The complexity of propositional implication

Beyersdorff, Olaf; Meier, Arne; Thomas, Michael E.; Vollmer, Heribert

doi:10.1016/j.ipl.2009.06.015

Cited by 24 publications

(43 citation statements)

References 13 publications

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“…Another problem which is amenable to the kind of complexity analysis we did in this paper is the identification of conflicts between two arguments (different notions for conflicts between arguments based on classical logic exist, see e.g. Gorogiannis and Hunter 2011) which is an intractable problem itself; however, since most conflicts are identified via the implication problem, we expect results similar to the one derived by Beyersdorff et al (2009a). A complexity analysis in that direction has already been undertaken by Wooldridge, Dunne, and Parsons (2006).…”

Section: Discussionmentioning

confidence: 71%

“…As For X ⊆ [B] ⊆ Y with X ∈ {S 00 , S 10 , D 2 } and Y ∈ {M, R 1 }, membership in NP follows from the facts that satisfiability is in Logspace (Lewis 1979), while entailment is in coNP (Beyersdorff et al 2009a). To prove the coNP-hardness of Arg(B), we give a reduction from the implication problem for B-formulae, which is coNP-hard if [B] contains one of the clones S 00 , S 10 , D 2 .…”

Section: The Same Classification Holds For Arg-disp(b)mentioning

confidence: 99%

“…We map this instance to ({ψ}, α) if ψ is satisfiable (which is easy to decide) and to a trivial positive instance otherwise. In all other cases, Logspace-membership follows from the fact that both problems, satisfiability and entailment, are contained in Logspace (see Beyersdorff et al 2009a) …”

Section: The Same Classification Holds For Arg-disp(b)mentioning

confidence: 99%

“…Several complexity classifications have already been obtained in this so-called Post's framework, such as the classical satisfiability problem (Lewis 1979), equivalence and implication problems (Reith 2003;Beyersdorff, Meier, Thomas, and Vollmer 2009a), satisfiability and model checking in modal and temporal logics (Bauland, Hemaspaandra, Schnoor, and Schnoor 2006;Bauland, Schneider, Schnoor, Schnoor, and Vollmer 2008), default logic (Beyersdorff, Meier, Thomas, and Vollmer 2009b), circumscription (Thomas 2009) and abduction (Creignou, Schmidt, and Thomas 2010).…”

Section: Introductionmentioning

confidence: 99%

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Complexity of logic-based argumentation in Post's framework

Creignou¹,

Schmidt²,

Thomas³

et al. 2011

Argument & Computation

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Many proposals for logic-based formalisations of argumentation consider an argument as a pair ( , α), where the support is understood as a minimal consistent subset of a given knowledge base which has to entail the claim α. In case the arguments are given in the full language of classical propositional logic reasoning in such frameworks becomes a computationally costly task. For instance, the problem of deciding whether there exists a support for a given claim has been shown to be p 2 -complete. In order to better understand the sources of complexity (and to identify tractable fragments), we focus on arguments given over formulae in which the allowed connectives are taken from certain sets of Boolean functions. We provide a complexity classification for four different decision problems (existence of a support, checking the validity of an argument, relevance and dispensability) with respect to all possible sets of Boolean functions. Moreover, we make use of a general schema to enumerate all arguments to show that certain restricted fragments permit polynomial delay. Finally, we give a classification also in terms of counting complexity.

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Section: Discussionmentioning

confidence: 71%

Section: The Same Classification Holds For Arg-disp(b)mentioning

confidence: 99%

Section: The Same Classification Holds For Arg-disp(b)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Complexity of logic-based argumentation in Post's framework

Creignou¹,

Schmidt²,

Thomas³

et al. 2011

Argument & Computation

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“…This approach has first been taken by Lewis, who classified the complexity of the satisfiability problem w. r. t. to all finite sets of Boolean functions [16]. Many problems have been studied in this way since then (see [24,23,13,1,3,2,18], amongst others).…”

Section: Introductionmentioning

confidence: 99%

The Complexity of Circumscriptive Inference in Post’s Lattice

Thomas

2009

Logic Programming and Nonmonotonic Reasoning

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Abstract. Circumscription is one of the most important formalisms for reasoning with incomplete information. It is equivalent to reasoning under the extended closed world assumption, which allows to conclude that the facts derivable from a given knowledge base are all facts that satisfy a given property. In this paper, we study the computational complexity of several formalizations of inference in propositional circumscription for the case that the knowledge base is described by a propositional theory using only a restricted set of Boolean functions. To systematically cover all possible sets of Boolean functions, we use Post's lattice. With its help, we determine the complexity of circumscriptive inference for all but two possible classes of Boolean functions. Each of these problems is shown to be either Π p 2 -complete, coNP-complete, or contained in L. In particular, we show that in the general case, unless P = NP, only literal theories admit polynomial-time algorithms, while for some restricted variants the tractability border is the same as for classical propositional inference.

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The Weight in Enumeration

Schmidt

2017

Language and Automata Theory and Applications

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In our setting enumeration amounts to generate all solutions of a problem instance without duplicates. We address the problem of enumerating the models of B-formulae. A B-formula is a propositional formula whose connectives are taken from a fixed set B of Boolean connectives. Without imposing any specific order to output the solutions, this task is solved. We completely classify the complexity of this enumeration task for all possible sets of connectives B imposing the orders of (1) non-decreasing weight, (2) non-increasing weight; the weight of a model being the number of variables assigned to 1. We consider also the weighted variants where a non-negative integer weight is assigned to each variable and show that this add-on leads to more sophisticated enumeration algorithms and even renders previously tractable cases intractable, contrarily to the constraint setting. As a by-product we obtain also complexity classifications for the optimization problems known as Min-Ones and Max-Ones which are in the B-formula setting two different tasks.

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The complexity of propositional implication

Cited by 24 publications

References 13 publications

Complexity of logic-based argumentation in Post's framework

Complexity of logic-based argumentation in Post's framework

The Complexity of Circumscriptive Inference in Post’s Lattice

The Weight in Enumeration

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