## A Complete Axiomatization of the Three-Valued Completion of Logic Programs (1991)

Venue: | Journal of Logic and Computation |

Citations: | 11 - 6 self |

### BibTeX

@ARTICLE{Stärk91acomplete,

author = {Robert F. Stärk},

title = {A Complete Axiomatization of the Three-Valued Completion of Logic Programs},

journal = {Journal of Logic and Computation},

year = {1991},

volume = {1},

pages = {1--6}

}

### OpenURL

### Abstract

We prove the completeness of extended SLDNF-resolution for the new class of #-programs with respect to the three-valued completion of a logic program. Not only the class of allowed programs but also the class of definite programs are contained in the class of #-programs. To understand better the three-valued completion of a logic program we introduce a formal system for three-valued logic in which one can derive exactly the three-valued consequences of the completion of a logic program. The system is proof theoretically interesting, since it is a fragment of Gentzen's sequent calculus LK. Keywords: Logic programming; three-valued logic; negation as failure; SLDNFresolution; sequent calculus. 1

### Citations

1921 |
Foundations of Logic Programming
- Lloyd
- 1987
(Show Context)
Citation Context ...d. Unfortunately it is much more difficult to describe and to understand what happens in a query evaluation process. The following two definitions are an adaptation of the definitions in §15 of Lloyd=-= [10]. Definition-=- 3 A goal Γ ′ is derived from Γ using (the selected atom) A, (the input clause) C and (the most general unifier) θ if Γ is of the form ∆0, A, ∆1 and C of the form B :- Π and θ is a most ge... |

978 |
Negation as Failure
- Clark
- 1978
(Show Context)
Citation Context ...A fails, the literal ¬A fails if A succeeds with answer the identity substitution. Formally it is defined as SLDNF-resolution which is SLD-resolution plus negation as failure. Clark has introduced in=-= [3]-=- the completion of a logic program as a declarative semantics for SLDNF-resolution (or his query evaluation procedure). He proved the result that if the query ?- L1, . . . , Ln succeeds from a program... |

634 |
Towards a theory of declarative knowledge
- Apt, Blair, et al.
- 1988
(Show Context)
Citation Context ...ication trees one can give a very short proof for the completeness of SLD-resolution for definite programs (see Stärk [15]). Closed implication trees were first introduced by Apt, Blair and Walker in=-= [1]-=-. Definition 8 Let L be a literal and P be a program. An implication tree for L with respect to P of rank k is a finite tree T whose nodes are literals and whose root is L such that (a+) if A is a pos... |

393 | Fixedpoint semantics for logic programming
- Fitting
(Show Context)
Citation Context ...ll predicates r, for all �a ∈ |A|: A(r(�a )) ≤ B(r(�a )). It is easy to see that this is a partial ordering. If A ≤ B and F [�x ] is a Kleene formula then A(F [�a ]) ≤ B(F [�a ]) f=-=or all �a ∈ |A|. In [5] Fitti-=-ng has defined for a program P an operator ΦP which is the threevalued analogue of the ‘immediate consequence operator’ TP of logic programming. The operator ΦP assigns to a three-valued structu... |

240 |
Negation in Logic Programming
- Kunen
- 1987
(Show Context)
Citation Context ...ion one wants also the opposite directions of these statements. But this is not possible in general since the completion of a program can be inconsistent or a query can flounder. Kunen has defined in =-=[8]-=- the three-valued completion of a logic program. This seems to be a good semantics for negation as failure, since the soundness results above of Clark remain valid if one replaces the classical conseq... |

132 |
Negation as failure using tight derivations for general logic programs
- Gelder
- 1988
(Show Context)
Citation Context ...the body of a clause of P the atom A does not unify with the head of any clause in P . Now if P is quasi-definite then P is an ε-program. 4 ◦ The programs which are safe for negation of Van Gelder =-=in [16] are �-=-�-programs. Definition 18 A weak implication tree T for L with respect to P is defined like an implication tree for L (Definition 8) but clause (a−) is replaced by (a−) if ¬A is a negative node o... |

103 |
Proof theory
- Schütte
- 1967
(Show Context)
Citation Context ...Γ ⊃ ∆ is provable in LK(P ) then Γ ⊃ ∆ is valid in any three-valued model of comp(P ). The proof of the completeness of LK(P ) goes similar to the proof of the completeness of Schütte valua=-=tions (see [11]). A ve-=-rsion with the same terminology and notions that we use can be found in Girard’s book ([6], Theorem 3.1.9, p. 164). Theorem 27 (Completeness of LK(P )) If a sequent Γ ⊃ ∆ is valid in any threev... |

88 |
Signed data dependencies in logic programming
- Kunen
- 1989
(Show Context)
Citation Context ...ins the class of allowed programs. It also contains the class of definite programs. Therefore our completeness result extends the results of Shepherdson in [13], Cavedon and Lloyd in [2] and Kunen in =-=[9]. -=-In the program above, the ε property means that if the goal ?- good(s, t) succeeds with answer ε, then the terms s and t have to be closed or the goal ?- member(s, t) has to fail. Having this large ... |

50 |
Proof theory and logical complexity
- Girard
- 1987
(Show Context)
Citation Context ...f the completeness of LK(P ) goes similar to the proof of the completeness of Schütte valuations (see [11]). A version with the same terminology and notions that we use can be found in Girard’s boo=-=k ([6], Theorem -=-3.1.9, p. 164). Theorem 27 (Completeness of LK(P )) If a sequent Γ ⊃ ∆ is valid in any threevalued model of comp(P ) then Γ ⊃ ∆ is provable in LK(P ) (with only atomic cuts). Proof. Assume t... |

29 |
A Completeness Theorem for SLDNF Resolution
- Cavedon, Lloyd
- 1989
(Show Context)
Citation Context ... ε-programs contains the class of allowed programs. It also contains the class of definite programs. Therefore our completeness result extends the results of Shepherdson in [13], Cavedon and Lloyd in=-= [2] a-=-nd Kunen in [9]. In the program above, the ε property means that if the goal ?- good(s, t) succeeds with answer ε, then the terms s and t have to be closed or the goal ?- member(s, t) has to fail. H... |

22 |
Negation as failure: a comparison of Clark's completed data base and Reiter's closed world assumption
- Shepherdson
- 1984
(Show Context)
Citation Context ...ection we give a formal definition of extended SLDNF-resolution and prove some basic facts about it. We will use these technical lemmas in the completeness proof of Section 4. In Shepherdson’s paper=-=s [12]-=- and [13] some of them are proved in detail, but the version of SLDNF-resolution that he uses is weaker than our version. We do not require that negative selected literals have to be closed. So we wil... |

18 |
IC-PROLOG – language features
- Clark, McCabe, et al.
- 1982
(Show Context)
Citation Context ...e steps (a−) and (b−) the literal ¬A has to be closed. But we think that there is no reason to require this. The ESLDNF derivation procedure corresponds to the implementation of negation in IC-Pr=-=olog [4]. We say -=-that a goal Γ succeeds with answer θ (from P ) or equivalently θ is a computed answer for Γ if there exists a k ≥ 0 and an ESLDNF proof of P/Γ of rank k with answer θ. A goal Γ is finitely fa... |

18 | Completeness of the negation as failure rule
- Jaffer, Lassez, et al.
- 1983
(Show Context)
Citation Context ...ular goal for P . We come to the main theorem of this section. It corresponds to the completeness of the negation as failure rule for definite programs which was proved by Jaffar, Lassez and Lloyd in =-=[7]. Theore-=-m 22 Let P be an ε-program. If the goal ?- L1, . . . , Lq is not finitely failed then there exists a countable three-valued structure M with (1) M is a model of comp(P ), (2) ¬∃(L1 ∧ . . . ∧ L... |

15 | A direct proof for the completeness of SLD-resolution
- Stärk
(Show Context)
Citation Context ...NF proofs. They are only a tool for proving properties of computations. Using implication trees one can give a very short proof for the completeness of SLD-resolution for definite programs (see Stärk=-= [15]-=-). Closed implication trees were first introduced by Apt, Blair and Walker in [1]. Definition 8 Let L be a literal and P be a program. An implication tree for L with respect to P of rank k is a finite... |

14 |
A sound and complete semantics for a version of negation as failure
- Shepherdson
- 1989
(Show Context)
Citation Context ...led or if ?- A, A, Γ succeeds with answer ε then ?- A, Γ succeeds with answer ε. It would be interesting to compare LK(P ) or one of its subsystems with the provability relation ⊢3I of Shepherds=-=on in [14]. -=-The relation ⊢3I is three-valued sound and intuitionistically sound. Therefore it is weaker than LK(P ). 23sAcknowledgment I am grateful to John Shepherdson and an anonymous referee for helpful comm... |

10 | Proof Theory and Logical Complexitiy - Girard - 1987 |

9 |
Negation as failure II
- Shepherdson
- 1985
(Show Context)
Citation Context ...e rules above. The class of ε-programs contains the class of allowed programs. It also contains the class of definite programs. Therefore our completeness result extends the results of Shepherdson in=-= [13], -=-Cavedon and Lloyd in [2] and Kunen in [9]. In the program above, the ε property means that if the goal ?- good(s, t) succeeds with answer ε, then the terms s and t have to be closed or the goal ?- m... |