## Sound and complete sld-resolution for bilattice-based annotated logic programs

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Venue: | In Proceedings of the International Conference INFORMATION-MFCSIT’06 |

Citations: | 2 - 2 self |

### BibTeX

@INPROCEEDINGS{Komendantskaya_soundand,

author = {Ekaterina Komendantskaya and Anthony Karel Seda},

title = {Sound and complete sld-resolution for bilattice-based annotated logic programs},

booktitle = {In Proceedings of the International Conference INFORMATION-MFCSIT’06},

year = {},

pages = {2006}

}

### OpenURL

### Abstract

We introduce the class of normal bilattice-based annotated first-order logic programs (BAPs) and develop declarative and operational semantics for them. In particular, SLD-resolution for these programs is defined and its soundness and completeness established.

### Citations

1856 |
Foundations of Logic Programming
- Lloyd
- 1987
(Show Context)
Citation Context ... it is continuous. We establish that TP computes the minimal Herbrand models for BAPs. Let D and v denote respectively a domain of interpretation and a variable assignment for a given language L, see =-=[22]-=-. An interpretation I for L consists of the following mappings. The first mapping I assigns |R|v : D n −→ B to each n-ary predicate symbol R in L. Further, for each element 〈α, β〉 of B, we define a ma... |

381 | Bilattices and the semantics of logic programming
- Fitting
- 1991
(Show Context)
Citation Context ...a well-known algebraic structure for reasoning about the sort of inconsistencies which arise when one formalises the process of accumulating information from different sources. In particular, Fitting =-=[6,7,8]-=- introduced quite general consequence operators for logic programs whose semantics are based on four-valued bilattices, and derived their basic properties. Annotated (sometimes called signed) language... |

215 | A logic for reasoning about probabilities
- Fagin, Halpern, et al.
- 1990
(Show Context)
Citation Context ...otated sorted programs [16] may be interesting. Another field of possible extension of our results is to relate BAPs to probabilistic logic programs, as they were defined and studied, for example, in =-=[1,5]-=- and other papers. Finally, it would be interesting to show how we can extend BAPs to logic programs with intervalbased annotations, and thereby establish linear programming for them. Such work would ... |

172 | Theory of Generalized Annotated Logic Programming and its Applications
- Kifer, Subrahmanian
- 1992
(Show Context)
Citation Context ... introduce sound and complete proof procedure for BAPs. Note that the semantic operators introduced earlier for annotated logic programs of this generality do not possess this important property, see =-=[4,15]-=-. Finally, we establish sound and complete SLD-resolution for BAPs. As far as we know, this is the first sound and complete proof procedure for first-order infinitely interpreted (bi)lattice-based ann... |

156 |
Multi-valued logics: A uniform approach to reasoning in AI
- Ginsberg
- 1988
(Show Context)
Citation Context ...s soundness and completeness established. Keywords: Annotated logic programming, bilattices, declarative and operational semantics, SLD-resolution. 1 Introduction Since their introduction by Ginsberg =-=[9]-=-, bilattices have become a well-known algebraic structure for reasoning about the sort of inconsistencies which arise when one formalises the process of accumulating information from different sources... |

133 | Probabilistic logic programming
- Ng, Subrahmanian
- 1992
(Show Context)
Citation Context ...erties. Annotated (sometimes called signed) languages are an alternative formal tool for handling, for example, the semantics of logic programs over many-valued logics and probabilistic programs, see =-=[4,11,15,23,26,31]-=-. Their use, however, gives rise to 1 The authors thank the Boole Centre for Research in Informatics at University College Cork for substantial support during the preparation of this paper. 2 Email: k... |

105 | Fixpoint semantics for logic programming — A survey
- Fitting
(Show Context)
Citation Context ...a well-known algebraic structure for reasoning about the sort of inconsistencies which arise when one formalises the process of accumulating information from different sources. In particular, Fitting =-=[6,7,8]-=- introduced quite general consequence operators for logic programs whose semantics are based on four-valued bilattices, and derived their basic properties. Annotated (sometimes called signed) language... |

95 | A logic for reasoning with inconsistency
- Kifer, Lozinskii
- 1992
(Show Context)
Citation Context ...heoretical Computer Science URL: www.elsevier.nl/locate/entcssKomendanskaya and Seda the obvious question of how annotated logic programming compares with logic programming based on (bi)lattices, see =-=[14,15,21,24,25,28]-=-. In this paper, we contribute to this discussion by combining the two approaches in that we make use of bilattice structures within the framework of annotated logic programs. Specifically, we introdu... |

91 |
Predicate logic as computational formalism
- Clark
- 1979
(Show Context)
Citation Context ...leteness. Throughout this section, we denote a BAP by P , and a BAP goal by G. We refer to [22] for a description of the unification process (originally defined by Herbrand and later refined by Clark =-=[3]-=-) for classical logic programming. Here, we follow this development, but involve annotation variables in the process of unification, see also [17]. The unification process for the individual variables... |

55 | Approximating the semantics of logic programs by recurrent neural networks
- Hölldobler, Kalinke, et al.
- 1999
(Show Context)
Citation Context ...mendanskaya and Seda We show in a companion paper [19] that the semantic operator for the logic programs we have introduced here can be computed by learning artificial neural networks in the style of =-=[13]-=-, but with learning functions embedded into connections between the layers. This shows that the automated proof procedure (SLD-resolution) we have introduced has its counterpart in the field of neural... |

42 | Bilattices in logic programming
- Fitting
- 1990
(Show Context)
Citation Context ...a well-known algebraic structure for reasoning about the sort of inconsistencies which arise when one formalises the process of accumulating information from different sources. In particular, Fitting =-=[6,7,8]-=- introduced quite general consequence operators for logic programs whose semantics are based on four-valued bilattices, and derived their basic properties. Annotated (sometimes called signed) language... |

27 | On a theory of probabilistic deductive databases. Theory and Practice of Logic Programming - Lakshmanan, Sadri - 2001 |

25 |
Logic programming with signs and annotations
- Lu
- 1996
(Show Context)
Citation Context ... an alternative to the constrained resolution for the general annotated logic programs of Kifer and Subrahmanian, see [15] and to resolutions for logics which are interpreted by linearly ordered sets =-=[12,23,31]-=- and/or finite sets [15,23,24,25]. 16sKomendanskaya and Seda We show in a companion paper [19] that the semantic operator for the logic programs we have introduced here can be computed by learning art... |

20 |
Approximate reasoning by similaritybased sld resolution
- Sessa
(Show Context)
Citation Context ...obinson, and was implemented (as SLD-resolution) in two-valued logic programming by Colmerauer et al. A detailed exposition of this can be found in [22], and for many-valued resolution procedures see =-=[12,14,15,23,27,29]-=-, for example. Kifer and Lozinskii show in [14] that unlike classical refutation procedures, where only the resolution rule is applied, latticebased theories need to have four procedures: resolution, ... |

19 | Sorted monotonic logic programs and their embeddings
- Damásio, Pereira
- 2004
(Show Context)
Citation Context ...erties. Annotated (sometimes called signed) languages are an alternative formal tool for handling, for example, the semantics of logic programs over many-valued logics and probabilistic programs, see =-=[4,11,15,23,26,31]-=-. Their use, however, gives rise to 1 The authors thank the Boole Centre for Research in Informatics at University College Cork for substantial support during the preparation of this paper. 2 Email: k... |

18 |
Automated deduction in multiple-valued logics, volume 10
- Hähnle
- 1993
(Show Context)
Citation Context ...erties. Annotated (sometimes called signed) languages are an alternative formal tool for handling, for example, the semantics of logic programs over many-valued logics and probabilistic programs, see =-=[4,11,15,23,26,31]-=-. Their use, however, gives rise to 1 The authors thank the Boole Centre for Research in Informatics at University College Cork for substantial support during the preparation of this paper. 2 Email: k... |

16 |
a logic for representing and reasoning with statistical knowledge
- Lp
- 1990
(Show Context)
Citation Context ...case, and allow negations unlike, for example, [4,23] and others. Let B = L1 ⊙ L2 denote a bilattice given as the product of two complete lattices L1, L2 each of which is a sublattice of the lattice (=-=[0, 1]-=-, ≤), where [0, 1] is the unit interval of real numbers and ≤ is the usual linear ordering on it. For the rest of the paper, we restrict our attention to bilattices with finite joins. We define an ann... |

11 | Query answering in normal logic programs under uncertainty
- Straccia
- 2005
(Show Context)
Citation Context ...heoretical Computer Science URL: www.elsevier.nl/locate/entcssKomendanskaya and Seda the obvious question of how annotated logic programming compares with logic programming based on (bi)lattices, see =-=[14,15,21,24,25,28]-=-. In this paper, we contribute to this discussion by combining the two approaches in that we make use of bilattice structures within the framework of annotated logic programs. Specifically, we introdu... |

10 | Signed formula logic programming: operational semantics and applications
- Calmet, Lu, et al.
- 1996
(Show Context)
Citation Context ...(possibly infinite) distributive bilattices with finite joins, and this framework considerably enriches that of propositional-based logic programs [4,21,26,28] based on finite sets or finite lattices =-=[2,10,14,23,24,25,31]-=-. Moreover, we allow annotations to be variables or to contain function symbols unlike, for example, the case of [2,10,14,23,24,25,31]. We use the fact proven in [20] that the semantic operator we int... |

10 |
A framework for automated reasoning in multiple-valued logics
- Lu, Murray, et al.
- 1994
(Show Context)
Citation Context ...heoretical Computer Science URL: www.elsevier.nl/locate/entcssKomendanskaya and Seda the obvious question of how annotated logic programming compares with logic programming based on (bi)lattices, see =-=[14,15,21,24,25,28]-=-. In this paper, we contribute to this discussion by combining the two approaches in that we make use of bilattice structures within the framework of annotated logic programs. Specifically, we introdu... |

8 | On approximation of the semantic operators determined by bilattice-based logic programs
- Komendantskaya, Seda, et al.
- 2005
(Show Context)
Citation Context ... finite lattices [2,10,14,23,24,25,31]. Moreover, we allow annotations to be variables or to contain function symbols unlike, for example, the case of [2,10,14,23,24,25,31]. We use the fact proven in =-=[20]-=- that the semantic operator we introduce is continuous. As usual, continuity of the semantic operator ensures that it reaches its least fixed point in at most ω iterations. This property is crucial fo... |

5 |
Deduction and search strategies for regular multiple-valued logics
- Lu, Murray, et al.
- 2005
(Show Context)
Citation Context ...ined resolution for the general annotated logic programs of Kifer and Subrahmanian, see [15] and to resolutions for logics which are interpreted by linearly ordered sets [12,23,31] and/or finite sets =-=[15,23,24,25]-=-. 16sKomendanskaya and Seda We show in a companion paper [19] that the semantic operator for the logic programs we have introduced here can be computed by learning artificial neural networks in the st... |

4 | Learning and Deduction in Neural Networks and Logic
- Komendantskaya
- 2007
(Show Context)
Citation Context ... is why, for the rest of the paper, we develop BAPs assuming that all the literals in the clauses are brought into positive form. A detailed analysis of properties of negation in BAPs can be found in =-=[17,18]-=-. Note that we use ⊗ to connect literals in the bodies of annotated clauses, and thus give priority to the lattice based on ≤k. This permits us to use monotonicity of this lattice relative to the nega... |

3 |
A many-sorted semantics for many-valued annotated logic programs
- Komendantskaya
(Show Context)
Citation Context ...al SLD-resolution. This happens because item (b) admits the use of a finite set of clauses at each step of the derivation. This item was not included in the classical definition of SLD-resolution. In =-=[16]-=-, we gave a many-sorted representation of BAPs. This translation allowed us to apply the classical definition of SLD-resolution to the many-sorted translation of BAPs. This result showed that in princ... |

3 | Logic programs with uncertainty: neural computations and automated reasoning
- Komendantskaya, Seda
- 2006
(Show Context)
Citation Context ...Subrahmanian, see [15] and to resolutions for logics which are interpreted by linearly ordered sets [12,23,31] and/or finite sets [15,23,24,25]. 16sKomendanskaya and Seda We show in a companion paper =-=[19]-=- that the semantic operator for the logic programs we have introduced here can be computed by learning artificial neural networks in the style of [13], but with learning functions embedded into connec... |

2 |
Towards an efficient tableaux proof procedure for multiple-valued logics
- Hähnle
- 1990
(Show Context)
Citation Context ...(possibly infinite) distributive bilattices with finite joins, and this framework considerably enriches that of propositional-based logic programs [4,21,26,28] based on finite sets or finite lattices =-=[2,10,14,23,24,25,31]-=-. Moreover, we allow annotations to be variables or to contain function symbols unlike, for example, the case of [2,10,14,23,24,25,31]. We use the fact proven in [20] that the semantic operator we int... |

2 | A sequent calculus for bilattice-based logic and its many-sorted representation - Komendantskaya - 2007 |

1 |
Deduction in many-valued logics: a survey. Mathware and soft computing
- Hähnle, Escalado-Imaz
- 1997
(Show Context)
Citation Context ...obinson, and was implemented (as SLD-resolution) in two-valued logic programming by Colmerauer et al. A detailed exposition of this can be found in [22], and for many-valued resolution procedures see =-=[12,14,15,23,27,29]-=-, for example. Kifer and Lozinskii show in [14] that unlike classical refutation procedures, where only the resolution rule is applied, latticebased theories need to have four procedures: resolution, ... |