Transcript Document

Negation: Declarative Interpretation
- An overview
• First Order Formulas and Logical Truth
• Completion of Programs
• SLDNF-resolution: Soundness and restricted completeness
• Extended Consequence Operator
• Standard Models
[Lloy87] J.W. Lloyd, Foundations of Logic Programming, Second Extended Edition,
Springer, 1987.
[ApBo94] Krzysztof Apt and Roland Bol, Logic Programming and Negation: A Survey,
Journal of Logic Programming, 19/20: 9-71, 1994.
11 November 2005
Foundations of Logic and Constraint Programming
1
First-Order Formulas
Given ranked alphabets F and  for functiona and predicate symbols,
respectively, and a set V of variables, the set of
(first-order) formulas (over , F and V ) is inductively defined as follows:

If atom A  TB  ,F,V , then A is a formula

If G1 and G2 are formulas, then G1 , G1  G2 (written G1, G2 ), G1  G2,
G1  G2 and G1  G2 are formulas

If G is a formula and x  V, then x G and x G are formulas
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Foundations of Logic and Constraint Programming
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Extended Notion of Logical Truth (1)
Given a formula G, and interpretation I with domain D, and a state s : V → D:
G is true in I under s , written I |=s G :

I |=s p(t1, ..., tn) : s(t1), ... , s(t1)  pI

I |=s G : I |s G

I |=s G1  G2 : I |=s G1 and I |=s G2

I |=s G1  G2 : I |=s G1 or I |=s G2

I |=s G1 G2 : if I |=s G2 then I |=s G1

I |=s G1 G2 : I |=s G2 iff I |=s G1

I |=s xG : for every d  D: I |=s’ G

I |=s xG : for some d  D: I |=s’ G
where s’ : V → D with s’(x) = d and s’(y) = s (y) for every y  V – {x}
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Foundations of Logic and Constraint Programming
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Extended Notion of Logical Truth (2)
Given a formula G, sets of Formulas S and T, and interpretation I ,
and where x1, ..., xk are the variables occurring in G

 x1, ..., xk G is the universal closure of G (abbreviated to G)

I |= G : I |=s G for every state s

G is true in I (or I is a model of G), written I |= G : I |= G

I is a model of S, written I |= S : I |=G for every G  S
•
T is a semantic (or logical) consequence of S written S |= T :
every model of S is a model of T ( I: I |= S implies I |= T)
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Foundations of Logic and Constraint Programming
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No Negative Consequences of (Extended) Programs (1)
Consider program Pmem:
member(x,[x|y]) 
member(x,[y|z])  member(x, z)
- Then, e.g
Pmem |= member(a, [a,b]) and Pmem | member(a, [ ]).
- But also, Pmem | member(a, [ ]) since
HB {member},{|, [], a} |= Pmem and HB {member},{|, [], a} | member(a, [ ]).
Nevertheless, the SLDNF tree of Pmem  {member(a, [ ])} is successful
member(a,[])
□
success
member(a,[])
failure
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Foundations of Logic and Constraint Programming
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No Negative Consequences of (Extended) Programs (2)
Problem:
For every extended program P, the corresponding Herbrand Base is a model.
-
Hence, there is no negative ground literal A as a logical consequence of P.
-
But SLDNF tree of Pmem  {A }, may be successful ! (?)
Hence, is SLDNF-resolution sound? What does soundness means?
Solution:
Strengthen P by completion (“replace implication by “equivalences”) to
comp(P) and compare SLDNF-resolution and comp(P), instead of P!
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Foundations of Logic and Constraint Programming
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Completed Definitions (Example 1)
P:
happy  snow, holidays
happy  sun
snow  cold, precipitation
cold 
precipitation 
Whereas P |= precipitation, cold, snow
comp(P):
% the least Herbrand Model
happy  ( snow, holidays )  sun
snow  cold, precipitation
cold  true
precipitation  true
holidays  false
sun  false
comp(P) |= precipitation, cold, snow,  holidays,  sun,  happy
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Foundations of Logic and Constraint Programming
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Completed Definitions (Example 2)
P: member(x,[x|y]) 
member(x,[y|z])  member(x,z)
disjoint([],x) 
disjoint([x|y],z)   member(x,z), disjoint(y,z)
comp(P): x1 x2 member(x1,x2) 
 x,y (x1 = x, x2 = [x|y]) 
 x,y,z (x1 = x, x2 = [y|z]), member(x,z)
x1 x2 disjoint(x1,x2) 
 x (x1 = [], x = x) 
 x,y,z (x1 = [x|y], x2 = z,
member(x,z), disjoint(y,z))
( plus the standard axioms for equality and inequality)
Then, e.g. comp(p) |= member(a,[a,b]), member(a,[ ]), member(a,[b,c])
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Foundations of Logic and Constraint Programming
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Completion (1)
-
The completion of extended program P (denoted by comp(P)), is the set of
formulas constructed from P by the following 6 steps:
1.
Associate with every n-ary predicate symbol p a sequence of pairwise distinct
variables x1, ... , xn which do not occur in P.
2.
Transform each clause c = p(t1, ... , tn)  B into
p(x1, ... , xn)  x1, = t1, ... , xn, = tn, B
3.
Transform each resulting formula p(x1, ... , xn)  G into
p(x1, ... , xn)   z G
where z is a sequence of the elements of Vars(c).
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Foundations of Logic and Constraint Programming
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Completion (2)
4.
For every n-ary predicate symbol p, let
p(x1, ... , xn)   z1 G1 ,
...
, p(x1, ... , xn)   zm Gm
be all implications obtained in step 3 (m  0).
4.a) If m >0, then replace these by the formula
 x1, ... , xn p(x1, ... , xn)   z1 G1  . . .   zm Gm
(if some Gi is empty, replace it by true.)
4.b) If m = 0, i.e predicate p had no defining clause, then add the formula
 x1, ... , xn p(x1, ... , xn)  false.
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Foundations of Logic and Constraint Programming
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Completion (3)
5.
Add the standard axioms of equality
5.1  [ x = x]
% reflexivity
5.2  [ x = y → y = x ]
% simetry
5.3  [ x = y , y = z → x = z ]
% transitivity
5.4  [ xi = y → f(x1, ..., xi, ... , xn) = f(x1, ..., y, ... , xn) ]
% substitutivity
5.5  [ xi = y → p(x1, ..., xi, ... , xn)  p(x1, ..., y, ... , xn) ] % substitutivity
6.
Add the standard axioms of inequality
6.1  [ x1  y1 ...  xm  ym → f(x1, ..., xi, ... , xn)  f(y1, ..., yi, ... , yn) ]
6.2  [ f(x1, ..., xi, ... , xn)  g(y1, ..., yi, ... , yn) ]
(when f  g )
6.3  [ x  t ] (when x is a proper subterm of t)
–
Notice that axioms 6.1 to 6.3 are a restriction of FO equality to the UNA
(Unique Names Assumptiom) – “different names denote different entities”
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Foundations of Logic and Constraint Programming
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Soundness of SLDNF-Resolution
Given extended program P, extended query Q and substitution q:

q | var(Q) is a correct answer substitution of Q : comp(P) |= Q q

Qq is a correct instance of Q : comp(P) |= Q q
Theorem (Lloyd, 1987)
If there exists a successful SLDNF-derivation of P  {Q} with CAS q, then
comp(P) |= Q q.
Corollary (Lloyd, 1987)
If there exists a successful SLDNF-derivation of P  {Q}, then
comp(P) |= Q.
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Foundations of Logic and Constraint Programming
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Incompleteness of SLDNF (Inconsistency)
P:
p  p
Thus, comp(P)  { p  p } “=“ { false }
Hence,

comp(p) |= p and comp(p) |=  p
In this case comp(P) is inconsistent, as it has no model, since for
every I, I | comp(P)
In this case of inconsistency, SLDNF-resolution is incomplete, since there is
neither a successful SLDNF-derivation for P  {p}, nor for P  { p},
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Foundations of Logic and Constraint Programming
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Incompleteness of SLDNF (Non Strictness)
P:
p  q
p  q
q  q
Thus, comp(P)  { p  q   q, q q } “=“ { p  true }
Hence,
comp(p) |= p.
In this case, of non-strictness (see later), SLDNF-resolution is incomplete, since
there is no successful SLDNF-derivation for P  {p}.
Notice that in the absence of one of the first two clauses, there would be no
incompleteness, since p  true would not be in comp(P).
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Foundations of Logic and Constraint Programming
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Incompleteness of SLDNF (Floundering)
P:
p(x)   q(x)
Thus, comp(P)  { x1 p(x1)   x (x1 = x,  q(x)) ,  x1 q(x1)  false } “=“
{ x1 p(x1)  true ,  x1 q(x1)  false }
Hence,
comp(p) |=  x1 p(x1) .
In this case, of floundering, SLDNF-resolution is incomplete, since there is no
successful SLDNF-derivation for P  {p(x1)}
Note: SLDNF blocks query  q(x), contrary to what Prolog does!
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Foundations of Logic and Constraint Programming
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Incompleteness of SLDNF (Unfairness)
P:
r  p, q
p  p
Thus, comp(P)  { r  p, q , p  p, q  false } “=“
{r  false , q  false }
Hence,
comp(P) |=  r .
In this case, SLDNF-resolution might be incomplete, since there is no successful
SLDNF-derivation for P  { r} when the leftmost selection rule (as in Prolog) is
adopted (unfairness of the selection rule).
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Foundations of Logic and Constraint Programming
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Dependency Graphs
A dependency graph Dp of an extended program P
:
directed graph with labelled edges, where

The nodes are the predicate symbols of P

The edges are either positive (labelled +) or negative (labelled -);

p →+ q edge in Dp :
P contains a clause p(s1, ..., sm)  L, q( t1, .., tn), R

p →- q edge in Dp :
P contains a clause p(s1, ..., sm)  L,  q( t1, .., tn), R
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Foundations of Logic and Constraint Programming
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Strict, Hierarchical and Stratified Programs
Given an extended program P, with dependency graph Dp, with predicate
symbols p, q, and an extended query Q:

p depends evenly / oddly on q :
There is a path in Dp from p to q with an even (including 0) / odd
number of negative edges.

P is strict wrt. Q :
No predicate symbol occurring in Q depends both evenly and oddly on
a predicate symbol in the head of a clause in P.

P is hierarchical :
No cycle exists in Dp

P is stratified :
No cycle with a negative edge exists in Dp
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Foundations of Logic and Constraint Programming
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Strict, Hierarchical and Stratified Examples (1)
P1:
p   p
p
-
• p depends on itself (oddly)
• P1 is strict (no even and odd dependency of p on any predicate)
• P1 is not hierarchical (there is a cycle in P1)
• P1 is not stratified (there is a cycle with a negative edge in P1)
P2:
p  q
p  q
q  q
q
p
+
+
• p depends evenly (0) and oddly (1) on q, which depends on itself (evenly)
• P2 is not strict (there is an even and an odd dependency of p on q
• P2 is not hierarchical (there is a cycle in P2,, as q depends on itself)
• P2 is stratified (there is no cycle with a negative edge in P2)
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Foundations of Logic and Constraint Programming
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Strict, Hierarchical and Stratified Examples (1)
P3:
p(x)   q(x)
p/1
q/1
• p depends oddly on q
• P3 is strict (no even and odd dependency of predicate p/1 on any predicate)
• P1 is hierarchical (there is no cycle in P3)
• P1 is stratified (there is no cycle with a negative edge in P3)
P4:
+
r  p, q
p  p
p
+
r
+
q
• r depends evenly on q and p, that depends evenly on itself
• P4 is strict (no predicate depends evenly and oddly on another predicate)
• P2 is not hierarchical (there is a cycle in P4, the self dependency of p)
• P2 is stratified (there is no cycle with a negative edge in P1)
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Foundations of Logic and Constraint Programming
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Restricted Completeness of SLDNF-resolution (1)
Theorem (Lloyd, 1987)
Let P be a hierarchical and allowed program and Q an allowed query.
If comp(P) |= Qq for some q such that Qq is ground, then there is a
successful SLDNF-derivation of P  {Q} with CAS q.
Note:
Theorem does not hold if an arbitrary selection rule is fixed.

The selection rule must be safe (does not select negative literals that
are not ground).
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Foundations of Logic and Constraint Programming
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Restricted Completeness of SLDNF-resolution (2)
Theorem (Lloyd, 1987)
Let P be a stratified and allowed program and Q an allowed query, such that
P is strict wrt. Q
If comp(P) |= Qq for some q such that Qq is ground, then there is a
successful SLDNF-derivation of P  {Q}. With CAS q.
Note:
Theorem does not hold if an arbitrary selection rule is fixed.

The selection rule must be safe and fair.
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Foundations of Logic and Constraint Programming
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Fair Selection Rule
An extended selection rule R is fair :
for every SLDNF-tree F via R and for every branch x in F:

Either x is failed; or

For every literal L occurring in a query of x, (some further instantiated
version of ) L is selected within a finite number of derivation steps
Examples:

Selection rule “select leftmost literal” is unfair (depth-first search)

Selection rule “select leftmost literal to the right of the literals introduced
in the previous derivation step, if it exists, otherwise select leftmost literal”
is fair (breadth-first search)
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Foundations of Logic and Constraint Programming
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Extended Consequence Operator
Let P be an extended program and I a Herbrand interpretation. Then
TP(I) : { H | H  B  ground(P), I |= B}
When P is a definite program, then
-

TP is monotonic

TP Is continuous

TP has the least fixpoint M(P)

M(P) = TP  w
In case of extended programs all these properties are lost, since TP is not
monotonic.
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Foundations of Logic and Constraint Programming
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Extended Consequence Operator is Not Continuous
Pa:
p  q
Let I = {q }
-
Then TP(I) = { },
-
Hence, it is not the case that I  TP(I).
-
Therefore, TP is not monotonic (nor continuous) for Pa (due to negation).
Pb:
p p
Let I1 = { } and I2 = {p }
-
Then TP(I1) = I1 = { }, and TP(I2) = I2 = {p},
-
Therefore, TP is monotonic (and continuous) for Pb. (and all with no negation)
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Foundations of Logic and Constraint Programming
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Extended TP-Characterization (1)
Lemma 4.3 ([ApBo94]):
Let P be an extended program and I a Herbrand interpretation. Then
I |= P iff TP (I)  I
Proof.
I |= P
iff for every H  B  ground(P) : I |= B implies I |= H
iff for every H  B  ground(P) : I |= B implies H  I
iff for every ground atom H : H  TP (I) implies H  I
iff TP (I)  I
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Foundations of Logic and Constraint Programming
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Extended TP-Characterization (2)
Definition
Let F and  be ranked alphabets of function and predicate symbols,
respectively, let =   be a binary predicate symbol (for “equality”), and let I
be a Herbrand interpretation for F and  .
Then I= :I  { t = t | t  HUF} is called a standardized Herbrand
interpretation for F and   {=}
Lemma 4.4 ([ApBo94]):
Let P be an extended program and I a Herbrand interpretation. Then
I |= comp(P) iff TP (I) = I
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Extended TP-Characterization (3)
Proof sketch of Lemma 4.4:
I= |= comp(P)
iff for every ground atom H:
I |= H   (H  B  ground(P)) B )
(since I= is a model for standard axioms of equality and inequality)
iff for every ground atom H:
H  I  I |= B for some H  B  ground(P)
iff for every ground atom H :
H  I implies H  TP (I)
iff TP (I) = I
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Foundations of Logic and Constraint Programming
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Extended TP-Characterization: Example (1)
Pa:
p  q
Let
I0 = { },
I1 = {p },
Then
TP(I0) = { p }, TP(I1) = { p },
comp(Pa) = {p,  q}
I2 = {q },
I3 = {p, q },
TP(I2) = { },
TP(I3) = { }.

TP (I1 ) = I1 .
Hence, I1 |= comp(Pa) and I1 |= Pa

TP (I2 )  I2. but TP (I2 )  I2.
Hence, I2 |= Pa, but I2 |  comp(Pa)

TP (I3 )  I3. but TP (I3 )  I3.
Hence, I3 |= Pa, but I3 |  comp(Pa)

TP (I0)  I0 .
Hence, I0 | Pa
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Foundations of Logic and Constraint Programming
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Extended TP-Characterization: Example (2)
Pb:
p  q
q  q
Let
I0 = { },
I1 = {p },
Then
TP(I0) = { p }, TP(I1) = { p },
comp(Pb) = {p   q}
I2 = {q },
I3 = {p, q },
TP(I2) = { q },
TP(I3) = { q }.

TP (I1 ) = I1 .
Hence, I1 |= comp(Pb) and I1 |= Pb

TP (I2 ) = I2 .
Hence, I2 |= comp(Pb) and I1 |= Pb

TP (I3 )  I3. but TP (I3 )  I3.
Hence, I3 |= Pb, but I3 |  comp(Pb)

TP (I0)  I0 .
Hence, I0 | Pa
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Foundations of Logic and Constraint Programming
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Completion may be Inadequate
ill   ill, infection
infection 
Thus, comp(P)  { ill   ill, infection , infection  true } “=“
{ill   ill, infection  true }
is inconsistent (it has no models).
Hence,
comp(P) |= healthy
But, I = { ill, infection } is the only Herbrand model of P.
Hence,
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P | healthy
Foundations of Logic and Constraint Programming
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Non-Intended Herbrand Models
Because Tp is not monotonic nor continuous, it is not possible, in general to get a
fixpoint, nor a least model from an extended program. Nevertheless, there are
minimal models, although not necessarily unique.
P1:
p  q
P1 has three Herbrand models
M1 = {p},
M2 = {q},
and
M3 = {p, q}
P1 has no least, but two minimal models: M1 and M2.
However: M1 and not M2 , is the “intended“ model of P1
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Foundations of Logic and Constraint Programming
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Supported Herbrand Interpretations
A Herbrand interpretation I is supported
:
For every H  I there exists some H  B  ground(P) such that I |= B
( intuitively: B is an explanation for H )
P1:
Example:
M1 = {p, q },
p  q
M2 = {p ,q},
and
M3 = {p, q}

M1 is a supported model of P1 ( q is the explanation for p)

M2 is no supported model of P1 ( there is no explanation for q)
Note that TP(M2) = { }  M1
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and that TP(M1) = M1.
Foundations of Logic and Constraint Programming
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Extended TP-Characterization (4)
Lemma 6.2 ([ApBo94]):
Let P be an extended program and I a Herbrand interpretation. Then
I |= P and I supported iff TP (I)= I
Proof (sketch):
I |= P and I supported
-
iff for every H  B  ground(P) : I |= B implies I |= H and
for every H  I: I |=  (H  B  ground(P)) B
-
iff for every ground atom H : I |= ( H   (H  B  ground(P)) B )
and I |= (H →  (H  B  ground(P)) B )
-
iff for every ground atom H : I |= ( H   (H  B  ground(P)) B )
-
iff I= is a model of comp(P)
-
iff TP (I) = I (cf. Lemma 4.4)
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Foundations of Logic and Constraint Programming
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Non-Intended Supported Models
P2:
p  q
q  q
- P2 has three Herbrand models
M1 = {p},
M2 = {q},
and
M3 = {p, q}
- P2 has two supported Herbrand models : M1 and M2.
In fact, both M1 and M2 are minimal models for comp(P2) = {p   q}
- However: M1 and not M2 , is the “intended“ model of P2.
M1 is called the standard model of P2 ( cf. later slide)
- In general, it is not possible to define standard models, unless the programs
are stratified!
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Foundations of Logic and Constraint Programming
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Stratifications
Let P be an extended program and Dp its dependency graph,

Predicate symbol p is defined in P : P contains a clause p(t1, ..., tm)  B

P1  ...  Pn = P is a stratification of P :

Pi   for every i  1.. n

Pi  Pj =  for every i,j  1.. n, with i  j

for every p defined in Pi and edge p →+ q in Dp :
q is not defined in  n j = i+1 Pj

for every p defined in Pi and edge p →- q in Dp :
q is not defined in  n j = i Pj
Lemma 6.5 ([ApBo94]):
An extended program is stratified iff it admits a (not necessarily unique)
stratification.
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Foundations of Logic and Constraint Programming
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Stratifications Example (1)
zero(0) 
positive(x)  num(x), zero(x)
num(0) 
num(s(x))  num(x)

P1  P2  P3 is a stratification of P , where

P1 = { num(0) 

P2 = { zero(0)  }

P3 = { positive(x)  num(x) ,  zero(x) }
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,
num(s(x))  num(x) }
Foundations of Logic and Constraint Programming
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Stratifications Example (2)
even(0) 
even(x)  odd(x), num(x)
odd(s(x))  even(x)
num(0) 
num(s(x))  num(x)
P admits no stratification
since
strat(even) > odd
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but strat(odd)  strat(even)
Foundations of Logic and Constraint Programming
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Standard Models (Stratified Programs)
Let
-
I be an Herbrand interpretation and  a set of predicate symbols:
-
I |  : I  { p(t1, ..., tm) | p  P, t1, ..., tm ground terms)
-
P1  ...  Pn be a stratification of a stratification of extended program P,
Then

M1 : least Herbrand model of P1 such that
M1 | { p | p not defined in P } = 

M2 : least Herbrand model of P2 such that
M2 | { p | p defined nowhere or in P1 } = M1
...

Mn : least Herbrand model of Pn such that
Mn | { p | p defined nowhere or in P1  ...  Pn-1 } = Mn-1
We call MP = Mn the standard model of P
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Standard Models: Example
zero(0) 
positive(x)  num(x), zero(x)
num(0) 
num(s(x))  num(x)
Let P1  P2  P3 with
- P1 = { num(0)  , num(s(x))  num(x) }
- P2 = { zero(0)  }
- P3 = { positive(x)  num(x) ,  zero(x) }
be a stratification of P. Then
 M1 = { num(t) | t  HU{s,0} }
 M2 = { num(t) | t  HU{s,0} }  {zero(0)}
 M3 = { num(t) | t  HU{s,0} }  {zero(0)}  {positive(t) | t  HU{s,0} – {0} }
Hence MP = M3 is the standard model of P
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Properties of Standard Models
Theorem 6.7 ([ApBo94]):
Consider a stratified program P. Then

MP does not depend on the chosen stratification of P,

MP is a minimal model of P,

MP is a supported model of P,
Corollary:
For a stratified program P, comp(P) admits a Herbrand Model.
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Foundations of Logic and Constraint Programming
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