907 research outputs found
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
On the Descriptive Complexity of Temporal Constraint Satisfaction Problems
Finite-domain constraint satisfaction problems are either solvable by
Datalog, or not even expressible in fixed-point logic with counting. The border
between the two regimes coincides with an important dichotomy in universal
algebra; in particular, the border can be described by a strong height-one
Maltsev condition. For infinite-domain CSPs, the situation is more complicated
even if the template structure of the CSP is model-theoretically tame. We prove
that there is no Maltsev condition that characterizes Datalog already for the
CSPs of first-order reducts of (Q;<); such CSPs are called temporal CSPs and
are of fundamental importance in infinite-domain constraint satisfaction. Our
main result is a complete classification of temporal CSPs that can be expressed
in one of the following logical formalisms: Datalog, fixed-point logic (with or
without counting), or fixed-point logic with the Boolean rank operator. The
classification shows that many of the equivalent conditions in the finite fail
to capture expressibility in Datalog or fixed-point logic already for temporal
CSPs.Comment: 57 page
The condensation phase transition in the regular -SAT model
Much of the recent work on random constraint satisfaction problems has been
inspired by ingenious but non-rigorous approaches from physics. The physics
predictions typically come in the form of distributional fixed point problems
that are intended to mimic Belief Propagation, a message passing algorithm,
applied to the random CSP. In this paper we propose a novel method for
harnessing Belief Propagation directly to obtain a rigorous proof of such a
prediction, namely the existence and location of a condensation phase
transition in the random regular -SAT model.Comment: Revised version based on arXiv:1504.03975, version
Gauge theory of things alive and universal dynamics
Positing complex adaptive systems made of agents with relations between them
that can be composed, it follows that they can be described by gauge theories
similar to elementary particle theory and general relativity. By definition, a
universal dynamics is able to determine the time development of any such system
without need for further specification. The possibilities are limited, but one
of them - reproduction fork dynamics - describes DNA replication and is the
basis of biological life on earth. It is a universal copy machine and a
renormalization group fixed point. A universal equation of motion in continuous
time is also presented.Comment: 13 pages, latex, uses fleqn.sty (can be removed without harm
Abstract Syntax and Variable Binding (Extended Abstract)
We develop a theory of abstract syntax with variable
binding. To every binding signature we associate a category
of models consisting of variable sets endowed with
compatible algebra and substitution structures. The syntax
generated by the signature is the initial model. This gives a
notion of initial algebra semantics encompassing the traditional
one; besides compositionality, it automatically veri-
fies the semantic substitution lemma
Clones with finitely many relative R-classes
For each clone C on a set A there is an associated equivalence relation
analogous to Green's R-relation, which relates two operations on A iff each one
is a substitution instance of the other using operations from C. We study the
clones for which there are only finitely many relative R-classes.Comment: 41 pages; proofs improved, examples adde
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