first-order logic (fol) - uoacgi.di.uoa.gr/~ys02/siteai2008/fol2spp.pdf · teqnht nohmosÔnh m....

Teqnht NohmosÔnh M. Koumpar�khc'

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First-Order Logic (FOL)

Ontological commitments of FOL:

• The world consists of objects i.e., things with individualidentities. Objects have properties that distinguish themfrom other objects.

• Objects participate in relations with other objects. Some ofthese relations are functions. Relations hold or do not hold.

These ontological commitments make FOL more powerful than PL.

FOL is here to stay!


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FOL: Syntax

The symbols of FOL (with equality) are the following:

• Parentheses: (, ).

• The logical connectives ¬,∧,∨,⇒ and ⇔.

• A countably infinite set of variables. This set will be denotedby V ars.

Examples: x, y, v, . . .


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FOL: Syntax (cont’d)

• The quantifier symbols: ∀, ∃• A countably infinite set of constant symbols.

Examples: John, Mary, 5, 6, Ball, . . .

• The equality symbol: =

• Predicate symbols: For each positive integer n, some set(possibly empty) of symbols, called n-place predicate symbols.

Examples: Happy(.), Brother(., .), Arrives(., ., .), . . .

• Function symbols: For each positive integer n, some set(possibly empty) of symbols, called n-place function symbols.

Examples: FatherOf(.), Cosine(.), . . .

Logicians usually introduce only the connectives ¬ and ∨ and oneof the quantifiers.


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Terms are expressions of FOL that refer to objects. The set of allterms will be denoted by Terms.

The following BNF grammar gives the syntax of terms:

Term → ConstantSymbol | V ariable

| FunctionSymbol(Term, . . . , T erm)

Examples:John, x, FatherOf(John), WifeOf(FatherOf(x)), . . .


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Atomic formulas are expressions of FOL that refer to simplefacts.

The following BNF grammar gives the syntax of atomic formulas:

AtomicFormula → Term = Term

| PredicateSymbol(Term, . . . , T erm)

Examples:John = ElderSonOf(FatherOf(John)), Happy(John),Lives(John, London), Arrives(John, Athens, Monday)


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Well-formed formulas (wffs) are the most complex kind ofexpressions in FOL. They can be used to refer to any complicated stateof affairs.

The following BNF grammar gives the syntax of wffs:

Wff → AtomicFormula | ( Wff ) | ¬ Wff

| Wff BinaryConnective Wff

| ( Quantifier V ariable ) Wff


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Examples of wffs:

• ¬Loves(Tony, Mary)

• Loves(Tony, Paula) ∨ Loves(Tony, F iona)

• Loves(John, Paula) ∧ Loves(John, F iona)

• (∀x)(SportsCar(x) ∧HasDriven(Mike, x) ⇒ Likes(Mike, x))

• (∃x)(SportsCar(x) ∧Owns(John, x))


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Free Variables

The following recursive definition defines the notion of free variablesof a wff.

• If φ is an atomic formula, x occurs free in φ iff x is a symbol of φ.

• x occurs free in ¬φ iff x is a symbol of φ.

• x occurs free in φ ∧ ψ iff x is a symbol of φ or ψ. Similarly for theremaining binary connectives.

• x occurs free in (∀v)φ iff x is a symbol of φ and x is different thanv. Similarly for ∃.

The opposite of free is bound.

Definition. If no variable occurs free in the wff φ, then φ is asentence.


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Free Variables (cont’d)

Examples:

• x is free in Brother(x, John) but not in

(∀x)(Cat(x) ⇒ Mammal(x)).

• y is free in

(∀x)(Friend(x, y) ⇒ Loves(x, y))

but not in

(∀x)(∀y)(Friend(x, y) ⇒ Loves(x, y)).


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FOL: Semantics

The meaning of FOL formulas is provided by interpretations.

An interpretation is a mapping between symbols of FOL andobjects, functions or relations in the world. More precisely:

• An interpretation maps each constant symbol to an object inthe world.

Example: In one particular interpretation the symbol John

might refer to John Major, the British PM. In anotherinterpretation it might refer to the evil King John, king ofEngland from 1199 to 1216.


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FOL: Semantics (cont’d)

• An interpretation maps each predicate symbol to a relation inthe world.

Example: In one particular interpretation the symbolBrother(., .) might refer to the relation of brotherhood. In aworld with three objects, King John, John Major, and Richardthe Lionheart, the relation of brotherhood is defined by thefollowing set of tuples:

{ 〈King John, Richard the Lionheart〉,〈Richard the Lionheart, King John〉 }


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FOL: Semantics (cont’d)

• An interpretation always maps the equality symbol to theidentity relation in the world. The identity relation is:

id = {〈o, o〉 : o is an object in the world}

• An interpretation maps each function symbol to a functionalrelation (or function) in the world.

Example: In one particular interpretation the symbolFatherOf(.) might refer to the relation of fatherhood.


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FOL: Formal Semantics

An interpretation I is a function which makes the followingassignments to the symbols of FOL:

1. I assigns to the quantifier symbol ∀ a non-empty set |I| calledthe universe or domain of I.

2. I assigns to each constant symbol c a member cI of theuniverse |I|.

3. I assigns to each n-place predicate symbol P an n-ary relationP I ⊆ |I|n; i.e., P I is a set of n-tuples of members of theuniverse.

4. I assigns to each n-place function symbol F an n-ary functionF I on |I|; i.e., F I : |I|n → |I|.


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Satisfaction

Definition. A variable assignment is a function s : V ars → |I|for some set of variables V ars and interpretation I.

Let φ be a wff of FOL, I an interpretation and s : V ars → |I| avariable assignment.

We will define what it means for I to satisfy φ with variableassignment s. This will be denoted by

|=I φ[s] or I |= φ[s].

Intuitively |=I φ[s] if and only if the state of affairs denoted by φ istrue according to I (where any variable x which occurs in φ, standsfor s(x) wherever it occurs free).


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Satisfaction (cont’d)

The formal definition of satisfaction proceeds as follows:

Terms. We define the function

s : Terms → |I|from the set of all terms Terms into the universe |I|. This functionis an extension of s, and maps each FOL term to the object in theuniverse denoted by this term:

• For each variable x, s(x) = s(x).

• For each constant symbol c, s(c) = cI .

• If t1, . . . , tn are terms and F is an n-place function symbol, then

s(F (t1, . . . , tn)) = F I(s(t1), . . . , s(tn)).


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Atomic formulas. The definition of satisfaction for atomicformulas is as follows:

• For atomic formulas involving the equality symbol,

|=I t1 = t2[s] iff s(t1) is identical to s(t2).

• For an n-place predicate symbol P ,

|=I P (t1, . . . , tn)[s] iff 〈s(t1), . . . , s(tn)〉 ∈ P I .


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Example: the WW in FOL

Breeze Breeze

Breeze

BreezeBreeze

Stench

Stench

BreezePIT

PIT

PIT

1 2 3 4

1

2

3

4

START

Gold

Stench


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Example (cont’d)

If we want to formalize the WW in FOL, we can use the followingsymbols:

• Constant symbols:

Agent, Wumpus,Gold, Breeze, Stench,Rm11, Rm12, . . . , Rm44

• Function symbols:

– The unary function symbol NorthOf to denote the uniqueroom which is north of the room denoted by the argumentof the function. For example, the room north of room 11 isroom 21.

– The unary function symbols SouthOf, WestOf, EastOf

with similar meanings.


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Example (cont’d)

• Predicate symbols:

– The binary predicate Location will be used to denote thelocation (i.e. room) of each object (agent, wumpus andgold).

– The binary predicate Percept will be used to denote thepercept (i.e., breeze or stench) in each room.

– The unary predicate Bottomless will be used to denote thata room contains a pit.


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Example (cont’d)

One could rightly object to having constant symbols R11, R12 tomodel the grid of rooms etc.

A better way to model the grid of rooms is to have a binaryfunction symbol Room(x, y) where x and y are the coordinates ofthe room.

Try this as an exercise!


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Example (cont’d)

Let us now provide an interpretation I of the above symbols whichcorresponds to the previous picture:

• The universe of I is the objects we see in the picture:

|I| = {agent, wumpus, gold, breeze, stench, rm11, . . . , rm44}.

• I makes the following assignments to constant symbols:

AgentI = agent, WumpusI = wumpus, GoldI = gold,

BreezeI = breeze, StenchI = stench,

Rm11I = rm11, . . . , Rm44I = rm44


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Example (cont’d)

• I assigns to the unary function symbol NorthOf the functionNorthOf I : |I| → |I| which is defined as follows:

NorthOf I(rm11) = rm21,

NorthOf I(rm21) = rm22, . . . , NorthOf I(rm34) = rm44

• I assigns to the unary function symbolsSouthOf,WestOf, EastOf the function symbolsSouthOf I , WestOf I , EastOf I that are defined similarly withNorthOf I .


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Example (cont’d)

• I assigns to the unary predicate symbol Bottomless thefollowing relation:

{〈rm13〉, 〈rm33〉, 〈rm44〉}

• I assigns to the binary predicate symbol Location the followingrelation:

{〈agent, rm11〉, 〈wumpus, rm31〉, 〈gold, rm32〉}


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Example (cont’d)

• I assigns to the binary predicate symbol Percept the following relation:

{〈rm12, breeze〉, 〈rm14, breeze〉, 〈rm21, stench〉, 〈rm23, breeze〉,〈rm32, breeze〉, 〈rm32, stench〉, 〈rm34, breeze〉, 〈rm41, stench〉,〈rm43, breeze〉}

Note: To describe interpretation I we used words like agent, breeze, etc.which start with a lowercase letter. These are not symbols of our FOLlanguage; they are just English words referring to what is in the picture.Instead, we could have drawn little pictures to describe the elements of theuniverse of the interpretation.


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Example (cont’d)

We now give examples of satisfaction:

• |=I x = y[s] for any variable assignment s which maps x and y

to identical objects of the universe (e.g.,s(x) = s(y) = wumpus). Why?

Because if s(x) = s(y) = wumpus then s(x) = s(x) = wumpus

is identical to s(y) = s(y) = wumpus.

• |=I Agent = Agent[s] for any variable assignments s.

This is trivial.


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Example (cont’d)

• |=I Rm21 = NorthOf(Rm11)[s] for any variable assignment s.Why?

Because s(Rm21) = Rm21I = rm21 and

s(NorthOf(Rm11)) = NorthOf I(s(Rm11)) =

= NorthOf I(Rm11I) = NorthOf I(rm11) = rm21.

• |=I Rm21 = NorthOf(x)[s] for any variable assignment s suchthat s(x) = rm11. Why?

Because s(Rm21) = Rm21I = rm21 and

s(NorthOf(x)) = NorthOf I(s(x)) =

NorthOf I(s(x)) = NorthOf I(rm11) = rm21.


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Example (cont’d)

• |=I Bottomless(x)[s] for any variable assignment s such thats(x) = rm13 or s(x) = rm33 or s(x) = rm44. Why?

Because if s(x) = rm13 then

〈s(x)〉 = 〈s(x)〉 = 〈rm13〉 ∈ BottomlessI .

Similarly, for the other cases.

• |=I Location(Agent, Rm11)[s] for any variable assignments s. Why?

Because

〈s(Agent), s(Rm11)〉 = 〈AgentI , Rm11I〉 = 〈agent, rm11〉 ∈ LocationI .


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Example (cont’d)

• |=I ¬Location(Gold, Rm44)[s] for any variable assignment s. Why?

Because

〈s(Gold), s(Rm44)〉 = 〈GoldI , Rm44I〉 = 〈gold, rm44〉 6∈ LocationI

therefore 6|=I Location(Gold, Rm44)[s] for any variable assignment s.

• |=I Location(Gold, Rm32) ∨ Location(Gold, Rm44)[s] for anyvariable assignment s. Why?

Because

〈s(Gold), s(Rm32)〉 = 〈GoldI , Rm32I〉 = 〈gold, rm32〉 ∈ LocationI

therefore |=I Location(Gold, Rm32)[s] for any variable assignment s.


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Example (cont’d)

• |=I (∃x)Location(x,Rm11)[s] for any variable assignment s. Why?

Because

〈s(Agent), s(Rm11)〉 = 〈AgentI , Rm11I〉 = 〈agent, rm11〉 ∈ LocationI

thus |=I Location(x,Rm11)[s(x|agent)].

• 6|=I (∀x)Location(Wumpus, x)[s] for any variable assignment s.Why?

Because

〈s(Wumpus), s(Rm11)〉 = 〈WumpusI , Rm11I〉 =

〈wumpus, rm11〉 6∈ LocationI

thus 6|=I Location(Wumpus, x)[s(x|rm11)].


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When we want to verify whether or not an interpretation satisfies awff φ with s, we do not really need all of the (infinite amount of)information that s gives us. All that matters are the values of thefunction s at the (finitely many) variables which occur free in s. Inparticular, if φ is a sentence, then s does not matter at all. This ismade formal by the following theorem.

Theorem. Let s1 and s2 be variable assignments from V ars into|I| which agree at all variables (if any) which occur free in the wffφ. Then

|=I φ[s1] iff |=I φ[s2].


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The previous theorem has the following corollary.

Corollary. Let φ be a sentence and I an interpretation. Then,either(a) I satisfies φ with every variable assignment s : V ars → |I|, or(b) I does not satisfy φ with any variable assignment.

The above corollary allows us to ignore variable assignmentswhenever we talk about satisfaction of sentences. Thus if φ is asentence and I an interpretation, we can just say that I satisfies(or does not satisfy) φ.


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Satisfiability

Definition. A formula φ is called satisfiable iff there exists aninterpretation I and variable assignment s such that |=I φ[s].Otherwise, the formula is called unsatisfiable.

Examples: The formulas

Location(Wumpus,Rm31), Location(Agent, Rm11), (∃x)R(y, x)

are satisfiable. The following formulas are unsatisfiable:

P (x) ∧ ¬P (x), (∀x)P (x) ∧ ¬P (A)

Can you write an algorithm which discovers whether a given wff issatisfiable?


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Truth and Models

Definition. Let φ be a sentence and I an interpretation. If I

satisfies φ then we will say that φ is true in I or I is a model of φ.

Example: The interpretation I defined in the WW example is amodel of the following sentences:

Location(Wumpus, Rm31), Location(Agent, Rm11),

(∃x)Percept(Breeze, x)

Definition. An interpretation I is a model of a set of sentencesKB iff it is a model of every member of KB.


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Entailment

Definition. Let KB be a set of wffs, and φ a wff. Then KB entailsor logically implies φ, denoted by KB |= φ, iff for everyinterpretation I and every variable assignment s : V ars → |I| such thatI satisfies every member of KB with s, I also satisfies φ with s.

Examples:

{ Happy(John), (∀x)(Happy(x) ⇒ Laughs(x)) } |= Laughs(John)

{WellPaid(John), ¬WellPaid(John)∨Happy(John) } |= Happy(John)

Can you give an algorithm that discovers whether a set of wffs entail awff?


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Validity and Equivalence

Definition. A wff φ is valid iff for every interpretation I andevery variable assignment s : V ars → |I|, I satisfies φ with s.

Examples: The formulas

P (A) ∨ ¬P (A), P (A) ⇒ P (A), (∀x)P (x) ⇒ (∃x)P (x)

are valid.

Can you write an algorithm which discovers whether a given wff isvalid?

Definition. Two formulas φ and ψ will be called logicallyequivalent, denoted by φ ≡ ψ, iff φ |= ψ and ψ |= φ.


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Satisfiability, Equivalence and Validity

In the following theorems, φ and ψ are arbitrary wffs.

Theorem. φ |= ψ iff φ ⇒ ψ is valid.

Proof.

“Only if”

Let I be an arbitrary interpretation and s an arbitrary variable assignments : V ars → |I|. We will prove that |=I (φ ⇒ ψ)[s].

We consider two cases depending on whether |=I φ[s] or not.

Let us first assume that |=I φ[s]. Since φ |= ψ, we have |=I ψ[s]. Thus,from the definition of satisfaction for implication, we have |=I (φ ⇒ ψ)[s].

Let us now assume that 6|=I φ[s]. From the definition of satisfaction forimplication, we can immediately conclude |=I (φ ⇒ ψ)[s].


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Proof (cont’d)

“If”

Let I be an arbitrary interpretation and s an arbitrary variableassignment s : V ars → |I|.Let us also assume that |=I φ[s]. We will prove that |=I ψ[s].

Since φ ⇒ ψ is valid, we have |=I (φ ⇒ ψ)[s]. Since we also knowthat |=I φ[s], we must have |=I ψ[s].

Thus, from the definition of entailment, we have φ |= ψ.


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Satisfiability, Equivalence and Validity (cont’d)

Theorem. φ is unsatisfiable iff ¬φ is valid. Proof?

Theorem. φ ≡ ψ iff φ ⇔ ψ is valid. Proof?


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Some Important Logical Equivalences

Let φ and ψ be wffs. Then:

1. ¬(φ ∧ ψ) ≡ ¬φ ∨ ¬ψ

2. ¬(φ ∨ ψ) ≡ ¬φ ∧ ¬ψ

3. φ ∧ ψ ≡ ¬(¬φ ∨ ¬ψ)

4. φ ∨ ψ ≡ ¬(¬φ ∧ ¬ψ)

5. φ ⇒ ψ ≡ ¬φ ∨ ψ

6. φ ⇔ ψ ≡ (φ ⇒ ψ) ∧ (ψ ⇒ φ)

Proofs?


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Proof of φ ⇒ ψ ≡ ¬φ ∨ ψ

Let us first prove thatφ ⇒ ψ |= ¬φ ∨ ψ.

Let I be an arbitrary interpretation and s an arbitrary variableassignment s : V ars → |I|.Let us assume that |=I (φ ⇒ ψ)[s]. We will prove that |=I (¬φ ∨ ψ)[s].

From the definition of satisfaction for implication, we have 6|=I φ[s] or|=I ψ[s].

If 6|=I φ[s] then, from the definition of satisfaction for negation, we have|=I ¬φ[s]. Thus, from the definition of satisfaction for disjunction, wehave |=I (¬φ ∨ ψ)[s].

If |=I ψ[s] then, from the definition of satisfaction for disjunction, wealso have |=I (¬φ ∨ ψ)[s].


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Proof (cont’d)

Let us now prove that¬φ ∨ ψ |= φ ⇒ ψ.

Let I be an arbitrary interpretation and s an arbitrary variableassignment s : V ars → |I|.Let us assume that |=I (¬φ ∨ ψ)[s]. We will prove that |=I (φ ⇒ ψ)[s].

From the definition of satisfaction for disjunction, we have |=I ¬φ[s] or|=I ψ[s]. We consider these two cases below.

If |=I ¬φ[s] then, from the definition of satisfaction for negation, wehave 6|=I φ[s]. Thus, from the definition of satisfaction for implication,we have |=I (φ ⇒ ψ)[s].

If |=I ψ[s] then we can immediately conclude |=I (φ ⇒ ψ)[s] from thedefinition of satisfaction for implication.


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Some Important Logical Equivalences (cont’d)

1. (∀x)φ ≡ ¬(∃x)¬φ

2. (∃x)φ ≡ ¬(∀x)¬φ

3. (∀x)¬φ ≡ ¬(∃x)φ

4. (∃x)¬φ ≡ ¬(∀x)φ

Proofs?


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Some Important Logical Equivalences (cont’d)

1. (∃x)(φ ∨ ψ) ≡ (∃x)φ ∨ (∃x)ψ

2. (∃x)(φ ∧ ψ) |= (∃x)φ ∧ (∃x)ψ

3. (∀x)φ ∨ (∀x)ψ |= (∀x)(φ ∨ ψ)

4. (∀x)(φ ∧ ψ) ≡ (∀x)φ ∧ (∀x)ψ

Proofs?


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One of the Proofs

Prove that

(∃x)(φ(x) ∧ ψ(x)) |= (∃x)φ(x) ∧ (∃x)ψ(x).

Proof: Let I be an interpretation and s a variable assignment suchthat

|=I (∃x)(φ(x) ∧ ψ(x))[s].

Then according to the definition of satisfaction for existentialstatements, there exists a d ∈ |I| such that

|=I (φ(x) ∧ ψ(x))[s(x|d)].

Then according to the definition of satisfaction for conjunctivestatements, we have

|=I φ(x)[s(x|d)]


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and|=I ψ(x)[s(x|d)].

Now from the definition of satisfaction for existential statementsagain, we have

|=I (∃x)φ(x)[s]

and|=I (∃x)ψ(x)[s].

Now from the definition of satisfaction for conjunctive statementswe have:

|=I (∃x)φ(x) ∧ (∃x)ψ(x)[s].

The proof is now finished.


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Representing Knowledge Using FOL

Definition. In knowledge representation, a domain is a sectionof the world about which we wish to express some knowledge.

• The domain of family relationships.

• The domain of sets.

• The wumpus domain.

• The domain of web resources (HTML pages, images, programsetc. on the WWW)

• ...


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Knowledge Engineering

The process of knowledge-base construction is called knowledgeengineering.

A knowledge engineer is someone who investigates a particulardomain, determines what concepts are important in that domain,and creates a formal representation of the objects and relations inthat domain.

If you want to see how FOL can be applied to represent knowledgein various sophisticated application domains, see Chapter 10 of theAIMA book (we will not cover it in the course).


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Other Logics in Computer Science

FOL is certainly the most important logic in use today bycomputer scientists. But there are others too:

• Second-order logic

• Modal logic (with operators such as “possible” and “certain”)

• Temporal logic (with operators such as “in the past”, etc.)

• Logics of knowledge and belief

• Logics for databases

• ....


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Readings

Chapter 8 of AIMA: First-Order Logic

Other formal presentations of FOL can be found in:

1. Any mathematical logic textbook. Most of the formal materialin these notes is from:

H.B. Enderton, “A Mathematical Introduction to Logic”,Academic Press, 1972.

See the Web page of the course for more details.

2. M.R. Genesereth and N.J. Nilsson, “Logical Foundations ofArtificial Intelligence”, Morgan Kaufmann, 1987.

first-order logic (fol) - uoacgi.di.uoa.gr/~ys02/siteai2008/fol2spp.pdf · teqnht nohmosÔnh m....

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