59 research outputs found
Parameterized complexity of DPLL search procedures
We study the performance of DPLL algorithms on parameterized problems. In particular, we investigate how difficult it is to decide whether small solutions exist for satisfiability and other combinatorial problems. For this purpose we develop a Prover-Delayer game which models the running time of DPLL procedures and we establish an information-theoretic method to obtain lower bounds to the running time of parameterized DPLL procedures. We illustrate this technique by showing lower bounds to the parameterized pigeonhole principle and to the ordering principle. As our main application we study the DPLL procedure for the problem of deciding whether a graph has a small clique. We show that proving the absence of a k-clique requires n steps for a non-trivial distribution of graphs close to the critical threshold. For the restricted case of tree-like Parameterized Resolution, this result answers a question asked in [11] of understanding the Resolution complexity of this family of formulas
Resolution and the binary encoding of combinatorial principles.
Res(s) is an extension of Resolution working on s-DNFs. We prove tight n (k) lower bounds for the size
of refutations of the binary version of the k-Clique Principle in Res(o(log log n)). Our result improves that of
Lauria, Pudlák et al. [27] who proved the lower bound for Res(1), i.e. Resolution. The exact complexity of
the (unary) k-Clique Principle in Resolution is unknown. To prove the lower bound we do not use any form of
the Switching Lemma [35], instead we apply a recursive argument specific for binary encodings. Since for the
k-Clique and other principles lower bounds in Resolution for the unary version follow from lower bounds in
Res(log n) for their binary version we start a systematic study of the complexity of proofs in Resolution-based
systems for families of contradictions given in the binary encoding.
We go on to consider the binary version of the weak Pigeonhole Principle Bin-PHPmn
for m > n. Using
the the same recursive approach we prove the new result that for any > 0, Bin-PHPmn
requires proofs of size
2n1− in Res(s) for s = o(log1/2 n). Our lower bound is almost optimal since for m 2
p
n log n there are
quasipolynomial size proofs of Bin-PHPmn
in Res(log n).
Finally we propose a general theory in which to compare the complexity of refuting the binary and unary
versions of large classes of combinatorial principles, namely those expressible as first order formulae in 2-form
and with no finite model
Incremental QBF Solving
We consider the problem of incrementally solving a sequence of quantified
Boolean formulae (QBF). Incremental solving aims at using information learned
from one formula in the process of solving the next formulae in the sequence.
Based on a general overview of the problem and related challenges, we present
an approach to incremental QBF solving which is application-independent and
hence applicable to QBF encodings of arbitrary problems. We implemented this
approach in our incremental search-based QBF solver DepQBF and report on
implementation details. Experimental results illustrate the potential benefits
of incremental solving in QBF-based workflows.Comment: revision (camera-ready, to appear in the proceedings of CP 2014,
LNCS, Springer
Knowledge Compilation of Logic Programs Using Approximation Fixpoint Theory
To appear in Theory and Practice of Logic Programming (TPLP), Proceedings of
ICLP 2015
Recent advances in knowledge compilation introduced techniques to compile
\emph{positive} logic programs into propositional logic, essentially exploiting
the constructive nature of the least fixpoint computation. This approach has
several advantages over existing approaches: it maintains logical equivalence,
does not require (expensive) loop-breaking preprocessing or the introduction of
auxiliary variables, and significantly outperforms existing algorithms.
Unfortunately, this technique is limited to \emph{negation-free} programs. In
this paper, we show how to extend it to general logic programs under the
well-founded semantics.
We develop our work in approximation fixpoint theory, an algebraical
framework that unifies semantics of different logics. As such, our algebraical
results are also applicable to autoepistemic logic, default logic and abstract
dialectical frameworks
Symmetry and complexity in propositional reasoning
We establish computational complexity results for a number of simple problem formulations connecting group action and prepositional formulas. The results are discussed
in the context of complexity results arising from established work in the area of automated reasoning techniques which exploit symmetry
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