# 6th Panhellenic Logic Symposium

Volos, Greece, 5-8 July 2007

### Invited Lectures

- Ayse Berkman (Middle East Technical University):
- Groups of Finite Morley Rank
I shall make a quick introduction to the subject, give examples and present the Borovik program that shaped around the main conjecture of the area:

*Algebraicity Conjecture*An infinite simple group of finite Morley rank is isomorphic to an algebraic group over an algebraically closed field.

In the second half of my talk, I shall present some recent results from the study of groups of finite Morley rank with pseudoreflection subgroups:

If*G*acts on an abelian group*V*, with an infinite definable connected abelian subgroup*R*such that*V*=[*V*,*R*] ⊕*CV*(*R*) and*R*acts transitively on the non-zero elements of [*V*,*R*], then*R*is called a pseudoreflection subgroup of*G*. - Stuart Barry Cooper (University of Leeds):
- The Interactive Structure of Information: Post's Program Revisited
Computability theory concerns information with a causal structure. As such, it provides a schematic analysis of many naturally occurring situations.

Emil Post was the first to focus on the close relationship between information, coded as real numbers, and its algorithmic infrastructure. Having characterized the close connection between the quantifier type of a real and the Turing jump operation, he looked for more subtle ways in which information entails a particular causal context. Specifically, he wanted to find simple relations on reals which produced richness of local computability-theoretic structure. To this extent, he was not just interested in causal structure as an abstraction, but in the way in which this structure emerges in natural contexts. "Post's program" was the genesis of a more far reaching research project.

In this talk we will firstly review the history of Post’s program, and look at two interesting developments of Post’s approach. The first of these developments concerns the extension of the core program, initially restricted to the Turing structure of the computably enumerable sets of natural numbers, to the Ershov hierarchy of sets. The second looks at how new types of information coming from the recent growth of research into randomness, and the revealing of unexpected new computability-theoretic infrastructure. We will conclude by viewing Post's program from a more general perspective. We will look at how algorithmic structure does not just emerge mathematically from information, but how that emergent structure can model the emergence of very basic aspects of the real world. - Anuj Dawar (University of Cambridge):
- On Preservation Theorems in Finite Model Theory
Among classical theorems of model theory, preservation theorems are results that relate the syntactic form of formulas to semantic closure properties of the structures they define. The status of preservation theorems in the finite has been an active area of research in finite model theory. More recent work in the area has shifted the focus from the class of all finite structures to classes of structures satisfying natural structural restrictions. In this talk I will examine various recent results relating to preservation under homomorphisms and extensions, both on the class of all finite structures and on more restricted classes.

- Thomas Eiter (Vienna University of Technology):
- Towards a Semantic Web: Combining Rules and Ontologies
In the last years, there have been increasing efforts to bridge the separated worlds of rule-based formalisms and of logic-based ontology languages for knowledge representation and reasoning. This is in particular motivated by the vision of an enhanced version of the World Wide Web, in which semantic access to contents is provided for machine-based information processing. While ontologies play here a key role, the most prominent formalisms and standards including the Resource Description Framework (RDF) and the Web Ontology Language (OWL), which is based on description logics, are insufficient to accommodate the needs for knowledge representation in practice. Several proposals for combining rule and ontology formalisms have been made so far, which differ in various respects. However no generally agreed upon solution to the problem has been found so far (and may not exist). In this talk, we address some approaches to combine rules and ontologies, and the problems which arise in this task. Special emphasis is given to non-monotonic combinations that are capable of dealing with missing information, which appears to be a necessary feature for advanced reasoning on the Web. The talk will conclude with some issues for future research.

- Ali Enayat (American University):
- Automorphisms of Models of Arithmetic
We discuss a recently developed method of constructing automorphisms of models of arithmetic that provides a unified, powerful approach to building desirable automorphisms of models of a variety of arithmetical theories, ranging from bounded arithmetic to second order arithmetic. As we shall explain, this new method can also be used to establish a long standing conjecture of Schmerl by showing that the isomorphism types of fixed point sets of countable arithmetically saturated model M of Peano arithmetic are precisely the isomorphism types of elementary submodels of M.

- Constantine Tsinakis (Vanderbilt University):
- Algebraic Methods in Logic
Algebraic logic studies classes of algebras that are related to logical systems, as well as the process by which a class of algebras becomes the algebraic counterpart (semantics)" of a logical system. A field practitioner usually approaches the solution of a problem in logic by first reformulating it in the language of algebra; then by using algebra to solve the reformulated problem; and lastly by expressing the result into the language of logic. A representative association of the preceding kind is the one between the class of Boolean algebras and classical propositional calculus.

The focus of this talk is substructural logics and their algebraic counterparts. Substructural logics are non-classical logics that are weaker than classical logic, in the sense that they lack one or more of the structural rules of contraction, weakening and exchange in their Genzen-style axiomatization. (It is, however, convenient to think of the classical logic and intuitionistic logic as substructural logics.) These logics encompass a large number of non-classical logics related to computer science (linear logic), linguistics (Lambek Calculus), philosophy (relevant logics), and multi-valued reasoning.

The following are among the objectives of the talk:

Propose a uniform framework for the study of the algebraic counter-parts of substructural propositional logics. These algebras, referred to as residuated lattices, have a recently discovered rich structure theory. (Note: The term "residuated lattice" has been used in the literature to refer to algebras that are integral, commutative and bounded. This class and its subclasses are not sufficiently general to provide semantics for all substructural logics.)

Show how the algebraic theory of residuated lattices can produce powerful tools for the comparative study of substructural logics.

Stress that the bridge algebraic logic builds is beneficial to both algebra and logic, and that an in depth study of residuated lattices is impossible without the use of logical and proof-theoretic techniques. This point is illustrated with a discussion of the close connection between interpolation theorems for a substructural logic and the amalgamation property for its algebraic counterpart.