Logics for Data and Knowledge Representation

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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
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Outline

• Introduction
• Syntax
• Semantics
• Reasoning Services

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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
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What we can express with PL and ClassL (I)
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• PL
• There is a monkey Monkey
• We use Intensional Interpretation the truth
  valuation ? of the proposition Monkey determines
  a truth-value ?(Monkey).
• ? tells us if a proposition P holds or not.
• ClassL
• Fausto is a Professor Fausto ?
  s(Professor)
• We use Extensional Interpretation the
  interpretation function s of Professor determines
  a class of objects s(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 ? s(Monkey) n
  s(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

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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 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

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Terms
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• Terms can be defined using the following BNF
  grammar
• lttermgt ltvariablegt ltconstantgt
• ltfunction symgt (lttermgt ,lttermgt)
• 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)
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Well formed formulas
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• Well formed formulas (wff) can be defined as
  follows
• ltatomic formulagt ltpredicate symgt (lttermgt
  ,lttermgt)
• lttermgt lttermgt
• ltwffgt ltatomic formulagt ltwffgt ltwffgt ?
  ltwffgt ltwffgt ? ltwffgt
• ltwffgt ? ltwffgt ? ltvariablegt ltwffgt ?
  ltvariablegt ltwffgt
• NOTE lttermgt lttermgt is optional. If it is
  included, we have a FO language with equality.
• NOTE We can also write ?x.P(x) or ?xP(x) as
  notation (with . or )

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Well formed formulas
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NL FOL
x is a man Man(x)
3 is an odd number Odd(3)
Each number is odd or even ?x (Odd(x) ? Even(x))
Each even number greater than 2 is the sum of two prime numbers ?x ( Even(x) ? GreaterThan(x,2) ? ?y ?z (Prime(y) ? Prime(z) ? x Sum(y,z)) )
There is (at least) a white dog ?x (Dog(x) ? White(x))
There are (at least) two dogs ?x ?y (Dog(x) ? Dog(y) ? ?(x y))
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Scope and index of logical operators
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• Given two wff a and ß
• Unary operators
• In a, ?xa and?xa,
• a is the scope and x is the index of the
  operator
• Binary operators
• In a ? ß, a ? ß and a ? ß,
• a and ß are the scope of the operator
• NOTE in the formula ?x1 A(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 ? if it is ?
  ?x a(x) or ?x a(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) ai for each constant ai
• I(An) ? Dn for 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 ? D
• a(xi) di ? D
• NOTE In countable domains (finite and
  enumerable) the elements of the domain D are
  given in an ordered sequence ltd1,,dngt 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 interpretation Ia for a FO language L over an
  assignment a and a domain D is an extended
  interpretation where
• Ia(x) a(x) for each variable x
• Ia(c) I(c) for each constant c
• 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
  x1
• 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) i1

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Satisfaction relation
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• We are now ready to provide the notion of
  satisfaction relation
• M ? ? a
• (to be read M satisfies ? under a or ? is true
  in M under a)
• where
• M is an interpretation function I over D
• M is a mathematical structure ltD, Igt
• a is an assignment x1, , xn ? D
• ? is a FO-formula
• NOTE if ? is a sentence with no free variables,
  we can simply write M ? ? (without the
  assignment a)

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Satisfaction relation for well formed formulas
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• ? atomic formula
• ? t1 t2 M ? (t1 t2) a iff Ia(t1)
  Ia(t2)
• ? 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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• ? well formed formula
• ? ? a M ? ? a a iff M ? a a
• ? a ? ß M ? a ? ß a iff M ? a a and M ?
  ß a
• ? a ? ß M ? a ? ß a iff M ? a a or M ?
  ß a
• ? a ? ß M ? a ? ß a iff M ? a a or M ? ß
  a
• ? ?xia M ? ?xia a iff M ? a s for all
  assignments
• s ltd1,, di,, dngt where s varies from a
  only
• for the i-th element (s is called an i-th
  variant of a)
• ? ?xia M ? ?xia a iff M ? a s for some
  assignment s ltd1,, di,, dngt 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 ? is true (w.r.t. an
  interpretation I) iff every assignment
• s ltd1,, dngt satisfies ?, i.e. M ? ? s for
  all s.
• NOTE under this definition, a formula ? might
  be neither true nor false w.r.t. an
  interpretation I (it depends on the assignment)
• If ? is true under I we say that I is a model
  for ?.
• Given a set of formulas G, M satisfies G iff M ?
  ? for all ? in G

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Satisfiability and Validity
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• We say that a formula ? is satisfiable iff there
  is a structure
• M ltD, Igt and an assignment a such that M ? ?
  a
• We say that a set of formulas G is satisfiable
  iff there is a structure M ltD, Igt and an
  assignment a such that
• M ? ? a for all ? in G
• We say that a formula ? is valid iff it is true
  for any structure and assignment, in symbols ? ?
• A set of formulas G is valid iff all formulas in
  G 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
• ?1 ?x (even(x) ? odd(succ(x)))
• ?2 even(zero)
• ?3 odd(zero)
• Is there any M and a such that M ? ?1 ? ?2 a?
• Ia(zero) I(zero) 0
• Ia(succ(x)) I(succ)(Ia(x)) S(n) 1 if n0,
  n1 otherwise
• I(even) 0, 2 I(odd) 1, 3
• Is there any M and a such that M ? ?1 ? ?3 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, ? a FO- formula,
  we say that
• ? ? ?
• (to be read ? entails ?)
• iff for all the interpretations M and
  assignments a,
• if M ? ? a then M ? ? a.

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Reasoning Services EVAL
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• Model Checking (EVAL)
• Is a FO-formula ? true under a structure M ltD,
  Igt and an assignment a?
• Check M ? ? a

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

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Reasoning Services SAT
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• Satisfiability (SAT)
• Given a FO-formula ?, is there any structure M
  ltD, Igt and an assignment a such that M ? ? a?
• Theorem (Church, 1936) SAT is undecidable
• However, it is decidable in finite domains.

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Reasoning Services VAL
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• Validity (VAL)
• Given a FO-formula ?, is ? true for all the
  interpretations M and assignments a, i.e. ? ??
• Theorem (Church, 1936) VAL is undecidable
• However, it is decidable in finite domains.

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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)

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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)
  )

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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.

G (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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