Foundations of Artificial Intelligence

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Foundations of Artificial Intelligence

• Chapter 7 The predicate calculus
• Luc De Raedt

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Predicate calculus

• Limitations of propositional calculus
• Language and its syntax
• Semantics (informally)
• Quantifiers
• Representing Knowledge
• Resolution
• Prolog
• This part is based on
• Nils Nilsson, AI a new synthesis, Morgan
  Kaufman, 1998, chapters 15, 16 and 17

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The language and its syntax (restricted version)

• Components
• An infinite set of object constants strings
  starting with a capital or a numeral
• E.g.
• An infinite set of function constants of all
  arities strings starting with lowercase and
  superscripted with arity
• E.g.
• An infinite set of relation constants of all
  arities strings beginning with uppercase and
  superscripted with arity
• E.g.
• Traditional connectives

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Terms (restricted)

• An object constant is a term
• A function constant f of arity n followed by n
  terms ti in parentheses is a term
• Examples

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Well-formed formulae (restricted)

• Atoms A relation constant F of arity n followed
  by n terms ti is an atom
• Examples
• Propositional wffs any expression formed with
  the propositional calculus and atoms as
  propositions.
• Example

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Semantics (restricted version)

• Worlds
• The world can have an infinite number of objects,
  called individuals, the domain
• The world can have an infinite number of
  functions f that map n-tuples of individuals into
  individuals
• The world can have an infinite number of
  relations over individuals,

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Interpretations

• An interpretation maps
• object constants to individuals
• n-ary function constants onto n-ary functions
• n-ary relation constants onto n-ary relations
• Given an interpretation, an atom has
  the value True iff the relation denoted by R
  holds among for those individuals denoted by the
  ti

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Models and related notions

• An interpretation satisfies a wff if the wff has
  the value True under the interpretation
• An interpretation that satisfies a wff is a model
  of that wff
• Any wff that has the value True under all
  interpretations is a tautology (it is valid)
• Any wff that does not have a model is
  unsatisfiable
• If a wff ? has value True under all those
  interpretations for which each of the of wffs in
  a set ? has value true then ? logically entails
  ?, notation
• Two wffs are equivalent iff their truth values
  are identical under all interpretations
• Essentially, the notions from propositional logic
  carry over

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Knowledge

• Does the following knowledge base (set of
  formulae) has a model ?

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Quantification

• How to express
• every object is red ?
• there is an object which is red ?
• For finite domains for infinite domains ?
• Use variables and quantifiers !

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Predicate calculus

• Now introduce
• An infinite set of variable symbols start with
  lowercase
• Each variable is also a term !
• The universal quantifier? and the existential
  one ?
• If ? is a wff and v is a variable then the
  following are also wffs
• Example
• Scoping ? is within the scope of the
  quantifier

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Variables, scoping, and quantifiers

• Scoping
• We will assume that all wffs are closed, i.e.
  that all variables are quantified, i.e. occur
  with the scope of a quantifier.
• closed
• Open
• Equivalent
• Not equivalent !

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Semantics

• has the value true (under a given
  interpretation) iff ?(v) has the value true for
  all assignments of the variable symbol v to
  individuals
• has the value true (under a given
  interpretation) iff ?(v) has the value true for
  an assignments of the variable symbol to
  individuals

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An example
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Rules of inference

• Some equivalences (De Morgan)
• Universal instantiation
• From infer where a is
  substituted for x throughout ?

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• Existential Generalization
• From infer where a has
  been replaced by x throughout ?
• Universal instantiation and Existential
  generalization are sound inference rules

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Wumpus

• Time could be incorporated as well

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Some sentences

• Not all students take both history and biology
• Only one student failed history (requires )
• No person likes a smart vegetarian
• No person likes a professor unless the professor
  is smart

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Clausal Form

• Clauses are universally quantified disjunctions
  of literals all variables in a clause are
  universally quantified

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Towards Resolution

• Examples
• We need to be able to work with variables !
• Unification of two expressions/literals

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Terms and instances

• Consider following atoms
• Ground expressions do not contain any variables

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Substitution
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Composing substitutions

• Composing substitutions s1 and s2 gives s1 s2
  which is that substitution obtained by first
  applying s2 to the terms in s1and adding
  remaining term/vars pairs to s1
• Apply to

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Properties of substitutions
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Unification

• Unifying a set of expressions wi
• Find substitution s such that
• Example
• The most general unifier g of wi has the
  property that if is any unifier of wi then
  there exists a substitution s such that
  wiswigs
• The common instance produced is unique up to
  alphabetic variants (variable renaming)

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Disagreement set

• The disagreement set of a set of expressions wi
  is the set of subterms ti of wi at the
  first position in wi for which the wi
  disagree

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Unification algorithm