First order logic in artificial Intelligence

1. Chapter 8
First Order Logic(FOL)

2. 3.7 Representation Revisited
Pros and cons of propositional logic
 Propositional logic is declarative: the syntax correspond to facts.
 Propositional logic allows partial/disjunctive/negated information.
 Propositional logic is compositional: meaning of B1,1 ∧ P1,2 is derived from meaning of B1,1 and of
P1,2.
 Meaning in propositional logic is context-independent (unlike natural language, where meaning
depends on context).
X Propositional logic has very limited expressive power (unlike natural language).
 E.g., cannot say “pits cause breezes in adjacent squares” except by writing one sentence for each
square.

3. First-order logic
 Propositional logic assumes the world contains facts.
 First-order logic (FOL), models the world in terms of
 Objects, which are things with individual identities.
 Properties of objects that distinguish them from other objects
 Relations that hold among sets of objects
 Functions, which are a subset of relations where there is only one “value” for any given “input”,
 Examples:
 Objects: Students, lectures, companies, cars ...
 Relations: Brother-of, bigger-than, outside, part-of, has-color, occurs-after, owns, visits, precedes, ...
 Properties: blue, oval, even, large, ...
 Functions: father-of, best-friend, second-half, one-more-than ...

4. Propositional Logic vs FOL
 The primary difference between propositional and first-order logic lies in the ontological
commitment made by each language—that is, what it assumes about the nature of reality.
 A logic can also be characterized by its epistemological commitments—the possible states of
knowledge that it allows with respect to each fact.

5. 3.8 Syntax and semantics of FOL
 Models for propositional logic links the proposition symbols to predefined truth values and every model
must provide the information required to determine if any given sentence is true or false.
 In case of FOL model, The domain(must be non empty) of a model contains the set of objects or domain
elements.
 The basic syntactic elements of first-order logic are objects, relations, and functions.

6.  The symbols are categorized into three kinds:
1. constant symbols, which stand for objects;
2. predicate symbols, which stand for relations; and
3. function symbols, which stand for functions.
 In addition to objects, relations, and functions, each model includes an interpretation that specifies exactly which
objects, relations and functions are referred to by the constant, predicate, and function symbol.
 In summary, a model in first-order logic consists of
a set of objects and an interpretation that maps constant symbols to objects,
predicate symbols to relations on those objects, and
function symbols to functions on those objects.
 For example, we might use the constant symbols Richard and John; the predicate symbols Brother , OnHead, Person,
King, and Crown; and the function symbol LeftLeg.

7. Term, Atomic sentence ,Complex Sentence
 A term is a logical expression that refers to an object.
 For example, in English ,we can use the expression “King John’s left leg” rather than giving a
name to his leg.
 This is what function symbols are for: instead of using a constant symbol, we use
LeftLeg(John).
 An atomic sentence (or atom for short) is formed from a predicate symbol optionally followed
by a parenthesized list of terms, such as Brother(Richard, John).
 Complex sentences are formed by the atomic sentences which are connected by the logical
connectives.
 Example: Brother (Richard, John) ∧ Brother (John, Richard).

8. Quantifiers
 Quantifiers are used to express the quantity of something. First-order logic contains two standard quantifiers,
called universal quantifiers, and existential quantifiers.
 Universal quantifiers(∀):
 The universal quantifier express that , the statements within its scope are true for everything, or every
instance of a specific thing.
 “All kings are persons” is written in first-order logic as ∀ x King(x) ⇒ Person(x) .
 Existential quantification (∃):
 The existential quantifier is a symbol of symbolic logic which expresses that the statements within its scope
are true for at least one instance of something.
 King John has a crown on his head, we write ∃ x Crown(x) ∧ OnHead(x, John) .
 A quantified sentence adds quantifiers  and .

9.  A well-formed formula(wff) is a sentence containing no “free” variables. That is, all variables are “bound” by universal or
existential quantifiers. For example: (x)P(x,y) has x bound as a universally quantified variable, but y is free.
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.
Equality
 term1 = term2 is true under a given interpretation if and only if term1 and term2 refer to the same object.
 Equality symbol can be used to signify that , the two terms refer to the same object.
 For example: ∃ x, y Brother (x, Richard) ∧ Brother (y, Richard)

10. Syntax of first-order logic

11.  Operator precedence are specified, from highest to lowest.

12. 3.9 Using FOL
Assertions and queries in first-order logic
 Sentences are added to a knowledge base using TELL, such sentences are called assertions(confident
and forceful statement of fact or belief).
 For example, we can assert that John is a king, Richard is a person, and all kings are persons:
 TELL(KB, King(John)) .
 TELL(KB, Person(Richard)) .
 TELL(KB, ∀ x King(x) ⇒ Person(x)) .
 We can ask questions to the knowledge base using ASK.
 Questions asked with ASK are called queries or goals.
 For example, ASK(KB, King(John)) QUERY returns true.

13. The Kinship Domain
 For example if we consider the domain of family relationships, or kinship.
 This domain includes facts such as
“Elizabeth is the mother of Charles” and
“Charles is the father of William” and
 rules such as “One’s grandmother is the mother of one’s parent.”
 Clearly, the objects in kinship domain are people.
 Here there are two unary predicates, Male and Female.
 Kinship relations—parenthood, brotherhood, marriage, and so on—are represented by binary predicates: Parent,
Sibling, Brother , Sister , Child, Daughter, Son, Spouse, Wife, Husband, Grandparent, Grandchild , Cousin, Aunt, and
Uncle.
 Functions can be used for Mother and Father , because every person has exactly one of each of these.

14. For example,
 one’s mother is one’s female parent:
∀ m, c Mother (c) = m ⇔ Female(m) ∧ Parent(m, c) .
 One’s husband is one’s male spouse:
∀ w, h Husband(h, w) ⇔ Male(h) ∧ Spouse(h, w) .
 Male and female are disjoint categories:
∀ x Male(x) ⇔ ¬Female(x) .
 Parent and child are inverse relations:
∀ p, c Parent(p, c) ⇔ Child(c, p) .
 A grandparent is a parent of one’s parent:
∀ g, c Grandparent(g, c) ⇔ ∃ p Parent(g, p) ∧ Parent(p, c) .
 A sibling is another child of one’s parents:
∀ x, y Sibling(x, y) ⇔ x = y ∧ ∃ p Parent(p, x) ∧ Parent(p, y) .

15. Sets
 Set can be defined as the collection of well-defined objects.
 We can represent individual sets, including the empty set ,which can be represented by { }.
 Sets can be build , by adding an element to a set or taking the union or intersection of two sets.
 There is one unary predicate, Set, which is true of sets.
 The binary predicates are
x∈ s (x is a member of set s) and
s1 ⊆ s2 (set s1 is a subset of set s2).
 The binary functions are
s1 ∩ s2 (the intersection of two sets),
s1 ∪ s2 (the union of two sets), and
{x|s} (the set resulting from adjoining element x to set s).

16. Set of Axioms
 The only sets are the empty set and those made by adjoining something to a set:
∀ s Set(s) ⇔ (s = { }) ∨ (∃ x, s2 Set(s2) ∧ s = {x|s2}) .
 The empty set has no elements adjoined into it(it can not be decomposed):
¬∃ x, s {x|s} = { } .
 Adjoining an element already in the set has no effect:
∀ x, s x∈ s ⇔ s = {x|s} .
 The only members of a set are the elements that were adjoined into it.(saying that x is a
member of s if and only if s is equal to some set s2 adjoined with some element y, where
either y is the same as x or x is a member of s2):
∀ x, s x∈ s ⇔ ∃ y, s2 (s = {y|s2} ∧ (x = y ∨ x∈ s2)) .

17.  A set is a subset of another set if and only if all of the first set’s members are members of
the second set:
∀ s1, s2 s1 ⊆ s2 ⇔ (∀ x x∈ s1 ⇒ x∈ s2) .
 Two sets are equal if and only if each is a subset of the other:
∀ s1, s2 (s1 = s2) ⇔ (s1 ⊆ s2 ∧ s2 ⊆ s1)
 An object is in the intersection of two sets if and only if it is a member of both sets:
∀ x, s1, s2 x∈ (s1 ∩ s2) ⇔ (x∈ s1 ∧ x∈s2) .
 An object is in the union of two sets if and only if it is a member of either set:
∀ x, s1, s2 x∈ (s1 ∪ s2) ⇔ (x∈ s1 ∨ x∈s2)

18. The wumpus world
 The agent percepts can be represented as five element list, [Stench, Breeze, Glitter, Bump, Scream].
 The corresponding first-order sentence stored in the knowledge base must include both the percept and the time
at which it occurred; otherwise, the agent will get confused about when it perceived what(integers time
stamps).
 For example the agent perceived ([Stench, Breeze, Glitter , None, None], 5) .
 Here, Percept is a binary predicate, and Stench and so on are constants placed in a list.
 The actions in the wumpus world can be : Turn(Right), Turn(Left), Forward, Shoot, Grab.
 To determine which is best, the agent program executes the query
ASKVARS(∃ a BestAction(a, 5)) ,
which returns a binding list such as {a/Grab}.
 The agent program can then return Grab as the action to take.

19. 3.10 Knowledge Engineering in FOL
 Knowledge engineering can be defined as the Construction of knowledge-base.
 Steps to construct the knowledge base
1. Identify the task
2. Assemble the relevant knowledge (knowledge acquisition: extract what they(expert) know)
3. Decide on a vocabulary of predicates, functions, and constants
4. Encode general knowledge about the domain
5. Encode a description of the specific problem instance
6. Pose queries to the inference procedure and get answers
7. Debug the knowledge base

20. The Electronic circuits domain
One-bit full adder

21. 1. Identify the task
Does the circuit actually add properly? (circuit verification)
2. Assemble the relevant knowledge
Composed of wires and gates; Types of gates (AND, OR, XOR, NOT)
Irrelevant: size, shape, color, cost of gates
3. Decide on a vocabulary
 Alternatives:
Type(X1) = XOR
Type(X1, XOR)
XOR(X1)

22. 4. Encode general knowledge of the domain

23. 5. 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))

24. 6. 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
7. Debug the knowledge base
May have omitted assertions like 1 ≠ 0