5,921 research outputs found
Dichotomy Results for Fixed-Point Existence Problems for Boolean Dynamical Systems
A complete classification of the computational complexity of the fixed-point
existence problem for boolean dynamical systems, i.e., finite discrete
dynamical systems over the domain {0, 1}, is presented. For function classes F
and graph classes G, an (F, G)-system is a boolean dynamical system such that
all local transition functions lie in F and the underlying graph lies in G. Let
F be a class of boolean functions which is closed under composition and let G
be a class of graphs which is closed under taking minors. The following
dichotomy theorems are shown: (1) If F contains the self-dual functions and G
contains the planar graphs then the fixed-point existence problem for (F,
G)-systems with local transition function given by truth-tables is NP-complete;
otherwise, it is decidable in polynomial time. (2) If F contains the self-dual
functions and G contains the graphs having vertex covers of size one then the
fixed-point existence problem for (F, G)-systems with local transition function
given by formulas or circuits is NP-complete; otherwise, it is decidable in
polynomial time.Comment: 17 pages; this version corrects an error/typo in the 2008/01/24
versio
Dichotomy Results for Fixed Point Counting in Boolean Dynamical Systems
We present dichotomy theorems regarding the computational complexity of
counting fixed points in boolean (discrete) dynamical systems, i.e., finite
discrete dynamical systems over the domain {0,1}. For a class F of boolean
functions and a class G of graphs, an (F,G)-system is a boolean dynamical
system with local transitions functions lying in F and graphs in G. We show
that, if local transition functions are given by lookup tables, then the
following complexity classification holds: Let F be a class of boolean
functions closed under superposition and let G be a graph class closed under
taking minors. If F contains all min-functions, all max-functions, or all
self-dual and monotone functions, and G contains all planar graphs, then it is
#P-complete to compute the number of fixed points in an (F,G)-system; otherwise
it is computable in polynomial time. We also prove a dichotomy theorem for the
case that local transition functions are given by formulas (over logical
bases). This theorem has a significantly more complicated structure than the
theorem for lookup tables. A corresponding theorem for boolean circuits
coincides with the theorem for formulas.Comment: 16 pages, extended abstract presented at 10th Italian Conference on
Theoretical Computer Science (ICTCS'2007
Solving constraint-satisfaction problems with distributed neocortical-like neuronal networks
Finding actions that satisfy the constraints imposed by both external inputs
and internal representations is central to decision making. We demonstrate that
some important classes of constraint satisfaction problems (CSPs) can be solved
by networks composed of homogeneous cooperative-competitive modules that have
connectivity similar to motifs observed in the superficial layers of neocortex.
The winner-take-all modules are sparsely coupled by programming neurons that
embed the constraints onto the otherwise homogeneous modular computational
substrate. We show rules that embed any instance of the CSPs planar four-color
graph coloring, maximum independent set, and Sudoku on this substrate, and
provide mathematical proofs that guarantee these graph coloring problems will
convergence to a solution. The network is composed of non-saturating linear
threshold neurons. Their lack of right saturation allows the overall network to
explore the problem space driven through the unstable dynamics generated by
recurrent excitation. The direction of exploration is steered by the constraint
neurons. While many problems can be solved using only linear inhibitory
constraints, network performance on hard problems benefits significantly when
these negative constraints are implemented by non-linear multiplicative
inhibition. Overall, our results demonstrate the importance of instability
rather than stability in network computation, and also offer insight into the
computational role of dual inhibitory mechanisms in neural circuits.Comment: Accepted manuscript, in press, Neural Computation (2018
Complexity of the Guarded Two-Variable Fragment with Counting Quantifiers
We show that the finite satisfiability problem for the guarded two-variable
fragment with counting quantifiers is in EXPTIME. The method employed also
yields a simple proof of a result recently obtained by Y. Kazakov, that the
satisfiability problem for the guarded two-variable fragment with counting
quantifiers is in EXPTIME.Comment: 20 pages, 3 figure
Datalog and Constraint Satisfaction with Infinite Templates
On finite structures, there is a well-known connection between the expressive
power of Datalog, finite variable logics, the existential pebble game, and
bounded hypertree duality. We study this connection for infinite structures.
This has applications for constraint satisfaction with infinite templates. If
the template Gamma is omega-categorical, we present various equivalent
characterizations of those Gamma such that the constraint satisfaction problem
(CSP) for Gamma can be solved by a Datalog program. We also show that
CSP(Gamma) can be solved in polynomial time for arbitrary omega-categorical
structures Gamma if the input is restricted to instances of bounded treewidth.
Finally, we characterize those omega-categorical templates whose CSP has
Datalog width 1, and those whose CSP has strict Datalog width k.Comment: 28 pages. This is an extended long version of a conference paper that
appeared at STACS'06. In the third version in the arxiv we have revised the
presentation again and added a section that relates our results to
formalizations of CSPs using relation algebra
Existential Second-Order Logic Over Graphs: A Complete Complexity-Theoretic Classification
Descriptive complexity theory aims at inferring a problem's computational
complexity from the syntactic complexity of its description. A cornerstone of
this theory is Fagin's Theorem, by which a graph property is expressible in
existential second-order logic (ESO logic) if, and only if, it is in NP. A
natural question, from the theory's point of view, is which syntactic fragments
of ESO logic also still characterize NP. Research on this question has
culminated in a dichotomy result by Gottlob, Kolatis, and Schwentick: for each
possible quantifier prefix of an ESO formula, the resulting prefix class either
contains an NP-complete problem or is contained in P. However, the exact
complexity of the prefix classes inside P remained elusive. In the present
paper, we clear up the picture by showing that for each prefix class of ESO
logic, its reduction closure under first-order reductions is either FO, L, NL,
or NP. For undirected, self-loop-free graphs two containment results are
especially challenging to prove: containment in L for the prefix and containment in FO for the prefix
for monadic . The complex argument by
Gottlob, Kolatis, and Schwentick concerning polynomial time needs to be
carefully reexamined and either combined with the logspace version of
Courcelle's Theorem or directly improved to first-order computations. A
different challenge is posed by formulas with the prefix : We show that they express special constraint satisfaction problems
that lie in L.Comment: Technical report version of a STACS 2015 pape
Tractable classes of binary CSPs defined by excluded topological minors
The binary Constraint Satisfaction Problem (CSP) is to decide whether there exists an assignment to a set of variables which satisfies specified constraints between pairs of variables. A CSP instance can be presented as a labelled graph (called the microstructure) encoding both the forms of the constraints and where they are imposed. We consider subproblems defined by restricting the allowed form of the microstructure. One form of restriction that has previously been considered is to forbid certain specified substructures (patterns). This captures some tractable classes of the CSP, but does not capture the well-known property of acyclicity. In this paper we introduce the notion of a topological minor of a binary CSP instance. By forbidding certain patterns as topological minors we obtain a compact mechanism for expressing several novel tractable classes, including new generalisations of the class of acyclic instances
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