5 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
Convergence of Opinion Diffusion is PSPACE-complete
We analyse opinion diffusion in social networks, where a finite set of
individuals is connected in a directed graph and each simultaneously changes
their opinion to that of the majority of their influencers. We study the
algorithmic properties of the fixed-point behaviour of such networks, showing
that the problem of establishing whether individuals converge to stable
opinions is PSPACE-complete
Complexity of fixed point counting problems in Boolean Networks
A Boolean network (BN) with components is a discrete dynamical system
described by the successive iterations of a function . This model finds applications in biology, where fixed points play a
central role. For example, in genetic regulations, they correspond to cell
phenotypes. In this context, experiments reveal the existence of positive or
negative influences among components: component has a positive (resp.
negative) influence on component meaning that tends to mimic (resp.
negate) . The digraph of influences is called signed interaction digraph
(SID), and one SID may correspond to a large number of BNs (which is, in
average, doubly exponential according to ). The present work opens a new
perspective on the well-established study of fixed points in BNs. When
biologists discover the SID of a BN they do not know, they may ask: given that
SID, can it correspond to a BN having at least/at most fixed points?
Depending on the input, we prove that these problems are in or
complete for , ,
\textrm{NP}^{\textrm{#P}} or . In particular, we prove
that it is -complete (resp. -complete) to
decide if a given SID can correspond to a BN having at least two fixed points
(resp. no fixed point).Comment: 43 page