## A theory of nonmonotonic rule systems I (1990)

### Cached

### Download Links

- [www.cs.engr.uky.edu]
- [www.cs.uky.edu]
- [ftp.ms.uky.edu]
- [www.cs.engr.uky.edu]
- [www.cs.uky.edu]
- DBLP

### Other Repositories/Bibliography

Citations: | 40 - 16 self |

### BibTeX

@MISC{Marek90atheory,

author = {W. Marek and A. Nerode and J. Remmel},

title = {A theory of nonmonotonic rule systems I},

year = {1990}

}

### Years of Citing Articles

### OpenURL

### Abstract

this paper. Here, drawing on all the research mentioned above for inspiration, we present a coherent unified theory of nonmonotonic formal systems. At the level of abstraction we achieve, we are finally able to see that nonmonotone systems pervade ordinary mathematical practice. There is no sign of any realization of the existence of such mathematical examples in the previous nonmonotonic logic literature. Perhaps these connections can only be seen by having a common abstract notion. What this commonality does for us is to make available known mathematical techniques from other areas of conventional mathematics for constructing and classifying belief sets (extensions) and, simultaneously, to make evident a common thread among disparate parts of mathematics and disparate nonmonotonic systems from artificial intelligence and computer science. On the level of Mathematical Philosophy there is a connection worth stating as well. Non-monotone reasoning takes place during the process of discovery of mathematical theorems, when one posits temporarily some proposition on the basis of no evidence against it, and explores the consequences of such a belief until new mathematical facts force their abandonment. These nonmonotone belief sets have their traces eradicated when final belief sets are achieved and demonstrative proofs are finished and published. The only hint of provisional belief sets left in mathematical papers is in the motivational remarks explaining what obstacles were overcome and by what changes in viewpoint the proof was achieved. Here is the main definition. A nonmonotone rule system consists of a set U and a set of triples (ff; fi; fl) called rules. Here ff = (ff 1 ; : : : ; ff n ) is a finite sequence of elements of U , called premises, and fi = (fi 1 ; : : : ...

### Citations

1519 |
A logic for default reasoning
- Reiter
- 1980
(Show Context)
Citation Context ...ons in computer science and AI. Among the other systems that have been studied are The theory of multiple believers of Hintikka, [18]. Truth Maintenance systems of Doyle, [8] Default logic of Reiter, =-=[40]-=- Autoepistemic logic of Moore, [34] Theory of individual and common knowledge and belief of Halpern and Moses [17] Logic programming with negation as failure [?]. This, by no means, exhausts the list.... |

905 | Jr, Theory of recursive functions and effective computability - Rogers - 1967 |

884 |
Circumscription—a form of non-monotonic reasoning
- McCarthy
- 1980
(Show Context)
Citation Context ...of mathematics. One can envisage making up non-monotone logics describing the mathematical nature of belief. The exact result depends on the definition chosen for “lack of evidence against”. McCarthy =-=[30]-=-, initiated the study of non-monotonicity with his notion of circumscription. With all relation symbols but one, ¯ R, of a model (the world we are discussing) held fixed, and given axioms ϕ( ¯ R) rela... |

664 |
A truth maintenance system
- Doyle
- 1979
(Show Context)
Citation Context ...tracted from different questions in computer science and AI. Among the other systems that have been studied are The theory of multiple believers of Hintikka, [18]. Truth Maintenance systems of Doyle, =-=[8]-=- Default logic of Reiter, [40] Autoepistemic logic of Moore, [34] Theory of individual and common knowledge and belief of Halpern and Moses [17] Logic programming with negation as failure [?]. This, b... |

634 |
Towards a theory of declarative knowledge
- Apt, Blair, et al.
- 1988
(Show Context)
Citation Context ...neral clauses, of the form: p ← q1,...,qn, ¬r1,...,¬rm We refer to [10] for the definition of a stable model of such a program. That concept is a generalization of the perfect models as introduced in =-=[3]-=-. Let U be the collection of atoms under consideration. Represent a general clause as a rule: q1,...,qn : r1,...rm p The translation tr(P) of a program P is the set of translations of its individual c... |

526 | Knowledge and Common Knowledge in a Distributed
- Halpern, Moses
- 1990
(Show Context)
Citation Context ...s of Hintikka, [18]. Truth Maintenance systems of Doyle, [8] Default logic of Reiter, [40] Autoepistemic logic of Moore, [34] Theory of individual and common knowledge and belief of Halpern and Moses =-=[17]-=- Logic programming with negation as failure [?]. This, by no means, exhausts the list. What issues in artificial intelligence or computer science motivates these systems? Suppose that we build a robot... |

509 | The Science of Programming - Gries - 1981 |

484 |
Knowledge and Belief
- Hintikka
- 1962
(Show Context)
Citation Context ...w many different non-monotonic system, abstracted from different questions in computer science and AI. Among the other systems that have been studied are The theory of multiple believers of Hintikka, =-=[18]-=-. Truth Maintenance systems of Doyle, [8] Default logic of Reiter, [40] Autoepistemic logic of Moore, [34] Theory of individual and common knowledge and belief of Halpern and Moses [17] Logic programm... |

474 |
An assumption-based tms
- Kleer
- 1986
(Show Context)
Citation Context ...es decisions on the basis of the absence of information in the database as well. A logic appropriate for describing a single such belief set is Doyle’s truth-maintenance system [8]. See also de Kleer =-=[6]-=-. Also stable models for logic programming with negation as failure ([10]) arise in this way. We expressed these examples informally in terms of the anthropomorphic notion of belief so as to bring out... |

465 |
Semantical considerations on nonmonotonic logic
- Moore
- 1985
(Show Context)
Citation Context ...ng the other systems that have been studied are The theory of multiple believers of Hintikka, [18]. Truth Maintenance systems of Doyle, [8] Default logic of Reiter, [40] Autoepistemic logic of Moore, =-=[34]-=- Theory of individual and common knowledge and belief of Halpern and Moses [17] Logic programming with negation as failure [?]. This, by no means, exhausts the list. What issues in artificial intellig... |

437 |
Introduction to Logic Programming
- Apt
- 1990
(Show Context)
Citation Context ...“marriage problem” for highly recursive societies, or finding of recursively presented formally real field ([31]), or finding an abcissa between 0 and 1 where a given recursive continuous function on =-=[0,1]-=- takes a maximum value ([19]). Now this recursion-theoretic methodology can be refined to give complexity-theoretic results on the same problems about extensions, as has been done in algebra by Nerode... |

354 |
On representatives of subsets
- Hall
- 1935
(Show Context)
Citation Context ...(b,g). That is, in a proper marriage each boy marries a girl he knows. A marriage M is symmetric if M maps B onto G. For a symmetric marriage, every girl is married. For finite societies Philip Hall (=-=[15]-=-) gave a necessary and sufficient condition for the existence of a proper marriage, namely: (*) For every finite set of boys B ′ ⊆ B, the set of girls that the boys of B ′ know altogether has cardinal... |

280 | A Decomposition Theorem for Partially Ordered Sets, The - Dilworth - 1950 |

152 |
On the relation between default and autoepistemic logic
- Konolige
- 1988
(Show Context)
Citation Context ...rms of the anthropomorphic notion of belief so as to bring out their common features. The actual nonmonotonic logics have much in common, and a number of translations between them have been proposed (=-=[23]-=-, [12],[13], [39],[28]). They have been investigated principally for propositional logic. Predicate versions suitable for actual applications are, up to now, pretty minimal. Study of monotonic rule sy... |

139 |
Π01 classes and degrees of theories
- Soare
- 1972
(Show Context)
Citation Context ...em ST =< UT,NT > such that there is an effective one-to-one degree preserving correspondence between the set of infinite paths through T and the set of extensions of ST. It follows from the result of =-=[21]-=- that any recursively bounded Π 0 1-class can be coded as the set of infinite paths through a recursive binary tree. We transfer all the results about degrees of elements of recursively bounded Π 0 1-... |

101 |
The stable semantics for logic programs
- Gelfond, Lifschitz
- 1988
(Show Context)
Citation Context ...as well. A logic appropriate for describing a single such belief set is Doyle’s truth-maintenance system [8]. See also de Kleer [6]. Also stable models for logic programming with negation as failure (=-=[10]-=-) arise in this way. We expressed these examples informally in terms of the anthropomorphic notion of belief so as to bring out their common features. The actual nonmonotonic logics have much in commo... |

94 |
On the relationship between Circumscription and Negation as Failure
- Gelfond, Przymusinska, et al.
- 1989
(Show Context)
Citation Context ... the anthropomorphic notion of belief so as to bring out their common features. The actual nonmonotonic logics have much in common, and a number of translations between them have been proposed ([23], =-=[12]-=-,[13], [39],[28]). They have been investigated principally for propositional logic. Predicate versions suitable for actual applications are, up to now, pretty minimal. Study of monotonic rule systems ... |

73 | Marketing programs - Brown - 1994 |

64 |
General logical databases and programs: default logic semantics and strati Information and Computation, 91:15{54
- Bidoit, Froidevaux
- 1991
(Show Context)
Citation Context ...nsideration. Represent a general clause as a rule: q1,...,qn : r1,...rm p The translation tr(P) of a program P is the set of translations of its individual clauses. The following result was proved in =-=[4]-=- and [29]: Proposition 3.2 A subset M ⊆ U is a stable model of P if and only if M is an extension of tr(P). Proposition 3.3 A subset M ⊆ U is a supported model of P if and only if M is a weak extensio... |

63 |
Formalizing nonmonotonic reasoning systems
- Etherington
(Show Context)
Citation Context ...or all ϕ ∈ L, S |= Fϕ. (2) S is a default extension of < D,W >if and only if: (i) for all ϑ ∈ S, S |= prϑ. (ii) for all ϑ /∈ S, S |= ¬prϑ. Compare this result with the earlier results of Etherington (=-=[9]-=-) who characterized default extensions by means of “most preferred models”. Here we use a different device. A careful inspection of our method indicates the following: First, imbed the language L into... |

49 |
Arithmetic Classification of Perfect Models of Stratified Programs
- Apt, Blair
- 1988
(Show Context)
Citation Context ...le models of logic programs by lifting Corollary 5.2, Theorem 5.5, Corollaries 5.6, 5.7, 5.8, 5.9, 5.10, 5.11, and Theorem 5.12. It is appropriate to compare the results of this section with those of =-=[2]-=-. They construct, for a given natural number n ≥ 1, a stratified finite program P (in particular its Herbrand expansion is a recursive propositional program) whose perfect model is a complete Σ 0 n se... |

40 |
Transversal Theory
- Mirsky
- 1970
(Show Context)
Citation Context ...he existence of marriages in an infinite society S as long as each boy knows only finitely many girls. Philip Hall’s theorem is a special case of the more general problem of finding transversals (see =-=[33]-=-). We claim that if S =< B,G,K > is a society satisfying (*) in which each boy knows only finitely many girls, then there is a nonmonotonic rule system Z =< U(S),N(S) > such that the collection of ext... |

38 | The Relationship Between Logic Program Semantics and Non-Monotonic Reasoning - Marek, Subrahmanian - 1989 |

35 |
Nerode; Effective content of field theory
- Metakides, A
- 1979
(Show Context)
Citation Context ... this case, the computational problems are of the same level of difficulty as (say) solving “marriage problem” for highly recursive societies, or finding of recursively presented formally real field (=-=[31]-=-), or finding an abcissa between 0 and 1 where a given recursive continuous function on [0,1] takes a maximum value ([19]). Now this recursion-theoretic methodology can be refined to give complexity-t... |

23 |
Degrees of members of Π01 classes
- Jockusch, Soare
- 1972
(Show Context)
Citation Context ... set of all recursively enumerable degrees b ≥T a. All of the above results follow from Theorem 5.5 plus the corresponding results for recursively bounded Π 0 1-classes due to Jockusch and Soare [21] =-=[22]-=- except of Corollary 5.10 which follows from the corresponding result for recursively bounded Π 0 1-classes due to Jockusch and McLaughlin [20]. Now we give a construction of a rule system < U,N > who... |

9 |
On the Relation between Truth Maintenance and Non-Monotonic Logics
- Reinfrank, Dressler
- 1989
(Show Context)
Citation Context ...pomorphic notion of belief so as to bring out their common features. The actual nonmonotonic logics have much in common, and a number of translations between them have been proposed ([23], [12],[13], =-=[39]-=-,[28]). They have been investigated principally for propositional logic. Predicate versions suitable for actual applications are, up to now, pretty minimal. Study of monotonic rule systems can be trac... |

8 |
Recursive Boolean Algebras
- Remmel
(Show Context)
Citation Context ...aracter of the set of maximal ideals containing given recursively enumerable ideal in a recursively presented free Boolean Algebra, a subject in which two of the authors have a lot of experience (see =-=[42]-=-, [35]). The simplest case covering many nonmonotonic systems arising from mathematics is that of “highly recursive” nonmonotone rule systems. There, it turns out that extensions can be, up to a one-t... |

5 |
Applications of Pure Recursion Theory to Recursive Analysis. Acta Sinica 28
- Huang, Nerode
(Show Context)
Citation Context ... recursive societies, or finding of recursively presented formally real field ([31]), or finding an abcissa between 0 and 1 where a given recursive continuous function on [0,1] takes a maximum value (=-=[19]-=-). Now this recursion-theoretic methodology can be refined to give complexity-theoretic results on the same problems about extensions, as has been done in algebra by Nerode and Remmel in [36], [37], a... |

5 |
Effective matchmaking
- Manaster, Rosenstein
- 1972
(Show Context)
Citation Context ... a construction of a rule system < U,N > whose extensions directly code infinite paths through a binary tree T and provide us with a more direct route to Theorem 5.5 which avoids using the results of =-=[24]-=- or [41]. Example 5.1 Paths through binary trees. Let T be a recursive binary tree contained in 2 <ω . Let U(T ) = {Pi,Pi:i ∈ ω}. Our idea is to have a set π such that | π ∩ {Pi,Pi} |= 1 for all i cor... |

4 |
Countable retracing functions and Π02 predicates
- Jockusch, McLaughlin
- 1969
(Show Context)
Citation Context ...ely bounded Π 0 1-classes due to Jockusch and Soare [21] [22] except of Corollary 5.10 which follows from the corresponding result for recursively bounded Π 0 1-classes due to Jockusch and McLaughlin =-=[20]-=-. Now we give a construction of a rule system < U,N > whose extensions directly code infinite paths through a binary tree T and provide us with a more direct route to Theorem 5.5 which avoids using th... |

4 |
Complexity-theoretic algebra. II. Boolean algebras
- Nerode, Remmel
- 1987
(Show Context)
Citation Context ...e ([19]). Now this recursion-theoretic methodology can be refined to give complexity-theoretic results on the same problems about extensions, as has been done in algebra by Nerode and Remmel in [36], =-=[37]-=-, and [38]. Since this is a more delicate matter than recursion theory, these developments are deferred again to a later paper. Next, we turn to investigations of the semantics for nonmonotonic rule s... |

3 | Decision procedure for default logic
- Marek, Nerode
(Show Context)
Citation Context ...ny rules which are, however, monotonic. There are infinitely many proof schemes, but the collection of formulas of form k(p) is finite anyway! This fact results in the finitary algorithm described in =-=[25]-=-. Our translation of propositional logic programs as rule systems provides us with an infinitary characterization of stable models of logic programs. The reason this is important is that the definitio... |

3 |
Relating Autoepistemic and Default Logics. In: Principles of Knowledge Representation and Reasoning
- Marek, Truszczyński
- 1989
(Show Context)
Citation Context ...phic notion of belief so as to bring out their common features. The actual nonmonotonic logics have much in common, and a number of translations between them have been proposed ([23], [12],[13], [39],=-=[28]-=-). They have been investigated principally for propositional logic. Predicate versions suitable for actual applications are, up to now, pretty minimal. Study of monotonic rule systems can be traced to... |

3 |
Complexity-theoretic Algebra: Vector Space Bases. In: Feasible mathematics
- Nerode, Remmel
- 1989
(Show Context)
Citation Context ...m value ([19]). Now this recursion-theoretic methodology can be refined to give complexity-theoretic results on the same problems about extensions, as has been done in algebra by Nerode and Remmel in =-=[36]-=-, [37], and [38]. Since this is a more delicate matter than recursion theory, these developments are deferred again to a later paper. Next, we turn to investigations of the semantics for nonmonotonic ... |

3 |
Graph colorings and recursively bounded Π0 1-classes
- Remmel
- 1986
(Show Context)
Citation Context ...ruction of a rule system < U,N > whose extensions directly code infinite paths through a binary tree T and provide us with a more direct route to Theorem 5.5 which avoids using the results of [24] or =-=[41]-=-. Example 5.1 Paths through binary trees. Let T be a recursive binary tree contained in 2 <ω . Let U(T ) = {Pi,Pi:i ∈ ω}. Our idea is to have a set π such that | π ∩ {Pi,Pi} |= 1 for all i correspond ... |

2 |
A survey of r.e. substructures
- Nerode, Remmel
- 1985
(Show Context)
Citation Context ...r of the set of maximal ideals containing given recursively enumerable ideal in a recursively presented free Boolean Algebra, a subject in which two of the authors have a lot of experience (see [42], =-=[35]-=-). The simplest case covering many nonmonotonic systems arising from mathematics is that of “highly recursive” nonmonotone rule systems. There, it turns out that extensions can be, up to a one-to-one ... |

1 | Logic Programming with Classical Negation. Unpublished Manuscript - Gelfond, Lifschitz |

1 |
Stable models for logic programs and default logic
- Marek, Truszczyński
- 1989
(Show Context)
Citation Context ...ion. Represent a general clause as a rule: q1,...,qn : r1,...rm p The translation tr(P) of a program P is the set of translations of its individual clauses. The following result was proved in [4] and =-=[29]-=-: Proposition 3.2 A subset M ⊆ U is a stable model of P if and only if M is an extension of tr(P). Proposition 3.3 A subset M ⊆ U is a supported model of P if and only if M is a weak extension of tr(P... |

1 |
A framework for representing knowledge. In: The Pychology of Computer Vision
- Minsky
(Show Context)
Citation Context ...aditionally studied, intuitionistic, modal, and classical. He did not qualify his definition, as we do, with the adjective “monotone,” because there were no other systems studied at that time. Minsky =-=[32]-=- suggested that there is another sort of reasoning which is not monotonic. This is reasoning in which we deduce a statement based on the absence of any evidence against the statement. Such a statement... |

1 |
Complexity-theoretic Algebra III: Bases of Vector Spaces
- Nerode, Remmel
(Show Context)
Citation Context ...Now this recursion-theoretic methodology can be refined to give complexity-theoretic results on the same problems about extensions, as has been done in algebra by Nerode and Remmel in [36], [37], and =-=[38]-=-. Since this is a more delicate matter than recursion theory, these developments are deferred again to a later paper. Next, we turn to investigations of the semantics for nonmonotonic rule systems. Th... |