Results 1 
9 of
9
Bounded Hyperset Theory and Weblike Data Bases
 Computational Logic and Proof Theory, 5th Kurt Gödel Colloquium, KGC’97, Springer LNCS
, 1997
"... this paper rather abstract, \static" settheoretic view on the WorldWide Web (WWW) or, more generally, on Weblike 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 ..."
Abstract

Cited by 18 (5 self)
 Add to MetaCart
this paper rather abstract, \static" settheoretic view on the WorldWide Web (WWW) or, more generally, on Weblike 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].
Number theory and elementary arithmetic
 Philosophia Mathematica
, 2003
"... Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder 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 t ..."
Abstract

Cited by 17 (5 self)
 Add to MetaCart
Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder 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
ΔLanguages for Sets and LOGSPACE Computable Graph Transformers
 Theoretical Computer Science
, 1997
"... We discuss several versions of a set theoretic \Deltalanguage as a reasonable prototype for "nested" data base query language where data base states and queries are considered, respectively, as hereditarilyfinite sets and set theoretic operations. In a previous work such a language exactly corresp ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
We discuss several versions of a set theoretic \Deltalanguage as a reasonable prototype for "nested" data base query language where data base states and queries are considered, respectively, as hereditarilyfinite sets and set theoretic operations. In a previous work such a language exactly corresponding to PTIMEcomputability was introduced. It is supposed that HFsets are naturally presented by vertices of acyclic graphs. Here we consider a number of languages for SubPTIME 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 ...
Constructibility and Decidability versus Domain Independence and Absoluteness
"... 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 domainindependence and absolut ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
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 domainindependence and absoluteness might be the key notions in a general theory of constructibility, predicativity, and computability. 1
The Descriptive Complexity of the FixedPoints of Bounded Formulas
 Computer Science Logic '2000, 14th Annual Conference of the EACSL, volume 1862 of Lecture Notes in Computer Science
, 2000
"... . We investigate the complexity of the fixedpoints 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 builtin BIT predicate, or equivalently, with a builtin membership relation between hereditaril ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
. We investigate the complexity of the fixedpoints 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 builtin BIT predicate, or equivalently, with a builtin 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 polynomialtime. Moreover, our results provide a natural characterization of the complexity class consisting of all languages computable by boundeddepth, polynomialsize circuits, and polylogarithmictime uniformity. As a byproduct, we see that this class coincides with LH(P), the logarithmictime hierarchy with an oracle to deterministic polynomialtime. Finally, we dis...
A New Approach to Predicative Set Theory
"... We suggest a new basic framework for the WeylFeferman 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 a ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
We suggest a new basic framework for the WeylFeferman 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 domainindependence from database theory, and Godel notion of absoluteness from set theory. The language of PZF is typefree, 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
Admissible closures of polynomial time computable arithmetic
, 2010
"... 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 s ..."
Abstract
 Add to MetaCart
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
ΔLanguages for Sets and LOGSPACE Computable Graph Transformers
 THEORETICAL COMPUTER SCIENCE
, 1997
"... We discuss several versions of a set theoretic \Deltalanguage as a reasonable prototype for "nested" data base query language where data base states and queries are considered, respectively, as hereditarilyfinite sets and set theoretic operations. In a previous work such a language exactly corresp ..."
Abstract
 Add to MetaCart
We discuss several versions of a set theoretic \Deltalanguage as a reasonable prototype for "nested" data base query language where data base states and queries are considered, respectively, as hereditarilyfinite sets and set theoretic operations. In a previous work such a language exactly corresponding to PTIMEcomputability was introduced. It is supposed that HFsets are naturally presented by vertices of acyclic graphs. Here we consider a number of languages for SubPTIME 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).
Hyperset Approach to Semistructured Databases and the Experimental Implementation of the Query Language Delta
"... This thesis presents practical suggestions towards the implementation of the hyperset approach to semistructured databases and the associated query language ∆ (Delta). This work can be characterised as part of a topdown approach to semistructured databases, from theory to practice. Over the last ..."
Abstract
 Add to MetaCart
This thesis presents practical suggestions towards the implementation of the hyperset approach to semistructured databases and the associated query language ∆ (Delta). This work can be characterised as part of a topdown approach to semistructured databases, from theory to practice. Over the last decade the rise of the WorldWide Web has lead to the suggestion for a shift from structured relational databases to semistructured databases, which can query distributed and heterogeneous data having unfixed/nonrigid structure in contrast to ordinary relational databases. In principle, the WorldWide Web can be considered as a large distributed semistructured database where arbitrary hyperlinking between Web pages can be interpreted as graph edges (inspiring the synonym ‘Weblike ’ for ‘semistructured ’ databases also called here WDB). In fact, most approaches to semistructured 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