this paper rather abstract, \static" set-theoretic view on the World-Wide Web (WWW) or, more generally, on Web-like Data Bases (WDB) and on the corresponding querying to WDB. Let us stress that it is not only about databases with an access via Web. The database itself should be organized in the same way as Web. I.e. it must consist of hyperlinked pages distributed among the computers participating either in global network like Internet or in some local, isolated from the outside world specic network based essentially on the same principles, except globality, and called also Intranet [15].

Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of first-order arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context. 1

We discuss several versions of a set theoretic \Delta-language as a reasonable prototype for "nested" data base query language where data base states and queries are considered, respectively, as hereditarily-finite sets and set theoretic operations. In a previous work such a language exactly corresponding to PTIME-computability was introduced. It is supposed that HF-sets are naturally presented by vertices of acyclic graphs. Here we consider a number of languages for Sub-PTIME computable set operations via corresponding graph transformers. Two such languages lead to a notion of NLOGSPACE and, respectively, DLOGSPACE computable queries over HF which appear the most natural, at our present knowledge, among others considered here. Unlike the "flat" relational data bases the problem of finding sufficiently good corresponding approach for HF proves to be more intricate and, furthermore, gives rise to some interesting questions in finite model theory (cf. Section 13). 1 General Introduction ...

We develop a unified framework for dealing with constructibility and absoluteness in set theory, decidability of relations in effective structures (like the natural numbers), and domain independence of queries in database theory. Our framework and results suggest that domain-independence and absoluteness might be the key notions in a general theory of constructibility, predicativity, and computability. 1

. We investigate the complexity of the fixed-points of bounded formulas in the context of finite set theory; that is, in the context of arbitrary classes of finite structures that are equipped with a built-in BIT predicate, or equivalently, with a built-in membership relation between hereditarily finite sets (input relations are allowed). We show that the iteration of a positive bounded formula converges in polylogarithmically many steps in the cardinality of the structure. This extends a previously known much weaker result. We obtain a number of connections with the rudimentary languages and deterministic polynomial-time. Moreover, our results provide a natural characterization of the complexity class consisting of all languages computable by bounded-depth, polynomialsize circuits, and polylogarithmic-time uniformity. As a byproduct, we see that this class coincides with LH(P), the logarithmic-time hierarchy with an oracle to deterministic polynomial-time. Finally, we dis...

We suggest a new basic framework for the Weyl-Feferman predicativist program by constructing a formal predicative set theory PZF which resembles ZF. The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they define in an absolute way, independent of the extension of the “surrounding universe”. This idea is implemented using syntactic safety relations between formulas and sets of variables. These safety relations generalize both the notion of domain-independence from database theory, and Godel notion of absoluteness from set theory. The language of PZF is type-free, and it reflects real mathematical practice in making an extensive use of statically defined abstract set terms. Another important feature of PZF is that its underlying logic is ancestral logic (i.e. the extension of FOL with a transitive closure operation). 1

We propose two admissible closures A(PTCA) and A(PHCA) of Ferreira’s system PTCA of polynomial time computable arithmetic and of full bounded arithmetic (or polynomial hierarchy computable arithmetic) PHCA. The main results obtained are: (i) A(PTCA) is conservative over PTCA with respect to ∀∃Σb 1 sentences, and (ii) A(PHCA) is conservative over full bounded arithmetic PHCA for ∀∃Σb ∞ sentences. This yields that (i) the Σb 1 definable functions of A(PTCA) are the polytime functions, and (ii) the Σb ∞ definable functions of A(PHCA) are the functions in the polynomial time hierarchy. 1

We discuss several versions of a set theoretic \Delta-language as a reasonable prototype for "nested" data base query language where data base states and queries are considered, respectively, as hereditarily-finite sets and set theoretic operations. In a previous work such a language exactly corresponding to PTIME-computability was introduced. It is supposed that HF-sets are naturally presented by vertices of acyclic graphs. Here we consider a number of languages for Sub-PTIME computable set operations via corresponding graph transformers. Two such languages lead to a notion of NLOGSPACE and, respectively, DLOGSPACE computable queries over HF which appear the most natural, at our present knowledge, among others considered here. Unlike the "flat" relational data bases the problem of finding sufficiently good corresponding approach for HF proves to be more intricate and, furthermore, gives rise to some interesting questions in finite model theory (cf. Section 13).

This thesis presents practical suggestions towards the implementation of the hyperset approach to semi-structured databases and the associated query language ∆ (Delta). This work can be characterised as part of a top-down approach to semi-structured databases, from theory to practice. Over the last decade the rise of the World-Wide Web has lead to the suggestion for a shift from structured relational databases to semi-structured databases, which can query distributed and heterogeneous data having unfixed/non-rigid structure in contrast to ordinary relational databases. In principle, the World-Wide Web can be considered as a large distributed semi-structured database where arbitrary hyperlinking between Web pages can be interpreted as graph edges (inspiring the synonym ‘Web-like ’ for ‘semi-structured ’ databases also called here WDB). In fact, most approaches to semi-structured databases are based on graphs, whereas the hyperset approach presented here represents such graphs as systems of set equations. This is more than just a style of notation, but rather a style of thought and the corresponding mathematical background leads to considerable differences with other approaches to