Definite Programs

1
Definite Programs
2
Definite Programs

• Definition A definite program clause is a
  clause of the form
• A ? B1 ? ? Bn
• which contains precisely one atom A in its
  consequent. A is called the head, and B1 ? ?
  Bn the body of the clause.
• Definition A definite program is a finite set
  of definite program clauses.
• Definition A definite goal is a clause of the
  form ? B1 ? ? Bn

3
Model Intersection Property

• Proposition Let P be a definite program and
  Mii?I a non-empty set of Herbrand models for P.
  Then ?i?IMi is an Herbrand model for P.If
  Mii?I contains all Herbrand models for p, then
• MP ?i?IMi is the least Herbrand model for P.
• Proof ?i?IMi is an Herbrand interpretation.
  Show, that it is a model.Each definite program
  has BP as model, hence I is not empty and one can
  show that MP is a model.
• Idea MP is the most natural model for P.

4
Emden Kowalski

• Theorem Let P be a definite program. ThenMP
  A?BP A is a logical consequence of P.
• Proof A is a logical consequence of P
  iffP?A is unsatisfiable iffP?A has no
  Herbrand model iffA is true wrt all Herbrand
  models of P iffA ? MP.

5
Herbrand-Interpretations
Herbrand-Base
Q(a) P(f(b))
Q(a) P(f(b))
I1
The true atoms of the Herbrand base correlate
with the corresponding interpretation.
!
I2
P(f(b))?
Q(a)?
P(f(b))?
Q(a)?
I3
6
Idea
BP

• (2BP, ?) is a lattice of all Herbrand
  interpretations of P with the bottom element ?
  and the top element BP
• The least upper bound (lub) of a set of
  interpretations is the union, the greatest lower
  bound is the intersection.

MP

• ?

7
Properties of Lattices

• Definition Let L be a lattice and TL?L a
  mapping. T is called monotonic, if x?y
  implicates, that T(x)?T(y).
• Definition Let L be a lattice and X?L, X is
  called directed,if each finite sub-set of X has
  an upper bound in X.
• Definition Let L be a lattice and TL?L a
  mapping. T is called continuous,if for each
  directed subset X T(lub(X))lub(T(X)).

8
Van Emden Kowalski The Semantics of Predicate
Logic as a Programming Language, J. ACM 23, 4,
1976, pp. 733-742.

• Definition Let P be a definite program. The
  mappingTP 2BP? 2BP is defined as follows Let I
  be an Herbrand interpretation. ThenTP(I)A?BP
  A ? A1 ? ? An is a ground instance of a clause
  in P and A1,,An ?I

9
Practice
Let P be a definite program. even(f(f(x)) ?
even(x). odd(f(x) ? even(x). even(0).
Let I0?. Then I1TP(I0)?
TP(I)A?BP A ? A1 ? ? An is a ground instance
of a clause in P and A1,,An ?I
10
TP links declarative and procedural semantics

• Let I0?.
• Then
• I1TP(I0)even(0),
• I2TP(I1)even(0), even(f(f(0)),
• I3TP(I2)even(0), even(f(f(0)),
  even(f(f(f(f(0)))),

• Example Let P be a definite program.
• even(f(f(x)) ? even(x).even(0).

TP(I)A?BP A ? A1 ? ? An is a ground instance
of a clause in P and A1,,An ?I
TP is monotonic.
11
Practice
Let P be a definite program. plus(x,f(y),f(z))
lt- plus(x,y,z)? plus(f(x),y,f(z)) lt- plus
(x,y,z)? plus(0,0,0)?

• Let I0?.
• Then
• I1TP(I0)?

TP(I)A?BP A ? A1 ? ? An is a ground instance
of a clause in P and A1,,An ?I
12
Practice
Let I0?. Then I1TP(I0)plus(0,0,0) I2TP(I1)
plus(0,0,0), plus(f(0),0,f(0),
plus(0,f(0),f(0)) I3TP(I2)plus(0,0,0),
plus(f(0),0,f(0), plus(0,f(0),f(0)),
plus(f(f(0)),0,f(f(0))), plus(0,f(f(0)),f(f(0))),
plus(f(0),f(0),f(f(0)), plus(f(0),f(0),f(f(0)))..
.
Let P be a definite program. plus(x,f(y),f(z))
lt- plus(x,y,z)? plus(f(x),y,f(z)) lt- plus
(x,y,z)? plus(0,0,0)?
TP(I)A?BP A ? A1 ? ? An is a ground instance
of a clause in P and A1,,An ?I
13
Continuity

• Proposition The mapping TP is continuous for
  each definite program P.
• Proof Let X be a directed subset of 2BP.Note
  first, that a I?X exists and that for all
  A1,,An A1,,An ?I iff A1,,An
  ?lub(X).We have to show, that
  TP(lub(X))lub(TP(X)) for each directed subset
  X.A?TP(lub(X)) iffA ? A1 ? ? An is a ground
  instance of a clause in P and A1,,An
  ?lub(X) iffA ? A1 ? ? An is a ground instance
  of a clause in P and A1,,An ?I, for some I?X
  iffA?TP(I) for some I?X iffA ? lub(TP(X))

14
Fixpoint-Model

• Proposition Let P be a definite program and I be
  an Herbrand interpretation of P. Then I is a
  model for P iff TP(I) ?I.
• ? ?
• Let I be an Herbrand interpretation, which is not
  a model of P and TP(I) ?I. Then there exist
  ground instances A, A1,,An?I and a clause A ?
  A1 ? ? An in P. Then A?TP(I) ?I. Refutation.?

15
Ordinal Numbers
1st infinite ordinal ?? 1,2,3,...

• finite ordinals
• 0 Ø
• 1 Ø
• 2 Ø,Ø
• 3 Ø, Ø,Ø,Ø
• ...
• if ??1 for some ?, ? is called a successor
  ordinal (e.g. finite ordinals gt 0), else ? is
  called a limit ordinal (e.g. 0, ?)?

16
Transfinite Induction

• Let P(?) be a property of ordinals. Assume that
  for all ordinals ?, if P(?) holds for all ?lt?,
  then P(?) holds. Then P(?) holds for all
  ordinals ?.

17
Ordinal Powers of T

• Definition
• let L be a complete lattice and TL?L be
  monotonic. Then we define
• T??????
• T?????T?T???????? if?? is a successor ordinal
• T?????lub?T?????????if?? is a limit ordinal
• T??????
• T?????T?T???????? if?? is a successor ordinal
• T?????lub?T?????????if?? is a limit ordinal
• Example

18
Fixpoint Characterisation of the Least Herbrand
Model

• Theorem Let P be a definite program. Then
  MPlfp(TP)TP??, whereT??T(T?(?-1)), is ? is a
  successor ordinal T??lubT?? ?lt?, if ? is a
  limit ordinal
• Proof MPglbI I is an Herbrand model for
  P glbI TP(I) ?I lfp(TP)
  (not shown here) TP??.

19
Answer

• Definition Let P be a definite program and G a
  definite goal. An answer for P?G is a
  substitution for variables of G.
• Definition Let P be a definite program, G a
  definite goal ? A1 ? ? An and ? an answer for
  P?G. We say that ? is a correct answer for
  P?G, if ?((A1 ? ? An)?) is a logical
  consequence of P.The answer no is correct if
  P?G is satisfiable.
• Example

20

• Theorem
• Let P be a definite program and G a definite goal
  
• ? A1 ? ? An. Suppose ? is an answer for P?G,
  such that (A1 ? ? An)? is ground. Then the
  following are equivalent
• ???? is correct
• 2. (A1 ? ? An)? is true wrt every Herbrand model
  of P
• 3. (A1 ? ? An)? is true wrt the least Herbrand
  model of P.

21

• Proofit suffices to show that 3 implies 1(A1 ?
  ? An)? is true wrt the least Herbrand model of P
  implies(A1 ? ? An)? is is true wrt all Herbrand
  models of P implies(A1 ? ? An)? is false wrt
  all Herbrand models of P implies
• P?(A1 ? ? An)? has no Herbrand
  modelsimplies P?(A1 ? ? An)? has no models.