The complexity of loop programs

Meyer, Albert R.; Ritchie, Dennis M.

doi:10.1145/800196.806014

Cited by 153 publications

(99 citation statements)

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“…Then, for all n ∈ N, the thread produced by the instruction sequence that computes f n is in essence a branching program and its size is polynomially bounded in n. As a consequence of this, IS 0 br \poly coincides with L/poly. The approaches to computational complexity based on loop programs [19], straight-line programs [16], and branching programs [13] appear to be the closest related to the approach followed in this paper.…”

Section: Discussionmentioning

confidence: 99%

Instruction Sequence Based Non-uniform Complexity Classes

Bergstra¹,

Middelburg²

2014

SACS

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Abstract. We present an approach to non-uniform complexity in which single-pass instruction sequences play a key part, and answer various questions that arise from this approach. We introduce several kinds of non-uniform complexity classes. One kind includes a counterpart of the well-known non-uniform complexity class P/poly and another kind includes a counterpart of the well-known non-uniform complexity class NP/poly. Moreover, we introduce a general notion of completeness for the non-uniform complexity classes of the latter kind. We also formulate a counterpart of the well-known complexity theoretic conjecture that NP ⊆ P/poly. We think that the presented approach opens up an additional way of investigating issues concerning non-uniform complexity.

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Section: Discussionmentioning

confidence: 99%

Instruction Sequence Based Non-uniform Complexity Classes

Bergstra¹,

Middelburg²

2014

SACS

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“…Analogously primitive recursive séquence functions can be used to give a meaning to Meyer-Ritchie LOOP programs (see [13]) in a version which allows more than one output variable. In this manner we obtain a relationship between the structural complexity of LOOP programs and the computational complexity of primitive recursive séquence functions.…”

Section: Complexity Classes Of Loop Programsmentioning

confidence: 99%

“…by programs with at most i nested LOOP-END instructions (see [13] [2,13,14,16,17]) we can draw the following diagram to show the relationships among the hiérarchies we have considered. In the diagram A ->B means that the class A is strictly enclosed in the class B; A---B means that the two classes are not comparable.…”

Section: Comparison With Other Hierarchies Of Primitive Recursive Funmentioning

confidence: 99%

“…We introducé LOOP programs (see Meyer and Ritchie [13]) with r input variables and s output variables. We consider the class of functions computed by these programs and prove that it coincides with the class of primitive recursive séquence functions.…”

Section: Introductionmentioning

confidence: 99%

“…If we consider the subclasses J^L+bi^ma.+ b of functions N r -> N we can compare our hierarchy with the known hiérarchies of primitive recursive functions defined by Axt, Cleave, Grzegorczyk, Meyer-Ritchie (see [2,4,11,13] respectively).…”

Section: Introductionmentioning

confidence: 99%

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A hierarchy of primitive recursive sequence functions

Fachini¹,

Maggiolo-Schettini²

1979

RAIRO. Inform. théor.

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A Classification of the Recursive Functions

Meyer

Ritchie

1972

Mathematical Logic Qtrly

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in Cambridge, Massachusetts (U.S.A.)I) 1. Introduction There have been numerous attempts to classify recursive functions into hierarchies. A well known example is that of GRZEGORCZYK who defines a hierarchy & ' I c b1 c c 6 s c . . -of the primitive recursive functions based on a form of ACKERMANN'S function [14].a) The classes of the GRZEGORCZYK hierarchy are closed under a variety of operations such as limited recursion and composition. They are intimately related to the computational complexity of functions, measured for example by the amount of time or space required to compute functions with TURING machines, and the classes can also be characterized syntactically by depth of nesting of primitive recursion [lo], [as], [19], [20], [23]. Several extensions of the GRZEGORZCYK hierarchy through the multiply recursive functions of PETER have been proposed [28], [ah], [ll]. Several refinements of the hierarchy have been considered as well [26], [9], [16]. I n these refinements certain closure properties of the original GRZEOORCZYK classeg are sacrificed ; in particular the refined classes are not closed under composition of functions. I n this paper we define elementary-honest classes of recursive functions which retain all of the closure properties of the original GRZEGORCZYK classes including closure under composition. Elementary-honest classes are the functions elementary in ffor recursive functions f whose graphs correspond to elementary predicates. We then show that between any two GRZEGORCZYK classes containing the elementary functions there are densely ordered elementary-honest classes and infinitely many such classes which are setwise incomparable.The existence of densely ordered chains of classes emphasizes the fact that ordinal length of a hierarchy is not a particularly significant measure of either the extent or the refinement of a hierarchy. One can embed a hierarchy of any countable order type into a dense chain. (A similar observation about primitive recursive degrees is made by FEFERMAN [ 131 .) Moreover, the existence of incomparable classes suggests that efforts to construct totally ordered hierarchies of recursive functions are not likely to lead to natural classifications. Grzegorczyk classes and Turing machinesWe assume the reader is familiar with the definition of the elementary functions 6' (which equals GRZEGORCZYR'S class as), and their closure properties under such operations as composition of functions, explicit transformation, limited recursion, limited sum and limited product [14]. We also assume familiarity with an arithmetization of TURING machines such as is carried out by R. W. RITCHIE, or DAVIS [26], [12]. 1) This research was supported in part by NSF Grant GP-2880, ARPA grant SD-146, and 2) We use L L C" to denote proper inclusion, and ''2" for improper inclusion. DSR 79457 under Project MAC.

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The complexity of loop programs

Cited by 153 publications

References 3 publications

Instruction Sequence Based Non-uniform Complexity Classes

Instruction Sequence Based Non-uniform Complexity Classes

A hierarchy of primitive recursive sequence functions

A Classification of the Recursive Functions

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