In the following we show how �L semantics and well-founded semantics yield the same models and recall the examples from the suppression task.
5.1 Claim
There is a relation between �L semantics and well-founded semantics for tight programs. In the following, we look at programs where negative facts are only formulated whenpis not the head of any other clause in P. Under �L semantics this does not restrict the expressions of our programs as by condition (�Lii), we can only conclude that p is in I⊥ iffor all clauses wherep is the head of, the body is in I⊥. Thus p ← ⊥ would not add any more information when
34 Emmanuelle-Anna Dietz and Steffen H¨olldobler
The Suppression Task under Three-Valued Logics 9 Table 6.Suppression Task with the corresponding �L and Well-founded models
P �L Model Well-founded Model
PAE �{e, l},{ab1}� �{e, l},{ab1}�
PABE �{e, l},{ab1, ab2}� �{e, l},{ab1, ab2}�
PACE �{e},{ab3}� �{e, ab1},{o, ab3, l}�
PACEmod �{e},{ab3}�
PAE �∅,{ab1, e, l}� �∅,{ab1, e, l}�
PABE �∅,{ab1, ab2, e}� �∅,{ab1, ab2, e, t, l}�
PABEmod �∅,{ab1, ab2, e}�
PACE �{ab3},{e, l}� �{ab3},{e, l}�
there is another clause withpin the head for which the body is not inI⊥. For well-founded models, we can omit negative facts because they are implicit when predicates are not defined.
Theorem 3. Let P be a tight logic program. We claim that the �L model ofP is the well-founded model ofPmod, where the modified programPmod=P−∪{A←
¬A|A∈nd(P−)}.
The proof is in the appendix.
5.2 The Suppression Task
Table 6 shows how the �L semantics and well-founded semantics are applied for the six cases of the suppression task. Programs PACE and PABE reflect the actual suppression of additional and alternative information and show that the corresponding �L model and well-founded model are different.PACEmod andPABEmod in the fourth and seventh row give the well-founded model of the modified programs as proposed in our claim and shows that the well-founded models are indeed the same as the corresponding �L models of the program.
6 Conclusion
We investigated �L semantics and well-founded semantics for the suppression task. Both approaches have different underlying three-valued interpretations. �L semantics adequately models the suppression task, whereas well-founded seman-tics does not. However, by modifying the programs under consideration for the well-founded semantics, both compute the same adequate models. We show that both approaches yield the same results for the class of tight programs.
7 Acknowledgments
The paper formalized an idea that was brought to our attention by Alexandre Miguel Pinto. Many thanks to Christoph Wernhard and Bertram Fronh¨ofer.
Modeling the Suppression Task under Three-Valued Lukasiewicz and Well-Founded Semantics 35
10 Emmanuelle-Anna Dietz, Steffen H¨olldobler
References
1. Stenning, K., van Lambalgen, M.: Human reasoning and cognitive science. Brad-ford Books. MIT Press (2008)
2. Byrne, R.M.J.: Suppressing valid inferences with conditionals. Cognition31(1989) 61–83
3. Fitting, M.: A Kripke-Kleene semantics for logic programs. J. Log. Program.2 (1985) 295–312
4. Kleene, S.C.: Introduction to metamathematics. Bibl. Matematica. North-Holland, Amsterdam (1952)
5. H¨olldobler, S., Kencana Ramli, C.D.: Logic programs under three-valued
�lukasiewicz semantics. In: Proceedings of the 25th International Conference on Logic Programming. ICLP ’09, Berlin, Heidelberg, Springer-Verlag (2009) 464–478 6. �Lukasiewicz: O logice tr´ojwarto´sciowej. Ruch Filozoficzny5(1920) 169–171 En-glish translation: On Three-Valued Logic. In: Jan �Lukasiewicz Selected Works.
(L. Borkowski, ed.), North Holland, 87-88, 1990.
7. Dietz, E.A., H¨olldobler, S., Ragni, M.: A Computational Approach to the Sup-pression Task. to appear in Proceedings of the 34th Cognitive Science Conference (2012)
8. Van Gelder, A., Ross, K.A., Schlipf, J.S.: The well-founded semantics for general logic programs. J. ACM38(1991) 619–649
9. Clark, K.L.: Negation as failure. In Minker, J., ed.: Logic and Data Bases. Vol-ume 1. Plenum Press, New York, London (1978) 293–322
10. Gelfond, M., Lifschitz, V.: The stable model semantics for logic programming. In Kowalski, R., Bowen, Kenneth, eds.: Proceedings of International Logic Program-ming Conference and Symposium, MIT Press (1988) 1070–1080
11. Przymusinski, T.: Well founded and stationary models of logic programs. Annals of Mathematics and Artificial Intelligence12(1994) 141–187
12. Gottwald, S.: A Treatise on Many-Valued Logics. Volume 9 of Studies in Logic and Computation. Research Studies Press, Baldock, UK (2001)
13. Przymusinski, T.C.: Every logic program has a natural stratification and an it-erated least fixed point model. In: Proceedings of the eighth ACM SIGACT-SIGMOD-SIGART symposium on Principles of database systems. PODS ’89, New York, NY, USA, ACM (1989) 11–21
14. Van Gelder, A., Ross, K., Schlipf, J.S.: Unfounded sets and well-founded seman-tics for general logic programs. In: Proceedings of the seventh ACM SIGACT-SIGMOD-SIGART symposium on Principles of database systems. PODS ’88, New York, NY, USA, ACM (1988) 221–230
15. Apt, K.R., Blair, H.A., Walker, A.: Foundations of deductive databases and logic programming. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (1988) 89–148
16. Przymusinski, T.C.: Foundations of deductive databases and logic programming.
Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (1988) 193–216 17. Fages, F.: Consistency of clark’s completion and existence of stable models. Meth.
of Logic in CS1(1994) 51–60
18. Erdem, E., Lifschitz, V.: Tight logic programs. CoRR (2003)
19. Kencana Ramli, C.D.: Logic programs and three-valued consequence operators.
Master’s thesis, Institute for Artificial Intelligence, Department of Computer Sci-ence, Technische Universit¨at Dresden (2009)
20. Hitzler, P., Wendt, M.: A uniform approach to logic programming semantics.
Theory and Practice of Logic Programming5(2005) 123–159
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The Suppression Task under Three-Valued Logics 11
8 Appendix
Proof of Theorem 3
A∈�L(P) if and only ifA∈W F(Pmod)
where �L(P) andW F(Pmod) are the �L model of P and the well-founded model ofPmod, respectively. Both are pairs of interpretations �I�, I⊥�:
(1)A∈�L(P)� if and only if A∈W F(Pmod)�and (2)A∈�L(P)⊥ if and only if A∈W F(Pmod)⊥
where �L(P)� (�L(P)⊥) and W F(Pmod)� (W F(Pmod)⊥) are the sets of atoms which are in I� (I⊥) in the �L model of P and in the well-founded model of Pmod, respectively.
Proof (1)
→AssumeA∈�L(P)�. This is only the case if condition (�Li) holds. Condition (�Li) and (WFi) are the same andP ⊆ Pmod, thusA∈W F(Pmod)�.
←If A∈W F(Pmod)�, then never because of some clause in the extension of P:{A← ¬A}, but because of some clause inP itself. Since conditions (�Li) and (WFi) are the same andA∈W F(Pmod)�thenA∈�L(P)�.
Proof (2)
→AssumeA∈�L(P)⊥. This is only the case if condition (�Lii) holds. Condition (�Lii) is stricter than condition (WFii) andP ⊆ Pmod, thusA∈W F(Pmod)⊥ as well.
←(By Contraposition) AssumeA�∈�L(P)⊥(to prove:A�∈W F(Pmod)⊥). Then for condition (�Lii) there are two possible cases:
i there is not such a clause A←A1, . . . , An,¬B1, . . . , Bm inP. That means Ais not the head of any clause: A∈nd(P). ThusA← ¬A∈ Pmod, which is also the only clause inPmod withAin the head.
AssumeA∈W F(Pmod)⊥. This is only the case if condition (WFiib) holds (as the only atom in the body of the clause A ← ¬A is a negated atom) which is only the case if for each clause withA in the head there is some negated atom Bj with 1 ≤ Bj ≤ Bm occurring in the body we have that Bj∈W F(Pmod)�andL(A)>L(Bj). We have exactly one clause withAin the head,A← ¬A, but thenL(A)>L(A). Contradiction.
ii for all clauses in P withA in the head neither condition �Liia nor �Liib hold for some clause withAin the head.
AssumeA ∈W F(Pmod)⊥. Then for each clause in P withA in the head, either (WFiia) or (WFiib) holds.
(a) Assume (WFiia) holds: then there exists i with 1 ≤ i ≤ n, Ai ∈ W F(Pmod)� and L(A) ≥ L(Ai). Since we restrict ourselves to tight programs, it can only be that 1≤i≤n,Ai∈W F(Pmod)� andL(A)>
L(Ai). Since we assume this for each clause in Pmod and P ⊆ Pmod (�Liia) holds forP as well and we have thatA∈�L(P)⊥. Contradiction.
Modeling the Suppression Task under Three-Valued Lukasiewicz and Well-Founded Semantics 37
12 Emmanuelle-Anna Dietz, Steffen H¨olldobler
(b) Assume (WFiib) holds: [...]. Then (�Liib) holds forP as well and we have thatA∈�L(P)⊥. Contradiction.
If we replace the clause A ← ¬A by the following two clauses and an addi-tional auxiliary atom e.g.A ← ¬not A, not A ← ¬A, then Pmod = P ∪ {A←
¬not A, not A← ¬A|A∈nd(P)}. The proof is the same except for part [i]:
i either there is not such a clause A←A1, . . . , An,¬B1, . . . , Bm inP. So we have that A is not the head of any clause and thus A ∈ nd(P). We in-troduce an additional auxiliary variable not A and two additional clauses A← ¬not Aandnot A← ¬AforPmod. These are the only two clauses in Pmod withAandnot Ain the head, respectively.
AssumeA∈W F(Pmod)⊥. This is only the case if condition (WFiib) holds (as the only atom in the body of the clauseA← ¬not Ais a negated atom).
That isnot A∈W F(Pmod)�andL(A)>L(not A).not A∈W F(Pmod)�if (WFi) holds. That is, there exists a clausenot A←A1, . . . An,¬B1, . . . ,¬Bm
inPmodsuch that for alliwith 1≤i≤n, Ai∈W F(Pmod)�andL(not A)>
L(Ai) and for all j with 1 ≤ j ≤ m, Bj ∈ W F(Pmod)⊥ and L(not A) >
L(Bj). As we know,not A← ¬Ais the only clause withnot Ain the head inPmod, thusA∈W F(Pmod)⊥andL(not A)>L(A) have to hold. Contra-diction.
38 Emmanuelle-Anna Dietz and Steffen H¨olldobler