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The Computational Complexity of Ideal Semantics
"... We analyse the computational complexity of the recently proposed ideal semantics within both abstract argumentation frameworks (AFs) and assumptionbased argumentation frameworks (ABFs). It is shown that while typically less tractable than credulous admissibility semantics, the natural decision prob ..."
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We analyse the computational complexity of the recently proposed ideal semantics within both abstract argumentation frameworks (AFs) and assumptionbased argumentation frameworks (ABFs). It is shown that while typically less tractable than credulous admissibility semantics, the natural decision problems arising with this extensionbased model can, perhaps surprisingly, be decided more efficiently than sceptical preferred semantics. In particular the task of finding the unique ideal extension is easier than that of deciding if a given argument is accepted under the sceptical semantics. We provide efficient algorithmic approaches for the class of bipartite argumentation frameworks and, finally, present a number of technical results which offer strong indications that typical problems in ideal argumentation are complete for the class P C   of languages decidable by polynomial time algorithms allowed to make nonadaptive queries to a C oracle, where C is an upper bound on the computational complexity of deciding credulous acceptance: C = NP for AFs and logic programming (LP) for ABFs modelling default theories. instantiations of ABFs; C = Σ p 2 Key words: Computational properties of argumentation; abstract argumentation
The computational complexity of ideal semantics I: abstract argumentation frameworks
, 2007
"... We analyse the computational complexity of the recently proposed ideal semantics within abstract argumentation frameworks. It is shown that while typically less tractable than credulous admissibility semantics, the natural decision problems arising with this extensionbased model can, perhaps surpri ..."
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We analyse the computational complexity of the recently proposed ideal semantics within abstract argumentation frameworks. It is shown that while typically less tractable than credulous admissibility semantics, the natural decision problems arising with this extensionbased model can, perhaps surprisingly, be decided more efficiently than sceptical admissibility semantics. In particular the task of finding the unique maximal ideal extension is easier than that of deciding if a given argument is accepted under the sceptical semantics. We provide efficient algorithmic approaches for the class of bipartite argumentation frameworks. Finally we present a number of technical results which offer strong indications that typical problems in ideal argumentation are complete for the class pnp   : languages decidable by polynomial time algorithms allowed to make nonadaptive queries to an np oracle.
Uncontested Semantics for ValueBased Argumentation
"... Abstract. We introduce an extensionbased semantics for valuebased argumentation frameworks (vafs) that provides a counterpart to the recently proposed ideal semantics in standard – i.e. value–free – argumentation frameworks. A significant motivation for this socalled “uncontested ..."
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Abstract. We introduce an extensionbased semantics for valuebased argumentation frameworks (vafs) that provides a counterpart to the recently proposed ideal semantics in standard – i.e. value–free – argumentation frameworks. A significant motivation for this socalled “uncontested
A note on satisfying truthvalue assignments of boolean formulas
 In Proceedings of the Seventh International Conference on Theory and Applications of Satisfiability Testing (SAT 2004
, 2004
"... Abstract. In this paper we define a class of truthvalue assignments, called bounded assignments, using a certain substitutional property. We show that every satisfiable Boolean formula has at least one bounded assignment. This allows us to show that satisfying truthvalue assignments of formulas in ..."
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Abstract. In this paper we define a class of truthvalue assignments, called bounded assignments, using a certain substitutional property. We show that every satisfiable Boolean formula has at least one bounded assignment. This allows us to show that satisfying truthvalue assignments of formulas in USAT can be syntactically defined in the language of classical propositional logic. We also discuss a possible application of bounded truthvalue assignments in local search and other methods for solving Boolean satisfiability problems.
Amplifying ZPP SAT[1] and the two queries problem
 IEEE Conference on Computational Complexity
, 2008
"... This paper shows a complete upward collapse in the Polynomial Hierarchy (PH) if for ZPP, two queries to a SAT oracle is equivalent to one query. That is, ZPPSAT[1] = ZPPSAT‖[2] = ⇒ ZPPSAT[1] = PH. These ZPP machines are required to succeed with probability at least 1/2 + 1/p(n) on inputs of lengt ..."
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This paper shows a complete upward collapse in the Polynomial Hierarchy (PH) if for ZPP, two queries to a SAT oracle is equivalent to one query. That is, ZPPSAT[1] = ZPPSAT‖[2] = ⇒ ZPPSAT[1] = PH. These ZPP machines are required to succeed with probability at least 1/2 + 1/p(n) on inputs of length n for some polynomial p(n). This result builds upon recent work by Tripathi [16] who showed a collapse of PH to SP2. The use of the probability bound of 1/2 + 1/p(n) is justified in part by showing that this bound can be amplified to 1−2−nk for ZPPSAT[1] computations. This paper also shows that in the deterministic case, PSAT[1] = PSAT‖[2] = ⇒ PH ⊆ ZPPSAT[1] where the ZPPSAT[1] machine achieves a probability of success of 1/2−2−nk. 1
ValiantVazirani Lemmata for Various Logics
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 63 (2008)
, 2008
"... We show analogues of a theorem due to Valiant and Vazirani [16] for intractable parameterized complexity classes such as W[P], W[SAT] and the classes of the Whierarchy as well as those of the Ahierarchy. We do so by proving a general “logical” version of it which may be of independent interest. ..."
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We show analogues of a theorem due to Valiant and Vazirani [16] for intractable parameterized complexity classes such as W[P], W[SAT] and the classes of the Whierarchy as well as those of the Ahierarchy. We do so by proving a general “logical” version of it which may be of independent interest.
A MACHINE MODEL FOR THE COMPLEXITY OF NPAPPROXIMATION PROBLEMS
"... This paper investigates a machinebased model for the complexity of approximating the CLIQUE problem. The model consists of nondeterministic polynomial time Turing machines with limited access to an NPcomplete oracle. Approximating the CLIQUE problem is complete for classes of functions computed by ..."
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This paper investigates a machinebased model for the complexity of approximating the CLIQUE problem. The model consists of nondeterministic polynomial time Turing machines with limited access to an NPcomplete oracle. Approximating the CLIQUE problem is complete for classes of functions computed by such machines. 1
Satisfiability parsimoniously reduces to the Tantrix rotation puzzle problem
 ACM COMPUTING RESEARCH REPOSITORY (CORR
, 2007
"... Holzer and Holzer [HH04] proved that the Tantrix rotation puzzle problem is NPcomplete. They also showed that for infinite rotation puzzles, this problem becomes undecidable. We study the counting version and the unique version of this problem. We prove that the satisfiability problem parsimoniousl ..."
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Holzer and Holzer [HH04] proved that the Tantrix rotation puzzle problem is NPcomplete. They also showed that for infinite rotation puzzles, this problem becomes undecidable. We study the counting version and the unique version of this problem. We prove that the satisfiability problem parsimoniously reduces to the Tantrix rotation puzzle problem. In particular, this reduction preserves the uniqueness of the solution, which implies that the unique Tantrix TM rotation puzzle problem is as hard as the unique satisfiability problem, and so is DPcomplete under polynomialtime randomized reductions, where DP is the second level of the boolean hierarchy over NP.
The ThreeColor and TwoColor Tantrix Rotation Puzzle Problems are NPComplete via Parsimonious Reductions
, 2007
"... Holzer and Holzer [HH04] proved the Tantrix TM rotation puzzle problem with four colors NPcomplete. Baumeister and Rothe [BR07] modified their construction to achieve a parsimonious reduction from satisfiability to this problem. Since parsimonious reductions preserve the number of solutions, it fol ..."
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Holzer and Holzer [HH04] proved the Tantrix TM rotation puzzle problem with four colors NPcomplete. Baumeister and Rothe [BR07] modified their construction to achieve a parsimonious reduction from satisfiability to this problem. Since parsimonious reductions preserve the number of solutions, it follows that the unique version of the fourcolor Tantrix TM rotation puzzle problem is DPcomplete under randomized reductions. In this paper, we study the threecolor and the twocolor Tantrix TM rotation puzzle problem. Restricting the number of allowed colors to three (respectively, to two) reduces the set of available Tantrix TM tiles from 56 to 14 (respectively, to 8). We prove that both the threecolor and the twocolor Tantrix TM rotation puzzle problem is NPcomplete, which answers a question raised by Holzer and Holzer [HH04] in the affirmative. Since both these reductions are parsimonious, it follows that both the unique threecolor and the unique twocolor Tantrix TM rotation puzzle problem is DPcomplete under randomized reductions. Finally, we prove that the infinite version of both the threecolor and the twocolor Tantrix TM rotation puzzle problem is undecidable.