Prove that any strictly monotone sequence (Uα)α<γ of open subsets of X has countable length, i.e. γ is countable. Hint: Use the same idea as in the proof of (a). (c) Show that every monotone sequence (Uα)α<ω1 open subsets of X eventually stabilizes, i.e. there is γ < ω1 such that for all α < ω1 with α ≥ γ, we have Uα = Uγ. Hint: Use the regularity of ω1. (d) Conclude that parts (a), (b) and (c)...

Infinitary logic, the logic of languages with infinitely long conjunctions, plays an important role in model theory, recursion theory and descriptive set theory. This book is the first modern introduction to the subject in 40 years, and will bring students and researchers in all areas of mathematical logic up to the threshold of modern research. The classical topics of back-and-forth systems, m...

The (effective) Suslin-Kleene Theorem is obtained as a corollary of a standard proof of the classical Suslin Theorem, by noticing that it is mostly constructive and applying to it a naive realizability interpretation. Effective Descriptive Set Theory is advertized as a refinement of the classical theory of definability (on Polish spaces) developed in the first half of the 20th century, for exam...

Clustering categorical sequences is currently a difficult problem due to the lack of an efficient representation model for sequences. Unlike the existing models, which mainly focus on the fixed-length tuples representation, in this paper, a new representation model on the variablelength tuples is proposed. The variable-length tuples are obtained using a pruning method applied to delete the redu...

We propose a new descriptive complexity notion of uniformity for branching programs solving problems defined on structured data. We observe that FO[=]-uniform (n-way) branching programs are unable to solve the tree evaluation problem studied by Cook, McKenzie, Wehr, Braverman and Santhanam [8] because such programs possess a variant of their thriftiness property. Similarly, FO[=]-uniform (n-way...

Our aim is to see which practices of Greek geometry can be expressed in various logics. Thus we refine Detlefsen’s notion of descriptive complexity by providing a scheme of increasing more descriptive formalizations of geometry Following Hilbert we argue that defining a field structure on a line in ‘Euclidean geometry’ provides a foundation for both geometry and algebra. In particular we prove ...

The descriptional complexity of semi-conditional grammars is studied. It is proved that every recursively enumerable language is generated by a semi-conditional grammar of degree (2, 1) with no more than seven conditional productions and eight nonterminals.

We outline a general theory of graph polynomials which covers all the examples we found in the vast literature, in particular, the chromatic polynomial, various generalizations of the Tutte polynomial, matching polynomials, interlace polynomials, and the cover polynomial of digraphs. We introduce the class of (hyper)graph polynomials definable in second order logic, and outline a research progr...

v Acknowledgements vii Chapter

This paper proves that every recursively enumerable language is generated by a scattered context grammar with no more than four nonterminals and three non-context-free productions. In its conclusion, it gives an overview of the results and open problems concerning scattered context grammars and languages.