40 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
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 #Pcomplete 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
Cluster Computing and the Power of Edge Recognition
We study the robustness--the invariance under definition changes--of the
cluster class CL#P [HHKW05]. This class contains each #P function that is
computed by a balanced Turing machine whose accepting paths always form a
cluster with respect to some length-respecting total order with efficient
adjacency checks. The definition of CL#P is heavily influenced by the defining
paper's focus on (global) orders. In contrast, we define a cluster class,
CLU#P, to capture what seems to us a more natural model of cluster computing.
We prove that the naturalness is costless: CL#P = CLU#P. Then we exploit the
more natural, flexible features of CLU#P to prove new robustness results for
CL#P and to expand what is known about the closure properties of CL#P.
The complexity of recognizing edges--of an ordered collection of computation
paths or of a cluster of accepting computation paths--is central to this study.
Most particularly, our proofs exploit the power of unique discovery of
edges--the ability of nondeterministic functions to, in certain settings,
discover on exactly one (in some cases, on at most one) computation path a
critical piece of information regarding edges of orderings or clusters
Permutation Inequalities for Walks in Graphs
Using spectral graph theory, we show how to obtain inequalities for the
number of walks in graphs from nonnegative polynomials and present a new family
of such inequalities
The Complexity of Computing the Size of an Interval
Given a p-order A over a universe of strings (i.e., a transitive, reflexive,
antisymmetric relation such that if (x, y) is an element of A then |x| is
polynomially bounded by |y|), an interval size function of A returns, for each
string x in the universe, the number of strings in the interval between strings
b(x) and t(x) (with respect to A), where b(x) and t(x) are functions that are
polynomial-time computable in the length of x.
By choosing sets of interval size functions based on feasibility requirements
for their underlying p-orders, we obtain new characterizations of complexity
classes. We prove that the set of all interval size functions whose underlying
p-orders are polynomial-time decidable is exactly #P. We show that the interval
size functions for orders with polynomial-time adjacency checks are closely
related to the class FPSPACE(poly). Indeed, FPSPACE(poly) is exactly the class
of all nonnegative functions that are an interval size function minus a
polynomial-time computable function.
We study two important functions in relation to interval size functions. The
function #DIV maps each natural number n to the number of nontrivial divisors
of n. We show that #DIV is an interval size function of a polynomial-time
decidable partial p-order with polynomial-time adjacency checks. The function
#MONSAT maps each monotone boolean formula F to the number of satisfying
assignments of F. We show that #MONSAT is an interval size function of a
polynomial-time decidable total p-order with polynomial-time adjacency checks.
Finally, we explore the related notion of cluster computation.Comment: This revision fixes a problem in the proof of Theorem 9.
Persistent Computations
We study computational effects of persistent Turing machines, independently introduced by Goldin and Wegner [GW98], and Kosub [Kos98]. Persistence is a mode of interaction which makes it possible to consider the computational behavior of a Turing machine as an infinite sequence of autonomous computations. We investigate different computability concepts such as conditional and essential computability. Furthermore, we give a characterization of all non-immune sets in terms of persistent computations.publishe
On Cluster Machines and Function Classes
We consider a special kind of non-deterministic Turing machines. Cluster machines are distinguished by a neighbourhood relationship between accepting paths. Based on a formalization using equivalence relations some subtle properties of these machines are proven. Moreover, by abstraction we gain the machine-independend concept of cluster sets which is the starting point to establish cluster operators. Cluster operators map complexity classes of sets into complexity classes of functions where for the domain classes only cluster sets are allowed. For the counting operator c#\Delta and the optimization operators cmax\Delta and cmin\Delta the structural relationships between images resulting from these operators on the polynomial-time hierarchy are investigated. Furthermore, we compare cluster operators with the corresponding common operators #\Delta, max\Delta and min\Delta [Tod90b, HW97]