3,971 research outputs found
The Complexity of Bisimulation and Simulation on Finite Systems
In this paper the computational complexity of the (bi)simulation problem over
restricted graph classes is studied. For trees given as pointer structures or
terms the (bi)simulation problem is complete for logarithmic space or NC,
respectively. This solves an open problem from Balc\'azar, Gabarr\'o, and
S\'antha. Furthermore, if only one of the input graphs is required to be a
tree, the bisimulation (simulation) problem is contained in AC (LogCFL). In
contrast, it is also shown that the simulation problem is P-complete already
for graphs of bounded path-width
Pure Nash Equilibria: Hard and Easy Games
We investigate complexity issues related to pure Nash equilibria of strategic
games. We show that, even in very restrictive settings, determining whether a
game has a pure Nash Equilibrium is NP-hard, while deciding whether a game has
a strong Nash equilibrium is SigmaP2-complete. We then study practically
relevant restrictions that lower the complexity. In particular, we are
interested in quantitative and qualitative restrictions of the way each players
payoff depends on moves of other players. We say that a game has small
neighborhood if the utility function for each player depends only on (the
actions of) a logarithmically small number of other players. The dependency
structure of a game G can be expressed by a graph DG(G) or by a hypergraph
H(G). By relating Nash equilibrium problems to constraint satisfaction problems
(CSPs), we show that if G has small neighborhood and if H(G) has bounded
hypertree width (or if DG(G) has bounded treewidth), then finding pure Nash and
Pareto equilibria is feasible in polynomial time. If the game is graphical,
then these problems are LOGCFL-complete and thus in the class NC2 of highly
parallelizable problems
Compact Labelings For Efficient First-Order Model-Checking
We consider graph properties that can be checked from labels, i.e., bit
sequences, of logarithmic length attached to vertices. We prove that there
exists such a labeling for checking a first-order formula with free set
variables in the graphs of every class that is \emph{nicely locally
cwd-decomposable}. This notion generalizes that of a \emph{nicely locally
tree-decomposable} class. The graphs of such classes can be covered by graphs
of bounded \emph{clique-width} with limited overlaps. We also consider such
labelings for \emph{bounded} first-order formulas on graph classes of
\emph{bounded expansion}. Some of these results are extended to counting
queries
Deciding Quantifier-Free Presburger Formulas Using Parameterized Solution Bounds
Given a formula in quantifier-free Presburger arithmetic, if it has a
satisfying solution, there is one whose size, measured in bits, is polynomially
bounded in the size of the formula. In this paper, we consider a special class
of quantifier-free Presburger formulas in which most linear constraints are
difference (separation) constraints, and the non-difference constraints are
sparse. This class has been observed to commonly occur in software
verification. We derive a new solution bound in terms of parameters
characterizing the sparseness of linear constraints and the number of
non-difference constraints, in addition to traditional measures of formula
size. In particular, we show that the number of bits needed per integer
variable is linear in the number of non-difference constraints and logarithmic
in the number and size of non-zero coefficients in them, but is otherwise
independent of the total number of linear constraints in the formula. The
derived bound can be used in a decision procedure based on instantiating
integer variables over a finite domain and translating the input
quantifier-free Presburger formula to an equi-satisfiable Boolean formula,
which is then checked using a Boolean satisfiability solver. In addition to our
main theoretical result, we discuss several optimizations for deriving tighter
bounds in practice. Empirical evidence indicates that our decision procedure
can greatly outperform other decision procedures.Comment: 26 page
Tree-like Queries in OWL 2 QL: Succinctness and Complexity Results
This paper investigates the impact of query topology on the difficulty of
answering conjunctive queries in the presence of OWL 2 QL ontologies. Our first
contribution is to clarify the worst-case size of positive existential (PE),
non-recursive Datalog (NDL), and first-order (FO) rewritings for various
classes of tree-like conjunctive queries, ranging from linear queries to
bounded treewidth queries. Perhaps our most surprising result is a
superpolynomial lower bound on the size of PE-rewritings that holds already for
linear queries and ontologies of depth 2. More positively, we show that
polynomial-size NDL-rewritings always exist for tree-shaped queries with a
bounded number of leaves (and arbitrary ontologies), and for bounded treewidth
queries paired with bounded depth ontologies. For FO-rewritings, we equate the
existence of polysize rewritings with well-known problems in Boolean circuit
complexity. As our second contribution, we analyze the computational complexity
of query answering and establish tractability results (either NL- or
LOGCFL-completeness) for a range of query-ontology pairs. Combining our new
results with those from the literature yields a complete picture of the
succinctness and complexity landscapes for the considered classes of queries
and ontologies.Comment: This is an extended version of a paper accepted at LICS'15. It
contains both succinctness and complexity results and adopts FOL notation.
The appendix contains proofs that had to be omitted from the conference
version for lack of space. The previous arxiv version (a long version of our
DL'14 workshop paper) only contained the succinctness results and used
description logic notatio
MALL proof equivalence is Logspace-complete, via binary decision diagrams
Proof equivalence in a logic is the problem of deciding whether two proofs
are equivalent modulo a set of permutation of rules that reflects the
commutative conversions of its cut-elimination procedure. As such, it is
related to the question of proofnets: finding canonical representatives of
equivalence classes of proofs that have good computational properties. It can
also be seen as the word problem for the notion of free category corresponding
to the logic.
It has been recently shown that proof equivalence in MLL (the multiplicative
with units fragment of linear logic) is PSPACE-complete, which rules out any
low-complexity notion of proofnet for this particular logic.
Since it is another fragment of linear logic for which attempts to define a
fully satisfactory low-complexity notion of proofnet have not been successful
so far, we study proof equivalence in MALL- (multiplicative-additive without
units fragment of linear logic) and discover a situation that is totally
different from the MLL case. Indeed, we show that proof equivalence in MALL-
corresponds (under AC0 reductions) to equivalence of binary decision diagrams,
a data structure widely used to represent and analyze Boolean functions
efficiently.
We show these two equivalent problems to be LOGSPACE-complete. If this
technically leaves open the possibility for a complete solution to the question
of proofnets for MALL-, the established relation with binary decision diagrams
actually suggests a negative solution to this problem.Comment: in TLCA 201
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