First-Order Logic

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First-Order Logic
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Conceptualization

• The formalization of knowledge begins with a
  conceptualization.
• This includes the objects presumed or
  hypothesized to exist in the world and their
  relationships.
• An object can be anything about we want to say
  something.
• Objects can be concrete (book, earth, sun, etc)
• Objects can be abstract (the number 2, the set of
  all integers, the justice etc)
• Objects can be primitive or composite (a circuit
  that contains many circuits)
• Objects can be fictional (Sherlock Holmes, Miss
  Right)
• A knowledge-representation task doesnt require
  that we consider all the objects in the world
  only those objects in a particular set are
  relevant.
• The set of objects about which knowledge is being
  expressed is called a universe of discourse.

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Example Blocks World
A
B
D
C
E

• Here we conceptualize the blocks. Some could
  conceptualize also the table, but we ignore it.
• The universe of discourse corresponding to this
  conceptualization is
• A,B,C,D,E
• Although in this example the universe of
  discourse is finite, this is not always the case.
  
• E.g. it is common in math to consider as the
  universe of discourse the set of all integers,
  the set of all real numbers etc.

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More about Conceptualization I

• A function is a kind of interrelationship among
  the objects in a universe of discourse.
• Although we can define many functions for a given
  set of objects, in conceptualizing a portion of
  the world we usually emphasize some functions and
  ignore others.
• The set of functions emphasized in a
  conceptualization is called the functional basis
  set.
• E.g. in our blocks world, it would make sense to
  define a (partial) function Hat that maps a block
  into the block on top of it, if such exists. The
  tuples corresponding to this function are
• ltB,Agt, ltC,Bgt, ltE,Dgt

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More about Conceptualization II

• A relation is the second kind of
  interrelationship among the objects in a universe
  of dicourse.
• The set of relations emphasized in a
  conceptualization is called the relational basis
  set.
• E.g. in our blocks world, we can consider the On
  relation
• ltA,Bgt, ltB,Cgt, ltD,Egt
• Also, we can consider the Above relation
• ltA,Bgt, ltB,Cgt, ltA,Cgt, ltD,Egt
• More, we can consider the unary Clear relation
• ltAgt, ltDgt
• Or, the unary Table relation
• ltCgt, ltEgt

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Formally

• A conceptualization is a triple consisting of
• a universe of discourse,
• a functional basis set for that universe of
  discourse and
• a relational basis set.
• E.g. ltA,B,C,D,E, Hat, On,Above,Clear,Tablegt
• No matter how we choose to conceptualize the
  world, there are other conceptualizations as
  well.
• What makes a conceptualization more appropriate
  than another for knowledge formalization?

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Reification

• E.g. consider a Blocks World conceptualization in
  which there are five blocks, no functions, and
  there are three unary relations
• ltA,B,C,D,E, , Red,White,Bluegt
• This conceptualization allows us to consider the
  color of blocks but not the properties of colors.
  
• In order to overcome this, we reify the relations
  and have instead the conceptualization
• ltA,B,C,D,E, Red,White,Blue, Color, Nicegt
• The advantage of this is that it allows us to
  consider properties of of properties.

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First-Order Logic

• Given a conceptualization of the world, we begin
  to formalize knowledge in a language appropriate
  to that conceptualization.
• E.g. we can express the fact that the block A is
  above block B by taking a relation symbol such as
  above and object symbols a and b and writing
• above(a,b)
• We can combine sentences by using logical
  operators (?,?,?,?,?,?) in order to create more
  complex sentences.
• above(a,b) ? above(b,a)
• The names we use, such as above, a, b are
  irrelevant.
• We could have said
• abnwyrcn(pjkj, oiuy) ? abnwyrcn(oiuy, pjkj)
• What really matters is that they should should
  have identifiable referents in the real world,
  (better saying in the conceptualization that we
  choose)

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Syntax
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Constants

• Constants refer to objects, functions and
  relationships.
• joe, mary, loves, happy,
• Simple sentences express relationships among
  objects.
• loves(joe,mary)
• They are called atoms.
• Compound sentences capture relationships among
  relations.
• loves(x,y) Þ loves(y,x)
• loves(x,y) Ù loves(y,x) Þ happy(x)
• Relations can be unary as well.
• tall(amy)

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Lets blur the distinction

• Constants refer to objects, functions and
  relationships.
• joe, mary, loves, happy, hat
• Ok, in order to be formal we should use object
  constant, function constant, and relation
  constant, but this is painful.
• So, lets say for
• Object constant object
• Function constant function
• Relation constant relation
• It should be clear that the ones on the left are
  different from ones on the right We are doing
  this for the ease of presentation.

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Example with Functions
E.g. Quincy loves his dog. loves(quincy, dog_of
(quincy)) Note We are allowed to relate
sentences only. So, we can say loves(quincy,
dog_of (quincy)) Ù loves(quincy, cat_of
(quincy)) But not, loves(quincy, dog_of
(quincy) Ù cat_of (quincy)) because dog_of
(quincy), cat_of (quincy) arent sentences, they
are objects.

• E.g. How about saying that Jimmy Durante has a
  big nose?
• durante is an object and
• nose_of (durante)
• is a function that constructs an object from the
  argument object.
• Then, we can write
• big(nose_of (durante))

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Full-blown First-Order Logic ?,?

• The language that we have described so far,
  consisting of atoms and the connectives
  (?,?,?,?,?,?) is typically called predicate
  logic.
• To extend it to first-order logic, we need to add
  quantifiers.
• The purpose of quantifiers is to allow us to say
  things about sets of objects.
• To say that Quincy loves everything we write
• ?x. loves (quincy, x)
• We can think of ? as a big conjunction. For
  example, if there are only three objects quincy,
  dog, and cat, what the above asserts is
• loves (quincy, dog) ? loves (quincy, cat) ?
  loves (quincy, quincy)
• To say that Quincy loves something we write
• ?x. loves (quincy, x)
• We can think of ? as a big disjunction. For
  example, if there are only three objects as
  above, then what we are asserting is
• loves (quincy, dog) ? loves (quincy, cat) ?
  loves (quincy, quincy)

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Mushrooms

• The universe of discourse is the set of all
  plants.
• There is a unary relation for being mushroom,
  another one for being purple, and a third for
  being poisonous.
• We designate these relations with the unary
  relation symbols mushroom, purple, and poisonous.
  
• All purple mushrooms are poisonous.
• ?x (purple(x) ? mushroom(x) ? poisonous(x))
• A mushroom is poisonous only if it is purple.
• ?x (mushroom(x) ? poisonous(x) ? purple(x))
• No purple mushroom is poisonous.
• ?x ?(purple(x) ? mushroom(x) ? poisonous(x))
• There is exactly one mushroom.
• ?x mushroom(x) ? (?y y?x ? ?mushroom(y))

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Reverse translation

• Translate the following into English.
• ?x hesitates(x) ? lost(x)
• He who hesitates is lost.
• ??x business(x) ? like(x,Showbusiness)
• There is no business like show business.
• ??x glitters(x) ? gold(x)
• Not everything that glitters is gold.
• ?x ?t person(x) ? time(t) ? canfool(x,t)
• You can fool some of the people all the time.

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Semantics
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Models

• This is all well and good, but there isnt much
  value to saying what sentences belong to a
  language unless we also say what those sentences
  mean.
• Simply said, we are interested in the truth value
  of given (possibly compound) sentences.
• Suppose that our first-order language had no
  logical connectives at all, just objects,
  functions, and relations.
• In that case, the sentences in our language would
  be simply what we called atoms.
• Let, A be the set of all atoms in our
  impoverished language.
• A model M is a subset of the set of atoms A.
• By a model M ? A, we will mean the state of
  affairs in which all of the atoms in M are true
  and all of the atoms not in M are false.
• Given a model it is obviously straightforward to
  decide whether or not some particular atom a is
  true in M its true if a?M and false if a?M.
• What makes first-order logic powerful, is that we
  can also determine whether compound sentences are
  true or false in M.

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Example

• Suppose that Tom is our only object, and that we
  have two relations, rich and poor.
• Now,
• A rich(tom), poor(tom)
• is the set of all atoms. So, we have 4 possible
  models.
• In M1?, Tom is neither rich nor poor.
• In M2rich(tom), Tom is rich and not poor.
• In M3poor(tom), tom is poor and not rich.
• In M4rich(tom), poor(tom), Tom is both rich
  and poor.
• Now, consider the sentence
• poor(tom) ? ?rich(tom)
• Using the truth table for the implication, the
  above sentence will be true in M1, M2, M3 and
  false only in M4. (try it)

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Entailment

• Definition.
• Given two sentences ? and ?, we will say that ?
  entails ?, writing ? ?, if ? holds in every
  model that ? holds.
• Example Consider
• ?? (???) ?
• For any sentences ? and ?, any model M in which
  ?? (???) holds is model in which ? holds and also
  a model in which ??? holds.
• But since ? holds in M, it must be the case that
  ? holds in M as well.
• By the definition of entailment, the claim
  follows.

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Entailment II

• A set of premises logically entails a conclusion
  if and only if every interpretation that
  satisfies the premises also satisfies conclusion.
• In the case of Propositional Logic, the number of
  interpretations is finite, and so it is possible
  to check logical entailment directly in finite
  time.
• In the case of Relational Logic, the number of
  models is infinite, and so a direct check of
  logical entailment is not feasible.

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Example

• p(a)
• q(a)
• p(a),q(a)
• p(b)
• q(b)
• p(b),q(b)
• p(a), p(b)
• p(a), p(b), q(a)
• p(a), p(b), q(b)
• p(a), p(b), q(a), q(b)
• ...
• Infinitely many models.

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Good News

• Given any set of sentences, there is a specially
  defined subset of interpretations called Herbrand
  interpretations.
• Under certain conditions, checking just the
  Herbrand models suffices to determine logical
  entailment.
• Checking just the Herbrand interpretations is
  less work than checking all interpretations.

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Herbrand

• The Herbrand universe for a set of sentences is
  the set of all ground terms that can be formed
  from just the constants used in those sentences.
• Ground terms are either object constants or
  functions applied to object constants.
• The Herbrand base for a set of sentences is the
  set of all ground atomic sentences that can be
  formed using just the elements in the Herbrand
  universe.
• A Herbrand model for a set of sentences is any
  subset of the Herbrand base for those sentences.

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Example

• Sentences
• "x. (r(a,x) Þ r(x,b))
• "x."y."z. (r(x,y) Ù r(x,y) Þ r(x,z))
• Herbrand Universe (constants used in sentences
  only)
• a,b
• Herbrand Base
• r(a,a), r(a,b), r(b,a), r(b,b)

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Herbrand Models

• r(a,a)
• r(a,b)
• r(b,a)
• r(b,b)
• r(a,a), r(a,b)
• r(a,a), r(b,a)
• r(a,a), r(b,b)
• r(a,b), r(b,a)
• r(a,b), r(b,b)
• r(b,a), r(b,b)
• r(a,a), r(a,b), r(b,a)
• r(a,a), r(a,b), r(b,b)
• r(a,a), r(b,a), r(b,b)
• r(a,b), r(b,a), r(b,b)
• r(a,a), r(a,b), r(b,a), r(b,b)
• 16 Herbrand interpretations in all. Note 16lt.

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Herbrand Theorem

• Herbrand Theorem A set of (pure) ?-free
  sentences is satisfiable if and only if it has a
  Herbrand model that satisfies it.
• Proof.
• If a set of ?-free sentences is satisfiable, then
  there is a model that satisfies it.
• Take the intersection of this model with the
  Herbrand base. This is a Herbrand model.
• Moreover, it is easy to see that it satisfies the
  sentences.
• If the sentences contain variables, the instances
  must all be true, including those in which the
  variables are replaced only by elements in the
  Herbrand universe.
• Note. With ?-free we mean pure ?-free. E.g.
  ??p(x), is nothing else but ??p(x). So, pure
  ?-free no negations preceding ?.

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Example

• Premises
• ?x. p(x) Þ q(x)
• p(a) Ú q(a)
• Conclusion
• q(a)

• Herbrand Models
• p(a)
• q(a)
• p(a),q(a)

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Herbrand Method

• Add negation of conclusion to the premises to
  form the satisfaction set.
• Loop over Herbrand interpretations. Cross out
  each interpretation that does not satisfy the
  sentences in the satisfaction set.
• If all Herbrand interpretations are crossed out,
  by the Herbrand Theorem, the set is
  unsatisfiable.
• Sound and Complete Negating the conclusion leads
  to a contradiction therefore, the premises
  logically entail the conclusion.
• Termination Since there are only finitely many
  Herbrand interpretations, the process halts.

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Example

• Premises
• ?x. p(x) Þ q(x)
• p(a) Ú q(a)
• Conclusion
• q(a)
• Satisfaction Set
• ?x. p(x) Þ q(x)
• p(a) Ú q(a)
• Øq(a)

• Herbrand Models
• p(a)
• q(a)
• p(a),q(a)
• By considering ? as a big conjunction we can
  verify that the satisfaction set is unsatisfiable
  in each Herbrand model.

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Herbrand for the rest of FOL

• In the presence of function constants, all ground
  functional terms are included.
• E.g. if we find in our sentences a,b,f,g then the
  Herbrand universe is
• a, b, f(a), f(b), g(a), g(b), f(f(a)), f(f(b)),
  f(g(a)), f(g(b)), ...
• Sad The size of the Herbrand universe for a
  functional language is infinite.
• Upshot Checking the Herbrand models for a
  language to determine logical entailment is not
  feasible in finite time.
• Solution Use formal proofs!