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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 187 (28 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.
Preferred Answer Sets for Extended Logic Programs
 ARTIFICIAL INTELLIGENCE
, 1998
"... In this paper, we address the issue of how Gelfond and Lifschitz's answer set semantics for extended logic programs can be suitably modified to handle prioritized programs. In such programs an ordering on the program rules is used to express preferences. We show how this ordering can be used ..."
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Cited by 152 (20 self)
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In this paper, we address the issue of how Gelfond and Lifschitz's answer set semantics for extended logic programs can be suitably modified to handle prioritized programs. In such programs an ordering on the program rules is used to express preferences. We show how this ordering can be used to define preferred answer sets and thus to increase the set of consequences of a program. We define a strong and a weak notion of preferred answer sets. The first takes preferences more seriously, while the second guarantees the existence of a preferred answer set for programs possessing at least one answer set. Adding priorities
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 106 (20 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.
Exact analysis of Dodgson elections: Lewis Carroll’s 1876 voting system is complete for parallel access to NP
 Journal of the ACM
, 1997
"... Abstract. In 1876, Lewis Carroll proposed a voting system in which the winner is the candidate who with the fewest changes in voters ’ preferences becomes a Condorcet winner—a candidate who beats all other candidates in pairwise majorityrule elections. Bartholdi, Tovey, and Trick provided a lower b ..."
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Cited by 72 (15 self)
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Abstract. In 1876, Lewis Carroll proposed a voting system in which the winner is the candidate who with the fewest changes in voters ’ preferences becomes a Condorcet winner—a candidate who beats all other candidates in pairwise majorityrule elections. Bartholdi, Tovey, and Trick provided a lower bound—NPhardness—on the computational complexity of determining the election winner in Carroll’s system. We provide a stronger lower bound and an upper bound that matches our lower bound. In particular, determining the winner in Carroll’s system is complete for parallel access to NP, that is, it is complete for � 2 p, for which it becomes the most natural complete problem known. It
On TruthTable Reducibility to SAT
, 2002
"... We show that polynomial time truthtable reducibility via Boolean circuits to SAT is the same as logspace truthtable reducibility via Boolean formulas to SAT and the same as logspace Turing reducibility to SAT . In addition, we prove that a constant number of rounds of parallel queries to SAT i ..."
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Cited by 54 (2 self)
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We show that polynomial time truthtable reducibility via Boolean circuits to SAT is the same as logspace truthtable reducibility via Boolean formulas to SAT and the same as logspace Turing reducibility to SAT . In addition, we prove that a constant number of rounds of parallel queries to SAT is equivalent to one round of parallel queries.
The Minimum Equivalent DNF Problem and Shortest Implicants
, 1998
"... We prove that the Minimum Equivalent DNF problem is \Sigma p 2 complete, resolving a conjecture due to Stockmeyer. The proof involves as an intermediate step a variant of a related problem in logic minimization, namely, that of finding the shortest implicant of a Boolean function. We also obtain ..."
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Cited by 50 (4 self)
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We prove that the Minimum Equivalent DNF problem is \Sigma p 2 complete, resolving a conjecture due to Stockmeyer. The proof involves as an intermediate step a variant of a related problem in logic minimization, namely, that of finding the shortest implicant of a Boolean function. We also obtain certain results concerning the complexity of the Shortest Implicant problem that may be of independent interest. When the input is a formula, the Shortest Implicant problem is \Sigma p 2  complete, and \Sigma p 2 hard to approximate to within an n 1=2\Gammaffl factor. When the input is a circuit, approximation is \Sigma p 2  hard to within an n 1\Gammaffl factor. However, when the input is a DNF formula, the Shortest Implicant problem cannot be \Sigma p 2 complete unless \Sigma p 2 = NP[log 2 n] NP . 1. Introduction Twolevel (DNF) logic minimization is a central practical problem in logic synthesis and also one of the more natural problems in the polynomial hierarchy....
Logical Definability of NP Optimization Problems
 Information and Computation
, 1994
"... : We investigate here NP optimization problems from a logical definability standpoint. We show that the class of optimization problems whose optimum is definable using firstorder formulae coincides with the class of polynomially bounded NP optimization problems on finite structures. After this, we ..."
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Cited by 42 (2 self)
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: We investigate here NP optimization problems from a logical definability standpoint. We show that the class of optimization problems whose optimum is definable using firstorder formulae coincides with the class of polynomially bounded NP optimization problems on finite structures. After this, we analyze the relative expressive power of various classes of optimization problems that arise in this framework. Some of our results show that logical definability has different implications for NP maximization problems than it has for NP minimization problems, in terms of both expressive power and approximation properties. To appear in Information and Computation. Research partially supported by NSF Grants CCR8905038 and CCR9108631. y email addresses: kolaitis@cse.ucsc.edu, thakur@cse.ucsc.edu z supersedes Technical report UCSCCRL9048 1 Introduction and Summary of Results It is well known that optimization problems had a major influence on the development of the theory of NPco...
Efficiency and envyfreeness in fair division of indivisible goods: Logical representation and complexity
 In Proceedings of the 19th International Joint Conference on Artificial Intelligence (IJCAI2005
, 2005
"... and complexity ..."
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The Complexity of Temporal Logic Model Checking
, 2002
"... Temporal logic. Logical formalisms for reasoning about time and the timing of events appear in several fields: physics, philosophy, linguistics, etc. Not surprisingly, they also appear in computer science, a field where logic is ubiquitous. Here temporal logics are used in automated reasoning, in pl ..."
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Cited by 40 (0 self)
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Temporal logic. Logical formalisms for reasoning about time and the timing of events appear in several fields: physics, philosophy, linguistics, etc. Not surprisingly, they also appear in computer science, a field where logic is ubiquitous. Here temporal logics are used in automated reasoning, in planning, in semantics of programming languages, in artificial intelligence, etc. There is one area of computer science where temporal logic has been unusually successful: the specification and verification of programs and systems, an area we shall just call programming for simplicity. In today's curricula, thousands of programmers first learn about temporal logic in a course on model checking!
Exact complexity of the winner problem for Young elections
 Theory Comput. Syst
"... Abstract. In 1977 Young proposed a voting scheme that extends the Condorcet Principle based on the fewest possible number of voters whose removal yields a Condorcet winner. We prove that both the winner and the ranking problem for Young elections is complete for PNP ‖ , the class of problems solvabl ..."
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Cited by 40 (7 self)
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Abstract. In 1977 Young proposed a voting scheme that extends the Condorcet Principle based on the fewest possible number of voters whose removal yields a Condorcet winner. We prove that both the winner and the ranking problem for Young elections is complete for PNP ‖ , the class of problems solvable in polynomial time by parallel access to NP. Analogous results for Lewis Carroll’s 1876 voting scheme were recently established by Hemaspaandra et al. In contrast, we prove that the winner and ranking problems in Fishburn’s homogeneous variant of Carroll’s voting scheme can be solved efficiently by linear programming. 1.