This thesis is a model-theoretic study of exponential differential equations in the context of differential algebra. I define the theory of a set of differential equations and give an axiomatization for the theory of the exponential differential equations of split semiabelian varieties. In particular, this includes the theory of the equations satisfied by the usual complex exponential function and the Weierstrass p-functions. The theory consists of a description of the algebraic structure on the solution sets together with necessary and sufficient conditions for a system of equations to have solutions. These conditions are stated in terms of a dimension theory; their necessity generalizes Ax’s differential field version of Schanuel’s conjecture and their sufficiency generalizes recent work of Crampin. They are shown to apply to the solving of systems of equations in holomorphic functions away from singularities, as well as in the abstract setting. The theory can also be obtained by means of a Hrushovski-style amalgamation construction, and I give a category-theoretic account of the method. Restricting to the usual exponential differential equation, I show that a “blurring” of Zilber’s pseudo-exponentiation satisfies the same theory. I conjecture that this theory also holds for a suitable blurring of the complex exponential maps and partially resolve the question, proving the necessity but not the sufficiency of the aforementioned conditions. As an algebraic application, I prove a weak form of Zilber’s conjecture on intersections with subgroups (known as CIT) for semiabelian varieties. This in turn is used to show that the necessary and sufficient conditions are expressible in the appropriate first order language.

A suitable subcategory of affine Azumaya algebras is defined and a functor from this category to the category of Zariski structures is constructed. The rudiments of a theory of presheaves of topological structures is developed and applied to construct examples of structures at a generic parameter. The category of equivariant algebras is defined and a first-order theory is associated to each object. For those theories satisfying a certain technical condition, uncountable categoricity and quantifier elimination results are established. Models are shown to be Zariski structures and a functor from the category of equivariant algebras to Zariski structures is constructed. The two functors obtained in the thesis are shown to agree on a nontrivial class of algebras.

This thesis explores the geometric principles underlying many of the known Trichotomy Theorems. The main aims are to unify the field construction in non-linear o-minimal structures and generalizations of Zariski Geometries as well as to pave the road for completely new results in this direction. In the first part of this thesis we introduce a new axiomatic framework in which all the relevant structures can be studied uniformly and show that these axioms are preserved under elementary extensions. A particular focus is placed on the study of a smoothness condition which generalizes the presmoothness condition for Zariski Geometries. We also modify Zilber's notion of universal specializations to obtain a suitable notion of infinitesimals. In addition, families of curves and the combinatorial geometry of one-dimensional structures are studied to prove a weak trichotomy theorem based on very weak one-basedness. It is then shown that under suitable additional conditions groups and group actions can be constructed in canonical ways. This construction is based on a notion of ``geometric calculus'' and can be seen in close analogy with ordinary differentiation. If all conditions are met, a definable distributive action of one one-dimensional type-definable group on another are obtained. The main result of this thesis is that both o-minimal structures and generalizations of Zariski Geometries fit into this geometric framework and that the latter always satisfy the conditions required in the group constructions. We also exhibit known methods that allow us to extract fields from this. In addition to unifying the treatment of o-minimal structures and Zariski Geometries, this also gives a direct proof of the Trichotomy Theorem for "type-definable" Zariski Geometries as used, for example, in Hrushovski's proof of the relative Mordell-Lang conjecture.

This thesis investigates the connections between henselian valuations and absolute Galois groups. There are fundamental links between these: On one hand, the absolute Galois group of a field often encodes information about (henselian) valuations on that field. On the other, in many cases a henselian valuation imposes a certain structure on an absolute Galois group which makes it easier to study. We are particularly interested in the question of when a field admits a non-trivial parameter-free definable henselian valuation. By a result of Prestel and Ziegler, this does not hold for every henselian valued field. However, improving a result by Koenigsmann, we show that there is a non-trivial parameter-free definable valuation on every henselian valued field. This allows us to give a range of conditions under which a henselian field does indeed admit a non-trivial parameter-free definable henselian valuation. Most of these conditions are in fact of a Galois-theoretic nature. Throughout the thesis, we discuss a number of applications of our results. These include fields elementarily characterized by their absolute Galois group, model complete henselian fields and henselian NIP fields of positive characteristic, as well as PAC and hilbertian fields.

Description logics (DLs) are knowledge representation formalisms with well-understood model-theoretic semantics and computational properties. The DL SROIQ provides the logical underpinning for the semantic web language OWL 2, which is quickly becoming the standard for knowledge representation on the web. A central component of most DL applications is an efficient and scalable reasoner, which provides services such as consistency testing and classification. Despite major advances in DL reasoning algorithms over the last decade, however, ontologies are still encountered in practice that cannot be handled by existing DL reasoners. We present a novel reasoning calculus for the description logic SROIQ which addresses two of the major sources of inefficiency present in the tableau-based reasoning calculi used in state-of-the-art reasoners: unnecessary nondeterminism and unnecessarily large model sizes. Further, we describe a new approach to classification which exploits partial information about the subsumption relation between concept names to reduce both the number of individual subsumption tests performed and the cost of working with large ontologies; our algorithm is applicable to the general problem of deducing a quasi-ordering from a sequence of binary comparisons. We also present techniques for extracting partial information about the subsumption relation from the models generated by constructive DL reasoning methods, such as our hypertableau calculus. Empirical results from a prototypical implementation demonstrate substantial performance improvements compared to existing algorithms and implementations.

In recent years the field of real numbers expanded by a multiplicative subgroup has been studied extensively. In this thesis, the known results will be extended to expansions of the real field. I will consider the structure R consisting of the field of real numbers and an irrational power function. Using Schanuel conditions, I will give a first-order axiomatization of expansions of R by a dense multiplicative subgroup which is a subset of the real algebraic numbers. It will be shown that every definable set in such a structure is a boolean combination of existentially definable sets and that these structures have o-minimal open core. A proof will be given that the Schanuel conditions used in proving these statements hold for co-countably many real numbers. The results mentioned above will also be established for expansions of R by dense multiplicative subgroups which are closed under all power functions definable in R. In this case the results hold under the assumption that the Conjecture on intersection with tori is true. Finally, the structure consisting of R and the discrete multiplicative subgroup 2^{Z} will be analyzed. It will be shown that this structure is not model complete. Further I develop a connection between the theory of Diophantine approximation and this structure.

The principal focus of this thesis is the study of the real numbers regarded as a structure endowed with its usual addition and multiplication and the operations of raising to real powers. For our first main result we prove that any statement in the language of this structure is equivalent to an existential statement, and furthermore that this existential statement can be chosen independently of the concrete interpretations of the real power functions in the statement; i.e. one existential statement will work for any choice of real power functions. This result we call uniform model completeness. For the second main result we introduce the first order theory of raising to an infinite power, which can be seen as the theory of a class of real closed fields, each expanded by a power function with infinite exponent. We note that it follows from the first main theorem that this theory is model-complete, furthermore we prove that it is decidable if and only if the theory of the real field with the exponential function is decidable. For the final main theorem we consider the problem of expanding an arbitrary o-minimal expansion of a field by a non-trivial exponential function whilst preserving o-minimality. We show that this can be done under the assumption that the structure already defines exponentiation on a bounded interval, and a further assumption about the prime model of the structure.

We investigate definability in henselian fields. Specifically, we are interested in those sets and substructures that are existentially definable or definable with `few' parameters. Our general approach is to use various versions of henselianity to understand the `local structure' of these definable sets. The fields in which we are most interested are those of positive characteristic, for example the local fields F_q((t)), but many of our methods and results also apply to p-adic and real closed fields. In positive characteristic we have to deal with inseparable field extensions and we develop the method of Λ-closure to `translate' inseparable field extensions into separable ones. In the first part of the thesis we focus on existentially definable sets, which are projections of algebraic sets. Our main tool is the Implicit Function Theorem (for polynomials) which is equivalent to t-henselianity, by work of Prestel and Ziegler. This enables us to prove that existentially definable sets are `large' in various senses. Using the Implicit Function Theorem, we also obtain a nonuniform local elimination of the existential quantifier. The non-uniformity and local character of this result at present forms an obstacle to full quantifier-elimination. From these technical statements we can deduce characterisations of, for example, existentially definable subfields and existentially definable transcendentals. We prove that a dense, regular extension of t-henselian fields is existentially closed which, in particular, implies the old result of Ershov that F_p(t)^h ≤_Ǝ F_p((t)). Using the existential closedness of large fields in henselian fields, we are able to apply many of these results to large fields. This answers questions for imperfect large fields that were answered in the perfect case by Fehm.</p> In the second part of the thesis, we work with power series fields F((t)) and subsets which are F- definable (and not contained in F). We use a `hensel-like' lemma to characterise F-orbits of (singleton) elements of F((t)). It turns out that all such orbits are Ǝ-t-definable. Consequently, we may apply our earlier results about existentially definable subsets to F-definable subsets. We can use this to characterise F-definable subfields of F((t)). As a further corollary, we obtain an Ǝ-0̸-definition of F_p[[t]] in F<sub>p<sub>((t)).

This thesis examines two approaches to Galois correspondences in formal logic. A standard result of classical first-order model theory is the observation that models of L-theories with a weak form of elimination of imaginaries hold a correspondence between their substructures and automorphism groups defined on them. This work applies the resultant framework to explore the practical consequences of a model-theoretic Galois theory with respect to certain first-order L-theories. The framework is also used to motivate an examination of its underlying model-theoretic foundations. The model-theoretic Galois theory of pure fields and valued fields is compared to the algebraic Galois theory of pure and valued fields to point out differences that may hold between them. The framework of this logical Galois correspondence is also applied to the theory of pseudoexponentiation to obtain a sketch of the Galois theory of exponential fields, where the fixed substructure of the complex pseudoexponential field B is an exponential field with the field Qrab as its algebraic subfield. This work obtains a partial exponential analogue to the Kronecker-Weber theorem by describing the pure field-theoretic abelian extensions of Qrab, expanding upon work in the twelfth of Hilbert’s problems. This result is then used to determine some of the model-theoretic abelian extensions of the fixed substructure of B. This work also incorporates the principles required of this model-theoretic framework in order to develop a model theory over substructural logics which is capable of expressing this Galois correspondence. A formal semantics is developed for quantified predicate substructural logics based on algebraic models for their propositional or nonquantified fragments. This semantics is then used to develop substructural forms of standard results in classical first-order model theory. This work then uses this substructural model theory to demonstrate the Galois correspondence that substructural first-order theories can carry in certain situations.

Logical deduction and abstraction from detail are fundamental, yet distinct aspects of reasoning about programs. This dissertation shows that the combination of logic and abstract interpretation enables a unified and simple treatment of several theoretical and practical topics which encompass the model theory of temporal logics, the analysis of satisfiability solvers, and the construction of Craig interpolants. In each case, the combination of logic and abstract interpretation leads to more general results, simpler proofs, and a unification of ideas from seemingly disparate fields. The first contribution of this dissertation is a framework for combining temporal logics and abstraction. Chapter 3 introduces trace algebras, a new lattice-based semantics for linear and branching time logics. A new representation theorem shows that trace algebras precisely capture the standard trace-based semantics of temporal logics. We prove additional representation theorems to show how structures that have been independently discovered in static program analysis, model checking, and algebraic modal logic, can be derived from trace algebras by abstract interpretation. The second contribution of this dissertation is a framework for proving when two lattice-based algebras satisfy the same logical properties. Chapter 5 introduces functions called subsumption and bisubsumption and shows that these functions characterise logical equivalence of two algebras. We also characterise subsumption and bisubsumption using fixed points and finitary logics. We prove a representation theorem and apply it to derive the transition system analogues of subsumption and bisubsumption. These analogues strictly generalise the well studied notions of simulation and bisimulation. Our fixed point characterisations also provide a technique to construct property preserving abstractions. The third contribution of this dissertation is abstract satisfaction, an abstract interpretation framework for the design and analysis of satisfiability procedures. We show that formula satisfiability has several different fixed point characterisations, and that satisfiability procedures can be understood as abstract interpreters. Our main result is that the propagation routine in modern sat solvers is a greatest fixed point computation involving abstract transformers, and that clause learning is an abstract transformer for a form of negation. The final contribution of this dissertation is an abstract interpretation based analysis of algorithms for constructing Craig interpolants. We identify and analyse a lattice of interpolant constructions. Our main result is that existing algorithms are two of three optimal abstractions of this lattice. A second new result we derive in this framework is that the lattice of interpolation algorithms can be ordered by logical strength, so that there is a strongest and a weakest possible construction.