## Goal-Directed Proof Search in Multiple-Conclusioned Intuitionistic Logic (2000)

Venue: | In Proceedings of the First International Conference on Computational Logic, volume LNAI 1861 |

Citations: | 1 - 0 self |

### BibTeX

@INPROCEEDINGS{Harland00goal-directedproof,

author = {James Harland and Tatjana Lutovac and Michael Winikoff},

title = {Goal-Directed Proof Search in Multiple-Conclusioned Intuitionistic Logic},

booktitle = {In Proceedings of the First International Conference on Computational Logic, volume LNAI 1861},

year = {2000},

pages = {254--268},

publisher = {Springer}

}

### OpenURL

### Abstract

. A key property in the definition of logic programming languages is the completeness of goal-directed proofs. This concept originated in the study of logic programming languages for intuitionistic logic in the (single-conclusioned) sequent calculus LJ, but has subsequently been adapted to multiple-conclusioned systems such as those for linear logic. Given these developments, it seems interesting to investigate the notion of goal-directed proofs for a multiple-conclusioned sequent calculus for intuitionistic logic, in that this is a logic for which there are both single-conclusioned and multiple-conclusioned systems (although the latter are less well known). In this paper we show that the language obtained for the multiple-conclusioned system differs from that for the single-conclusioned case, show how hereditary Harrop formulae can be recovered, and investigate contraction-free fragments of the logic. 1 Introduction Logic programming is based upon the observation that if ...

### Citations

992 |
Negation as Failure
- Clark
- 1978
(Show Context)
Citation Context ...ogramming language. For example, analysis in intuitionistic logic has uncovered extensions to Horn clauses such as allowing implications, universal quantifiers, and negations in the bodies of clauses =-=[3, 16]-=-, incorporation of higher-order facilities [16], and negations and disjunctions in the heads of clauses [17]. Secondly, this process can be applied to logics other than classical (or intuitionistic) l... |

622 |
Untersuchungen über das logische Schließen. Mathematische Zeitschrift
- Gentzen
- 1935
(Show Context)
Citation Context ... multiple-conclusioned system for intuitionistic logic, and to compare the results with the single-conclusioned case. 2 Preliminaries 2.1 Sequent Calculi Sequent calculi are due originally to Gentzen =-=[7]-=- and are often used in the analysis of proof systems. This is because sequent calculus rules are local (and hence conceptually straightforward to implement) and there is a natural distinction between ... |

393 | Uniform Proofs as a Foundation for Logic Programming
- Miller, Nadathur, et al.
- 1991
(Show Context)
Citation Context ...ogramming language. For example, analysis in intuitionistic logic has uncovered extensions to Horn clauses such as allowing implications, universal quantifiers, and negations in the bodies of clauses =-=[3, 16]-=-, incorporation of higher-order facilities [16], and negations and disjunctions in the heads of clauses [17]. Secondly, this process can be applied to logics other than classical (or intuitionistic) l... |

362 | Logic programming with focusing proofs in linear logic
- Andreoli
- 1992
(Show Context)
Citation Context ...an be applied to logics other than classical (or intuitionistic) logic. For example, a number of logic programming languages have been derived from linear logic including Lygon[11], Forum[15], LinLog =-=[1]-=-, LO [2], Lolli [12], ACL [13], and LC [21]. A popular proof search strategy is the notion of goal-directed proof [16], which, roughly speaking, requires that the goal be decomposed before the program... |

312 | Logic programming in a fragment of intuitionistic linear logic
- Hodas, Miller
- 1992
(Show Context)
Citation Context ...gics other than classical (or intuitionistic) logic. For example, a number of logic programming languages have been derived from linear logic including Lygon[11], Forum[15], LinLog [1], LO [2], Lolli =-=[12]-=-, ACL [13], and LC [21]. A popular proof search strategy is the notion of goal-directed proof [16], which, roughly speaking, requires that the goal be decomposed before the program, and hence the comp... |

210 | Linear objects: Logical processes with built-in inheritance
- Andreoli, Pareschi
- 1991
(Show Context)
Citation Context ...plied to logics other than classical (or intuitionistic) logic. For example, a number of logic programming languages have been derived from linear logic including Lygon[11], Forum[15], LinLog [1], LO =-=[2]-=-, Lolli [12], ACL [13], and LC [21]. A popular proof search strategy is the notion of goal-directed proof [16], which, roughly speaking, requires that the goal be decomposed before the program, and he... |

145 |
Contraction-Free Sequent Calculi for Intuitionistic Logic
- Dyckhoff
- 1992
(Show Context)
Citation Context ... the propositional fragment does not require contraction on the right. It is then interesting to pursue the question of whether contraction on the left is required in the propositional case. Dyckhoff =-=[4]-=- has shown that it is possible to use a more intricate proof system in which contraction is not needed at all for any propositional fragment. In our case, we are interested in determining whether the ... |

91 | Forum: A multiple-conclusion specification logic
- Miller
- 1996
(Show Context)
Citation Context ...his process can be applied to logics other than classical (or intuitionistic) logic. For example, a number of logic programming languages have been derived from linear logic including Lygon[11], Forum=-=[15]-=-, LinLog [1], LO [2], Lolli [12], ACL [13], and LC [21]. A popular proof search strategy is the notion of goal-directed proof [16], which, roughly speaking, requires that the goal be decomposed before... |

90 |
Automated deduction in nonclassical logic
- Wallen
- 1990
(Show Context)
Citation Context ...entification of logic programming languages in intuitionistic logic is a solved problem. However, it is less widely known that there are multiple-conclusioned sequent calculi for intuitionistic logic =-=[22]-=-. Whilst these are not as well known as LJ, they have been of some interest for the relationship between intuitionistic and classical inference [20]. Given such inference systems, the question natural... |

70 | A Uniform Proof-theoretic Investigation of Linear Logic Programming
- Pym, Harland
- 1994
(Show Context)
Citation Context ...s is a particularly interesting question given that there has been a significant amount of investigation of notions of goal-directed provability for multiple-conclusioned systems such as linear logic =-=[1, 15, 19, 21]-=- and classical logic [10, 18]. Thus it seems appropriate to investigate the design of logic programming languages via goal-directed provability for a multiple-conclusioned system for intuitionistic lo... |

55 |
A fixpoint semantics for disjunctive logic programs
- Minker, Rajasekar
- 1990
(Show Context)
Citation Context ... as allowing implications, universal quantifiers, and negations in the bodies of clauses [3, 16], incorporation of higher-order facilities [16], and negations and disjunctions in the heads of clauses =-=[17]-=-. Secondly, this process can be applied to logics other than classical (or intuitionistic) logic. For example, a number of logic programming languages have been derived from linear logic including Lyg... |

47 | ACL–A Concurrent Linear Logic Programming Paradigm
- Kobayashi, Yonezawa
- 1993
(Show Context)
Citation Context ... than classical (or intuitionistic) logic. For example, a number of logic programming languages have been derived from linear logic including Lygon[11], Forum[15], LinLog [1], LO [2], Lolli [12], ACL =-=[13]-=-, and LC [21]. A popular proof search strategy is the notion of goal-directed proof [16], which, roughly speaking, requires that the goal be decomposed before the program, and hence the computation us... |

41 | Programming in Lygon: An overview
- Harland, Pym, et al.
- 1996
(Show Context)
Citation Context ...Secondly, this process can be applied to logics other than classical (or intuitionistic) logic. For example, a number of logic programming languages have been derived from linear logic including Lygon=-=[11]-=-, Forum[15], LinLog [1], LO [2], Lolli [12], ACL [13], and LC [21]. A popular proof search strategy is the notion of goal-directed proof [16], which, roughly speaking, requires that the goal be decomp... |

21 | On the intuitionistic force of classical search
- Ritter, Pym, et al.
(Show Context)
Citation Context ...nclusioned sequent calculi for intuitionistic logic [22]. Whilst these are not as well known as LJ, they have been of some interest for the relationship between intuitionistic and classical inference =-=[20]-=-. Given such inference systems, the question naturally arises as to what logic programming languages would look like in such systems, and what the results of the previous analysis would be. This is a ... |

14 | A deterministic terminating sequent calculus for Gödel-Dummett logic
- Dyckhoff
- 1999
(Show Context)
Citation Context ...! (D 1 _ D 2 ) is not intuitionistically equivalent to (G ! D 1 ) _ (G ! D 2 ). However, it should be noted that this equivalence does hold in a slightly stronger logic (called Godel-Dummett logic in =-=[5]-=-), in which this is one of the Independence of Premise rules. This logic is also relevant to issues of program equivalence [8], and so an investigation of the proof theory of such a logic and its rela... |

14 | A proof-theoretic analysis of goal-directed provability
- Harland
- 1994
(Show Context)
Citation Context ...ary Harrop formulae are a logic programming language in intuitionistic logic, using goal-directed proof search in LJ. Further, there is evidence that this class of formulae is, in some sense, maximal =-=[9]-=- (at least for the first-order case). Thus it would seem that the identification of logic programming languages in intuitionistic logic is a solved problem. However, it is less widely known that there... |

8 | On Normal Forms and Equivalence for Logic Programs
- Harland
- 1992
(Show Context)
Citation Context ...nce does hold in a slightly stronger logic (called Godel-Dummett logic in [5]), in which this is one of the Independence of Premise rules. This logic is also relevant to issues of program equivalence =-=[8]-=-, and so an investigation of the proof theory of such a logic and its relation to LM would be particularly interesting. 7 Acknowledgements The authors are grateful to Pablo Armelin, Roy Dyckhoff and D... |

8 | Concurrent Logic Programming as Uniform Linear Proofs
- Volpe
- 1994
(Show Context)
Citation Context ...al (or intuitionistic) logic. For example, a number of logic programming languages have been derived from linear logic including Lygon[11], Forum[15], LinLog [1], LO [2], Lolli [12], ACL [13], and LC =-=[21]-=-. A popular proof search strategy is the notion of goal-directed proof [16], which, roughly speaking, requires that the goal be decomposed before the program, and hence the computation uses the progra... |

7 | Uniform provability in classical logic
- Nadathur
- 1998
(Show Context)
Citation Context ...tion given that there has been a significant amount of investigation of notions of goal-directed provability for multiple-conclusioned systems such as linear logic [1, 15, 19, 21] and classical logic =-=[10, 18]-=-. Thus it seems appropriate to investigate the design of logic programming languages via goal-directed provability for a multiple-conclusioned system for intuitionistic logic, and to compare the resul... |

6 | Logic Programming with Linear Logic - Winikoff - 1997 |

5 | On Goal-directed Provability in Classical Logic
- Harland, Pym
- 1992
(Show Context)
Citation Context ...tion given that there has been a significant amount of investigation of notions of goal-directed provability for multiple-conclusioned systems such as linear logic [1, 15, 19, 21] and classical logic =-=[10, 18]-=-. Thus it seems appropriate to investigate the design of logic programming languages via goal-directed provability for a multiple-conclusioned system for intuitionistic logic, and to compare the resul... |

4 | Towards the Automation of the Design of Logic Programming Languages.Technical Report 97-30
- Lutovac, Harland
- 1997
(Show Context)
Citation Context ...a of an inference is the formula which is present in the conclusion, but not in the premise(s). Intuitively, the inference converts the active formulae into the principal formula (but as discussed in =-=[14]-=-, this is sometimes too simplistic). When looking to permute the order of two inferences, it is necessary to check that the principal formula of the upper inference is not an active formula of the low... |