Allowing templates with infinite domains greatly expands the range of problems that can be formulated as a non-uniform constraint satisfaction problem. It turns out that many CSPs over infinite templates can be formulated with templates that are ω-categorical. We survey examples of such problems in temporal and spatial reasoning, infinite-dimensional algebra, acyclic colorings in graph theory, artificial intelligence, phylogenetic reconstruction in computational biology, and tree descriptions in computational linguistics. We then give an introduction to the universal-algebraic approach to infinite-domain constraint satisfaction, and discuss how cores, polymorphism clones, and pseudovarieties can be used to study the computational complexity of CSPs with ω-categorical templates. The theoretical results will be illustrated by examples from the mentioned application areas. We close with a series of open problems and promising directions of future research.

Abstract. We introduce differential arc spaces in analogy to the algebraic arc spaces and show that a differential variety is determined by its arcs at a point. Using differential arcs, we show that if (K, +, ×, δ1,..., δn) is a differentially closed field of characteristic zero with n commuting derivations and p ∈ S(K) is a regular type over K, then either p is locally modular or there is a definable subgroup G ≤ (K, +) of the additive group having a regular generic type that is nonorthogonal to p. 1.

Abstract. We show that every locally compact Polish group is isomorphic to the isometry group of a proper separable metric space. This answers a question of Gao and Kechris. We also analyze the natural action of the isometry group of a separable ultrametric space on the space. This leads us to a structure theorem representing an arbitrary separable ultrametric space as a bundle with an ultrametric base and with ultrahomogeneous fibers which are invariant under the action of the isometry group. 1.

It is shown that if K is an algebraically closed valued field with valuation ring R, then Th(K) has elimination of imaginaries if sorts are added whose elements are certain cosets in K n of certain definable R-submodules of K n (for all n ≥ 1). The proof involves the development of a theory of independence for unary types, which play the role of 1-types, followed by an analysis of germs of definable functions from unary sets to the sorts. Contents.

The current solution for ecient high-level parallel programming in the industry is to use directives to the compiler. However these directives pose two problems: rst, they are often designed in an ad hoc manner and their subtleties are less easy to understand than the rest of the language; second, the degree of stringency of the directives is not xed, so that evaluating the eciency of the directives he writes is not easy for the programmer. This article proposes a methodology for addressing these issues. The data mapping directives of the language High Performance Fortran are used as an example; in particular, it is shown how the use of formal semantics can help clarify and structure the issues.

In this paper, we describe some central mathematical problems in medical imaging. The subject has been undergoing rapid changes driven by better hardware and software. Much of the software is based on novel methods utilizing geometric partial di#erential equations in conjunction with standard signal/image processing techniques as well as computer graphics facilitating man/machine interactions. As part of this enterprise, researchers have been trying to base biomedical engineering principles on rigorous mathematical foundations for the development of software methods to be integrated into complete therapy delivery systems. These systems support the more effective delivery of many image-guided procedures such as radiation therapy, biopsy, and minimally invasive surgery. We will show how mathematics may impact some of the main problems in this area including image enhancement, registration, and segmentation.

This paper builds a decision-theoretic framework to examine the relationship between language and knowledge. A decision maker describes the world through theories. A theory consists of universal propositions called patterns, and it is formulated in some language. I look at two characteristics of a successful theory. A theory is informative if it allows agents to precisely predict outcomes of some process. A theory is brief if it consists of finitely many patterns. The main result of the paper identifies languages for which there is no trade-off between both characteristics: Any informative theory logically implies a theory that is informative as well as brief. I illustrate the main result on specific problems of reasoning under uncertainty: recommendation problems, binary preferences of a customer, or a stylized example of a chemical research.

We give some sufficient conditions for a predicate P in a complete theory T to be “stably embedded”. Let P be P with its “induced ∅-definable structure”. The conditions are that P (or rather its theory) is “rosy”, P has NIP in T and that P is stably 1-embedded in T. This generalizes a recent result of Hasson and Onshuus [6] which deals with the case where P is o-minimal in T. Our proofs make use of the theory of strict nonforking and weight in NIP theories ([3], [10]). 1 Introduction and

Definable sets in algebraically closed valued fields: elimination of imaginaries. (English summary) J. Reine Angew. Math. 597 (2006), 175–236. In this fundamental paper, the authors shed light on elimination of imaginaries in algebraically closed valued fields. The importance of imaginaries for a better understanding of the model theory of any structure cannot be underestimated since they deal with structures interpretable in the given one. Already the techniques of this paper have been applied to other valued fields [T. Mellor, Ann. Pure Appl. Logic 139 (2006), no. 1-3, 230–279; MR2206257 (2006k:03064)]. An imaginary in a first-order structure is an equivalence class of a definable equivalence relation. Let T be a multisorted complete first-order theory; then T is said to have elimination of imaginaries if for any model M of T, any sorts M1,..., Mk in M, any ∅-definable set S ⊂ M1 × · · · × Mk and any ∅-definable equivalence relation E on S, there is an ∅-definable function f from S into a product of sorts such that for any a, b ∈ S we have Eab iff f(a) = f(b). In a way, f(a) acts as a code for the E-class of a. An oversimplified illustration of this phenomenon is M = C, S = C2 � {(0, 0)}, E(a1, a2)(b1, b2) iff (a1, a2), (b1, b2) define the same line through (0, 0), and f(a1, a2) =

1 Descriptive use of logic and Intended models 1 1.1 Standard models of arithmetic.......................... 1 1.2 Axiomatics and Formal theories......................... 3 1.3 Hintikka and the two uses of logic in mathematics.............. 5