31 research outputs found
The Reducts of the Homogeneous Binary Branching C-relation
Let (L;C) be the (up to isomorphism unique) countable homogeneous structure
carrying a binary branching C-relation. We study the reducts of (L;C), i.e.,
the structures with domain L that are first-order definable in (L;C). We show
that up to existential interdefinability, there are finitely many such reducts.
This implies that there are finitely many reducts up to first-order
interdefinability, thus confirming a conjecture of Simon Thomas for the special
case of (L;C). We also study the endomorphism monoids of such reducts and show
that they fall into four categories.Comment: 39 pages, 4 figure
A complexity dichotomy for poset constraint satisfaction
In this paper we determine the complexity of a broad class of problems that
extends the temporal constraint satisfaction problems. To be more precise we
study the problems Poset-SAT(), where is a given set of
quantifier-free -formulas. An instance of Poset-SAT() consists of
finitely many variables and formulas
with ; the question is
whether this input is satisfied by any partial order on or
not. We show that every such problem is NP-complete or can be solved in
polynomial time, depending on . All Poset-SAT problems can be formalized
as constraint satisfaction problems on reducts of the random partial order. We
use model-theoretic concepts and techniques from universal algebra to study
these reducts. In the course of this analysis we establish a dichotomy that we
believe is of independent interest in universal algebra and model theory.Comment: 29 page
Canonical functions: a proof via topological dynamics
Canonical functions are a powerful concept with numerous applications in the study of groups, monoids, and clones on countable structures with Ramsey-type properties. In this short note, we present a proof of the existence of canonical functions in certain sets using topological dynamics, providing a shorter alternative to the original combinatorial argument. We moreover present equivalent algebraic characterisations of canonicity
On the Scope of the Universal-Algebraic Approach to Constraint Satisfaction
The universal-algebraic approach has proved a powerful tool in the study of
the complexity of CSPs. This approach has previously been applied to the study
of CSPs with finite or (infinite) omega-categorical templates, and relies on
two facts. The first is that in finite or omega-categorical structures A, a
relation is primitive positive definable if and only if it is preserved by the
polymorphisms of A. The second is that every finite or omega-categorical
structure is homomorphically equivalent to a core structure. In this paper, we
present generalizations of these facts to infinite structures that are not
necessarily omega-categorical. (This abstract has been severely curtailed by
the space constraints of arXiv -- please read the full abstract in the
article.) Finally, we present applications of our general results to the
description and analysis of the complexity of CSPs. In particular, we give
general hardness criteria based on the absence of polymorphisms that depend on
more than one argument, and we present a polymorphism-based description of
those CSPs that are first-order definable (and therefore can be solved in
polynomial time).Comment: Extended abstract appeared at 25th Symposium on Logic in Computer
Science (LICS 2010). This version will appear in the LMCS special issue
associated with LICS 201
Tameness in least fixed-point logic and McColm's conjecture
We investigate four model-theoretic tameness properties in the context of
least fixed-point logic over a family of finite structures. We find that each
of these properties depends only on the elementary (i.e., first-order) limit
theory, and we completely determine the valid entailments among them. In
contrast to the context of first-order logic on arbitrary structures, the order
property and independence property are equivalent in this setting. McColm
conjectured that least fixed-point definability collapses to first-order
definability exactly when proficiency fails. McColm's conjecture is known to be
false in general. However, we show that McColm's conjecture is true for any
family of finite structures whose limit theory is model-theoretically tame