Ranking Primitive Recursions: The Low Grzegorczyk Classes Revisited

Bellantoni, Stephen J.; Niggl, Karl-Heinz

doi:10.1137/s009753979528175x

Cited by 17 publications

(13 citation statements)

References 17 publications

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“…The present paper continues research in implicit computational complexity as initiated by Simmons [33], Bellantoni and Cook [4], and Leivant [18], [19], [20], which have led to resource-free, purely functional characterisations of many complexity classes, such as fptime [4], [21], [23], [5], [28], flinspace [2], [20], [5], [28], NP and the polynomial-time hierarchy [3], the Kalmár-elementary functions [30] and fpspace [15], [31], the exponential time functions of linear growth [9], and the Grzegorczyk hierarchy at and above the linearspace level [5], [28], [16], [17], among many others. As well, implicit characterisations through higher type recursion have been given for the Kalmár-elementary functions [22], [15], [1], for polynomial space [24], and for fptime [6], [13].…”

Section: Introductionmentioning

confidence: 65%

Certifying Polynomial Time and Linear/Polynomial Space for Imperative Programs

Niggl¹,

Wunderlich²

2006

SIAM J. Comput.

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In earlier work of Kristiansen and Niggl the polynomial-time computable functions were characterised by stack programs of µ-measure 0, and the linear-space computable functions by loop programs of µ-measure 0. Until recently, an open problem was how to extend these characterisations to programs with user-friendly basic instructions, such as assignment statements, and with mixed data structures. It is shown how to strengthen the above characterisations to imperative programs built from arbitrary basic instructions by sequencing, if-then-else and for-do statements. These programs operate on variables, each of which may represent any data structure such as stacks, registers, trees or graphs. The paper presents a new method of certifying "polynomial size boundedness" of such imperative programs under the natural assumption that the basic instructions used are polynomial size bounded, too. The certificate for a program P with variables among X 1 ,. .. , X n will be an (n + 1) × (n + 1) matrix M (P) over the finite set {0, 1, ∞}. It is shown that certified string programs (i.e., stack programs built from any polynomial-time computable basic instructions) exactly compute the functions in fptime. Accordingly, certified general loop programs (using any linear-space computable basic instructions) exactly compute the functions in flinspace. Furthermore, it is shown that certified power string programs (i.e., string programs built from polynomial-space computable basic instructions, and extended by power loop statements) compute exactly the polynomial-space computable functions fpspace. In addition, examples of certified "natural" (implementations of) algorithms, such as insertion sort or binary addition and multiplication, are given.

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Section: Introductionmentioning

confidence: 65%

Certifying Polynomial Time and Linear/Polynomial Space for Imperative Programs

Niggl¹,

Wunderlich²

2006

SIAM J. Comput.

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“…To see this, rst observe that one can show: CLO ⊆ CLO It suces to consider any f := WBRN (g, h, B), assuming inductively g, h, B ∈ CLO. Accordingly, the y-section we need is dened by (5) y{i} := H i . −1 (y).…”

Section: Theorem 43 Clo = Clomentioning

confidence: 99%

“…By (5), (6) the y-section implementation in CLO we need this time is To complete the denition off , it remains to dene a boundB ∈ CLO, and again we run into a problem. To see this, rst observe that one can show: [2]), the main goal being to give a higher type characterization of NC, building on ideas and techniques presented in [6].…”

Section: Theorem 43 Clo = Clomentioning

confidence: 99%

“…Since then, it has been applied directly or in adapted form to many characterizations of complexity classes, e.g. the Kálmar-elementary functions and Pspace are treated in [25], in [20], [5] the method is extended to all levels of the Grzegorczyk hierarchy, and in [15] that method is adapted so as to compute all functions at Grzegorczyk level n+2 by loop programs of µ-measure n.…”

Section: Introductionmentioning

confidence: 99%

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Implicit characterizations of FPTIME and NC revisited

Niggl

Wunderlich

2010

The Journal of Logic and Algebraic Programming

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Various simplied or improved, and partly corrected well-known implicit characterizations of the complexity classes FPTIME and NC are presented. Primarily, the interest is in simplifying the required simulations of various recursion schemes in the corresponding (implicit) framework, and in developing those simulations in a more uniform way, based on a step-by-step comparison technique, thus consolidating groundwork in implicit computational complexity.

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“…predicative recursion (Bellantoni-Cook [1]), tiered definition schemes (Leivant [17]), measures on lambda terms (Niggl [25]) and measures on primitive recursive definitions (Bellantoni-Niggl [5]). In contrast to these formalism, the imperative languages introduced in this report are flexible programming languages closely related to a von Neumann computer architecture.…”

Section: Uniform Translations Between Formalismsmentioning

confidence: 99%

The Implicit Computational Complexity of Imperative Programming Languages

Kristiansen¹

2001

BRICS

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During the last decade Cook, Bellantoni, Leivant and others have developed the theory of implicit computational complexity, i.e. the theory of predicative recursion, tiered definition schemes, etcetera. We extend and modify this theory to work in a context of imperative programming languages, and characterise complexity classes like <tt>P</tt>, <small>LINSPACE</small>, <small>PSPACE</small> and the classes in the Grzegorczyk hiearchy. Our theoretical framework seems promising with respect to applications in engineering.

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Ranking Primitive Recursions: The Low Grzegorczyk Classes Revisited

Cited by 17 publications

References 17 publications

Certifying Polynomial Time and Linear/Polynomial Space for Imperative Programs

Certifying Polynomial Time and Linear/Polynomial Space for Imperative Programs

Implicit characterizations of FPTIME and NC revisited

The Implicit Computational Complexity of Imperative Programming Languages

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