19 research outputs found
Quantified Constraints in Twenty Seventeen
I present a survey of recent advances in the algorithmic and computational complexity theory of non-Boolean Quantified Constraint Satisfaction Problems, incorporating some more modern research directions
QCSP on reflexive tournaments
We give a complexity dichotomy for the Quantified Constraint Satisfaction Problem QCSP(H) when H is a reflexive tournament. It is well known that reflexive tournaments can be split into a sequence of strongly connected components H1,…,Hn so that there exists an edge from every vertex of Hi to every vertex of Hj if and only if
Dismantlability, connectedness, and mixing in relational structures
The Constraint Satisfaction Problem (CSP) and its counting counterpart
appears under different guises in many areas of mathematics, computer science,
and elsewhere. Its structural and algorithmic properties have demonstrated to
play a crucial role in many of those applications. For instance, in the
decision CSPs, structural properties of the relational structures
involved---like, for example, dismantlability---and their logical
characterizations have been instrumental for determining the complexity and
other properties of the problem. Topological properties of the solution set
such as connectedness are related to the hardness of CSPs over random
structures. Additionally, in approximate counting and statistical physics,
where CSPs emerge in the form of spin systems, mixing properties and the
uniqueness of Gibbs measures have been heavily exploited for approximating
partition functions and free energy.
In spite of the great diversity of those features, there are some eerie
similarities between them. These were observed and made more precise in the
case of graph homomorphisms by Brightwell and Winkler, who showed that
dismantlability of the target graph, connectedness of the set of homomorphisms,
and good mixing properties of the corresponding spin system are all equivalent.
In this paper we go a step further and demonstrate similar connections for
arbitrary CSPs. This requires much deeper understanding of dismantling and the
structure of the solution space in the case of relational structures, and new
refined concepts of mixing introduced by Brice\~no. In addition, we develop
properties related to the study of valid extensions of a given partially
defined homomorphism, an approach that turns out to be novel even in the graph
case. We also add to the mix the combinatorial property of finite duality and
its logic counterpart, FO-definability, studied by Larose, Loten, and Tardif.Comment: 27 pages, full version of the paper accepted to ICALP 201
Effective lambda-models vs recursively enumerable lambda-theories
A longstanding open problem is whether there exists a non syntactical model
of the untyped lambda-calculus whose theory is exactly the least lambda-theory
(l-beta). In this paper we investigate the more general question of whether the
equational/order theory of a model of the (untyped) lambda-calculus can be
recursively enumerable (r.e. for brevity). We introduce a notion of effective
model of lambda-calculus calculus, which covers in particular all the models
individually introduced in the literature. We prove that the order theory of an
effective model is never r.e.; from this it follows that its equational theory
cannot be l-beta or l-beta-eta. We then show that no effective model living in
the stable or strongly stable semantics has an r.e. equational theory.
Concerning Scott's semantics, we investigate the class of graph models and
prove that no order theory of a graph model can be r.e., and that there exists
an effective graph model whose equational/order theory is minimum among all
theories of graph models. Finally, we show that the class of graph models
enjoys a kind of downwards Lowenheim-Skolem theorem.Comment: 34