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L ogics for D ata and K nowledge R epresentationPowerPoint Presentation

L ogics for D ata and K nowledge R epresentation

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### Logics for Data and KnowledgeRepresentation

First Order Logics (FOL)

Originally by Alessandro Agostini and Fausto Giunchiglia

Modified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese

Propositions

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- All knowledge representation languages deal with sentences
- Sentences denote propositions (by definition), namely they express something true or false (the mental image of what you mean when you write the sentence)
- Logic languages deal with propositions
- Logic languages have different expressiveness, i.e. they have different expressive power.

“All men are mortals” “Obama is the president of the USA”

What we can express with PL and ClassL (I)

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- PL
“There is a monkey” Monkey

We useIntensional Interpretation: the truth valuation ν of the proposition Monkey determines a truth-value ν(Monkey).

ν tells us if a proposition P “holds” or not.

- ClassL
“Fausto is a Professor” Fausto ∈ σ(Professor)

We useExtensional Interpretation: the interpretation function σ of Professor determines a class of objects σ(Professor) that includes Fausto. In other words, given x ∈ P:

“x belongs to P” or “x in P” or “x is an instance of P”

4

What we can express with PL and ClassL (II)

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- PL
Monkey ∧Banana

We mean that there is a monkey and there is a banana.

- ClassL
Monkey ⊓ Banana

We mean that there is an x ∈ σ(Monkey) ∩ σ(Banana)

5

What we cannot express in PL and ClassL

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- n-ary relations
They express objects in Dn

Near(Kimba,Simba) LessThan(x,3) Connects(x,y,z)

- Functions
They return a value of the domain, Dn → D

Sum(2,3) Multiply(x,y) Exp(3,6)

- Universal quantification
∀x Man(x) → Mortal(x) “All men are mortal”

- Existential quantification
∃x (Dog(x) ∧ Black(x)) “There exists a dog which is black”

The need for greater expressive power

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- We need FOL for a greater expressive power. In FOL we have:
- constants/individuals (e.g. 2)
- variables (e.g. x)
- Unary predicates (e.g. Man)
- N-ary predicates (eg. Near)
- functions (e.g. Sum, Exp)
- quantifiers (∀, ∃)
- equality symbol = (optional)
NOTE:

P(a) → ∃x P(x)

∀x P(x) → ∃x P(x)

∀x P(x) → P(x)

7

Example of what we can express in FOL

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constants

Cita

Monkey

1-ary predicates

n-ary predicates

Eats

Hunts

Kimba

Simba

Lion

Near

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The use of FOL in mathematics

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- FOL has being introduced to express mathematical properties
- The set of axioms describing the properties of equality between natural numbers (by Peano):
Axioms about equality

- ∀x1 (x1 = x1) reflexivity
- ∀x1 ∀x2 (x1 = x2 → x2 = x1) symmetricity
- ∀x1 ∀x2 ∀x3 (x1 = x2x2 = x3 → x1 = x3) transitivity
- ∀x1 ∀x2 (x1 = x2 → S(x1) = S(x2)) successor
NOTE: Other axioms can be given for the properties of the successor, the addition (+) and the multiplication (x).

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Alphabet of symbols

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- Variables x1, x2, …, y, z
- Constants a1, a2, …, b, c
- Predicate symbols A11, A12, …, Anm
- Function symbols f11, f12, …, fnm
- Logical symbols ∧, ∨, , → , ∀, ∃
- Auxiliary symbols ( )
- Indexes on top are used to denote the number of arguments, called arity, in predicates and functions.
- Indexes on the bottom are used to disambiguate between symbols having the same name.
- Predicates of arity =1 correspond to properties or concepts

Terms

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- Terms can be defined using the following BNF grammar:
<term> ::= <variable> | <constant> | <function> (<term> {,<term>}*)

- A term is called a closed term iff it does not contain variables

x, 2, SQRT(x), Sum(2, 3), Sum(2, x), Average(x1, x2, x3)

Sum(2, 3)

Well formed formulas

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- Well formed formulas (wff) can be defined as follows:
<atomic formula> ::= <predicate> (<term> {,<term>}*) |

<term> = <term>

<wff> ::= <atomic formula> | ¬<wff> |<wff> ∧ <wff> | <wff> ∨ <wff> |

<wff> → <wff> | ∀ <variable> <wff> | ∃ <variable> <wff>

NOTE: <term> = <term> is optional. If it is included, we have a FO language with equality.

NOTE: We can also write ∃x.P(x) or ∃x:P(x) as notation (with ‘.’ or “:”)

Scope and index of logical operators

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Given two wff α and β

- Unary operators
In ¬α, ∀xα and∃xα,

α is the scope and x is the index of the operator

- Binary operators
In α β, α β and α β,

α and β are the scope of the operator

NOTE: in the formula ∀x1A(x2), x1 is the index but x1 is not in the scope, therefore the formula can be simplified to A(x2).

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Free and bound variables

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- A variable x is bound in a formula γ if it is γ = ∀x α(x) or ∃x α(x) that is x is both in the index and in the scope of the operator.
- A variable is free otherwise.
- A formula with no free variables is said to be a sentence or closed formula.
- A FO theory is any set of FO-sentences.
NOTE: we can substitute the bound variables without changing the meaning of the formula, while it is in general not true for free variables.

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Interpretation function

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- An interpretation I for a FO language L over a domain D is a function such that:
- I(ai) = di ∈Dfor each constant ai
- I(An) Dnfor each predicate A of arity n
- I(fn) is a function f: Dn D Dn +1 for each function f of arity n

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Assignment

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- An assignment for the variables {x1, …, xn} of a FO language L over a domain D is a mapping function a: {x1, …, xn} Dn
a(xi) = di ∈ D

NOTE: In countable domains (finite and enumerable) the elements of the domain D are given in an ordered sequence <d1,…,dn> such that the assignment of the variables xi follows the sequence.

NOTE: the assignment a can be defined on free variables only.

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Interpretation over an assignment a

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- An interpretationIa for a FO language L over an assignment a and a domain D is an extended interpretation where:
- Ia(c) = I(c) for each constant c
- Ia(x) = a(x) for each variable x
- Ia(fn(t1,…, tn)) = I(fn)(Ia(t1),…, Ia(tn)) for each function f of arity n
NOTE: Ia is defined on terms only

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Interpretation over an assignment a

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- D = the set N of natural numbers
- Succ(x) = the function that returns the successor x+1
- zero = we could assign the value 0 to it.
Ia(xi) = a(xi) = i ∈ N

Ia(zero) = I(zero) = 0 ∈ N

Ia(Succ(x)) = I(Succ)(Ia(x)) = S(a(x)) such that S(a(x)) = a(x)+1

Ia(Succ(xi)) = I(Succ)(Ia(xi)) = S(i) = i+1

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Satisfaction relation

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- We are now ready to provide the notion of satisfaction relation:
M ⊨ γ [a]

(to be read: M satisfies γ under a or γ is true in M under a)

where:

- M is an interpretation function I over D
M is a mathematical structure <D, I>

- a is an assignment {x1, …, xn} Dn
- γ is a FO-formula
NOTE: if γ is a sentence with no free variables, we can simply write: M ⊨ γ (without the assignment a)

- M is an interpretation function I over D

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Satisfaction relation for well formed formulas

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- γ atomic formula:
- γ: t1= t2 M ⊨ (t1= t2)[a] iff Ia(t1) = Ia(t2)
- γ: An(t1,…, tn) M ⊨ An(t1,…, tn)[a] iff (Ia(t1), …, Ia(tn)) ∈ I(An)

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Satisfaction relation for well formed formulas

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- γ well formed formula:
- γ: α M ⊨ α[a] iff M ⊭α[a]
- γ: α β M ⊨ α β[a] iff M ⊨ α[a] and M ⊨ β[a]
- γ: α β M ⊨ α β[a] iff M ⊨ α[a] or M ⊨ β[a]
- γ: α β M ⊨ α β[a] iff M ⊭α[a] or M ⊨ β[a]
- γ: ∀xiα M ⊨ ∀xiα[a] iff M ⊨ α[s] for all assignments
s = <d1,…, d’i,…, dn> where s varies from a only

for the i-th element (s is called an i-th variant of a)

- γ: ∃xiα M ⊨ ∃xiα[a] iff M ⊨ α[s] for some assignment s = <d1,…, d’i,…, dn> i-th variant of a

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Satisfaction relation for a set of formulas

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- We say that a formula γ is true (w.r.t. an interpretation I) iff every assignment
a = <d1,…, dn> satisfies γ, i.e. M ⊨ γ [a] for all a.

NOTE: under this definition, a formula γ might be neither true nor false w.r.t. an interpretation I (it depends on the assignment)

- If γ is true under I we say that I is a model for γ.
- Given a set of formulas Γ, M satisfies Γ iff M ⊨ γ for all γ in Γ

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Satisfiability and Validity

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- We say that a formula γ is satisfiable iff there is a structure
M = <D, I> and an assignment a such that M ⊨ γ [a]

- We say that a set of formulas Γ is satisfiable iff there is a structure M = <D, I> and an assignment a such that
M ⊨ γ [a] for all γ in Γ

- We say that a formula γ is valid iff it is true for any structure and assignment, in symbols⊨ γ
- A set of formulas Γ is valid iff all formulas in Γ are valid.

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Example of satisfiability

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- D = {0, 1, 2, 3}
constants = {zero} variables = {x}

predicate symbols = {even, odd} functions = {succ}

γ1 : ∀x (even(x) odd(succ(x)))

γ2 : even(zero)

γ3 : odd(zero)

Is there any M and a such that M ⊨ γ1 γ2 [a]?

Ia(zero) = I(zero) = 0

Ia(succ(x)) = I(succ)(Ia(x)) = S(n) = {1 if n=0, n+1 otherwise}

I(even) = {0, 2} I(odd) = {1, 3}

Is there any M and a such that M ⊨ γ1 γ3 [a]?

Yes! Just invert the interpretation of even and odd.

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Example of valid formulas

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- ∀x1 (P(x1) Q(x1)) ∀x1 P(x1) ∀x1 Q(x1)
- ∃x1 (P(x1) Q(x1)) ∃x1 P(x1) ∃x1 Q(x1)
- ∀x1 P(x1) ∃x1 P(x1)
- ∀x1 ∃x1 P(x1) ∃x1 P(x1)
- ∃x1 ∀x1 P(x1) ∀x1 P(x1)

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Entailment

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- Let be a set of FO- formulas, γ a FO- formula, we say that
⊨ γ

(to be read entailsγ)

iff for all the interpretations M and assignments a,

if M ⊨ [a] then M ⊨ γ [a].

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Reasoning Services: EVAL

Yes

EVAL

γ, M, a

No

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- Model Checking (EVAL)
Is a FO-formula γ true under a structure M = <D, I> and an assignment a?

Check M ⊨ γ [a]

- Undecidable in infinite domains (in general)
- Decidable in finite domains

Reasoning Services: SAT

M, a

SAT

γ

No

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- Satisfiability (SAT)
Given a FO-formula γ, is there any structure M = <D, I> and an assignment a such that M ⊨ γ [a]?

Theorem (Church, 1936): SAT is undecidable

However, it is decidable in finite domains.

Reasoning Services: VAL

Yes

VAL

γ

No

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- Validity (VAL)
Given a FO-formula γ, is γ true for all the interpretations M and assignments a, i.e. ⊨ γ?

Theorem (Church, 1936): VAL is undecidable

However, it is decidable in finite domains.

How to reason on finite domains (I)

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- ⊨ ∀x P(x) [a] D = {a, b, c}
we have only 3 possible assignments a(x) = a, a(x) = b, a(x) = c

we translate in ⊨ P(a) P(b) P(c)

- ⊨ ∃x P(x) [a] D = {a, b, c}
we have only 3 possible assignments a(x) = a, a(x) = b, a(x) = c

we translate in ⊨ P(a) P(b) P(c)

How to reason on finite domains (II)

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- ⊨ ∀x ∃y R(x,y) [a] D = {a, b, c}
we have 9 possible assignments, e.g. a(x) = a, a(y) = b

we translate in ⊨ ∃y R(a,y) ∃y R(b,y) ∃y R(c,y)

and then in⊨ (R(a,a) R(a,b) R(a,c) )

(R(b,a) R(b,b) R(b,c) )

(R(c,a) R(c,b) R(c,c) )

Calculus

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- Semantic tableau is a decision procedure to determine the satisfiability of finite sets of formulas.
- There are rules for handling each of the logical connectives thus generating a truth tree.
- A branch in the tree is closed if a contradiction is present along the path (i.e. of an atomic formula, e.g. B and B)
- If all branches close, the proof is complete and the set of formulas are unsatisfiable, otherwise are satisfiable.
- With refutation tableaux the objective is to show that the negation of a formula is unsatisfiable.

Γ = {(A B), B}

B

A B

A

B

closed

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Rules of the semantic tableaux in FOL (I)

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- The initial set of formulas are considered in conjunction and are put in the same branch of the tree
- Conjunctions lie on the same branch
- Disjunctions generate new branches
- Propositional rules:
() A B () A B

--------- ---------

A A | B

B

() A A

--------- ---------

A A

(A B) B

B

(A B)

A B

A

B

closed

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Rules of the semantic tableaux in FOL (II)

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- Universal quantifiers are instantiated with a terminal t.
- Existential quantifiers are instantiated with a new constant symbol c.
(∀) ∀x.P(x)

---------

P(t)

(∃) ∃x.P(x)

---------

P(c)

NOTE: a constant is a terminal

NOTE: Multiple applications of the ∀ rule might be necessary (as in the example)

{∀x.P(x), ∃x.(¬P(x) ¬P(f(x)))

∀x.P(x)

∃x.(¬P(x) ¬P(f(x))

(∃)

¬P(c) ¬P(f(c))

P(c)

(∀)

¬P(c)

¬P(f(c))

()

closed

(∀)

P(f(c))

closed

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Rules of the semantic tableaux in FOL (III)

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- Correctness: the two rules for universal and existential quantifiers together with the propositional rules are correct:
closed tableau → unsatisfiable set of formulas

- Completeness: it can also be proved:
unsatisfiable set of formulas → ∃ closed tableau

- However, the problem is finding such a closed tableau. An unsatisfiable set can in fact generate an infinite-growing tableau.
- ∀ rule is non-deterministic: it does not specify which term to instantiate with. It also may require multiple applications.
- In tableaux with unification, the choice of how to instantiate the ∀ is delayed until all the other formulas have been expanded. At that point the ∀ is applied to all unbounded formulas.

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