Computational Logic: today

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Computational Logic todays menu

• Practicalities
• Clause form of first-order logic syntax and
  semantics, clause instance, universe of
  discourse, interpretations, resolution,
  unification, consistency of a set of clauses.
  Syntactic (resolution-based) vs. semantic
  (reasoning on interpretations) proofs of
  (in)consistency.
• Logic Programs (based on Horn clause form)
  general properties, syntax, declarative
  semantics, procedural semantics.

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Practicalities

• Lab this Fri _at_ 330 in ASB 9840
• Sicstus runs on CSIL license only covers
  installation on SFU owned systems.
• However, students can run Sicstus from their home
  by ssh into any CSIL workstations via
  fraser.sfu.ca. i.e.
• 1) ssh to fraser.sfu.ca
• 2) ssh to any of the CSIL Linux workstations, as
  shown on
• http//www.cs.sfu.ca/CC/CSILPC/CSIL_Layout.pdf

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Some Definitions

• Instance of an expression a (usually more
  specific) form of that expression, obtained by
  applying a substitution to the expression(i.e.,
  by assigning terms to variables). E.g.
• Clause C ancestor(X,Z)- parent(X,Y),
• ancestor(Y,Z).
• Substitution ? Xmother(adam),Yadam
• Instance C?
• ancestor(mother(adam)),Z)-
• parent(mother(adam),adam), ancestor(adam,Z).

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Clause form Building blocks Syntax Semantics

• Predicate states that the relation
• p(t1,,tn), ngt0 called p holds among the
  individuals called ti
• Term either an denotes an individual
• atom (constant or ) in the universe
• variable
• structure f(t1,,tm)
• (or functional expression)

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Use of functional expressions

• loves(mother(X),X).
• ?- loves(Who,frankenstein).
• Who will UNIFY with mother(frankenstein), which
  never evaluates it simply stands, as is, for the
  individual mother of frankenstein
• MORALE Only one name is allowed for each
  individual (NO SYNONYMS!)

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General vs. Horn Clause Form Syntax Semantics

• General Clause Form
• h1,hn- b1,,bm. For all variables
• in the clause,
• (h1 oror hn) if (b1 andbm).
• Horn Clause Form (at most one conclusion)
• RULE For all variables in the clause, h is
  T if b1 andbm are all T
• h- b1,,bm. ,

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Horn Clauses two meanings Syntax Semantics

• Horn Clause Form (at most one conclusion)
• RULE Declarative meaning For all
• h- b1,,bm. variables in the clause,
• h is T if b1 andbm are all T
• Procedural meaning To solve an instance of
  h, solve b1 andbm affected by the same
  substitutions which gave the instance.

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Horn clauses special cases Syntax Semantics

• (D declarative interpretation, P
  procedural interpretation)
• FACT Condition-less clause (only a head- no
  body)
• h. D For all variables in the clause,
• h is unconditionally true
• P All instances of h are trivially solved
• QUERY Conclusion-less clause
• ?- b1,,bm. D For all variables in the clause,
• not(b1 andbm), or equivalently, For NO
  variables in the clause, b1 andbm
• P Find an instance of the body which solves
  all its conjoints, or say no if they cant be
  solved.

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Clause Resolution in Prolog

• CLAUSE 1 h- b1,,bm.
• CLAUSE2 ?- q1, q2 , qn.
• 1. Find a substitution that unifies h with q1
• 2. Apply that same substitution to ?-
  b1,,bm,q2, , qn.
• You obtain the resolvent of both clauses.

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Proof by refutation, or contradiction- the
intuition

• PROGRAM
• p(mary). (p(mary) is true)
• QUERY
• ?- p(X). (there is no value of X for which
  p(X) is true)
• Proof by contradiction Assume that there is no
  value of X for which p(X) is true. If the
  resulting set of clauses is inconsistent,
  conclude there is the value of X which makes the
  set inconsistent.

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Resolving our fact with our query

• 1. Find a substitution that unifies the head in
  one with a body item in the other
• Xmary
• 2. Generate the resolvent we get the empty
  clause (no conditions, no conclusion). And the
  empty clause is inherently contradictory. The
  contradiction came from assuming there is no
  value for X that satisfies p(X). Therefore our
  assumption is wrong, and the counterexample that
  proves it wrong is precisely Xmary. In other
  words, there exists an X (namely, Xmary)
  satisfying p(X).

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Morale

• Thats why procedurally, a query may be viewed as
  a request for Prolog to find values for the
  variables() (if such exist) that make the body
  of a query true (i.e., that make the query
  itself false- since the query reads not (b1,bn),
  or modulo notation, ?- b1,,bn).
• If we can apply resolution repeatedly on our
  query until we generate the empty clause, weve
  proved that the querys body does have values for
  its variables which makes it true those that led
  to contradiction, or the empty clause.
• () These values are obtained through composition
  of all substitutions used in the process

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Syntactic proof of inconsistency

• For the assignment, be sure to provide, as well
  as a semantic proof of inconsistency (i.e.,
  reasoning on possible interpretations), a
  syntactic one (i.e., through resolution) resolve
  the query with some clause in the program whose
  head matches the first item in the query, and
  repeat until the empty clause is generated.
  Generating the empty clause through resolution
  proves that the set of clauses is inconsistent.

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Logic Program description of knowledge in terms
of

• - Facts p.
• - Rules p- p1, p2, , pn.
• - Queries ?- q1, q2 , qm.
• ., - and ?- stand for if,
• , means and
• p, pi and qi are predicates-- of the form
  pred(t1,,tr).
• ti are terms-- either atoms (constants or
  numbers), variables, or functional expressions
  (of the form f(t1,,ts))

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Prolog based on Horn clauses

• Relational
• No explicit types or classes
• High expressiveness each program line has its
  own meaning
• Ideal for prototyping
• Recursiveness, automatic inference,
  non-determinism, unification (two-way matching)
• --------- features not present in functional
  programming

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Definition Herbrand Universe

• Universe of Discourse, or Herbrand Universe
  (given a set of clauses) the set of all
  variable-free terms that can be built from the
  predicates and function symbols occurring in the
  set of clauses. Example 1- given the set of
  clauses S1
• ancestor(X,Z)- parent(X,Y),
  ancestor(Y,Z).
• ancestor(X,Z)- parent(X,Z).
• parent(adam,abel).
• parent(adam,cain).
• Whats the universe of discourse for S1?
• H(S1) adam,abel,cain

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Example 2- an infinite Herbrand Universe

• S2
• number(0).
• number(s(X))- number(X).
• H(S2) 0, s(0), s(s(0)),
• N.B.
• If there are no constants in your set of clauses
  (this will in practice rarely happen) you need to
  add an arbitrary one into its Herbrand universe
• As soon as you use any functional expression as
  an argument of a predicate, the universe becomes
  infinite.

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Definitions

• Interpretation any assignment
• To each term, of an individual in the universe
• to each n-ary predicate symbol in the set of
  clauses, of an n-ary relation over the universe.
• To specify an interpretation specify its effect
  on the truth or falsity of variable-free
  predicates.

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Example recall S2

• number(0).
• number(s(X))- number(X).
• H(S2) 0,s(0),s(s(0)), (Herbrand Universe)
• Predicates I1 I2 I3 I4
• number(0) T F T T T
• number(s(0)) T T F T T
• number(s(s(0))) T T T F T
• number(s(s(s(0)))) T T T T F ..
  T ..

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Definitions

• A variable-free (ground) clause is true in an
  interpretation I ? a) whenever all its conditions
  are T in I, at least one of its conclusions is T
  in I, or equivalently,
• ? b) at least one of its conditions is
  false in I or at least one of its conclusions is
  T in I.
• Example mortal(socrates)- human(socrates)
  .
• is T in any interpretation in which
  mortal(socrates) is T, and also in any
  interpretation in which human(socrates) is F.

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Definitions

• A set of clauses S is inconsistent ? it is not
  consistent.
• A set of clauses S is consistent ? all its
  clauses are T in some interpretation of S.
• A clause is T in an interpretation I of a set of
  clauses S ? every variable-free instance of the
  clause obtained by replacing variables by terms
  from H(S) is true in I. Otherwise the clause is F
  in I.

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Semantic proof of consistency

• Given a set of clauses, e.g.
• S3 (1) human(socrates).
• (2) snores(X)- human(X).
• Is there an interpretation in which all its
  clauses are true? (If there is at least one, the
  set is consistent if there are none, it is
  inconsistent)

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Answer

• YES, there is the one that assigns the value T
  both to human(socrates) and to snores(socrates).
  This interpretation satisfies clause 1) (which is
  already ground) and also the only possible
  ground instance of 2), namely
• snores(socrates)- human(socrates).
• Therefore, in this interpretation, all ground
  instances of the set of clauses are T, and
  therefore, all its clauses are T. S3 is
  consistent.

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The same semantic proof, in step-by-step detail,
for finite universes

• 1. Determine the Herbrand universe of your set of
  clauses. In our example, H(S3)socrates
• 2. List all ground instances (i.e., instances
  with NO variables) of the predicates. We have
  only two predicates (human/1, snores/1), and H
  has only one element , so all ground instances of
  the predicates are, in our case human(socrates),
  snores(socrates).
• 3. List all posible interpretations of these
  ground instances.

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Result of step 3 for our example

• All possible interpretations (I1, I2, I3, I4)
• human(socrates) snores(socrates)
• I1 T T
• I2 T F
• I3 F T
• I4 F F

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Intuitively

• A set of clauses represents our knowledge about
  some world- usually what we intend to represent
  is the real world, or some subset thereof.
• The possible interpretations of that knowledge
  are the possible worlds in which we can reason
  about that knowledge.
• In our example, we have four possible worlds in
  one of these worlds, Socrates is both human and
  snores, in another one he is human but does not
  snore, etc.

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Steps of the semantic proof (cont.)

• 4. List all ground instances of each clause.
• In our example, we have
• human(socrates).
• snores(socrates)- human(socrates).
• 5. Add these to your table, and calculate their
  truth values in each of the interpretations
  listed

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Result for step 5 in our example

• ( We now note humanhu, socratess, snoressn)
• INTERPRETs. TRUTH VALUE OF CLAUSE
• INSTANCES IN EACH INTER.
• hu(s) sn(s) hu(s). sn(s)- hu(s).
• I1 T T T T
• I2 T F T F
• I3 F T F T
• I4 F F F T

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Detailed steps of the semantic proof (cont.)

• 6. Interpret these results in the light of the
  definitions given
• S3 is consistent, since there is an
  interpretation (namely, I1) satisfying all
  variable-free instances of its clauses-- i.e.,
  satisfying all its clauses
• N.B. For proving consistency, we just need to
  provide one interpretation that makes all clauses
  true. For proving inconsistency, we have to show
  that NO interpretation does. Either we list them
  all (if small enough) or we reason about truth
  values , as we exemplify next.

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If the universe is infinite or too big

• A proof by enumeration becomes impossible for
  clauses with infinite or huge Herbrand universes.
  In this case, just show one interpretation that
  satisfies all clauses (N.B. this is also always
  an option for clauses with small universes- we
  just went through detailed step-by-step for
  studying purposes).
• E.g. to prove that S2 is consistent (transparency
  19), we show that in interpretation I1, in which
  all elements of H(S2) are T, both clauses of S2
  are T. Therefore there is an interpretation (I1)
  in which all clauses of S2 are T.

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Semantic proof of inconsistency

• S4 (1) human(bob).
• (2) snores(X)- human(X).
• (3) ?-snores(bob).
• In any interpretation satisfying (1),
• human(bob) must be T (4).
• In any interpretation satisfying (1) and (2), all
  instances of (2) must be T in particular the
  instance with Xbob
• snores(bob)- human(bob) must be T (5). And
• snores(bob) must be T (6), because of (4) and
  (5).
• In any interpretation satisfying (3),
• snores(bob) must be F (7).

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Semantic proof of inconsistency (cont.)

• If an interpretation existed satisfying (1), (2)
  and (3) simultaneously, it would have to satisfy
  both (6) and (7). But this is impossible
  snores(bob) cannot be both T and F.
• Therefore, no interpretation exists which can
  simultaneously satisfy all three clauses in S4.
  This set is inconsistent.

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More detail about matching (unifying) two terms

• If both are constants or numbers they match ltgt
  they are the same constant or the same number
• If one is a variable they always match, by
  instantiating the variable to the other term.
• If they are complex terms, they match ltgt they
  have the same functor and all their corresponding
  arguments match
• Two terms match ltgt it follows from the above
  that they do.

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Remember to study one chapter per night of the
online notes

• http//www.learnprolognow.org/
• (Learn Prolog now! By Blackburn et al.)
• (you should have done four chapters by now)
• Do all exercises as you go along, and run them.

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Todays question

• 1) Show an instance of the clause
• grandparent(X,Y)- parent(X,Z), parent(Z,Y).
• which can be used to resolve this clause with the
  query
• ?- grandparent(mary,W).
• 2) Show the substitution used to obtain your
  instance
• 3) Show the resolvent of both clauses.