Learning via queries in [+, <]

1992 ◽  
Vol 57 (1) ◽  
pp. 53-81 ◽  
Author(s):  
William I. Gasarch ◽  
Mark G. Pleszkoch ◽  
Robert Solovay
Keyword(s):  
Active Learning ◽  
Query Language ◽  
Independent Interest ◽  
Decidability Result ◽  
Presburger Arithmetic ◽  
First Order ◽  
Recursive Functions ◽  
Bounded Number ◽  
Open Question ◽  
Primitive Recursive

AbstractWe prove that the set of all recursive functions cannot be inferred using first-order queries in the query language containing extra symbols [+ , <]. The proof of this theorem involves a new decidability result about Presburger arithmetic which is of independent interest. Using our machinery, we show that the set of all primitive recursive functions cannot be inferred with a bounded number of mind changes, again using queries in [+, <]. Additionally, we resolve an open question in [7] about passive versus active learning.

A formalization of the theory of ordinal numbers

1965 ◽  
Vol 30 (3) ◽  
pp. 295-317 ◽  
Author(s):  
Gaisi Takeuti
Keyword(s):  
Set Theory ◽  
Natural Extension ◽  
Mathematical Induction ◽  
Predicate Calculus ◽  
Transfinite Induction ◽  
First Order ◽  
Recursive Functions ◽  
Ordinal Numbers ◽  
Primitive Recursive ◽  
Order Predicate Calculus

Although Peano's arithmetic can be developed in set theories, it can also be developed independently. This is also true for the theory of ordinal numbers. The author formalized the theory of ordinal numbers in logical systems GLC (in [2]) and FLC (in [3]). These logical systems which contain the concept of ‘arbitrary predicates’ or ‘arbitrary functions’ are of higher order than the first order predicate calculus with equality. In this paper we shall develop the theory of ordinal numbers in the first order predicate calculus with equality as an extension of Peano's arithmetic. This theory will prove to be easy to manage and fairly powerful in the following sense: If A is a sentence of the theory of ordinal numbers, then A is a theorem of our system if and only if the natural translation of A in set theory is a theorem of Zermelo-Fraenkel set theory. It will be treated as a natural extension of Peano's arithmetic. The latter consists of axiom schemata of primitive recursive functions and mathematical induction, while the theory of ordinal numbers consists of axiom schemata of primitive recursive functions of ordinal numbers (cf. [5]), of transfinite induction, of replacement and of cardinals. The latter three axiom schemata can be considered as extensions of mathematical induction.In the theory of ordinal numbers thus developed, we shall construct a model of Zermelo-Fraenkel's set theory by following Gödel's construction in [1]. Our intention is as follows: We shall define a relation α ∈ β as a primitive recursive predicate, which corresponds to F′ α ε F′ β in [1]; Gödel defined the constructible model Δ using the primitive notion ε in the universe or, in other words, using the whole set theory.

scholarly journals Algebraic Stream Processing

1995 ◽  
Author(s):  
◽  
S. Stephens
Keyword(s):  
Broad Class ◽  
Stream Processing ◽  
General Purpose ◽  
Second Order ◽  
First Order ◽  
Recursive Functions ◽  
Sound Theory ◽  
Alternative Approach ◽  
Develop Software ◽  
Primitive Recursive

We identify and analyse the typically higher-order approaches to stream processing in the literature. From this analysis we motivate an alternative approach to the specification of SPSs as STs based on an essentially first-order equational representation. This technique is called Cartesian form specification. More specifically, while STs are properly second-order objects we show that using Cartesian forms, the second-order models needed to formalise STs are so weak that we may use and develop well-understood first-order methods from computability theory and mathematical logic to reason about their properties. Indeed, we show that by specifying STs equationally in Cartesian form as primitive recursive functions we have the basis of a new, general purpose and mathematically sound theory of stream processing that emphasises the formal specification and formal verification of STs. The main topics that we address in the development of this theory are as follows. We present a theoretically well-founded general purpose stream processing language ASTRAL (Algebraic Stream TRAnsformer Language) that supports the use of modular specification techniques for full second-order STs. We show how ASTRAL specifications can be given a Cartesian form semantics using the language PREQ that is an equational characterisation of the primitive recursive functions. In more detail, we show that by compiling ASTRAL specifications into an equivalent Cartesian form in PREQ we can use first-order equational logic with induction as a logical calculus to reason about STs. In particular, using this calculus we identify a syntactic class of correctness statements for which the verification of ASTRAL programmes is decidable relative to this calculus. We define an effective algorithm based on term re-writing techniques to implement this calculus and hence to automatically verify a very broad class of STs including conventional hardware devices. Finally, we analyse the properties of this abstract algorithm as a proof assistant and discuss various techniques that have been adopted to develop software tools based on this algorithm.

scholarly journals Primitive recursive functions. II

1955 ◽  
Vol 6 (4) ◽  
pp. 663-663 ◽  
Author(s):  
Raphael M. Robinson
Keyword(s):  
Recursive Functions ◽  
Primitive Recursive

Designing Document SQL (DSQL)

2009 ◽  
Vol 20 (4) ◽  
pp. 26-53 ◽  
Author(s):  
Arijit Sengupta ◽  
V. Ramesh
Keyword(s):  
Ad Hoc ◽  
Query Language ◽  
Optimization Techniques ◽  
Order Logic ◽  
Conservative Extension ◽  
Tree Structures ◽  
First Order ◽  
Easy Integration ◽  
Query Semantics ◽  
Theoretical Foundations

This article presents DSQL, a conservative extension of SQL, as an ad-hoc query language for XML. The development of DSQL follows the theoretical foundations of first order logic, and uses common query semantics already accepted for SQL. DSQL represents a core subset of XQuery that lends well to optimization techniques, while at the same time allows easy integration into current databases and applications that useSQL. The intent of DSQL is not to replace XQuery, the current W3C recommended XML query language, but to serve as an ad-hoc querying frontend to XQuery. Further, the authors present proofs for important query language properties such as complexity and closure. An empirical study comparing DSQL and XQuery for the purpose of ad-hoc querying demonstrates that users perform better with DSQL for both flat and tree structures, in terms of both accuracy and efficiency.

A Hierarchy on the Class of Primitive Recursive Ordinal Functions

1976 ◽  
Vol 28 (6) ◽  
pp. 1205-1209
Author(s):  
Stanley H. Stahl
Keyword(s):  
Natural Numbers ◽  
Recursive Functions ◽  
Set Functions ◽  
Primitive Recursive ◽  
Class Of Functions ◽  
Recursive Set

The class of primitive recursive ordinal functions (PR) has been studied recently by numerous recursion theorists and set theorists (see, for example, Platek [3] and Jensen-Karp [2]). These investigations have been part of an inquiry concerning a larger class of functions; in Platek's case, the class of ordinal recursive functions and in the case of Jensen and Karp, the class of primitive recursive set functions. In [4] I began to study PR in depth and this paper is a report on an attractive analogy between PR and its progenitor, the class of primitive recursive functions on the natural numbers (Prim. Rec).

scholarly journals PRESBURGER ARITHMETIC, RATIONAL GENERATING FUNCTIONS, AND QUASI-POLYNOMIALS

2015 ◽  
Vol 80 (2) ◽  
pp. 433-449 ◽  
Author(s):  
KEVIN WOODS
Keyword(s):  
Generating Functions ◽  
Counting Function ◽  
Order Theory ◽  
Characteristic Functions ◽  
Presburger Arithmetic ◽  
First Order ◽  
First Order Theory ◽  
Rational Generating Function ◽  
Rational Generating Functions

AbstractPresburger arithmetic is the first-order theory of the natural numbers with addition (but no multiplication). We characterize sets that can be defined by a Presburger formula as exactly the sets whose characteristic functions can be represented by rational generating functions; a geometric characterization of such sets is also given. In addition, ifp= (p1, . . . ,pn) are a subset of the free variables in a Presburger formula, we can define a counting functiong(p) to be the number of solutions to the formula, for a givenp. We show that every counting function obtained in this way may be represented as, equivalently, either a piecewise quasi-polynomial or a rational generating function. Finally, we translate known computational complexity results into this setting and discuss open directions.

Sign in / Sign up

Export Citation Format

Share Document