineering in FOL
Identify the task
Assemble the relevant knowledge
Decide on a vocabulary of predicates, functions, and constants
Encode general knowledge about the domain
Encode a description of the specific problem instance
Pose queries to the inference procedure and get answers
Debug the knowledge base

50. The electronic circuits domain
One-bit full adder

51. The electronic circuits domain
Identify the task
Does the circuit actually add properly? (circuit verification)
Assemble the relevant knowledge
Composed of wires and gates; Types of gates (AND, OR, XOR, NOT)
Irrelevant: size, shape, color, cost of gates
Decide on a vocabulary
Alternatives:
Type(X1) = XOR
Type(X1, XOR)
XOR(X1)

52. The electronic circuits domain
Encode general knowledge of the domain
t1,t2 Connected(t1, t2)  Signal(t1) = Signal(t2)
t Signal(t) = 1  Signal(t) = 0
1 ≠ 0
t1,t2 Connected(t1, t2)  Connected(t2, t1)
g Type(g) = OR  Signal(Out(1,g)) = 1  n Signal(In(n,g)) = 1
g Type(g) = AND  Signal(Out(1,g)) = 0  n Signal(In(n,g)) = 0
g Type(g) = XOR  Signal(Out(1,g)) = 1  Signal(In(1,g)) ≠ Signal(In(2,g))
g Type(g) = NOT  Signal(Out(1,g)) ≠ Signal(In(1,g))

53. The electronic circuits domain
Encode the specific problem instance
Type(X1) = XOR Type(X2) = XOR
Type(A1) = AND Type(A2) = AND
Type(O1) = OR
Connected(Out(1,X1),In(1,X2)) Connected(In(1,C1),In(1,X1))
Connected(Out(1,X1),In(2,A2)) Connected(In(1,C1),In(1,A1))
Connected(Out(1,A2),In(1,O1)) Connected(In(2,C1),In(2,X1))
Connected(Out(1,A1),In(2,O1)) Connected(In(2,C1),In(2,A1))
Connected(Out(1,X2),Out(1,C1)) Connected(In(3,C1),In(2,X2))
Connected(Out(1,O1),Out(2,C1)) Connected(In(3,C1),In(1,A2))

54. The electronic circuits domain
Pose queries to the inference procedure
What are the possible sets of values of all the terminals for the adder circuit?
i1,i2,i3,o1,o2 Signal(In(1,C_1)) = i1  Signal(In(2,C1)) = i2  Signal(In(3,C1)) = i3  Signal(Out(1,C1)) = o1  Signal(Out(2,C1)) = o2
Debug the knowledge base
May have omitted assertions like 1 ≠ 0

55. Monotonicity of First Order Logic
Once you deduce something in first order logic you cannot get rid of it.
Engine won’t start Wont-start(E)
petrol guage reads empty Reads-empty(PG)
petrol tank empty (inferred) Reads-empty(PG) Empty(PT)
petrol guage faulty Faulty(PG)
Can be handled by qualification:
Reads-empty(PG)  Faulty(PG)  Empty(PT)

56. Example Suppose we have the following text:
The law says that it is a crime for an American to sell weapons to hostile nations. The country Nono, an enemy of America, has some missiles, and all of its missiles were sold to it by Colonel West, who is an American.
How can we use FOL to represent the knowledge in the text?
How can we use the inference rules of FOL to infer that “West is a criminal” ?

57. Representing Knowledge in FOL
“it is a crime for an American to sell weapons to hostile nations”
( x,y,z) American(x)  Weapon(y)  Nation(z)  Hostile(z)  Sells(x,z,y)  Criminal(x)
“Nono … has some missiles”
( x) Owns(Nono,x)  Missile(x)
“all of its missiles were sold to it by Colonel West”
( x) Owns(Nono,x)  Missile(x)  Sells(West,Nono,x)

58. Representing Knowledge in FOL
Missiles are weapons
( x) Missile(x)  Weapon(x)
An enemy of America is a hostile nation
( x) Enemy(x,America)  Hostile(x)
“West, who is an American”
American(West)
“The country Nono”
Nation(Nono)
“Nono, an enemy of America”
Enemy(Nono,America) , Nation(America)

59. Proof
From ( x) Owns(Nono,x)  Missile(x) with existential elimination:
Owns(Nono,M1)  Missile(M1)
From Owns(Nono,M1)  Missile(M1) with and-elimination:
Owns(Nono,M1) , Missile(M1)
From ( x) Missile(x)  Weapon(x) with universal elimination:
Missile(M1)  Weapon(M1)
From Missile(M1) , Missile(M1)  Weapon(M1) with modus ponens:
Weapon(M1)

60. Proof
From ( x) Owns(Nono,x)  Missile(x)  Sells(West,Nono,x) with universal elimination:
Owns(Nono,M1)  Missile(M1)  Sells(West,Nono,M1)
From Owns(Nono,M1)  Missile(M1) ,
Owns(Nono,M1)  Missile(M1)  Sells(West,Nono,M1) with modus ponens:
Sells(West,Nono,M1)
From ( x,y,z) American(x)  Weapon(y)  Nation(z)  Hostile(z)  Sells(x,z,y)  Criminal(x) with universal elimination (3 times):
American(West)  Weapon(M1)  Nation(Nono)  Hostile(Nono)  Sells(Westr,Nono,M1)  Criminal(West)

61. Proof
From ( x) Enemy(x,America)  Hostile(x) with universal elimination:
Enemy(Nono,America)  Hostile(Nono)
From Enemy(Nono,America) ,
Enemy(Nono,America)  Hostile(Nono) with modus ponens:
Hostile(Nono)
From American(West) , Weapon(M1) , Nation(Nono) , Hostile(Nono) , Sells(West,Nono,M1) with and introduction:
American(West)  Weapon(M1)  Nation(Nono)  Hostile(Nono)  Sells(West,Nono,M1)

62. Proof American(West)  Weapon(M1)  Nation(Nono)  Hostile(Nono) 
From American(West)  Weapon(M1)  Nation(Nono)  Hostile(Nono)  Sells(West,Nono,M1) ,
American(West)  Weapon(M1)  Nation(Nono)  Hostile(Nono) 
Sells(Westr,Nono,M1)  Criminal(West)
with modus ponens:
Criminal(West)

63. Summary Propositional Logic Add predicates and quantifiers
limited expressive power
Add predicates and quantifiers
Add functions and equals
First Order Logic
Examples