L ogics for d ata and k nowledge r epresentation
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L ogics for D ata and K nowledge R epresentation. First Order Logics (FOL) . Originally by Alessandro Agostini and Fausto Giunchiglia Modified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese. Outline. Introduction Syntax Semantics Reasoning Services. 2. Propositions.

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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


Outline

  • Introduction

  • Syntax

  • Semantics

  • Reasoning Services

2


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”

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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)

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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)

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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 = x2x2 = 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 symbolsf11, 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 “:”)


Well formed formulas

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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)

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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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