666 research outputs found
Complexity Results for Modal Dependence Logic
Modal dependence logic was introduced recently by V\"a\"an\"anen. It enhances
the basic modal language by an operator =(). For propositional variables
p_1,...,p_n, =(p_1,...,p_(n-1);p_n) intuitively states that the value of p_n is
determined by those of p_1,...,p_(n-1). Sevenster (J. Logic and Computation,
2009) showed that satisfiability for modal dependence logic is complete for
nondeterministic exponential time. In this paper we consider fragments of modal
dependence logic obtained by restricting the set of allowed propositional
connectives. We show that satisfibility for poor man's dependence logic, the
language consisting of formulas built from literals and dependence atoms using
conjunction, necessity and possibility (i.e., disallowing disjunction), remains
NEXPTIME-complete. If we only allow monotone formulas (without negation, but
with disjunction), the complexity drops to PSPACE-completeness. We also extend
V\"a\"an\"anen's language by allowing classical disjunction besides dependence
disjunction and show that the satisfiability problem remains NEXPTIME-complete.
If we then disallow both negation and dependence disjunction, satistiability is
complete for the second level of the polynomial hierarchy. In this way we
completely classify the computational complexity of the satisfiability problem
for all restrictions of propositional and dependence operators considered by
V\"a\"an\"anen and Sevenster.Comment: 22 pages, full version of CSL 2010 pape
Minimization for Generalized Boolean Formulas
The minimization problem for propositional formulas is an important
optimization problem in the second level of the polynomial hierarchy. In
general, the problem is Sigma-2-complete under Turing reductions, but
restricted versions are tractable. We study the complexity of minimization for
formulas in two established frameworks for restricted propositional logic: The
Post framework allowing arbitrarily nested formulas over a set of Boolean
connectors, and the constraint setting, allowing generalizations of CNF
formulas. In the Post case, we obtain a dichotomy result: Minimization is
solvable in polynomial time or coNP-hard. This result also applies to Boolean
circuits. For CNF formulas, we obtain new minimization algorithms for a large
class of formulas, and give strong evidence that we have covered all
polynomial-time cases
A Fine-Grained Hierarchy of Hard Problems in the Separated Fragment
Recently, the separated fragment (SF) has been introduced and proved to be
decidable. Its defining principle is that universally and existentially
quantified variables may not occur together in atoms. The known upper bound on
the time required to decide SF's satisfiability problem is formulated in terms
of quantifier alternations: Given an SF sentence
in which is quantifier free, satisfiability can be decided in
nondeterministic -fold exponential time. In the present paper, we conduct a
more fine-grained analysis of the complexity of SF-satisfiability. We derive an
upper and a lower bound in terms of the degree of interaction of existential
variables (short: degree)}---a novel measure of how many separate existential
quantifier blocks in a sentence are connected via joint occurrences of
variables in atoms. Our main result is the -NEXPTIME-completeness of the
satisfiability problem for the set of all SF sentences that have
degree or smaller. Consequently, we show that SF-satisfiability is
non-elementary in general, since SF is defined without restrictions on the
degree. Beyond trivial lower bounds, nothing has been known about the hardness
of SF-satisfiability so far.Comment: Full version of the LICS 2017 extended abstract having the same
title, 38 page
Generalising unit-refutation completeness and SLUR via nested input resolution
We introduce two hierarchies of clause-sets, SLUR_k and UC_k, based on the
classes SLUR (Single Lookahead Unit Refutation), introduced in 1995, and UC
(Unit refutation Complete), introduced in 1994.
The class SLUR, introduced in [Annexstein et al, 1995], is the class of
clause-sets for which unit-clause-propagation (denoted by r_1) detects
unsatisfiability, or where otherwise iterative assignment, avoiding obviously
false assignments by look-ahead, always yields a satisfying assignment. It is
natural to consider how to form a hierarchy based on SLUR. Such investigations
were started in [Cepek et al, 2012] and [Balyo et al, 2012]. We present what we
consider the "limit hierarchy" SLUR_k, based on generalising r_1 by r_k, that
is, using generalised unit-clause-propagation introduced in [Kullmann, 1999,
2004].
The class UC, studied in [Del Val, 1994], is the class of Unit refutation
Complete clause-sets, that is, those clause-sets for which unsatisfiability is
decidable by r_1 under any falsifying assignment. For unsatisfiable clause-sets
F, the minimum k such that r_k determines unsatisfiability of F is exactly the
"hardness" of F, as introduced in [Ku 99, 04]. For satisfiable F we use now an
extension mentioned in [Ansotegui et al, 2008]: The hardness is the minimum k
such that after application of any falsifying partial assignments, r_k
determines unsatisfiability. The class UC_k is given by the clause-sets which
have hardness <= k. We observe that UC_1 is exactly UC.
UC_k has a proof-theoretic character, due to the relations between hardness
and tree-resolution, while SLUR_k has an algorithmic character. The
correspondence between r_k and k-times nested input resolution (or tree
resolution using clause-space k+1) means that r_k has a dual nature: both
algorithmic and proof theoretic. This corresponds to a basic result of this
paper, namely SLUR_k = UC_k.Comment: 41 pages; second version improved formulations and added examples,
and more details regarding future directions, third version further examples,
improved and extended explanations, and more on SLUR, fourth version various
additional remarks and editorial improvements, fifth version more
explanations and references, typos corrected, improved wordin
On SAT representations of XOR constraints
We study the representation of systems S of linear equations over the
two-element field (aka xor- or parity-constraints) via conjunctive normal forms
F (boolean clause-sets). First we consider the problem of finding an
"arc-consistent" representation ("AC"), meaning that unit-clause propagation
will fix all forced assignments for all possible instantiations of the
xor-variables. Our main negative result is that there is no polysize
AC-representation in general. On the positive side we show that finding such an
AC-representation is fixed-parameter tractable (fpt) in the number of
equations. Then we turn to a stronger criterion of representation, namely
propagation completeness ("PC") --- while AC only covers the variables of S,
now all the variables in F (the variables in S plus auxiliary variables) are
considered for PC. We show that the standard translation actually yields a PC
representation for one equation, but fails so for two equations (in fact
arbitrarily badly). We show that with a more intelligent translation we can
also easily compute a translation to PC for two equations. We conjecture that
computing a representation in PC is fpt in the number of equations.Comment: 39 pages; 2nd v. improved handling of acyclic systems, free-standing
proof of the transformation from AC-representations to monotone circuits,
improved wording and literature review; 3rd v. updated literature,
strengthened treatment of monotonisation, improved discussions; 4th v. update
of literature, discussions and formulations, more details and examples;
conference v. to appear LATA 201
The Complexity of Reasoning for Fragments of Autoepistemic Logic
Autoepistemic logic extends propositional logic by the modal operator L. A
formula that is preceded by an L is said to be "believed". The logic was
introduced by Moore 1985 for modeling an ideally rational agent's behavior and
reasoning about his own beliefs. In this paper 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 counting the number of stable expansions of a given knowledge
base. To the best of our knowledge this is the first paper analyzing the
counting problem for autoepistemic logic
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