## Defaults as restrictions on classical Hilbert-style proofs (1995)

Venue: | Journal of Logic Language and Information |

Citations: | 4 - 3 self |

### BibTeX

@ARTICLE{Amati95defaultsas,

author = {Gianni Amati and Luigia Carlucci Aiello and Fiora Pirri},

title = {Defaults as restrictions on classical Hilbert-style proofs},

journal = {Journal of Logic Language and Information},

year = {1995},

volume = {3}

}

### OpenURL

### Abstract

Since the earliest formalisation of default logic by Reiter many contributions to this appealing approach to nonmonotonic reasoning have been given. The different formalisations are here presented in a general framework that gathers the basic notions, concepts and constructions underlying default logic. Our view is to interpret defaults as special rules that impose a restriction on the juxtaposition of monotonic Hilbert-style proofs of a given logic L. We propose to describe default logic as a logic where the juxtaposition of default proofs is subordinate to a restriction condition \Psi. Hence a default logic is a pair (L; \Psi) where properties of the logic L, like compactness, can be interpreted through the restriction condition \Psi. Different default systems are then given a common characterization through a specific condition on the logic L. We also prove cumulativity for any default logic (L; \Psi) by slightly modifying the notion of default proof. We extend, in fact, the language of L in a way close to that followed by Brewka in the formulation of his cumulative default system. Finally we show the existence of infinitely many intermediary default logics, depending on \Psi and called linear logics, which lie between Reiter's and / Lukaszewicz' versions of default logic.

### Citations

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Citation Context ...ons of default logic. Key words: Artificial Intelligence, Knowledge Representation, Non-Monotonic Reasoning, Default Logic, Proof Theory. 1. Introduction Default logic, first introduced by Reiter in (=-=Reiter, 1980), is one -=-of the main formalisations of nonmonotonic reasoning. The basic idea was to allow reasoning with patterns such as "In the absence of any information to the contrary assume A" or any paraphra... |

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Citation Context ... properties of the nonmonotonic consequence operator, is considered for arbitrary set of premises; see also (Makinson, 1993). Successively cumulativity has been studied by Kraus, Lehmann and Magidor (=-=Kraus et al., 1990-=-), by (Brewka, 1991) and finally in the general framework for nonmonotonic inference presented in (Gardenfors and Makinson, 1994), where cumulativity follows from the set of given postulates. Cumulati... |

185 |
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Citation Context ...n fact relation \Psi is not affected by the extension of the language. In Section 4 we study cumulativity. The cumulativity postulate for the nonmonotonic consequence relation j was first studied in (=-=Gabbay, 1985-=-; Clarke and Gabbay, 1988) and in (Makinson, 1988), where cumulativity, together with other properties of the nonmonotonic consequence operator, is considered for arbitrary set of premises; see also (... |

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(Show Context)
Citation Context ...ion of defaults such as ordered theories, or on the language such as Horn theories, etc.; see among the others (Reiter, 1980; Besnard et al., 1983; Etherington, 1987; Poole et al., 1988; Poole, 1988; =-=Poole, 1989-=-; Poole, 1989; Junker, 1989; Schwind, 1990; MacNish, 1991; Junker and Brewka, 1991; Poole, 1991; Schwind and Risch, 1991; Levy, 1991). A general proof-theoretical method is proposed in (Amati et al., ... |

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Citation Context ...blem. A further example is the lack of cumulativity of the system, that is, the set of consequences of a default theory may not be preserved when some consequence is considered as a new premise; see (=-=Makinson, 1988-=-; Brewka, 1991; Giordano and Martelli, 1994) among the others. Given such a variety of contributions one may ask whether there is a canonical approach to default logic which subsumes the different var... |

101 |
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Citation Context ... a fix-point construction. The idea of defining a restriction relation on the Hilbert-style proofs is not novel in logic. In Relevance Logic, for example, the Relevant Deduction Theorem of MohChurch (=-=Dunn, 1986-=-) asserts that the relevant proofs from a set of assumptions (a set of proper axioms) are those, among the set of standard proofs, in which all the axioms are actually used at least once in the proof.... |

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(Show Context)
Citation Context ...lations of default logic have been proposed, see for instance (/Lukaszewicz, 1984; / Lukaszewicz, 1988; Guerreiro et al., 1990; Delgrande and Jackson, 1991; Brewka, 1991; Rychlik, 1991; Schaub, 1991; =-=Gelfond et al., 1991-=-; Doherty and / Lukaszewic, 1992; Giordano and Martelli, 1994). Each of them tries to overcome some unintuitive behaviour of Reiter's system. One flaw of Reiter's formulation is the lack of semimonoto... |

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(Show Context)
Citation Context ...the presentation of defaults such as ordered theories, or on the language such as Horn theories, etc.; see among the others (Reiter, 1980; Besnard et al., 1983; Etherington, 1987; Poole et al., 1988; =-=Poole, 1988-=-; Poole, 1989; Poole, 1989; Junker, 1989; Schwind, 1990; MacNish, 1991; Junker and Brewka, 1991; Poole, 1991; Schwind and Risch, 1991; Levy, 1991). A general proof-theoretical method is proposed in (A... |

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Citation Context ...ion of defaults such as ordered theories, or on the language such as Horn theories, etc.; see among the others (Reiter, 1980; Besnard et al., 1983; Etherington, 1987; Poole et al., 1988; Poole, 1988; =-=Poole, 1989-=-; Poole, 1989; Junker, 1989; Schwind, 1990; MacNish, 1991; Junker and Brewka, 1991; Poole, 1991; Schwind and Risch, 1991; Levy, 1991). A general proof-theoretical method is proposed in (Amati et al., ... |

36 |
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(Show Context)
Citation Context ..., for example, a default theory may have either none, one or more extensions. Following Reiter many different formulations of default logic have been proposed, see for instance (/Lukaszewicz, 1984; / =-=Lukaszewicz, 1988-=-; Guerreiro et al., 1990; Delgrande and Jackson, 1991; Brewka, 1991; Rychlik, 1991; Schaub, 1991; Gelfond et al., 1991; Doherty and / Lukaszewic, 1992; Giordano and Martelli, 1994). Each of them tries... |

30 | Compiling a default reasoning system into prolog. New Generation Computing 9(1):3–38
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(Show Context)
Citation Context ...mong the others (Reiter, 1980; Besnard et al., 1983; Etherington, 1987; Poole et al., 1988; Poole, 1988; Poole, 1989; Poole, 1989; Junker, 1989; Schwind, 1990; MacNish, 1991; Junker and Brewka, 1991; =-=Poole, 1991-=-; Schwind and Risch, 1991; Levy, 1991). A general proof-theoretical method is proposed in (Amati et al., 1995) where tableaux are used to provide an alternative characterisation to the fix-point defin... |

23 |
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(Show Context)
Citation Context ...o the default systems. In other words, our formulation of (L; \Psi) forces the proof theory of default logic to behave, as much as possible, as that of L. This view is close to the idea expressed in (=-=Gardenfors and Makinson, 1994-=-) that there is no need for a special formalism to express default reasoning - though expectation inference operations are used in (Gardenfors and Makinson, 1994) instead of restriction conditions on ... |

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(Show Context)
Citation Context ...perty of Reiter's, Brewka's and / Lukaszewicz' default systems. This notion of compactness can be related to the one given by Freund, Lehmann and Makinson (Freund et al., 1989) and deeply studied in (=-=Freund and Lehmann, 1993-=-) and (Freund and Lehmann, 1994). In Section 2 we discuss the relation between these two notions of compactness. A formulation of default logic without resorting to fix-points is also presented in Fro... |

16 |
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(Show Context)
Citation Context ...fault logic by using the property of anti-monotonicity of Reiter's MAIN-Kluwer.tex; 16/01/1995; 12:19; no v.; p.7 Defaults as restrictions on classical Hilbert-style proofs 7 fix-point operator; see (=-=Dix, 1992-=-). In particular he shows that the class of normal prerequisite-free defaults is cumulative, though it does not satisfy the property of negation rationality which implies cautious monotony. The proof ... |

15 |
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(Show Context)
Citation Context ...the language of L to the language AL in which a formula ' is equipped with a set of formulae (the support for ') to formulate cumulativity for (L; \Psi). The notion of support was earlier devised in (=-=Sandewall, 1985-=-) and developped in (/Lukaszewicz, 1988) and in (Brewka, 1991). In particular we introduce the definition of supported default proof, that is, a \Psi-proof considered together with the set of justific... |

13 |
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(Show Context)
Citation Context ...iter's default system, for example, a default theory may have either none, one or more extensions. Following Reiter many different formulations of default logic have been proposed, see for instance (/=-=Lukaszewicz, 1984-=-; / Lukaszewicz, 1988; Guerreiro et al., 1990; Delgrande and Jackson, 1991; Brewka, 1991; Rychlik, 1991; Schaub, 1991; Gelfond et al., 1991; Doherty and / Lukaszewic, 1992; Giordano and Martelli, 1994... |

10 | Nonmonotonic reasoning: from finitary relations to infinitary inference operations
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(Show Context)
Citation Context ... / Lukaszewicz' default systems. This notion of compactness can be related to the one given by Freund, Lehmann and Makinson (Freund et al., 1989) and deeply studied in (Freund and Lehmann, 1993) and (=-=Freund and Lehmann, 1994-=-). In Section 2 we discuss the relation between these two notions of compactness. A formulation of default logic without resorting to fix-points is also presented in Froidevaux and Mengin; see (Froide... |

9 |
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(Show Context)
Citation Context ...ugh a generalisation of this compactness property of Reiter's, Brewka's and / Lukaszewicz' default systems. This notion of compactness can be related to the one given by Freund, Lehmann and Makinson (=-=Freund et al., 1989-=-) and deeply studied in (Freund and Lehmann, 1993) and (Freund and Lehmann, 1994). In Section 2 we discuss the relation between these two notions of compactness. A formulation of default logic without... |

9 |
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- Schaub
- 1991
(Show Context)
Citation Context ...ifferent formulations of default logic have been proposed, see for instance (/Lukaszewicz, 1984; / Lukaszewicz, 1988; Guerreiro et al., 1990; Delgrande and Jackson, 1991; Brewka, 1991; Rychlik, 1991; =-=Schaub, 1991-=-; Gelfond et al., 1991; Doherty and / Lukaszewic, 1992; Giordano and Martelli, 1994). Each of them tries to overcome some unintuitive behaviour of Reiter's system. One flaw of Reiter's formulation is ... |

8 |
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- 2000
(Show Context)
Citation Context ...n \Psi is not affected by the extension of the language. In Section 4 we study cumulativity. The cumulativity postulate for the nonmonotonic consequence relation j was first studied in (Gabbay, 1985; =-=Clarke and Gabbay, 1988-=-) and in (Makinson, 1988), where cumulativity, together with other properties of the nonmonotonic consequence operator, is considered for arbitrary set of premises; see also (Makinson, 1993). Successi... |

7 |
Cumulative default logic: in defence of nonmonotonic inference rules
- Brewka
- 1991
(Show Context)
Citation Context ...ions. Following Reiter many different formulations of default logic have been proposed, see for instance (/Lukaszewicz, 1984; / Lukaszewicz, 1988; Guerreiro et al., 1990; Delgrande and Jackson, 1991; =-=Brewka, 1991-=-; Rychlik, 1991; Schaub, 1991; Gelfond et al., 1991; Doherty and / Lukaszewic, 1992; Giordano and Martelli, 1994). Each of them tries to overcome some unintuitive behaviour of Reiter's system. One fla... |

7 |
A correct non-monotonic ATMS
- Junker
- 1989
(Show Context)
Citation Context ...dered theories, or on the language such as Horn theories, etc.; see among the others (Reiter, 1980; Besnard et al., 1983; Etherington, 1987; Poole et al., 1988; Poole, 1988; Poole, 1989; Poole, 1989; =-=Junker, 1989-=-; Schwind, 1990; MacNish, 1991; Junker and Brewka, 1991; Poole, 1991; Schwind and Risch, 1991; Levy, 1991). A general proof-theoretical method is proposed in (Amati et al., 1995) where tableaux are us... |

6 | A proof theoretical approach to default reasoning I: tableaux for default logic
- Amati, Aiello, et al.
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(Show Context)
Citation Context ...8; Poole, 1989; Poole, 1989; Junker, 1989; Schwind, 1990; MacNish, 1991; Junker and Brewka, 1991; Poole, 1991; Schwind and Risch, 1991; Levy, 1991). A general proof-theoretical method is proposed in (=-=Amati et al., 1995-=-) where tableaux are used to provide an alternative characterisation to the fix-point definition for any default logic (Reiter's default system included). This approach is inspired by a generalisation... |

5 | Nml3, a non-monotonic logic with explicit default - Doherty, Lukaszewicz - 1992 |

5 |
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- 1994
(Show Context)
Citation Context ...nstance (/Lukaszewicz, 1984; / Lukaszewicz, 1988; Guerreiro et al., 1990; Delgrande and Jackson, 1991; Brewka, 1991; Rychlik, 1991; Schaub, 1991; Gelfond et al., 1991; Doherty and / Lukaszewic, 1992; =-=Giordano and Martelli, 1994-=-). Each of them tries to overcome some unintuitive behaviour of Reiter's system. One flaw of Reiter's formulation is the lack of semimonotonicity 1 of the system: in some cases, when adding new defaul... |

5 |
Handling partially ordered defaults in TMS
- Junker, Brewka
- 1991
(Show Context)
Citation Context ...orn theories, etc.; see among the others (Reiter, 1980; Besnard et al., 1983; Etherington, 1987; Poole et al., 1988; Poole, 1988; Poole, 1989; Poole, 1989; Junker, 1989; Schwind, 1990; MacNish, 1991; =-=Junker and Brewka, 1991-=-; Poole, 1991; Schwind and Risch, 1991; Levy, 1991). A general proof-theoretical method is proposed in (Amati et al., 1995) where tableaux are used to provide an alternative characterisation to the fi... |

5 |
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- Rychlik
- 1991
(Show Context)
Citation Context ...g Reiter many different formulations of default logic have been proposed, see for instance (/Lukaszewicz, 1984; / Lukaszewicz, 1988; Guerreiro et al., 1990; Delgrande and Jackson, 1991; Brewka, 1991; =-=Rychlik, 1991-=-; Schaub, 1991; Gelfond et al., 1991; Doherty and / Lukaszewic, 1992; Giordano and Martelli, 1994). Each of them tries to overcome some unintuitive behaviour of Reiter's system. One flaw of Reiter's f... |

4 |
Contribution to a proof theory for generic defaults
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- 1990
(Show Context)
Citation Context ...fault theory may have either none, one or more extensions. Following Reiter many different formulations of default logic have been proposed, see for instance (/Lukaszewicz, 1984; / Lukaszewicz, 1988; =-=Guerreiro et al., 1990-=-; Delgrande and Jackson, 1991; Brewka, 1991; Rychlik, 1991; Schaub, 1991; Gelfond et al., 1991; Doherty and / Lukaszewic, 1992; Giordano and Martelli, 1994). Each of them tries to overcome some unintu... |

4 | Hierarchical default logic - MacNish - 1991 |

3 |
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- Etherington
- 1987
(Show Context)
Citation Context ..., mainly based on some restrictions on the presentation of defaults such as ordered theories, or on the language such as Horn theories, etc.; see among the others (Reiter, 1980; Besnard et al., 1983; =-=Etherington, 1987-=-; Poole et al., 1988; Poole, 1988; Poole, 1989; Poole, 1989; Junker, 1989; Schwind, 1990; MacNish, 1991; Junker and Brewka, 1991; Poole, 1991; Schwind and Risch, 1991; Levy, 1991). A general proof-the... |

2 |
Computing extensions of default theories
- Levy
- 1991
(Show Context)
Citation Context ... et al., 1983; Etherington, 1987; Poole et al., 1988; Poole, 1988; Poole, 1989; Poole, 1989; Junker, 1989; Schwind, 1990; MacNish, 1991; Junker and Brewka, 1991; Poole, 1991; Schwind and Risch, 1991; =-=Levy, 1991-=-). A general proof-theoretical method is proposed in (Amati et al., 1995) where tableaux are used to provide an alternative characterisation to the fix-point definition for any default logic (Reiter's... |