## On Goal-Directed Provability in Classical Logic (1994)

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### BibTeX

@MISC{Harland94ongoal-directed,

author = {James Harland and Royal Melbourne},

title = {On Goal-Directed Provability in Classical Logic},

year = {1994}

}

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### Abstract

this paper we explore the possibilities for a notion of goal-directed proof in classical logic. The technical point to consider is how to deal with the multipleconclusioned nature of classical sequents, i.e. that classical succedents may contain more than one formula. This means that there may be more than one "candidate" right rule, as there may be several non-atomic formulae in the succedent, and so a choice has to be made as to which formula is to be reduced in the next step. The question of whether this choice may be free or restricted is one of the key decisions to be made. The free choice, i.e. that the order in which the formulae are reduced does not matter, will clearly constrain the logic programming language more than the restricted, one, and is arguably more declarative; on the other hand, the weaker notion is arguably more goal-directed, and there is no obvious reason to insist on the stronger version. We will refer to the free choice as right-reductive proofs, and to the restricted one as right-directed proofs. Thus there seems to be more than one notion of goal-directed proof in classical logic, and clearly the corresponding logic programming languages may differ according to which class of proofs is used. However, as we shall see, there are do not seem to be any "interesting" languages for which the weaker notion is complete but the stronger one is not, and so it appears the stronger version (which requires that all right rules permute over each other) is the more useful notion.

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Citation Context ...ons between this work and the notion of goal-directness in intuitionistic logic, and possibilities for further work. 2 Preliminaries Below we review the sequent calculus for classical logic (see e.g. =-=[8]-=-). We will refer to this proof system as LK. We write sequents as \Gamma \Gamma! \Delta, where both \Gamma and \Delta may contain an arbitrary number of formulae. In a sequent \Gamma \Gamma! \Delta, \... |

592 |
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Citation Context ... sequent is defined. We will often refer to the rules -L, -L, 9-L, 8-L and oe-L as left rules, and the rules -R, -R, 9-R, 8-R and oe-R as right rules. The cut rule, by a famous theorem due to Gentzen =-=[4]-=-, is eliminable. Theorem 2.1 (Cut elimination for LK) \Gamma `LK \Delta iff \Gamma \Gamma! \Delta has a proof not containing the cut rule. The significance of this result is that we may omit considera... |

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(Show Context)
Citation Context .... Hence, for either conception of goal-directed provability, there seems to be no inherent reason why programs cannot contain clauses of the form psq. Thus we will show how disjunctive logic programs =-=[12]-=- may be given a proof-theoretic basis. We show how both notions of goal-directed provability may be used to determine classes of formulae for which these notions are complete. In particular, for the w... |

47 | The uniform proof-theoretic foundation of linear logic programming
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- 1994
(Show Context)
Citation Context ...ructure of the succedent, i.e. the goal. This particular notion of goal directed provability has been studied in various ways, [11, 10, 6], and has also lead to similar investigations in linear logic =-=[7]-=-. One thing that should be noted about such investigations is that they are all concerned with the completeness properties of certain classes of proofs. In particular, for the intuitionistic case, a u... |

28 | Proof search in first-order linear logic and other cut-free sequent calculi, in - Lincoln, Shankar - 1994 |

22 |
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(Show Context)
Citation Context ...quence of steps in the search for a uniform proof is determined by the structure of the succedent, i.e. the goal. This particular notion of goal directed provability has been studied in various ways, =-=[11, 10, 6]-=-, and has also lead to similar investigations in linear logic [7]. One thing that should be noted about such investigations is that they are all concerned with the completeness properties of certain c... |

19 |
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Citation Context ...e succedent until it can be reduced no further, and then reducing the antecedent. The best known characterization of goal-directed provability is the notion of a uniform proof in intuitionistic logic =-=[11]-=-. An intuitionistic proof is uniform if for any sequent in the proof for which the succedent is not an atom, the rule used to derive the sequent is the introduction rule for the principle connective o... |

14 | On Hereditary Harrop Formulae as a Basis for Logic Programming - Harland - 1991 |

14 | A proof-theoretic analysis of goal-directed provability - Harland - 1994 |

8 |
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Citation Context ... Negation as Failure, in that we will be able to give NAF a purely proof-theoretic foundation. We discuss some possibilities of the application of these results in this direction, similar to those of =-=[14]-=-. This paper is organised as follows. In Section 2 we briefly review the classical sequent calculus and discuss some technical preliminaries. In Section 3 we discuss the notions of goal-directed prova... |

7 |
The permutability of rules in the classical inferential calculus
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(Show Context)
Citation Context ...erties of the logic, i.e. when the order of two rules in a proof can be interchanged. An exhaustive analysis of the permutation properties of classical logic has been provided by Kleene [9] and Curry =-=[3]-=-. In particular, the following lemma is a trivial consequence of the main theorem of [9]. Lemma 3.2 (Kleene's Permutation Lemma) If there is a proof in which an occurrence of a right rule occurs immed... |

6 |
Permutability of inferences in Gentzen's calculi
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- 1952
(Show Context)
Citation Context ...rmutation properties of the logic, i.e. when the order of two rules in a proof can be interchanged. An exhaustive analysis of the permutation properties of classical logic has been provided by Kleene =-=[9]-=- and Curry [3]. In particular, the following lemma is a trivial consequence of the main theorem of [9]. Lemma 3.2 (Kleene's Permutation Lemma) If there is a proof in which an occurrence of a right rul... |

6 | Private Communication - McCarty |

2 |
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- 1990
(Show Context)
Citation Context ...quence of steps in the search for a uniform proof is determined by the structure of the succedent, i.e. the goal. This particular notion of goal directed provability has been studied in various ways, =-=[11, 10, 6]-=-, and has also lead to similar investigations in linear logic [7]. One thing that should be noted about such investigations is that they are all concerned with the completeness properties of certain c... |

1 |
Herbrand Methods in Linear Logic, to appear
- Cerrito
- 1992
(Show Context)
Citation Context ...rch strategies, and so are vital to the soundness of the Negation as Failure (NAF) rule. Clearly if we use a proof search 1 Hence we remove all quantifiers of universal strength in the terminology of =-=[1]-=-. method which is incomplete, then we cannot apply the NAF rule with any surety. However, given a complete set of reduction properties, it is possible to use the unprovability of the reduced sequents ... |