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On the Complexity of Propositional Knowledge Base Revision, Updates, and Counterfactuals
 ARTIFICIAL INTELLIGENCE
, 1992
"... We study the complexity of several recently proposed methods for updating or revising propositional knowledge bases. In particular, we derive complexity results for the following problem: given a knowledge base T , an update p, and a formula q, decide whether q is derivable from T p, the updated (or ..."
Abstract

Cited by 186 (12 self)
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We study the complexity of several recently proposed methods for updating or revising propositional knowledge bases. In particular, we derive complexity results for the following problem: given a knowledge base T , an update p, and a formula q, decide whether q is derivable from T p, the updated (or revised) knowledge base. This problem amounts to evaluating the counterfactual p > q over T . Besides the general case, also subcases are considered, in particular where T is a conjunction of Horn clauses, or where the size of p is bounded by a constant.
The Complexity of LogicBased Abduction
, 1993
"... Abduction is an important form of nonmonotonic reasoning allowing one to find explanations for certain symptoms or manifestations. When the application domain is described by a logical theory, we speak about logicbased abduction. Candidates for abductive explanations are usually subjected to minima ..."
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Cited by 163 (26 self)
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Abduction is an important form of nonmonotonic reasoning allowing one to find explanations for certain symptoms or manifestations. When the application domain is described by a logical theory, we speak about logicbased abduction. Candidates for abductive explanations are usually subjected to minimality criteria such as subsetminimality, minimal cardinality, minimal weight, or minimality under prioritization of individual hypotheses. This paper presents a comprehensive complexity analysis of relevant decision and search problems related to abduction on propositional theories. Our results indicate that abduction is harder than deduction. In particular, we show that with the most basic forms of abduction the relevant decision problems are complete for complexity classes at the second level of the polynomial hierarchy, while the use of prioritization raises the complexity to the third level in certain cases.
Propositional Circumscription and Extended Closed World Reasoning are $\Pi^P_2$complete
 Theoretical Computer Science
, 1993
"... Circumscription and the closed world assumption with its variants are wellknown nonmonotonic techniques for reasoning with incomplete knowledge. Their complexity in the propositional case has been studied in detail for fragments of propositional logic. One open problem is whether the deduction prob ..."
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Cited by 99 (22 self)
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Circumscription and the closed world assumption with its variants are wellknown nonmonotonic techniques for reasoning with incomplete knowledge. Their complexity in the propositional case has been studied in detail for fragments of propositional logic. One open problem is whether the deduction problem for arbitrary propositional theories under the extended closed world assumption or under circumscription is $\Pi^P_2$complete, i.e., complete for a class of the second level of the polynomial hierarchy. We answer this question by proving these problems $\Pi^P_2$complete, and we show how this result applies to other variants of closed world reasoning.
Randomised Techniques in Combinatorial Algorithmics
, 1999
"... ix Chapter 1 Introduction 1 1.1 Algorithmic Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Technical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 ..."
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Cited by 20 (7 self)
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ix Chapter 1 Introduction 1 1.1 Algorithmic Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Technical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.2 Parallel Computational Complexity . . . . . . . . . . . . . . . . . . . . . 7 1.2.3 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.4 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.5 Random Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.6 Group Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Chapter 2 Parallel Uniform Generation of Unlabelled Graphs 25 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2 Sampling O...
Maintaining the transitive closure of graphs in SQL
 In Int. J. Information Technology
, 1999
"... It is common knowledge that relational calculus and even SQL are not expressive enough to express recursive queries such as the transitive closure. In a real database system, one can overcome this problem by storing a graph together with its transitive closure and maintaining the latter whenever upd ..."
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Cited by 7 (3 self)
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It is common knowledge that relational calculus and even SQL are not expressive enough to express recursive queries such as the transitive closure. In a real database system, one can overcome this problem by storing a graph together with its transitive closure and maintaining the latter whenever updates to the former occur. This leads to the concept of an incremental evaluation system, or IES. Much is already known about the theory of IES but very little has been translated into practice. The purpose of this paper is to ll in this gap by providing a gentle introduction to and an overview of some recent theoretical results on IES. The introduction is through the translation into SQL of three interesting positive maintenance results that have practical importance { the maintenance of the transitive closure of acyclic graphs, of undirected graphs, and of arbitrary directed graphs. Interestingly, these examples also allow ustoshow the relationship between power and cost in the incremental maintenance of database queries. 1
Generating Boolean ยตExpressions
, 1995
"... In this paper, we consider the class of Boolean μfunctions, which are the Boolean functions de nable by μexpressions (Boolean expressions in which no variable occurs more than once). We present an algorithm which transforms a Boolean formula E into an equivalent μexpression  if possibl ..."
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Cited by 4 (0 self)
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In this paper, we consider the class of Boolean μfunctions, which are the Boolean functions de nable by μexpressions (Boolean expressions in which no variable occurs more than once). We present an algorithm which transforms a Boolean formula E into an equivalent μexpression  if possible  in time linear in E times 2^nm, where E is the size of E and nm is the number of variables that occur more than once in E. As an application, we obtain a polynomial time algorithm for Mundici's problem of recognizing μfunctions from kformulas [17]. Furthermore, we show that recognizing Boolean μfunctions is coNPcomplete for functions essentially dependent on all variables and we give a bound close to coNP for the general case.
SQL Can Maintain PolynomialHierarchy Queries
, 1997
"... this report, we only consider queries from flat relations to flat relations; and the criteria for permissible update is restricted to the insertion and deletion of a single tuple. A restriction is also imposed so that the constants that appear in the auxiliary database must also appear in the databa ..."
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Cited by 2 (2 self)
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this report, we only consider queries from flat relations to flat relations; and the criteria for permissible update is restricted to the insertion and deletion of a single tuple. A restriction is also imposed so that the constants that appear in the auxiliary database must also appear in the database or in the answer or in some fixed set. In this report, this fixed set is Q , the set of rational numbers. We use the firstorder incremental evaluation system, IES(FO)(called FOIES in [10]), to illustrate the concept. IES(FO) uses firstorder logic to express update functions [9, 11]. For each relation symbol R,
Maintaining Transitive Closure of Graphs in SQL
 In Int. J. Information Technology
, 1999
"... It is common knowledge that relational calculus and even SQL are not expressive enough to express recursive queries such as the transitive closure. In a real database system, one can overcome this problem by storing a graph together with its transitive closure and maintaining the latter whenever ..."
Abstract
 Add to MetaCart
It is common knowledge that relational calculus and even SQL are not expressive enough to express recursive queries such as the transitive closure. In a real database system, one can overcome this problem by storing a graph together with its transitive closure and maintaining the latter whenever updates to the former occur. This leads to the concept of an incremental evaluation system, or IES. Much is already known about the theory of IES but very little has been translated into practice. The purpose of this paper is to fill in this gap by providing a gentle introduction to and an overview of some recent theoretical results on IES. The introduction is through the translation into SQL of three interesting positive maintenance results that have practical importance  the maintenance of the transitive closure of acyclic graphs, of undirected graphs, and of arbitrary directed graphs. Interestingly, these examples also allow us to show the relationship between power and cost in ...