Results 1  10
of
10
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].
The strength of Mac Lane set theory
 ANNALS OF PURE AND APPLIED LOGIC
, 2001
"... SAUNDERS MAC LANE has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system ZBQC of set theory of which the axioms are Extensionality, Null Set, Pairing, ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
SAUNDERS MAC LANE has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system ZBQC of set theory of which the axioms are Extensionality, Null Set, Pairing,
On Bounded Set Theory
"... We consider some Bounded Set Theories (BST), which are analogues to Bounded Arithmetic. Corresponding provablyrecursive operations over sets are characterized in terms of explicit definability and PTIME or LOGSPACEcomputability. We also present some conservativity results and describe a relation ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
We consider some Bounded Set Theories (BST), which are analogues to Bounded Arithmetic. Corresponding provablyrecursive operations over sets are characterized in terms of explicit definability and PTIME or LOGSPACEcomputability. We also present some conservativity results and describe a relation between BST, possibly with AntiFoundation Axiom, and a Logic of Inductive Definitions (LID) and Finite Model Theory.
A Bounded Set Theory with AntiFoundation Axiom and Inductive Definability
 Computer Science Logic, 8th Workshop, CSL'94 Kazimierz
, 1995
"... this paper let \Deltaformulas and \Deltaterms be those defined as \Delta R with the construct theleast omitted. (They define predicates and operations also known as basic [Gan74] or rudimentary [Jen72] ones.) Note, that provablytotal \Sigma definable operations in KP 0 coincide with those defin ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
this paper let \Deltaformulas and \Deltaterms be those defined as \Delta R with the construct theleast omitted. (They define predicates and operations also known as basic [Gan74] or rudimentary [Jen72] ones.) Note, that provablytotal \Sigma definable operations in KP 0 coincide with those definable by \Deltaterms [Saz85, Saz85a, Saz87]. Therefore, we may use the name KP 0 also for the subtheory of KPR 0 , based on this \Deltalanguage, which does not involve both the term construct theleast and the corresponding axiom. Analogously, \Delta R (\Delta
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
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
A Framework for Formalizing Set Theories Based on the Use of Static Set Terms
"... To Boaz Trakhtenbrot: a scientific father, a friend, and a great man. Abstract. We present a new unified framework for formalizations of axiomatic set theories of different strength, from rudimentary set theory to full ZF. It allows the use of set terms, but provides a static check of their validity ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
To Boaz Trakhtenbrot: a scientific father, a friend, and a great man. Abstract. We present a new unified framework for formalizations of axiomatic set theories of different strength, from rudimentary set theory to full ZF. It allows the use of set terms, but provides a static check of their validity. Like the inconsistent “ideal calculus ” for set theory, it is essentially based on just two settheoretical principles: extensionality and comprehension (to which we add ∈induction and optionally the axiom of choice). Comprehension is formulated as: x ∈{x  ϕ} ↔ϕ, where {x  ϕ} is a legal set term of the theory. In order for {x  ϕ} to be legal, ϕ should be safe with respect to {x}, where safety is a relation between formulas and finite sets of variables. The various systems we consider differ from each other mainly with respect to the safety relations they employ. These relations are all defined purely syntactically (using an induction on the logical structure of formulas). The basic one is based on the safety relation which implicitly underlies commercial query languages for relational database systems (like SQL). Our framework makes it possible to reduce all extensions by definitions to abbreviations. Hence it is very convenient for mechanical manipulations and for interactive theorem proving. It also provides a unified treatment of comprehension axioms and of absoluteness properties of formulas. 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).
From Constructibility and Absoluteness to Computability and Domain Independence
"... Abstract. Gödel’s main contribution to set theory is his proof that GCH is consistent with ZFC (assuming that ZF is consistent). For this proof he has introduced the important ideas of constructibility of sets, and of absoluteness of formulas. In this paper we show how these two ideas of Gödel natur ..."
Abstract
 Add to MetaCart
Abstract. Gödel’s main contribution to set theory is his proof that GCH is consistent with ZFC (assuming that ZF is consistent). For this proof he has introduced the important ideas of constructibility of sets, and of absoluteness of formulas. In this paper we show how these two ideas of Gödel naturally lead to a simple unified framework for dealing with computability of functions and relations, domain independence of queries in relational databases, and predicative set theory. 1