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, the mgu, g of wi has
  the property that if s 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
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Predicate calculus Resolution

• John Allan Robinson (1965)

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Example
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A stronger version of resolution

• Use more than one literal per clause

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Resolution
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Resolution

• Properties
• Resolution is sound
• Incomplete
• But fortunately it is refutation complete
• If KB is unsatisfiable then KB - ?

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Refutation Completeness

• To decide whether a formula KB w do
• Convert KB to clausal form KB
• Convert ?w to clausal form ?w
• Combine ?w and KB to give ?
• Iteratively apply resolution to ? and add the
  results back to ? until either no more resolvents
  can be added, or until the empty clause is
  produced.

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Converting to clausal form

• To convert a formula KB into clausal form
• Eliminate implication signs
• Reduce scope of negation signs
• Standardize variables
• Eliminate existential quantifiers using
  skolemization
• Same as in prop. logic

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Skolemization

• General rule is that each occurrence of an
  existentially quantified variable is replaced by
  a skolem function whose arguments are those
  universally quantified variables whose scopes
  includes the scope of the existentially
  quantified one
• Skolem functions do not yet occur elsewhere !
• Resulting formula is not logically equivalent !

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Skolemization examples

• A well formed formula and its skolem form are not
  logically equivalent. However, a set of formulae
  is (un)satisfiable if and only if its skolem
  form is (un)satisfiable.

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Converting to clausal form

• 5. Convert to prenex form
• Move all universal quantifiers to the front
• 6. Put the matrix in conjunctive normal form
• Use distribution rule
• 7. Eliminate universal quantifiers
• 8. Eliminate conjunction symbol
• 9. Rename variables so that no variable in more
  than one clause.

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Using resolution to prove theorems
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Resolution Search Strategies

• Ordering strategies
• In what order to perform resolution ?
• Breadth-first, depth-first, iterative deepening ?
• Unit-preference strategy
• Prefer those resolution steps in which at least
  one clause is a unit clause (containing a single
  literal)
• Refinement strategies
• Linear Input at least one of the clauses being
  resolved is a member of the original set of
  clauses

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• Refinement strategies (ctd.)
• Ancestor c2 is a descendant of c1 iff c2 is a
  resolvent of c1 (and another clause) or if c2 is
  a resolvent of a descendant of c1 (and another
  clause) c1 is an ancestor of c2
• Set of support the set of clauses coming from
  the negation of the theorem (to be proven) and
  their descendants
• Set of support strategy require that at least
  one of the clauses in each resolution step
  belongs to the set of support
• Ancestry filtering require that at least one of
  the clauses in each resolution step belongs to
  original set of clauses or is an ancestor of the
  other clause being resolved.

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Answer extraction

• Suppose we wish to prove whether KB (?w)f(w)
• We are probably interested in knowing the w for
  which f(w) holds.
• Add Ans(w) literal to each clause coming from the
  negation of the theorem to be proven stop
  resolution process when there is a clause
  containing only Ans literal

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Horn clause logic and Prolog

• SAT for propositional clausal theories is
  NP-complete, but for Horn-clauses it is
  polynomial
• Inference with first order Horn clauses is also
  more efficient/easier than with full clauses
• Horn clauses are the basis of Computational
  Logic, Logic Programming and the Prolog
  programming language
• Horn clauses correspond to rules in a knowledge
  base
• Very effective model for knowledge representation
• We just the basics, so called pure Prolog

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Horn logic/Prolog
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Proof tree

• - Moves
• Moves - Bat_ok, Liftable
• - Bat_ok, Liftable
• Bat_ok -
• - Liftable
• Liftable -
• -

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SLD-derivation

• - Moves
• - Bat_ok, Liftable
• - Liftable
• -

• Contains the sequence of goals that arise during
  the computation

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p.4
These slides are adapted from Peter Flachs
book Simply Logical, John Wiley, 1994.
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London Underground in Prolog (1)
p.3

• Connected(Bond_street,Oxfd_circus,Central)
  -Connected(Oxfd_circus,Tottenham_court_road,Cent
  ral)-Connected(Bond_street,Green_park,Jubilee)-
  Connected(Green_park,Vharing_cross,Jubilee)-Conn
  ected(Green_park,Piccadilly_circus,Piccadilly)-C
  onnected(Piccadilly_circus,Leicester_square,Piccad
  illy)-Connected(Green_park,Oxfd_circus,Victoria)
  -Connected(Oxfd_circus,Piccadilly_circus,Bakerlo
  o)-Connected(Piccadilly_circus,Charing_cross,Bak
  erloo)-Connected(Tottenham_court_road,Leicester_
  square,Northern)-Connected(Leicester_square,Char
  ing_cross,Northern)-

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London Underground in Prolog (2)
p.3-4

• Two stations are nearby if they are on the same
  line with at most one other station in between
• Nearby(Bond_street,Oxfd_circus).Nearby(Oxfd_circu
  s,Tottenham_court_road).Nearby(Bond_street,Totten
  ham_court_road).Nearby(Bond_street,Green_park).N
  earby(Green_park,Charing_cross).Nearby(Bond_stree
  t,Charing_cross).Nearby(Green_park,Piccadilly_cir
  cus).
• or better
• Nearby(x,y)-Connected(x,y,l).Nearby(x,y)-Connec
  ted(x,z,l),Connected(z,y,l).

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p.5

• Compare
• Nearby(x,y)-Connected(x,y,l).Nearby(x,y)-Conne
  cted(x,z,l),Connected(z,y,l).
• with
• Not_too_far(x,y)-Connected(x,y,l).Not_too_far(x
  ,y)- Connected(x,z,l1),Connected(z,y,l2).

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Proof tree
Fig.1.2, p.7
-Nearby(Tottenham_court_road,w)
Nearby(x1,y1)-Connected(x1,u1,l1)
Connected(Tottenham_court_road,Leicester_square,N
orthern)-
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p.7
-Nearby(w,Charing_cross)
Nearby(x1,y1)-Connected(x1,z1,l1),Connected(z1,y
1,l1)
Connected(Bond_street,Green_park,Jubilee)
Connected(Green_park,Charing_cross,Jubilee)
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Recursion (1)
p.8

• A station is Reachable from another if they are
  on the same line, or with one, two, changes
• Reachable(x,y)-Connected(x,y,l).Reachable(x,y)-
  Connected(x,z,l1),Connected(z,y,l2).Reachable(x,
  y)- Connected(x,z1,l1),Connected(z1,z2,l2),
  Connected(z2,y,l3).
• or better
• Reachable(x,y)-Connected(x,y,l).Reachable(x,y)-
  Connected(x,z,l),Reachable(z,y).

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Fig. 1.3, p.9
-Reachable(Bond_street,w)
Reachable(x1,y1)-Connected(x1,z1,l1),
Feachable(z1,y1)
Connected(Bond_street, Oxfd_circus,Central)
Reachable(x2,y2)-Connected(x2,z2,l2),
Reachable(z2,y2)
Connected(Oxfd_circus, Tottenham_court_road, Centr
al)
Reachable(x3,y3)-Connected(x3,y3,l3)
Connected(Tottenham_court_road, Leicester_square,
Northern)
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p.12
Reachable(x,y,Noroute)-Connected(x,y,l). Reachabl
e(x,y,route(z,r))-Connected(x,z,l),
Reachable(z,y,r). -Reachable(Oxfd_
circus,Charing_cross,r).r route(Tottenham_court
_road,route(Leicester_square,Noroute))r
route(piccadilly_circus,Noroute)r
route(Picadilly_circus,route(Leicester_square,Noro
ute))
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Proof tree

• -Above(A,C)
• Above(x,y)-On(x,z),Above(z,y)
• -On(A,z),Above(z,C)
• On(A,B)-
• -Above(B,C)
• Above(x1,y1)-On(x1,y1).
• -On(B,C)
• On(B,C)-
• -

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SLD-derivation

• -Above(A,C)
• -On(A,z),Above(z,C)
• -Above(B,C)
• -On(B,C)
• -

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SLD-tree

• -Above(A,C)
• / \
• -On(A,C) -On(A,z),Above(z,C)
• -Above(B,C)
• / \
• -On(B,C) -On(B,z),Above(z,C)
• - -Above(C,C)
• / \
• -On(C,C) -On(C,z),Above(z,C)

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SLD-derivation

• SLD
• Linear resolution with Selection rule for
  Definite clauses
• SLD-derivation
• Summarizes one proof path
• SLD-tree
• Summarizes all possible proofs
• Substitutions
• May be written explicitly

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SLD-tree

• Summarizes all possible proofs of the goal
• Prolog selects leftmost literal in goal to
  resolve upon !
• Clauses to resolve the literal with are chosen
  according to clause order !
• Prolog traverses the tree depth-first in order
• Prolog is incomplete (can get trapped into
  infinite paths in the tree)
• Prolog is memory efficient (depth-first)
• Alternatively, iterative deepening or breadth
  first (which would be complete)
• Do not forget to standardize the variables apart
  in each resolution step !

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p.44-5
Student_of(x,t)-Follows(x,c),Teaches(t,c).Follow
s(Paul,Computer_science).Follows(Paul,Expert_syst
ems).Follows(Maria,Ai_techniques).Teaches(Adrian
,Expert_systems).Teaches(Peter,Ai_techniques).Te
aches(Peter,Computer_science).
-Student_of(s,Peter)
-Follows(s,c),Teaches(Peter,c)
-Teaches(Peter,Ai_techniques)
-Teaches(Peter,Computer_science)
-Teaches(Peter,Expert_systems)
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List processing examples
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• - M(B,c(A,c(B,nil)))
• - M(B,c(B,nil))
• -
• Is B a member of A,B ?

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• - M(w,c(A,c(B,nil)))
• \
• - - M(w,c(B,nil)
• wA / \
• - -M(w,nil)
• wB
• Generate members w of A,B
• Backtracking returns all solutions !
• Prolog returns
• Yes wA Yes wB Fail.

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• - M(B,w)
• / \
• - -M(B,w2)
• wc(B,w1) \
• - -M(B,w4)
• wc(u1,c(B,u2))
• infinite tree
• Generate lists w that B is a member of ?