ISWeb - Information Systems & Semantic Web Steffen Staab [email protected] Foundations of Logic Programming 1 of 48 Three-Valued Models of Program Completion

ISWeb - Information Systems & Semantic Web

Steffen [email protected]

Foundations of Logic Programming1 of 48

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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.

Completion:p(pq)q.

Program has no 2-valued model.

p p, q.p p.

Completion:p(pq)p.q.

Program has 2-valued model I={q,p}




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Truth Tables for 3-Valued Logic (Extensions Only!)

11

0

ABBA

1

00

A BBA

10

1

1

11

0

ABBA

1

1

0

A BBA

AA

Attention::AB BA

no longer holds!




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Example: Completion and 3-Valued Logic

p p, q.

Completion:p(pq)q.

Program has 3-valued model for p=I={q}

p p, q.p p.

Completion:p(pq)p.q.

Program has 3-valued model forp= I={q} I={q,p}




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Uniqueness

p p, q.q rq sr rs s

Completion:p(pq)qrsrrss

2-Valued Models{p,q,r,s}{p,q,r,s}{p,q,r,s}

3-Valued Model{}

Not comparable




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Undefined

DefinitionLiteral 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, iffor 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.




<is web>Unfounded Sets

DefinitionLet 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

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).

the atoms {p(d), q(a), q(b), q(c)} are an unfounded set wrt I=Ø.

For the instantiated normal program P:

(2):

None of the heads can be derived first.

There exists no definition for q(c). Thus (1) is satisfied.




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Example

the atoms {p(a), p(b)} are no unfounded set wrt I=Ø!

Neither (1) nor (2) are satisfied.

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).

For the instantiated normal program P:




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Notation: Let S be a set of literals. Then we write S for {p : pS}




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Union of Unfounded Sets

DefinitionThe 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

DefinitionLet P be a normal program.Transformations TP, UP and WP are defined as follows:

– pTP(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).

LemmaTP, UP and WP are monotonic transformations.




<is web>I and I

DefinitionThe 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 ,

Note that 0 is a limit ordinal, and I0 = .(2) For a successor ordinal = + 1,

(3) Finally, define

II

II PW1

II




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Stufe

DefinitionFor any literal p in I, we define the stage of p to be the least ordinal such that p I.

LemmaI is a monotonic sequence of partial interpretations.

Proof SketchInduction over




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Closure Ordinal

DefinitionThe closure ordinal for the sequence I is the least ordinal such that: I = I.




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DefinitionThe 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.




<is web>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).

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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LemmaLet I be defined as above. Then I does not weakly falsify any instantiated clause R of P.

Proof SketchThe 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

TheoremFor every countable ordinal , I in the sequence described above is a partial model of P.

ProofUsing the first and the previous lemma.




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Well-Founded (Partial) Model

DefinitionSuppose that for each pBP,

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

TheoremEvery 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.




<is web>Example: Suppose all Valid Moves From one

Position to an other are in the EDB

F

T

F F F

T

F

T

F

T

T

F

F F

(a) (b) (c)

winning(X) move(X,Y), not winning(Y).

player 1 moves

player 2 moves

player 1loses

player 1loses player 1 zieht

player 2loses

player 2 moves




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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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• 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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• 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 well-founded 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 NP-complete 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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ISWeb - Information Systems & Semantic Web Steffen Staab [email protected] Foundations of Logic Programming 1 of 48 Three-Valued Models of Program Completion