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25
Mathematically Strong Subsystems of Analysis With Low Rate of Growth of Provably Recursive Functionals
, 1995
"... This paper is the first one in a sequel of papers resulting from the authors Habilitationsschrift [22] which are devoted to determine the growth in proofs of standard parts of analysis. A hierarchy (GnA # )n#I N of systems of arithmetic in all finite types is introduced whose definable objects of ..."
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Cited by 34 (21 self)
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This paper is the first one in a sequel of papers resulting from the authors Habilitationsschrift [22] which are devoted to determine the growth in proofs of standard parts of analysis. A hierarchy (GnA # )n#I N of systems of arithmetic in all finite types is introduced whose definable objects of type 1 = 0(0) correspond to the Grzegorczyk hierarchy of primitive recursive functions. We establish the following extraction rule for an extension of GnA # by quantifierfree choice ACqf and analytical axioms # having the form #x # #y ## sx#z # F0 (including also a `non standard' axiom F  which does not hold in the full settheoretic model but in the strongly majorizable functionals): From a proof GnA # +ACqf + # # #u 1 , k 0 #v ## tuk#w 0 A0(u, k, v, w) one can extract a uniform bound # such that #u 1 , k 0 #v ## tuk#w # #ukA0 (u, k, v, w) holds in the full settheoretic type structure. In case n = 2 (resp. n = 3) #uk is a polynomial (resp. an elementary recursive function) in k, u M := #x. max(u0, . . . , ux). In the present paper we show that for n # 2, GnA # +ACqf+F  proves a generalization of the binary Knig's lemma yielding new conservation results since the conclusion of the above rule can be verified in G max(3,n) A # in this case. In a subsequent paper we will show that many important ine#ective analytical principles and theorems can be proved already in G2A # +ACqf+# for suitable #. 1
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 29 (15 self)
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We study a restricted version of Shannon's General . . .
The Expressive Power of Higherorder Types or, Life without CONS
, 2001
"... Compare firstorder functional programs with higherorder programs allowing functions as function parameters. Can the the first program class solve fewer problems than the second? The answer is no: both classes are Turing complete, meaning that they can compute all partial recursive functions. In pa ..."
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Cited by 24 (1 self)
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Compare firstorder functional programs with higherorder programs allowing functions as function parameters. Can the the first program class solve fewer problems than the second? The answer is no: both classes are Turing complete, meaning that they can compute all partial recursive functions. In particular, higherorder values may be firstorder simulated by use of the list constructor ‘cons’ to build function closures. This paper uses complexity theory to prove some expressivity results about small programming languages that are less than Turing complete. Complexity classes of decision problems are used to characterize the expressive power of functional programming language features. An example: secondorder programs are more powerful than firstorder, since a function f of type [Bool]〉Bool is computable by a consfree firstorder functional program if and only if f is in PTIME, whereas f is computable by a consfree secondorder program if and only if f is in EXPTIME. Exact characterizations are given for those problems of type [Bool]〉Bool solvable by programs with several combinations of operations on data: presence or absence of constructors; the order of data values: 0, 1, or higher; and program control structures: general recursion, tail recursion, primitive recursion.
A system for the static analysis of XPath
 ACM TOIS
, 2006
"... XPath is the standard language for navigating XML documents and returning a set of matching nodes. We present a sound and complete decision procedure for containment of XPath queries, as well as other related XPath decision problems such as satisfiability, equivalence, overlap, and coverage. The con ..."
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Cited by 16 (3 self)
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XPath is the standard language for navigating XML documents and returning a set of matching nodes. We present a sound and complete decision procedure for containment of XPath queries, as well as other related XPath decision problems such as satisfiability, equivalence, overlap, and coverage. The considered XPath fragment covers most of the language features used in practice. Specifically, we propose a unifying logic for XML, namely, the alternationfree modal μcalculus with converse. We show how to translate major XML concepts such as XPath and regular XML types (including DTDs) into this logic. Based on these embeddings, we show how XPath decision problems, in the presence or absence of XML types, can be solved using a decision procedure for μcalculus satisfiability. We provide a complexity analysis of our system together with practical experiments to illustrate the efficiency of the approach for realistic scenarios.
Ranking primitive recursions: The low grzegorczyk classes revisited
 SIAM Journal of Computing
, 1998
"... Abstract. Traditional results in subrecursion theory are integrated with the recent work in “predicative recursion ” by defining a simple ranking ρ of all primitive recursive functions. The hierarchy defined by this ranking coincides with the Grzegorczyk hierarchy at and above the linearspace level. ..."
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Cited by 10 (1 self)
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Abstract. Traditional results in subrecursion theory are integrated with the recent work in “predicative recursion ” by defining a simple ranking ρ of all primitive recursive functions. The hierarchy defined by this ranking coincides with the Grzegorczyk hierarchy at and above the linearspace level. Thus, the result is like an extension of the Schwichtenberg/Müller theorems. When primitive recursion is replaced by recursion on notation, the same series of classes is obtained except with the polynomial time computable functions at the first level.
Subrecursion as Basis for a Feasible Programming Language
 Proceedings of CSL'94, number 933 in LNCS
, 1994
"... We are motivated by finding a good basis for the semantics of programming languages and investigate small classes in subrecursive hierarchies of functions. We do this with the help of pairing functions because in this way we can explore the amazing coding powers of Sexpressions of LISP within t ..."
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Cited by 9 (8 self)
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We are motivated by finding a good basis for the semantics of programming languages and investigate small classes in subrecursive hierarchies of functions. We do this with the help of pairing functions because in this way we can explore the amazing coding powers of Sexpressions of LISP within the domain of natural numbers. In the process of doing this we introduce a missing stage in Grzegorczykbased hierarchies which solves the longstanding open problem of what is the precise relation between the small recursive classes and those of complexity theory. 1 Introduction We investigate subrecursive hierarchies based on pairing functions and solve a longstanding open problem in small recursive classes of what is the relationship between these and computational complexity classes (see [11]). The problem is solved by discovering that there is a missing stage in Grzegorczykbased hierarchies [7, 11]. The motivation for this research comes from our search for a good programming langu...
Upper and Lower Bounds on ContinuousTime Computation
"... We consider various extensions and modifications of Shannon's General Purpose Analog Computer, which is a model of computation by differential equations in continuous time. We show that several classical computation classes have natural analog counterparts, including the primitive recursive function ..."
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Cited by 8 (2 self)
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We consider various extensions and modifications of Shannon's General Purpose Analog Computer, which is a model of computation by differential equations in continuous time. We show that several classical computation classes have natural analog counterparts, including the primitive recursive functions, the elementary functions, the levels of the Grzegorczyk hierarchy, and the arithmetical and analytical hierarchies.
Properties of Monoids That Are Presented By Finite Convergent StringRewriting Systems  a Survey
, 1997
"... In recent years a number of conditions has been established that a monoid must necessarily satisfy if it is to have a presentation through some finite convergent stringrewriting system. Here we give a survey on this development, explaining these necessary conditions in detail and describing the rela ..."
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Cited by 6 (5 self)
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In recent years a number of conditions has been established that a monoid must necessarily satisfy if it is to have a presentation through some finite convergent stringrewriting system. Here we give a survey on this development, explaining these necessary conditions in detail and describing the relationships between them. 1 Introduction Stringrewriting systems, also known as semiThue systems, have played a major role in the development of theoretical computer science. On the one hand, they give a calculus that is equivalent to that of the Turing machine (see, e.g., [Dav58]), and in this way they capture the notion of `effective computability' that is central to computer science. On the other hand, in the phrasestructure grammars introduced by N. Chomsky they are used as sets of productions, which form the essential part of these grammars [HoUl79]. In this way stringrewriting systems are at the very heart of formal language theory. Finally, they are also used in combinatorial semig...
An analog characterization of the subrecursive functions
 PROC. 4TH CONFERENCE ON REAL NUMBERS AND COMPUTERS
, 2000
"... We study a restricted version of Shannon’s General Purpose Analog Computer in which we only allow the machine to solve linear differential equations. This corresponds to only allowing local feedback in the machine’s variables. We show that if this computer is allowed to sense inequalities in a dif ..."
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Cited by 6 (1 self)
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We study a restricted version of Shannon’s General Purpose Analog Computer in which we only allow the machine to solve linear differential equations. This corresponds to only allowing local feedback in the machine’s variables. We show that if this computer is allowed to sense inequalities in a differentiable way, then it can compute exactly the elementary functions. Furthermore, we show that if the machine has access to an oracle which computes a function f(x) with a suitable growth as x goes to infinity, then it can compute functions on any given level of the Grzegorczyk hierarchy. More precisely, we show that the model contains exactly the nth level of the Grzegorczyk hierarchy if it is allowed to solve n − 3 nonlinear differential equations of a certain kind. Therefore, we claim that there is a close connection between analog complexity classes, and the dynamical systems that compute them, and classical sets of subrecursive functions.
Deciding XPath containement with MSO
 Data & Knowledge Engineering
, 2007
"... XPath is the standard language for addressing parts of an XML document. We present a sound and complete decision procedure for containment of XPath queries. The considered XPath fragment covers most of the language features used in practice. Specifically, we show how XPath queries can be translated ..."
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Cited by 5 (2 self)
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XPath is the standard language for addressing parts of an XML document. We present a sound and complete decision procedure for containment of XPath queries. The considered XPath fragment covers most of the language features used in practice. Specifically, we show how XPath queries can be translated into equivalent formulas in monadic secondorder logic. Using this translation, we construct an optimized logical formulation of the containment problem, which is decided using tree automata. When the containment relation does not hold between two XPath expressions, a counterexample XML tree is generated. We provide practical experiments that illustrate the efficiency of the decision procedure for realistic scenarios.