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Beyersdorff O.,Leibniz University of Hanover | Beyersdorff O.,University of Leeds | Datta S.,Chennai Mathematical Institute | Krebs A.,University of Tubingen | And 7 more authors.
ACM Transactions on Computation Theory | Year: 2013

In this paper we initiate the study of proof systems where verification of proofs proceeds by NC0 circuits. We investigate the question which languages admit proof systems in this very restricted model. Formulated alternatively, we ask which languages can be enumerated by NC0 functions. Our results show that the answer to this problem is not determined by the complexity of the language. On the one hand, we construct NC0 proof systems for a variety of languages ranging from regular to NP complete. On the other hand, we show by combinatorial methods that even easy regular languages such as Exact-OR do not admit NC0 proof systems. We also show that Majority does not admit NC0 proof systems. Finally, we present a general construction of NC0 proof systems for regular languages with strongly connected NFA's. © 2013 ACM. Source


Creignou N.,Aix - Marseille University | Schmidt J.,Aix - Marseille University | Thomas M.,TWT GmbH | Woltran S.,Vienna University of Technology
Argument and Computation | Year: 2011

Many proposals for logic-based formalisations of argumentation consider an argument as a pair (Φ,α), where the support Φ 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 Σ 2 p-complete. In order to better understand the sources of complexity (and to identify tractable fragments), we focus on arguments given over formul 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. © 2011 Copyright Taylor and Francis Group, LLC. Source


Thomas M.,TWT GmbH
Information Processing Letters | Year: 2012

For decision problems Π(B) defined over Boolean circuits using gates from a restricted set B only, we have Π(B)≤m AC0Π( B′) for all finite sets B and B′ of gates such that all gates from B can be computed by circuits over gates from B′. In this note, we show that a weaker version of this statement holds for decision problems defined over Boolean formulae, namely that Π(B)≤m NC2Π( B′∪{∧,∨}) and Π(B)≤m NC2Π( B′∪{0,1}) for all finite sets B and B′ of Boolean functions such that all f∈B can be defined in B′. © 2012 Elsevier B.V. © 2012 Elsevier B.V. All rights reserved. Source


Creignou N.,Aix - Marseille University | Meier A.,Leibniz University of Hanover | Vollmer H.,Leibniz University of Hanover | Thomas M.,TWT GmbH
ACM Transactions on Computational Logic | Year: 2012

Autoepistemic logic extends propositional logic by the modal operator L. A formula φ that is preceded by an L is said to be "believed." The logic was introduced by Moore in 1985 for modeling an ideally rational agent's behavior and reasoning about his own beliefs. In this article we analyze all Boolean fragments of autoepistemic logic with respect to the computational complexity of the three most common decision problems expansion existence, brave reasoning and cautious reasoning. As a second contribution we classify the computational complexity of checking that a given set of formulae characterizes a stable expansion and that of counting the number of stable expansions of a given knowledge base. We improve the best known Δ 2 p-upper bound on the former problem to completeness for the second level of the Boolean hierarchy. To the best of our knowledge, this is the first paper analyzing counting problem for autoepistemic logic. © 2012 ACM 1529-3785/2012/04-ART17 $10.00. Source


Beyersdorff O.,Leibniz University of Hanover | Datta S.,Chennai Mathematical Institute | Mahajan M.,Chennai Mathematical Institute | Scharfenberger-Fabian G.,University of Greifswald | And 3 more authors.
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) | Year: 2011

In this paper we initiate the study of proof systems where verification of proofs proceeds by NC0 circuits. We investigate the question which languages admit proof systems in this very restricted model. Formulated alternatively, we ask which languages can be enumerated by NC0 functions. Our results show that the answer to this problem is not determined by the complexity of the language. On the one hand, we construct NC0 proof systems for a variety of languages ranging from regular to NP-complete. On the other hand, we show by combinatorial methods that even easy regular languages such as Exact-OR do not admit NC0 proof systems. We also present a general construction of proof systems for regular languages with strongly connected NFA's. © 2011 Springer-Verlag GmbH. Source

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