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26
Efficient First Order Functional Program Interpreter With Time Bound Certifications
, 2000
"... We demonstrate that the class of rst order functional programs over lists which terminate by multiset path ordering and admit a polynomial quasiinterpretation, is exactly the class of function computable in polynomial time. The interest of this result lies (i) on the simplicity of the conditions on ..."
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Cited by 25 (10 self)
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We demonstrate that the class of rst order functional programs over lists which terminate by multiset path ordering and admit a polynomial quasiinterpretation, is exactly the class of function computable in polynomial time. The interest of this result lies (i) on the simplicity of the conditions on programs to certify their complexity, (ii) on the fact that an important class of natural programs is captured, (iii) and on potential applications on program optimizations. 1 Introduction This paper is part of a general investigation on the implicit complexity of a specication. To illustrate what we mean, we write below the recursive rules that computes the longest common subsequences of two words. More precisely, given two strings u = u1 um and v = v1 vn of f0; 1g , a common subsequence of length k is dened by two sequences of indices i 1 < < i k and j1 < < jk satisfying u i q = v j q . lcs(; y) ! 0 lcs(x; ) ! 0 lcs(i(x); i(y)) ! lcs(x; y) + 1 lcs(i(...
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 18 (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.
Theories With SelfApplication and Computational Complexity
 Information and Computation
, 2002
"... Applicative theories form the basis of Feferman's systems of explicit mathematics, which have been introduced in the early seventies. In an applicative universe, all individuals may be thought of as operations, which can freely be applied to each other: selfapplication is meaningful, but not ne ..."
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Cited by 12 (9 self)
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Applicative theories form the basis of Feferman's systems of explicit mathematics, which have been introduced in the early seventies. In an applicative universe, all individuals may be thought of as operations, which can freely be applied to each other: selfapplication is meaningful, but not necessarily total. It has turned out that theories with selfapplication provide a natural setting for studying notions of abstract computability, especially from a prooftheoretic perspective.
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.
On Proofs About Threshold Circuits and Counting Hierarcies (Extended Abstract)
, 1998
"... ) Jan Johannsen Chris Pollett Department of Mathematics Department of Computer Science University of California, San Diego Boston University La Jolla, CA 910930112 Boston, MA 02215 Abstract We dene theories of Bounded Arithmetic characterizing classes of functions computable by constantdepth t ..."
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Cited by 9 (2 self)
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) Jan Johannsen Chris Pollett Department of Mathematics Department of Computer Science University of California, San Diego Boston University La Jolla, CA 910930112 Boston, MA 02215 Abstract We dene theories of Bounded Arithmetic characterizing classes of functions computable by constantdepth threshold circuits of polynomial and quasipolynomial size. Then we dene certain secondorder theories and show that they characterize the functions in the Counting Hierarchy. Finally we show that the former theories are isomorphic to the latter via the socalled RSUV isomorphism. 1 Introduction A phenomenon that is commonly observed in Complexity Theory is that proofs of results about counting complexity classes (#P , Mod p P etc.) can often be scaled down to yield results about small depth circuit classes with the corresponding counting gates. For example, Toda's result [17] that every problem in the Polynomial Hierarchy can be solved in polynomial time with an oracle for #P correspond...
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 9 (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.
A Safe Recursion Scheme for Exponential Time
 In Sergei I. Adian and Anil Nerode, editors, LFCS
, 1997
"... Using a function algebra characterization of exponential time due to Monien [5], in the style of BellantoniCook [2], we characterize exponential time functions of linear growth via a safe courseofvalues recursion scheme. 1 Introduction In 1991 [2], S. Bellantoni and S.A. Cook characterized the c ..."
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Cited by 7 (1 self)
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Using a function algebra characterization of exponential time due to Monien [5], in the style of BellantoniCook [2], we characterize exponential time functions of linear growth via a safe courseofvalues recursion scheme. 1 Introduction In 1991 [2], S. Bellantoni and S.A. Cook characterized the class FP of polynomial time computable functions as the smallest class of functions containing certain initial functions, and closed under safe composition and safe recursion on notation. In 1965, A. Cobham had earlier characterized FP in a similar manner using composition and bounded recursion on notation, where f is defined by bounded recursion on notation from g, h 0 , h 1 , k, if f(0; ~y) = g(~y) f(2x; ~y) = h 0 (x; ~y; f(x; ~y)); if x 6= 0 f(2x + 1; ~y) = h 1 (x; ~y; f(x; ~y)); provided that f(x; ~y) k(x; ~y). In addition to removing an initial function required by Cobham for polynomial growth rate, the novelty of the BellantoniCook construction was to remove the bounding requiremen...
A characterization of alternating log time by first order functional programs
 In LPAR 2006, volume 4246 of LNAI
, 2006
"... Abstract. We a give an intrinsic characterization of the class of functions which are computable in NC 1 that is by a uniform, logarithmic depth and polynomial size family circuit. Recall that the class of functions in ALogTime, that is in logarithmic time on an Alternating Turing Machine, is NC 1. ..."
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Cited by 7 (5 self)
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Abstract. We a give an intrinsic characterization of the class of functions which are computable in NC 1 that is by a uniform, logarithmic depth and polynomial size family circuit. Recall that the class of functions in ALogTime, that is in logarithmic time on an Alternating Turing Machine, is NC 1. Our characterization is in terms of first order functional programming languages. We define measuretools called Supinterpretations, which allow to give space and time bounds and allow also to capture a lot of program schemas. This study is part of a research on static analysis in order to predict program resources. It is related to the notion of Quasiinterpretations and belongs to the implicit computational complexity line of research. 1
A ProofTheoretic Characterization of the Basic Feasible Functionals
 Theoretical Computer Science
, 2002
"... We provide a natural characterization of the type two MehlhornCookUrquhart basic feasible functionals as the provably total type two functionals of our (classical) applicative theory PT introduced in [27], thus providing a proof of a result claimed in the conclusion of [27]. ..."
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Cited by 7 (6 self)
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We provide a natural characterization of the type two MehlhornCookUrquhart basic feasible functionals as the provably total type two functionals of our (classical) applicative theory PT introduced in [27], thus providing a proof of a result claimed in the conclusion of [27].
Continuous Time Computation with Restricted Integration Capabilities
, 2004
"... Recursion theory on the reals, the analog counterpart of recursive function theory, is an approach to continuoustime computation inspired by the models of Classical Physics. In recursion theory on the reals, the discrete operations of standard recursion theory are replaced by operations on cont ..."
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Cited by 4 (1 self)
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Recursion theory on the reals, the analog counterpart of recursive function theory, is an approach to continuoustime computation inspired by the models of Classical Physics. 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 di#erential equations like indefinite integrals, linear di#erential equations and more general Cauchy problems. We define classes of real recursive functions in a manner similar to the standard recursion theory and we study their complexity. We prove both upper and lower bounds for several classes of real recursive functions, which lie inside the elementary functions, and can be characterized in terms of space complexity. In particular, we show that hierarchies of real recursive classes closed under restricted integration operations are related to the exponential space hierarchy. The results