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Three-Valued Models of Program Completion
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Three-Valued Interpretation
To any partial interpretation I (in 2-valued logic),
there corresponds the obvious 3-valued
interpretation, in which atoms missing from I are
assigned the truth value .
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Example: Problems with Completion in 2-Valued Logic
p  p, q.
p  p, q.
p  p.
Completion:
p(pq)
q.
Completion:
p(pq)p.
q.
Program has no
2-valued model.
Program has
2-valued model
I={q,p}
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Truth Tables for 3-Valued Logic (Extensions Only!)
A
A
A
B
AB
A
B
AB
A
B
AB


0

0
0


0


1


1


1

1





1

0
1

1



1
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A
B
AB
0


1

1
Attention::
AB  BA



no longer holds!
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Example: Completion and 3-Valued Logic
p  p, q.
p  p, q.
p  p.
Completion:
p(pq)
q.
Completion:
p(pq)p.
q.
Program has 3-valued
model for
p= , I={q}
Program has 3-valued
model for
p=, I={q} I={q,p}
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Uniqueness
p  p, q.
qr
qs
rr
ss
2-Valued Models
{p,q,r,s}
Not comparable
{p,q,r,s}
{p,q,r,s}
3-Valued Model
{}
Completion:
p(pq)
qrs
rr
ss
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Undefined
Definition
Literal q is called undefined in I, denoted by , if
neither q nor its complement is in I.
A conjunction of literals evaluates to undefined in
I if no literal in the conjunction is false in I and at
least one is undefined in I. (cmp. truth table).
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Stratified Programs
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Stratification (rep.)
Definition (rep.)
A normal program P is stratified, if all its
predicate symbols have a level such that:
– no predicate symbol positively depends on a
predicate symbol of a higher level
– no predicate symbol negatively depends on a
predicate symbol of greater or equal level.
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Local Stratification
Definition
A normal program is stratified if each atom in BP
can be assigned a countable ordinal level such
that no atom
– positively depends of an atom of greater level
– negatively depends of an atom of equal or
greater level.
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Example for Local Stratification
even(s(X))  even(X).
even(0).
BP:
{even(0)0, even(s(0))1, even(s(s(0)))2,
even(…)3, …}
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Perfect Model
Definition
Let P be a normal program and I a model. I is a perfect
model for a given level of BP, if
for every other model J,
if a positive literal p is the atom of least level in one model,
but not in the other, then p is in J.
In other words, atoms of higher level are preferred for the
perfect model.
Przymusinski: All locally stratified programs have a perfect
model, which is independent of the ranking system chosen.
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Examples for Local Stratification
even(s(X))  even(X).
even(0).
even(0)  q(X).
BP:
J={q(0)0,even(0)1, even(s(s(0)))3, …}
I={even(0)0, even(s(s(0)))2, …}
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Well-Founded Semantics
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Theorem:
If a program P is locally stratified, then is has a
well-founded model, which is identical to the
perfect model.
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Motivation
Problems:
• Not all completed programs are consistent
• SLDNF is only complete if the refutation does not
flounder.
• Stratification limits recursion.
Technical Solution: 3-Valued Logic
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Redefinition of „Interpretation“
Definition:
Let P be a normal program. A partial
interpretation I is a consistent set of literals
whose atoms are in BP.
A total interpretation I is a partial interpretation
containing every atom in BP or its negation.
A conjunction of ground literals is true in I, if all
its literals are true in I. It is false, if any of its
literals is false in I.
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Satisfied / Falsified / Weakly Falsified
Definition:
An ground clause is satisfied in a partial or
complete interpretation I if the head is true in I or
some subgoal is false in I.
The clause is falsified if the head is false and all
subgoal are true.
If the head is false in I, but no subgoal is false in I
then we say the clause is weakly falsified in I.
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Partial / Total Model
Definition:
A total model of a program P is a total
interpretation such that every
instantiated clause of P is satisfied.
A partial model of P is a partial
interpretation that can be extended to
a total model of P
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Lemma:
Let P be a normal program and I a partial
interpretation. If I weakly falsifies no clause from P,
then I is a partial model of P.
Proof:
Normal programs have BP as a model. For each
atom from BP, if its negation is not in I, add the
atom to I.
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Unfounded Sets
Definition
Let P be a normal program and I be a partial
interpretation.
We say A  BP is an unfounded set (of P) wrt I, if each
atom p  A satisfies the following:
For each instantiated clause R from P, whose head is p,
(at least) one of the following holds:
(1) Some (positive or negative) subgoal q of the body is
false in I.
(2) Some positive subgoal of the body
occurs in A.
A literal satisfying (1) or (2) is called a witness of
unusability for clause R (wrt I)
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Example
For the instantiated normal program P:
p(a)  p(c),  p(b).
p(b)   p(a).
p(e)   p(d).
p(c).
p(d)  q(a),  q(b).
(2):
p(d)  q(b),  q(c).
None of the heads can
be derived first.
q(a)  p(d).
q(b)  q(a).
the atoms {p(d), q(a), q(b), q(c)} are an
unfounded set wrt I=Ø.
There exists no definition for q(c). Thus (1) is satisfied.
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Example
For the instantiated normal program P:
p(a)  p(c),  p(b).
Neither (1) nor (2) are
satisfied.
p(b)   p(a).
p(e)   p(d).
p(c).
p(d)  q(a),  q(b).
p(d)  q(b),  q(c).
q(a)  p(d).
q(b)  q(a).
the atoms {p(a), p(b)} are no unfounded set wrt I=Ø!
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Notation:
Let S be a set of literals.
Then we write S for {p : pS}
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Union of Unfounded Sets
Definition
The greatest unfounded set (of P) wrt I,
denoted UP(I), is the union of all sets that are
unfounded wrt I.
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Well-Founded Partial Models
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Transformations
• A transformation is a transformation between
sets of literals, whose atoms are elements of the
Herbrand-base of a program P.
• A transformation T is called monotonic, if
T(I)  T(J), whenever I  J.
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Transformations
Definition
Let P be a normal program.
Transformations TP, UP and WP are defined as follows:
– pTP(I) iff there is some instantiated clause
R of P, such that R has head p, and each subgoal literal
in the body of R is true in I.
– UP(I) is the greatest unfounded set of P wrt I.
– WP(I) = TP(I)    UP(I).
Lemma
TP, UP and WP are monotonic transformations.
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I and I
Definition
The sets I ( ranges over all countable ordinals) and I,
whose elements are literals in the Herbrand-base of a
program P, are defined recursively by:
(1) For limit ordinal ,
I 
I

 

Note that 0 is a limit ordinal, and I0 = .
(2) For a successor ordinal  =  + 1,
I  1  WP I  
(3) Finally, define
I    I

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Stufe
Definition
For any literal p in I, we define the stage of p to be
the least ordinal  such that p  I.
Lemma
I is a monotonic sequence of partial
interpretations.
Proof Sketch
Induction over 
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Closure Ordinal
Definition
The closure ordinal for the sequence I
is the least ordinal  such that: I = I.
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Well-Founded Semantics
Definition
The well-founded semantics of a program P is the
“meaning” represented by the least fixed point of
WP or the limit I.
Every positive literal denotes that its atom is true.
Every negative literal denotes that its atom is
false.
Missing atoms have no truth value assigned by
the semantics.
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Example
p(a)  p(c), p(b).
p(b)  p(a).
p(e)  p(d).
p(c).
p(d)  q(a), q(b).
p(d)  q(b), q(c).
q(a)  p(d).
q(b)  q(a).
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I0=Ø
TP(I0)={p(c)}
UP(I0)={p(d),q(a),q(b),q(c)}
WP(I0)={p(c),p(d),q(a),
q(b), q(c)}
TP(I1)={p(e),p(c)}
UP(I1)={p(d),q(a),q(b),q(c)}
WP(I1)={p(e),p(c),p(d),
q(a),q(b),q(c)}
TP(I2)={p(e),p(c)}
…
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Well-Founded Semantics
Lemma
Let I be defined as above. Then I does not weakly
falsify any instantiated clause R of P.
Proof Sketch
The definition of WP only makes the head L of R false,
which was unfounded in previous iterations. Hence
either the body of R was false or some atom of R was
in the unfounded set. In both cases the body of R is
now wrong.
The monotony of WP is important for this lemma.
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Well-Founded (Partial) Model
Theorem
For every countable ordinal , I in the sequence
described above is a partial model of P.
Proof
Using the first and the previous lemma.
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Well-Founded (Partial) Model
Definition
Suppose that for each pBP,
I contains either p or p, i.e. I is a total
interpretation.
Then by the above theorem I is a total model
and we call this the well-founded model;
Otherwise, we call I the well founded
partial model.
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Minimal Model
Theorem
Every Horn program has a well-founded model
I, which is the minimum model in the sense of
Van Emden and Kowalski, that is, its positive
literals are contained in every Herbrand model.
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Theorem:
If a program P is locally stratified, then is has a
well-founded model, which is identical to the
perfect model.
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Example: Suppose all Valid Moves From one Position to an
other are in the EDB
winning(X)  move(X,Y), not winning(Y).
F

player 1 moves
T
T
player 2 moves
F
F
F
player 1
loses
player 1
loses
T
T
F
player 1 zieht
F
(a)
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F
player 2
loses
T
F
F
(b)
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(c)
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Example
The program is not locally stratified, because the
Herbrand-instantiation contains a clause, in which
winning negatively depends on itself:
winning(a)  move(a,a), not winning(a).
This destroys the perfect model, even if move(a,a) is
not contained in the EDB.
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Computational Complexity
• The set of statements derivable shall be
“reasonably computable”.
• We need to show that the data-complexity of
well-founded semantics is polynomial.
• Well-founded semantics is competitive with
other methods, such as stratified semantics and
fitting model.
• For logic programs without functions the
Herbrand-universe is finite and its construction
effective. (Class Datalog).
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Computational Complexity
• Limit the discussion to logic programs without
function symbols.
• The Herbrand universe of a program is the set of
constants appearing in the program.
• Consider a fixed IDB PI consisting of a set
inference rules, which might be applied to
various EDBs or sets of facts.
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Computational Complexity
• Predicates that appear as subgoals in PI, but not in
the head of any rule, constitute the EDB predicates.
• The EDB PE is a set of positive ground literals
ranging over the EDB predicates.
(The constants in PE may or may not appear in PI.)
• given an EDB PE:
– P(PE) = PI PE is a Logic program and its well-founded
partial model is denoted by I(PE).
– PI defines the transformation from PE to I(PE).
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Data Complexity
Definition:
The data complexity of an IDB is defined as the
computational complexity of deciding the
answer to a ground atomic query as a function
of the size of the EDB.
In the context of well-founded semantics, this
means deciding whether the ground atom is
positive in the well-founded partial model.
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Data Complexity
- Since the IDB is fixed, the predicates in the wellfounded model have fixed number and arity.
- The Herbrand-base has a size that is polynomial
in the size of the EDB.
- Also since the IDB is fixed, the size of the
Herbrand instantiation of the program is
polynomial in the size of the EDB.
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Data Complexity
Theorem:
The data complexity of the well-founded
semantics for function-free programs is
polynomial time.
Comments:
• The Fitting model has polynomial data
complexity for function free programs.
• Determining whether P has a stable model is NPcomplete for general propositional logic
programs.
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Advantages of Well-Founded Semantics
The well-founded semantics for normal
programs extends earlier proposals and has
advantages over them in that
(1)it is applicable to all logic programs
(2)compared to other methods a larger part of
the Herbrand base tends to be classified as
either true or false
(3)truth values are assigned in a reasonably
predictable and intuitively satisfying way.
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