Syntax and semantics of multi-adjoint normal logic programming. (English) Zbl 1397.68026
Summary: Multi-adjoint logic programming is a general framework with interesting features, which involves other positive logic programming frameworks such as monotonic and residuated logic programming, generalized annotated logic programs, fuzzy logic programming and possibilistic logic programming. One of the most interesting extensions of this framework is the possibility of considering a negation operator in the logic programs, which will improve its flexibility and the range of real applications. This paper introduces multi-adjoint normal logic programming, which is an extension of multi-adjoint logic programming including a negation operator in the underlying lattice. Beside the introduction of the syntax and semantics of this paradigm, we provide sufficient conditions for the existence of stable models defined on a convex compact set of an euclidean space. Finally, we consider a particular algebraic structure in which sufficient conditions can be given in order to ensure the unicity of stable models of multi-adjoint normal logic programs.
References:
| [1] | Alcántara, J.; Damásio, C. V.; Pereira, L. M., Paraconsistent logic programs, Lecture Notes in Artificial Intelligence, vol. 2424, 345-356, (2002) · Zbl 1013.68212 |
| [2] | Alcántara, J.; Damásio, C. V.; Pereira, L. M., An encompassing framework for paraconsistent logic programs, A Paraconsistent Decagon, J. Appl. Log., 3, 1, 67-95, (2005) · Zbl 1063.03015 |
| [3] | Alsinet, T.; Chesñevar, C.; Godo, L.; Sandri, S.; Simari, G., Formalizing argumentative reasoning in a possibilistic logic programming setting with fuzzy unification, Int. J. Approx. Reason., 48, 3, 711-729, (2008) · Zbl 1184.68497 |
| [4] | Banach, S., Sur LES opérations dans LES ensembles abstraits et leur application aux équations intégrales, Fundam. Math., 3, 1, 133-181, (1922) · JFM 48.0201.01 |
| [5] | Baral, C.; Gelfond, M.; Rushton, N., Probabilistic reasoning with answer sets, Theory Pract. Log. Program., 9, 01, 57-144, (jan 2009) · Zbl 1170.68003 |
| [6] | Bauters, K.; Schockaert, S.; Cock, M. D.; Vermeir, D., Characterizing and extending answer set semantics using possibility theory, Theory Pract. Log. Program., 15, 01, 79-116, (jan 2014) · Zbl 1379.68041 |
| [7] | Cornejo, M. E.; Lobo, D.; Medina, J., Stable models in normal residuated logic programs, (Kacprzyk, J.; Koczy, L.; Medina, J., 7th European Symposium on Computational Intelligence and Mathematices, ESCIM 2015, (2015)), 150-155 |
| [8] | Cornejo, M. E.; Lobo, D.; Medina, J., Towards multi-adjoint logic programming with negations, (Koczy, L.; Medina, J., 8th European Symposium on Computational Intelligence and Mathematics, ESCIM 2016, (2016)), 24-29 |
| [9] | Cornejo, M. E.; Medina, J.; Ramírez-Poussa, E., A comparative study of adjoint triples, Fuzzy Sets Syst., 211, 1-14, (2013) · Zbl 1272.03111 |
| [10] | Cornejo, M. E.; Medina, J.; Ramírez-Poussa, E., Multi-adjoint algebras versus non-commutative residuated structures, Int. J. Approx. Reason., 66, 119-138, (2015) · Zbl 1350.06003 |
| [11] | Damásio, C.; Medina, J.; Ojeda-Aciego, M., Termination of logic programs with imperfect information: applications and query procedure, J. Appl. Log., 5, 435-458, (2007) · Zbl 1122.68025 |
| [12] | Damásio, C. V.; Pereira, L. M., Hybrid probabilistic logic programs as residuated logic programs, (Logics in Artificial Intelligence, JELIA’00, Lecture Notes in Artificial Intelligence, vol. 1919, (2000)), 57-73 · Zbl 0998.68030 |
| [13] | Damásio, C. V.; Pereira, L. M., Monotonic and residuated logic programs, (Symbolic and Quantitative Approaches to Reasoning with Uncertainty, ECSQARU’01, Lecture Notes in Artificial Intelligence, vol. 2143, (2001)), 748-759 · Zbl 1001.68545 |
| [14] | Dubois, D.; Lang, J.; Prade, H., Automated reasoning using possibilistic logic: semantics, belief revision, and variable certainty weights, IEEE Trans. Knowl. Data Eng., 6, 1, 64-71, (feb 1994) |
| [15] | Fitting, M., The family of stable models, J. Log. Program., 17, 2-4, 197-225, (1993) · Zbl 0798.68096 |
| [16] | Gelfond, M.; Lifschitz, V., The stable model semantics for logic programming, (ICLP/SLP, vol. 88, (1988)), 1070-1080 |
| [17] | Gelfond, M.; Lifschitz, V., Logic programs with classical negation, (Warren, D. H.; Szeredi, P., Logic Programming, (1990), MIT Press Cambridge, MA, USA), 579-597 |
| [18] | Gelfond, M.; Lifschitz, V., Classical negation in logic programs and disjunctive databases, New Gener. Comput., 9, 3, 365-385, (1991) · Zbl 0735.68012 |
| [19] | Julián, P.; Moreno, G.; Penabad, J., On fuzzy unfolding: a multi-adjoint approach, Fuzzy Sets Syst., 154, 1, 16-33, (2005) · Zbl 1099.68017 |
| [20] | Kifer, M.; Subrahmanian, V. S., Theory of generalized annotated logic programming and its applications, J. Log. Program., 12, 335-367, (1992) |
| [21] | Kowalski, R. A.; Sadri, F., Logic programs with exceptions, New Gener. Comput., 9, 3, 387-400, (Aug 1991) · Zbl 0737.68015 |
| [22] | Krajči, S.; Lencses, R.; Vojtáš, P., A comparison of fuzzy and annotated logic programming, Fuzzy Sets Syst., 144, 1, 173-192, (2004) · Zbl 1065.68024 |
| [23] | Leborgne, D., Calcul différentiel et géométrie, (1982), Presses Universitaires de France · Zbl 0498.58001 |
| [24] | Lloyd, J., Foundations of logic programming, (1987), Springer Verlag · Zbl 0668.68004 |
| [25] | Loyer, Y.; Straccia, U., Epistemic foundation of stable model semantics, Theory Pract. Log. Program., 6, 355-393, (2006) · Zbl 1110.68070 |
| [26] | Madrid, N.; Ojeda-Aciego, M., Towards a fuzzy answer set semantics for residuated logic programs, (Web Intelligence/IAT Workshops, (2008)), 260-264 |
| [27] | Madrid, N.; Ojeda-Aciego, M., On coherence and consistence in fuzzy answer set semantics for residuated logic programs, (Lect. Notes Comput. Sci., vol. 5571, (2009)), 60-67 · Zbl 1246.68082 |
| [28] | Madrid, N.; Ojeda-Aciego, M., Measuring inconsistency in fuzzy answer set semantics, IEEE Trans. Fuzzy Syst., 19, 4, 605-622, (Aug. 2011) |
| [29] | Madrid, N.; Ojeda-Aciego, M., On the existence and unicity of stable models in normal residuated logic programs, Int. J. Comput. Math., 89, 310-324, (2012) · Zbl 1238.68045 |
| [30] | Medina, J., Adjoint pairs on interval-valued fuzzy sets, (Hüllermeier, E.; Kruse, R.; Hoffmann, F., Information Processing and Management of Uncertainty in Knowledge-Based Systems, Commun. Comput. Inf. Sci., vol. 81, (2010), Springer), 430-439 · Zbl 1209.68547 |
| [31] | Medina, J.; Mérida-Casermeiro, E.; Ojeda-Aciego, M., A neural implementation of multi-adjoint logic programming, J. Appl. Log., 2, 3, 301-324, (2004) · Zbl 1073.68025 |
| [32] | Medina, J.; Ojeda-Aciego, M.; Vojtáš, P., A multi-adjoint logic approach to abductive reasoning, (Logic Programming, ICLP’01, Lecture Notes in Computer Science, vol. 2237, (2001)), 269-283 · Zbl 1053.68707 |
| [33] | Medina, J.; Ojeda-Aciego, M.; Vojtáš, P., Multi-adjoint logic programming with continuous semantics, (Logic Programming and Non-Monotonic Reasoning, LPNMR’01, Lecture Notes in Artificial Intelligence, vol. 2173, (2001)), 351-364 · Zbl 1007.68023 |
| [34] | Minker, J.; Rajasekar, A., A fixpoint semantics for disjunctive logic programs, J. Log. Program., 9, 1, 45-74, (1990) · Zbl 0713.68016 |
| [35] | Nieuwenborgh, D. V.; Cock, M. D.; Vermeir, D., An introduction to fuzzy answer set programming, Ann. Math. Artif. Intell., 50, 3-4, 363-388, (jul 2007) · Zbl 1125.68118 |
| [36] | Nieves, J. C.; Osorio, M.; Cortés, U., Semantics for possibilistic disjunctive programs, Theory Pract. Log. Program., 13, 01, 33-70, (jul 2011) · Zbl 1272.68081 |
| [37] | Pavelka, J., On fuzzy logic I, II, III, Z. Math. Log. Grundl. Math., 25, (1979) · Zbl 0435.03020 |
| [38] | Pereira, L. M.; Alferes, J. J., Well founded semantics for logic programs with explicit negation, (European Conference on Artificial Intelligence, (1992), John Wiley & Sons), 102-106 |
| [39] | Przymusinski, T., Well-founded semantics coincides with three-valued stable semantics, Fundam. Inform., 13, 445-463, (1990) · Zbl 0706.68029 |
| [40] | Straccia, U., Query answering in normal logic programs under uncertainty, (Lect. Notes Comput. Sci., vol. 3571, (2005)), 687-700 · Zbl 1122.68672 |
| [41] | Straccia, U., Query answering under the any-world assumption for normal logic programs, (Lect. Notes Comput. Sci., vol. 3571, (2006)), 687-700 · Zbl 1122.68672 |
| [42] | Straccia, U., A top-down query answering procedure for normal logic programs under the any-world assumption, (Proc. of the 10th International Conference on Principles of Knowledge Representation, (2006)), 329-339 |
| [43] | Straccia, U.; Ojeda-Aciego, M.; Damásio, C. V., On fixed-points of multivalued functions on complete lattices and their application to generalized logic programs, SIAM J. Comput., 38, 5, 1881-1911, (2009) · Zbl 1183.68158 |
| [44] | Tarski, A., A lattice-theoretical fixpoint theorem and its applications, Pac. J. Math., 5, 2, 285-309, (1955) · Zbl 0064.26004 |
| [45] | Tsoy-Wo, M., Classical analysis on normed spaces, (1995), World Scientific Publishing · Zbl 0839.46003 |
| [46] | Van Gelder, A.; Ross, K. A.; Schlipf, J. S., The well-founded semantics for general logic programs, J. ACM, 38, 3, 619-649, (July 1991) |
| [47] | Vojtáš, P., Fuzzy logic programming, Fuzzy Sets Syst., 124, 3, 361-370, (2001) · Zbl 1015.68036 |
| [48] | Wagner, G., A database needs two kinds of negation, 357-371, (1991), Springer Berlin Heidelberg Berlin, Heidelberg |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
