## Nonmonotonic Reasoning (1993)

Venue: | In Proc |

Citations: | 5 - 0 self |

### BibTeX

@INPROCEEDINGS{Marek93nonmonotonicreasoning,

author = {V. W. Marek and A. Nerode},

title = {Nonmonotonic Reasoning},

booktitle = {In Proc},

year = {1993},

publisher = {Springer}

}

### OpenURL

### Abstract

Classical logic is the study of ”safe ” formal reasoning. Western Philosophers de-veloped classical logic over a period of thirty-three centuries after its introduction in the form of syllogistic by Aristotle [1] in the third century B. C. Beginning in the nineteenth century with De Morgan [2] and Boole [3], responsibility for the develop-ment of classical logic moved from the philosophical to the mathematical community.

### Citations

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A logic for default reasoning
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- 1980
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Citation Context ...rlier philosophers stemming from those mentioned above, showed how to formalize at least some commonsense reasoning [23]. This discovery formed the impetus for pioneering investigations of of Reiter, =-=[24, 25]-=-, McDermott and Doyle, [26, 27] and others. All this led to the development of what is now a popular area of research, Nonmonotonic Logics – which we shall outline in this article. We believe that the... |

1326 |
On computable numbers with an application to the Entscheidungsproblem
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Citation Context ...cal discipline which led to model theory, set theory, recursion theory, and proof theory. The discovery of recursive function theory by Church [11], Gödel [12], Herbrand [13], Kleene [14], and Turing =-=[15]-=- in the nineteen thirties, the development of digital computers with a Von Neumann architecture in the 1950’s, the interest in machine simulation of hu1sman intelligence (AI) in the 1960’s, and the de... |

986 |
A machine-oriented logic based on the resolution principle
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Citation Context ...ovability. This relationship is denoted by T ⊢ ϕ. There are various techniques for the syntactic entailment. These include: natural deduction, [28], Hilbert-style systems, [8], resolution refutation, =-=[29]-=- and semantic tableaux, [30, 31]. It always involves manipulation of formulas and derivation of additional formulas by syntactic means. The key result in such techniques is always a completeness prope... |

978 |
Negation as failure
- Clark
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(Show Context)
Citation Context ...eed to be minimal as a model of a smaller program. For complexity results on minimal models see Eiter and Gottlob, [47]. 5.2 Supported models One can assign to a general program its completion Clark, =-=[48]-=-. Completion of the program is a first order theory obtained from P by the following procedure. For each predicate p a formula ∀ ¯ Xp( ¯ (X)) ≡ (B1 ∨ ... ∨ Bk) is the completion of p. Here formulas Bj... |

867 |
Circumscription - a form of non-monotonic reasoning
- McCarthy
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Citation Context ...cond-order technique for minimizing extensions of predicates. In its simplest and most natural form circumscription scheme says that a theory implicitly defines a predicate it circumscribes McCarthy, =-=[72]-=-. Specifically, by circumscribing a predicate P in a theory T we assume that all the information about the predicate is given by T. Thus only those models where P is minimal need to be consider. Forma... |

774 |
Introduction to Metamathematics
- Kleene
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Citation Context ... into a mathematical discipline which led to model theory, set theory, recursion theory, and proof theory. The discovery of recursive function theory by Church [11], Gödel [12], Herbrand [13], Kleene =-=[14]-=-, and Turing [15] in the nineteen thirties, the development of digital computers with a Von Neumann architecture in the 1950’s, the interest in machine simulation of hu1sman intelligence (AI) in the 1... |

743 | The semantics of predicate logic as a programming language
- Emdem, Kowalski
- 1976
(Show Context)
Citation Context ... program P we associate an operator TP, mapping subsets of Herbrand base into subsets of Herbrand base, by a the following TP(I) = {p : ∃ I |= body(C) ∧ p = head(C)} C∈groundP van Emden and Kowalski, =-=[46]-=-. Operator TP is monotone and compact. Hence, by Knaster-Tarski Lemma, TP possesses the least fixpoint. This fixpoint coincides with the least Herbrand model MP of P. Moreover, if P1, P2 are two progr... |

673 |
Modal Logic: an introduction
- Chellas
- 1980
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Citation Context ...rch effort in logics for computer science than there ever was in traditional logics. Some of the needed logics are extensions of classical logic by new operators. Example are modal and temporal logic =-=[16, 17]-=-. These are useful as logics of programs and logics of agents. Some needed logics are restrictions of classical logic. Examples are intuitionistic logic which can be used to extract programs from proo... |

631 |
Towards a theory of declarative knowledge
- APT, BLAIR, et al.
- 1988
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Citation Context ...s of the operator TP. 5.3 Perfect model Some programs allow for identification of a particular minimal model which has particularly nice properties. Call a program P stratified Apt, Blair and Walker, =-=[50]-=- if there is a function rank on the set of predicates of the program P such that whenever we have a clause of the form (5) then for all i rank(qi) ≤ rank(p) and for all j, rank(rj) < rank(p). The intu... |

461 |
Semantical considerations on non-monotonic logic
- Moore
- 1985
(Show Context)
Citation Context ...of S5-models of I with universal 24srelations. This implies that the formulas true in all S5-expansions of I are precisely S5-consequences of I. This was considered counterintuitive at a time. Moore, =-=[60]-=- considered a different operator, explicitly taking into account both positive and negative introspection with respect to T Mo(T) = Cn(I ∪ {Lϕ : ϕ ∈ T } ∪ {¬Lϕ : ϕ /∈ T }) and argued that this operato... |

436 |
Logic programming
- Apt
- 1990
(Show Context)
Citation Context ... of P. This implies 10sthat the answers to the ground atomic queries to P coincide with the results of SLD resolution with respect to P. More information on CWA and Logic Programming can be found in, =-=[37]-=-. 4 Default Logic Default logic is one of the better understood formalisms in Nonmonotonic Reasoning. It admits various interpretations, but the most popular is that it assigns meaning to the quantifi... |

430 |
Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik, 38:173–198
- Gödel
- 1931
(Show Context)
Citation Context ... how to make classical logic into a mathematical discipline which led to model theory, set theory, recursion theory, and proof theory. The discovery of recursive function theory by Church [11], Gödel =-=[12]-=-, Herbrand [13], Kleene [14], and Turing [15] in the nineteen thirties, the development of digital computers with a Von Neumann architecture in the 1950’s, the interest in machine simulation of hu1sma... |

383 |
Proofs and Types
- Girard, Lafont, et al.
- 1990
(Show Context)
Citation Context ...er logics. These received little attention previously. These include Huet-Coquand’s Theory of Constructions [19], based on Girard’s strong normalization theorem for higher order intuitionistic logics =-=[20]-=-. Another is Constable’s NUPRL [21], based on the normalization theorem of Martin-Löf’s predicative higher order logics [22]. All thus far mentioned are extensions or restrictions of classical logics.... |

358 |
Intuitionistic Type Theory
- Martin-Löf
- 1984
(Show Context)
Citation Context ...irard’s strong normalization theorem for higher order intuitionistic logics [20]. Another is Constable’s NUPRL [21], based on the normalization theorem of Martin-Löf’s predicative higher order logics =-=[22]-=-. All thus far mentioned are extensions or restrictions of classical logics. An outstanding feature of classical logics and their extensions and restrictions is that once the premises of an argument h... |

334 | Circumscription
- LIFSCHITZ
- 1994
(Show Context)
Citation Context ...cription reduces to first-order sentence. For instance theory T consisting of a single axiom ∀xQ(x) ∧ R(x) ⊃ P(x) after circumscribing P gives ∀x(Q(x) ∧ R(x)) ≡ P(x). Lifschitz, in a series of papers =-=[73, 74, 75, 76]-=- analyzed the most important properties of circumscription. When the theory T is a propositional theory, circumscribing theory T reduces to minimizing models of T (in general we can minimize some or a... |

302 |
First-Order Logic
- Smullyan
- 1968
(Show Context)
Citation Context ... is denoted by T ⊢ ϕ. There are various techniques for the syntactic entailment. These include: natural deduction, [28], Hilbert-style systems, [8], resolution refutation, [29] and semantic tableaux, =-=[30, 31]-=-. It always involves manipulation of formulas and derivation of additional formulas by syntactic means. The key result in such techniques is always a completeness property which says that T |= ϕ if an... |

283 |
On closed world data bases
- REITER
- 1978
(Show Context)
Citation Context ...rlier philosophers stemming from those mentioned above, showed how to formalize at least some commonsense reasoning [23]. This discovery formed the impetus for pioneering investigations of of Reiter, =-=[24, 25]-=-, McDermott and Doyle, [26, 27] and others. All this led to the development of what is now a popular area of research, Nonmonotonic Logics – which we shall outline in this article. We believe that the... |

258 |
On the declarative semantics of deductive databases and logic programs
- Przymusinski
- 1988
(Show Context)
Citation Context ...t a 21sprogram with n strata can compute a Σ 0 n complete set. Conversely the perfect model of a stratified program with n strata is Σ 0 n. A “local” version of stratification is due to Przymusinski, =-=[52]-=-. Here we require that the rank is defined not on the set of all predicates of the language of the program but rather on the Herbrand base. The stratification conditions are similar, but pertain to cl... |

238 |
Non-monotonic logic
- McDermott, Doyle
- 1980
(Show Context)
Citation Context ...m those mentioned above, showed how to formalize at least some commonsense reasoning [23]. This discovery formed the impetus for pioneering investigations of of Reiter, [24, 25], McDermott and Doyle, =-=[26, 27]-=- and others. All this led to the development of what is now a popular area of research, Nonmonotonic Logics – which we shall outline in this article. We believe that the development and computer imple... |

196 |
Autoepistemic logic
- Marek, Truszczynski
- 1991
(Show Context)
Citation Context ... [55], Marek and Truszczynski, [56]. There exist programs without stable models. The existence problem for stable models of propositional general logic programs is NPcomplete, Marek and Truszczynski, =-=[57]-=-. For finite predicate general logic programs existence of a stable model is a Σ 1 1-complete problem Marek, Nerode and Remmel, [58]. 6 Modal nonmonotonic logics The modal flavor of nonmonotonic reaso... |

191 | On the complexity of propositional knowledge base revision, updates and counterfactual
- Eiter, Gottlob
- 1992
(Show Context)
Citation Context ...program changes semantics drastically. A minimal model of a larger program does not need to be minimal as a model of a smaller program. For complexity results on minimal models see Eiter and Gottlob, =-=[47]-=-. 5.2 Supported models One can assign to a general program its completion Clark, [48]. Completion of the program is a first order theory obtained from P by the following procedure. For each predicate ... |

191 |
Contributions to the theory of logic programming
- APT, EMDEN
- 1982
(Show Context)
Citation Context ... 20sequality (so called Clark Equational Theory) A supported structure for P is a model of completion of P. Such structure is a model of P and so it is called supported model of P. Apt and van Emden, =-=[49]-=- proved that Herbrand models of completion of P are characterized as fixpoints of the operator TP. 5.3 Perfect model Some programs allow for identification of a particular minimal model which has part... |

182 |
Complexity results for nonmonotonic logics
- Gottlob
- 1992
(Show Context)
Citation Context ...lt theories. All of these are inefficient because the complexity problems associated with default logic are located, generally, on the second level of the polynomial hierarchy. Specifically, Gottlob, =-=[39]-=- found the complexity of basic problems for default logic 1. (Existence) The problem: Given a default theory 〈D,W 〉 does 〈D,W 〉 possesses an extension, is Σ P 2 -complete. 2. (Membership in some exten... |

179 |
Nonmonotonic logic: Context-dependent reasoning
- Marek, Truszczynski
- 1993
(Show Context)
Citation Context ...so discuss some nonmonotonic aspects of Logic Programming and a general mechanism for treating nonmonotonicity in logic. The subject of this article is discussed in several monographs. These include: =-=[32, 33, 34, 35, 36, 31]-=-. The theory of nonmonotonic reasoning is, at present, an active area of research in Artificial Intelligence and many monographs will, undoubtedly, follow. 3 Closed World Assumption Closed World Assum... |

152 |
On the relation between default and autoepistemic logic
- Konolige
- 1988
(Show Context)
Citation Context ...re are many representations of nonmonotonic reasonings in modal nonmonotonic logics with various degree of faithfulness. These include Konolige interpretation of default logic in autoepistemic logic, =-=[64]-=-, Truszczynski interpretation of default logic in nonmonotonic S4F, [65] and many others. Stable semantics of logic programs can be faithfully represented in both autoepistemic logic, [66] and in refl... |

148 |
Nonmonotonic Reasoning: Logical Foundations of Commonsense
- Brewka
- 1991
(Show Context)
Citation Context ...ault logic is, currently, best understood among several modes of nonmonotonic reasoning. There are several monographs devoted to the subject. These include: Besnard, [32], ̷Lukaszewicz, [36], Brewka, =-=[45]-=- and Marek and Truszczynski, [35]. 5 Nonmonotonic Aspects of Logic Programming A Horn logic program is a finite set of expressions of the form p( ¯ X) ← q1( ¯ (X)),...,qm( ¯ X) (4) Here ¯ X is a strin... |

141 |
On indefinite databases and the closed world assumption
- MINKER
- 1982
(Show Context)
Citation Context ...variables in T). Minimization of all propositional variables at once is known as extended generalized closed world assumption (GCWA) and has been studied by Yahia and Henschen, [77], see also Minker, =-=[78]-=-. Clearly the entailment problem for propositional circumscription is decidable. An algorithm has been proposed by Przymusinski, [79]. In the predicate case the circumscription is very complex. Schlip... |

110 |
On stratified autoepistemic theories
- Gelfond
- 1987
(Show Context)
Citation Context ...emic logic, [64], Truszczynski interpretation of default logic in nonmonotonic S4F, [65] and many others. Stable semantics of logic programs can be faithfully represented in both autoepistemic logic, =-=[66]-=- and in reflexive autoepistemic logic, [67]. Similarly, supported semantics for logic programs can be faithfully represented in autoepistemic logic, [68]. Lifschitz, [69] introduced a general mechanis... |

109 |
Principia Mathematica
- Whitehead, Russell
- 1910
(Show Context)
Citation Context ... De Morgan [2] and Boole [3], responsibility for the development of classical logic moved from the philosophical to the mathematical community. Boole [3], Peirce [4], Frege [5], Schröder [6], Russell =-=[7]-=-, Hilbert [8], Gödel [9], and Tarski [10] showed how to make classical logic into a mathematical discipline which led to model theory, set theory, recursion theory, and proof theory. The discovery of ... |

106 |
A note on the Entscheidungsproblem
- Church
- 1936
(Show Context)
Citation Context ... [10] showed how to make classical logic into a mathematical discipline which led to model theory, set theory, recursion theory, and proof theory. The discovery of recursive function theory by Church =-=[11]-=-, Gödel [12], Herbrand [13], Kleene [14], and Turing [15] in the nineteen thirties, the development of digital computers with a Von Neumann architecture in the 1950’s, the interest in machine simulati... |

100 |
The stable semantics for logic programs
- Gelfond, Lifschitz
- 1988
(Show Context)
Citation Context ...rsive program is hyperarithmetical and all hyperarithmetical sets are extensions of predicates in perfect model of suitably chosen locally stratified program. 5.4 Stable models Gelfond and Lifschitz, =-=[54]-=- defined the notion of a stable model of a program. Let M be a subset of the Herbrand base of P. Reduce the program ground(P) as follows. Given a clause C p ← q1,....qm, ¬r1,...,¬rn in ground(P) elimi... |

98 |
Nonmonotonic logic II: Nonmonotonic modal theories
- McDermott
- 1982
(Show Context)
Citation Context ...m those mentioned above, showed how to formalize at least some commonsense reasoning [23]. This discovery formed the impetus for pioneering investigations of of Reiter, [24, 25], McDermott and Doyle, =-=[26, 27]-=- and others. All this led to the development of what is now a popular area of research, Nonmonotonic Logics – which we shall outline in this article. We believe that the development and computer imple... |

93 |
Reasoning with Incomplete Information
- Etherington
- 1988
(Show Context)
Citation Context ...so discuss some nonmonotonic aspects of Logic Programming and a general mechanism for treating nonmonotonicity in logic. The subject of this article is discussed in several monographs. These include: =-=[32, 33, 34, 35, 36, 31]-=-. The theory of nonmonotonic reasoning is, at present, an active area of research in Artificial Intelligence and many monographs will, undoubtedly, follow. 3 Closed World Assumption Closed World Assum... |

90 | Dualities between alternative semantics for logic programming and nonmonotonic reasoning
- Baral, Subrahmanian
- 1993
(Show Context)
Citation Context ...s suggested changing the mechanism for computing objects associated with default theories. Other structures associated with default theories include well-founded extensions of Baral and Subrahmanian, =-=[43]-=-, weak extensions Marek and Truszczynski, [44] and many others. Default logic is, currently, best understood among several modes of nonmonotonic reasoning. There are several monographs devoted to the ... |

86 |
An Investigation of the Laws of Thought, on Which are Founded the Mathematical Theories of Logic and Probabilities
- Boole
- 1854
(Show Context)
Citation Context ... over a period of thirty-three centuries after its introduction in the form of syllogistic by Aristotle [1] in the third century B. C. Beginning in the nineteenth century with De Morgan [2] and Boole =-=[3]-=-, responsibility for the development of classical logic moved from the philosophical to the mathematical community. Boole [3], Peirce [4], Frege [5], Schröder [6], Russell [7], Hilbert [8], Gödel [9],... |

82 |
Collected Papers of Charles Sanders Peirce. Volume 5
- Peirce
- 1935
(Show Context)
Citation Context ...ginning in the nineteenth century with De Morgan [2] and Boole [3], responsibility for the development of classical logic moved from the philosophical to the mathematical community. Boole [3], Peirce =-=[4]-=-, Frege [5], Schröder [6], Russell [7], Hilbert [8], Gödel [9], and Tarski [10] showed how to make classical logic into a mathematical discipline which led to model theory, set theory, recursion theor... |

80 |
An introduction to default logic
- Besnard
- 1989
(Show Context)
Citation Context ...so discuss some nonmonotonic aspects of Logic Programming and a general mechanism for treating nonmonotonicity in logic. The subject of this article is discussed in several monographs. These include: =-=[32, 33, 34, 35, 36, 31]-=-. The theory of nonmonotonic reasoning is, at present, an active area of research in Artificial Intelligence and many monographs will, undoubtedly, follow. 3 Closed World Assumption Closed World Assum... |

78 |
Implementing Mathematics with the Nuprl Development System
- Constable, Allen, et al.
- 1986
(Show Context)
Citation Context ...tention previously. These include Huet-Coquand’s Theory of Constructions [19], based on Girard’s strong normalization theorem for higher order intuitionistic logics [20]. Another is Constable’s NUPRL =-=[21]-=-, based on the normalization theorem of Martin-Löf’s predicative higher order logics [22]. All thus far mentioned are extensions or restrictions of classical logics. An outstanding feature of classica... |

76 |
Un systeme de communication homme-machine en francais. Technical report, Groupe de Intelligence Artificielle Universitae de Aix-Marseille
- Colmerauer, Kanoui, et al.
- 1973
(Show Context)
Citation Context ...s logics of programs and logics of agents. Some needed logics are restrictions of classical logic. Examples are intuitionistic logic which can be used to extract programs from proofs and ideal PROLOG =-=[18]-=-, where the program specification is the program itself. Some needed logics develop aspects of higher-order logics. These received little attention previously. These include Huet-Coquand’s Theory of C... |

76 |
Formalization of common sense, papers by John McCarthy edited by
- McCarthy
- 1990
(Show Context)
Citation Context ... as such has been systematically formalized. McCarthy, building on the efforts of earlier philosophers stemming from those mentioned above, showed how to formalize at least some commonsense reasoning =-=[23]-=-. This discovery formed the impetus for pioneering investigations of of Reiter, [24, 25], McDermott and Doyle, [26, 27] and others. All this led to the development of what is now a popular area of res... |

74 | Minimal belief and negation as failure
- Lifschitz
- 1994
(Show Context)
Citation Context ... in both autoepistemic logic, [66] and in reflexive autoepistemic logic, [67]. Similarly, supported semantics for logic programs can be faithfully represented in autoepistemic logic, [68]. Lifschitz, =-=[69]-=- introduced a general mechanism for uniform treatment of modal nonmonotonic logics in a single bimodal logic MBNF. Computational mechanisms associated with modal nonmonotonic logics vary greatly in co... |

65 |
Cumulative Default Logic: In defense of nonmonotonic inference rules
- Brewka
- 1991
(Show Context)
Citation Context |

57 |
Non-Monotonic Reasoning. Formalization of Commonsense Reasoning
- Lukaszewicz
- 1990
(Show Context)
Citation Context |

57 |
Relating autoepistemic and default logics
- Marek, Truszczynski
- 1989
(Show Context)
Citation Context ...ng objects associated with default theories. Other structures associated with default theories include well-founded extensions of Baral and Subrahmanian, [43], weak extensions Marek and Truszczynski, =-=[44]-=- and many others. Default logic is, currently, best understood among several modes of nonmonotonic reasoning. There are several monographs devoted to the subject. These include: Besnard, [32], ̷Lukasz... |

56 |
Negation by default and unstratifiable logic programs
- Bidoit, Froidevaux
- 1991
(Show Context)
Citation Context ... stratified then P possesses a unique stable model. It is its perfect model. It turns out that stable models of general logic programs are closely connected with Default Logic, Bidoit and Froidevaux, =-=[55]-=-, Marek and Truszczynski, [56]. There exist programs without stable models. The existence problem for stable models of propositional general logic programs is NPcomplete, Marek and Truszczynski, [57].... |

53 |
Stable semantics for logic programs and default reasoning
- Marek, Truszczynski
- 1999
(Show Context)
Citation Context ... unique stable model. It is its perfect model. It turns out that stable models of general logic programs are closely connected with Default Logic, Bidoit and Froidevaux, [55], Marek and Truszczynski, =-=[56]-=-. There exist programs without stable models. The existence problem for stable models of propositional general logic programs is NPcomplete, Marek and Truszczynski, [57]. For finite predicate general ... |

51 |
Vorlesungen über die algebra der logik (exakte logic). Dritter Band: Algebra und Logik der Relative
- Schroder
- 1895
(Show Context)
Citation Context ... century with De Morgan [2] and Boole [3], responsibility for the development of classical logic moved from the philosophical to the mathematical community. Boole [3], Peirce [4], Frege [5], Schröder =-=[6]-=-, Russell [7], Hilbert [8], Gödel [9], and Tarski [10] showed how to make classical logic into a mathematical discipline which led to model theory, set theory, recursion theory, and proof theory. The ... |

49 |
Arithmetic classification of perfect models of stratified programs
- Apt, Blair
- 1988
(Show Context)
Citation Context ...ct model of P is a model of completion of P as well. If P is a propositional program that is stratified than its perfect model can be computed in time linear in |P |. In predicate case Apt and Blair, =-=[51]-=- show that a 21sprogram with n strata can compute a Σ 0 n complete set. Conversely the perfect model of a stratified program with n strata is Σ 0 n. A “local” version of stratification is due to Przym... |

46 | Modal Nonmonotonic Logics: Ranges, Characterization, Computation
- Marek, Schwarz
- 1993
(Show Context)
Citation Context ...e K), K5, K45, and KD45. Many other areas (called ranges) where expansions coincide are known presently. One such area is between S4 and S4F for all finite theories I Marek, Schwarz and Truszczynski, =-=[63]-=-. The degree of freedom (the choice of S) present in modal nonmonotonic logics makes them a very powerful mechanism for representation of other forms of nonmonotonic reasoning. There are many represen... |

45 |
Die Vollständigkeit der Axiome des logischen Funktionenkalküls
- Gödel
- 1930
(Show Context)
Citation Context ... [3], responsibility for the development of classical logic moved from the philosophical to the mathematical community. Boole [3], Peirce [4], Frege [5], Schröder [6], Russell [7], Hilbert [8], Gödel =-=[9]-=-, and Tarski [10] showed how to make classical logic into a mathematical discipline which led to model theory, set theory, recursion theory, and proof theory. The discovery of recursive function theor... |