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62
Some Properties of Query Languages for Bags
 IN PROCEEDINGS OF 4TH INTERNATIONAL WORKSHOP ON DATABASE PROGRAMMING LANGUAGES
, 1993
"... In this paper we study the expressive power of query languages for nested bags. We define the ambient bag language by generalizing the constructs of the relational language of BreazuTannen, Buneman and Wong, which is known to have precisely the power of the nested relational algebra. Relative s ..."
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Cited by 39 (26 self)
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In this paper we study the expressive power of query languages for nested bags. We define the ambient bag language by generalizing the constructs of the relational language of BreazuTannen, Buneman and Wong, which is known to have precisely the power of the nested relational algebra. Relative strength of additional polynomial constructs is studied, and the ambient language endowed with the strongest combination of those constructs is chosen as a candidate for the basic bag language, which is called BQL (Bag Query Language). We prove that achieveing the power of BQL in the relational language amounts to adding simple arithmetic to the latter. We show that BQL has shortcomings of the relational algebra: it can not express recursive queries. In particular, parity test is not definable in BQL. We consider augmenting BQL with powerbag and structural recursion to overcome this deficiency. In contrast to the relational case, where powerset and structural recursion are equivalent...
An analog characterization of the Grzegorczyk hierarchy
 Journal of Complexity
, 2002
"... We study a restricted version of Shannon's General . . . ..."
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Cited by 34 (16 self)
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We study a restricted version of Shannon's General . . .
Linear Logic & Elementary Time
 Information and Computation
, 2001
"... Introduction Think of elementary linear logic as an idealized functional programming language with a severe typing mechanism. Definition by recursion is, of course, forbidden, but some sort of iteration still is possible and the purpose of this paper is to show that enough computing power remains s ..."
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Cited by 30 (0 self)
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Introduction Think of elementary linear logic as an idealized functional programming language with a severe typing mechanism. Definition by recursion is, of course, forbidden, but some sort of iteration still is possible and the purpose of this paper is to show that enough computing power remains so that elementary recursive functions can be implemented. Actually, the whole paper can be considered an exercise in programming elegantly with a rather desolate language. To zero in on an interesting class of functions, one usually tries to weaken in the given logic whatever corresponds to induction or iteration. Here we're following a di#erent strand, rather specific to the linear logic decomposition of the implication as !A#B, by fiddling with the rules handling `!'. The standard rules are enough to embed the full power of intuitionistic computations. So the game is to find a sensible way to make them harder to use than in full linear logic. There ar
Recursive analysis characterized as a class of real recursive functions
 Fundamenta Informaticae
, 2006
"... Recently, using a limit schema, we presented an analog and machine independent algebraic characterization of elementary functions over the real numbers in the sense of recursive analysis. In a different and orthogonal work, we proposed a minimalization schema that allows to provide a class of real r ..."
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Cited by 22 (8 self)
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Recently, using a limit schema, we presented an analog and machine independent algebraic characterization of elementary functions over the real numbers in the sense of recursive analysis. In a different and orthogonal work, we proposed a minimalization schema that allows to provide a class of real recursive functions that corresponds to extensions of computable functions over the integers. Mixing the two approaches we prove that computable functions over the real numbers in the sense of recursive analysis can be characterized as the smallest class of functions that contains some basic functions, and closed by composition, linear integration, minimalization and limit schema.
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 ..."
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Cited by 19 (6 self)
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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
A classification of rapidly growing Ramsey functions
 PROC. AMER. MATH. SOC
, 2003
"... Let f be a numbertheoretic function. A finite set X of natural numbers is called flarge if card(X) ≥ f(min(X)). Let PHf be the Paris Harrington statement where we replace the largeness condition by a corresponding flargeness condition. We classify those functions f for which the statement PHf i ..."
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Cited by 17 (6 self)
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Let f be a numbertheoretic function. A finite set X of natural numbers is called flarge if card(X) ≥ f(min(X)). Let PHf be the Paris Harrington statement where we replace the largeness condition by a corresponding flargeness condition. We classify those functions f for which the statement PHf is independent of first order (Peano) arithmetic PA.Iffis a fixed iteration of the binary length function, then PHf is independent. On the other hand PHlog ∗ is provable in PA. More precisely let fα(i):=i  H −1 α (i) where  i h denotes the htimes iterated binary length of i and H−1 α denotes the inverse function of the αth member Hα of the Hardy hierarchy. Then PHfα is independent of PA (for α ≤ ε0) iffα = ε0.
An analog characterization of elementarily computable functions over the real numbers
 31st Int. Col. on Automata, Languages and Programming (ICALP ’04
, 2004
"... Abstract We present an analog and machineindependent algebraic characterization of elementarily computable functions over the real numbers in the sense of recursive analysis: we prove that they correspond to the smallest class of functions that contains some basic functions, and closed by composit ..."
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Cited by 12 (2 self)
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Abstract We present an analog and machineindependent algebraic characterization of elementarily computable functions over the real numbers in the sense of recursive analysis: we prove that they correspond to the smallest class of functions that contains some basic functions, and closed by composition, linear integration, and a simple limit schema. We generalize this result to all higher levels of the Grzegorczyk Hierarchy. Concerning recursive analysis, our results provide machineindependent characterizations of natural classes of computable functions over the real numbers, allowing to dene these classes without usual considerations on higherorder (type 2) Turing machines. Concerning analog models, our results provide a characterization of the power of a natural class of analog models over the real numbers. 1
A ModelTheoretic Approach to Ordinal Analysis
 Bulletin of Symbolic Logic
, 1997
"... . We describe a modeltheoretic approach to ordinal analysis via the finite combinatorial notion of an #large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in no ..."
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Cited by 12 (3 self)
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. We describe a modeltheoretic approach to ordinal analysis via the finite combinatorial notion of an #large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in nonstandard instances, give rise to models of the theory being analyzed. This method is applied to obtain ordinal analyses of a number of interesting subsystems of first and secondorder arithmetic. x1. Introduction. Two of proof theory's defining goals are the justification of classical theories on constructive grounds, and the extraction of constructive information from classical proofs. Since Gentzen, ordinal analysis has been a major component in these pursuits, and the assignment of recursive ordinals to theories has proven to be an illuminating way of measuring their constructive strength. The traditional approach to ordinal analysis, which uses cutelimination procedures to transfor...
The Complexity of Real Recursive Functions
 Unconventional Models of Computation (UMC'02), LNCS 2509
, 2002
"... We explore recursion theory on the reals, the analog counterpart of recursive function theory. In recursion theory on the reals, the discrete operations of standard recursion theory are replaced by operations on continuous functions, such as composition and various forms of differential equations. W ..."
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Cited by 11 (5 self)
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We explore recursion theory on the reals, the analog counterpart of recursive function theory. In recursion theory on the reals, the discrete operations of standard recursion theory are replaced by operations on continuous functions, such as composition and various forms of differential equations. We define classes of real recursive functions, in a manner similar to the classical approach in recursion theory, and we study their complexity. In particular, we prove both upper and lower bounds for several classes of real recursive functions, which lie inside the primitive recursive functions and, therefore, can be characterized in terms of standard computational complexity.