## A Proof-Theoretic Analysis of Goal-Directed Provability (1992)

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Venue: | Journal of Logic and Computation |

Citations: | 14 - 7 self |

### BibTeX

@ARTICLE{Harland92aproof-theoretic,

author = {James Harland},

title = {A Proof-Theoretic Analysis of Goal-Directed Provability},

journal = {Journal of Logic and Computation},

year = {1992},

volume = {4},

pages = {4--1}

}

### Years of Citing Articles

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### Abstract

One of the distinguishing features of logic programming seems to be the notion of goal-directed provability, i.e. that the structure of the goal is used to determine the next step in the proof search process. It is known that by restricting the class of formulae it is possible to guarantee that a certain class of proofs, known as uniform proofs, are complete with respect to provability in intuitionistic logic. In this paper we explore the relationship between uniform proofs and classes of formulae more deeply. Firstly we show that uniform proofs arise naturally as a normal form for proofs in first-order intuitionistic sequent calculus. Next we show that the class of formulae known as hereditary Harrop formulae are intimately related to uniform proofs, and that we may extract such formulae from uniform proofs in two different ways. We also give results which may be interpreted as showing that hereditary Harrop formulae are the largest class of formulae for which uniform proo...

### Citations

1855 | Foundations of Logic Programming - Lloyd - 1987 |

696 | A Framework for Defining Logics
- Harper, Honsel, et al.
- 1993
(Show Context)
Citation Context ...uch properties seem precisely what is desired by the above notion of input-output behaviour. Indeed, there is growing interest in the application of constructive logics to computer science in general =-=[24, 16, 5]-=-. When applying intuitionistic logic to programming tasks, we are usually more interested in consequence than theoremhood, in that it is not so important to determine whether a given formula is true, ... |

686 | Introduction to Metamathematics - Kleene - 1952 |

620 | The Art of Prolog - Sterling, Shapiro - 1986 |

304 |
An Introduction To Mathematical Logic and Type Theory: To Truth Through Proof
- Andrews
- 1986
(Show Context)
Citation Context ...on-free formula. Then for any definite condition D of F and definite consequence G of F 1. D ` I F 2. F ` I G 16 The proof is straightforward, and similar to that of the Substitutivity of Implication =-=[1]-=-, and hence is omitted. We may think of this as stating that D has more explicit information than F , so that if we were to consider an ordering of formulae in which F 1sF 2 iff F 2 ` I F 1 , then the... |

186 | Computability and logic - Boolos, Burgess, et al. - 2007 |

82 | N-prolog: an extension of prolog with hypothetical implications. i - Gabbay, Reyle - 1984 |

65 | Lexical scoping as universal quantification - Miller - 1989 |

61 | Higher-Order Horn Clauses - Nadathur, Miller |

46 | The uniform proof-theoretic foundation of linear logic programming (extended abstract - Harland, Pym - 1991 |

21 | A Logical Analysis of Modules - Miller - 1989 |

20 | Computational foundations of basic recursive function theory
- Constable, Smith
- 1993
(Show Context)
Citation Context ...uch properties seem precisely what is desired by the above notion of input-output behaviour. Indeed, there is growing interest in the application of constructive logics to computer science in general =-=[24, 16, 5]-=-. When applying intuitionistic logic to programming tasks, we are usually more interested in consequence than theoremhood, in that it is not so important to determine whether a given formula is true, ... |

19 | Clausal intuitionistic logic I. fixed point semantics - McCarty - 1988 |

19 |
Uniform Proofs as a Foundation for Logic
- Miller, Nadathur, et al.
(Show Context)
Citation Context ...rhaps not surprising that this class of formulae has such a relatively strong property. There have been various schemes proposed for logic programming languages which 1 are extensions of Horn clauses =-=[2, 8, 9, 22, 23, 28, 29, 31]-=-. Given these various extensions, it seems natural to ask whether there is a maximal class of formulae which may be used as a programming language. Moreover, there does not seem to be universally agre... |

13 | On Hereditary Harrop Formulae as a basis for Logic Programming - Harland - 1991 |

13 |
Some applications of Gentzenâ€™s proof theory to automated deduction
- Beeson
- 1991
(Show Context)
Citation Context ...rhaps not surprising that this class of formulae has such a relatively strong property. There have been various schemes proposed for logic programming languages which 1 are extensions of Horn clauses =-=[2, 8, 9, 22, 23, 28, 29, 31]-=-. Given these various extensions, it seems natural to ask whether there is a maximal class of formulae which may be used as a programming language. Moreover, there does not seem to be universally agre... |

8 | On Normal Forms and Equivalence for Logic Programs
- Harland
- 1992
(Show Context)
Citation Context ...succeeds for every t 2 fa; f(a); f(f(a)); : : :g, but p(y) fails. 3 . 2 In fact, the natural logic in which to interpret hereditary Harrop formulae is slightly stronger than intuitionistic logic; see =-=[11, 13]-=- for details. 3 Note, however, that the goal 9xp(x) succeeds 20 p(a) 8x p(f(x))p(x) Hence we may wish for a slightly stronger rule for universal quantification, in that if P ` u p(y), then P ` u p(t) ... |

7 | The permutability of rules in the classical inferential calculus - Curry - 1952 |

5 | Some Uses of Higher-Order Logic - Miller, Nadathur - 1986 |

5 |
An Overview of
- Nadathur, Miller
- 1988
(Show Context)
Citation Context ... limitations of uniform proofs and some possibilities for further work. 2 Preliminaries First we define hereditary Harrop formulae. Applications of these formulae to logic programming may be found in =-=[28, 25, 26, 29, 30]-=-. We assume the existence of a finite set of constant and function symbols, and a countable set of variables. Definition 2.1 We refer to the set of all ground terms as the Herbrand universe, denoted b... |

3 | An Intermediate Logic for Logic Programs - Harland - 1990 |

3 | Uniform Proofs as a Foundation for - Miller, Nadathur, et al. - 1991 |

2 | A Proof-Theoretic Analysis of Logic Programming - Harland - 1990 |

1 | Hypothetical Reasoning and Definitional Reflection - Schroeder-Heister - 1989 |

1 | A Intuitionistic Sequent Calculus B;C - Tarnlund - 1977 |

1 |
Logic Programming in a Fragment of Intuitonistic Linear Logic
- Hodas, Miller
- 1991
(Show Context)
Citation Context ... of goal-directed provability may used to identify logic programming languages independently of the logic in use; such a strategy has been used to identify logic programming languages in linear logic =-=[14, 15, 17]-=-. 3 In this way we may think of the permutation properties of the proof theory of the logic in question (in conjunction with the notion of goal-directed search) as determining what fragments of the lo... |

1 |
Intuitionism: An Introduction to a Seminar
- McCarty
- 1983
(Show Context)
Citation Context ...t merely that p(x) is true, but the terms t for which p(t) is true. This immediately suggests that we are interested in constructive proofs, for which a natural starting place is intuitionistic logic =-=[7, 18, 10, 21]-=-. This accords well with the work of Miller et al. [28], in that in all their logic programming languages, a goal is derivable iff it is provable intuitionistically. A well-known property of intuition... |

1 |
Higher-Order Horn Clauses Journal of the Association for Computing Machinery 37:4:777-814, ctober
- Nadathur, Miller
- 1990
(Show Context)
Citation Context ... limitations of uniform proofs and some possibilities for further work. 2 Preliminaries First we define hereditary Harrop formulae. Applications of these formulae to logic programming may be found in =-=[28, 25, 26, 29, 30]-=-. We assume the existence of a finite set of constant and function symbols, and a countable set of variables. Definition 2.1 We refer to the set of all ground terms as the Herbrand universe, denoted b... |