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12
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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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.
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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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
Sharply Bounded Alternation within P
, 1996
"... We define the sharply bounded hierarchy, SBH (QL), a hierarchy of classes within P , using quasilineartime computation and quantification over values of length log n. It generalizes the limited nondeterminism hierarchy introduced by Buss and Goldsmith, while retaining the invariance properties. T ..."
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Cited by 5 (3 self)
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We define the sharply bounded hierarchy, SBH (QL), a hierarchy of classes within P , using quasilineartime computation and quantification over values of length log n. It generalizes the limited nondeterminism hierarchy introduced by Buss and Goldsmith, while retaining the invariance properties. The new hierarchy has several alternative characterizations. We define both SBH (QL) and its corresponding hierarchy of function classes, FSBH(QL),and present a variety of problems in these classes, including ql m complete problems for each class in SBH (QL). We discuss the structure of the hierarchy, and show that certain simple structural conditions on it would imply P 6= PSPACE. We present characterizations of SBH (QL) relations based on alternating Turing machines and on firstorder definability, as well as recursiontheoretic characterizations of function classes corresponding to SBH (QL).
Sharply bounded alternation and quasilinear time
 Theory of Computing Systems
, 1998
"... We de ne the sharply bounded hierarchy, SBH (QL), a hierarchy of classes within P, using quasilineartime computation and quanti cation over strings of length log n. It generalizes the limited nondeterminism hierarchy introduced by Buss and Goldsmith, while retaining the invariance properties. The n ..."
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Cited by 4 (0 self)
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We de ne the sharply bounded hierarchy, SBH (QL), a hierarchy of classes within P, using quasilineartime computation and quanti cation over strings of length log n. It generalizes the limited nondeterminism hierarchy introduced by Buss and Goldsmith, while retaining the invariance properties. The new hierarchy hasseveral alternative characterizations. We de ne both SBH (QL) and its corresponding hierarchy of function classes, ql and present a variety of problems in these classes, including mcomplete problems for each class in SBH (QL). We discuss the structure of the hierarchy, and show that determining its precise relationship to deterministic time classes can imply P 6 = PSPACE. We present characterizations of SBH (QL) relations based on alternating Turing machines and on rstorder de nability, aswell as recursiontheoretic characterizations of function classes corresponding to SBH (QL).
Alternating function classes within P
 University of Manitoba Computer Science Dept
, 1992
"... We de ne the notion of adding \small amounts " of nondeterminism to a deterministic function class, and give a machine model � the result is a functional AC 0 closure of the deterministic class. We characterize, by the \safe parameters " technique, the classes of functions computable in li ..."
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Cited by 3 (3 self)
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We de ne the notion of adding \small amounts " of nondeterminism to a deterministic function class, and give a machine model � the result is a functional AC 0 closure of the deterministic class. We characterize, by the \safe parameters " technique, the classes of functions computable in linear and in quasilinear time on a multitape Turing machine. We thencombine these two results by extending the \safe parameters " characterizations to the functions computable in (quasi)linear time with small amounts of nondeterminism, and discuss implications for both sequential and parallel complexity.
Linear Ramified Higher Type Recursion and Parallel Complexity
"... A typed lambda calculus with recursion in all finite types is defined such that the first order terms exactly characterize the parallel complexity class NC. This is achieved by use of the appropriate forms of recursion (concatenation recursion and logarithmic recursion), a ramified type structure an ..."
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A typed lambda calculus with recursion in all finite types is defined such that the first order terms exactly characterize the parallel complexity class NC. This is achieved by use of the appropriate forms of recursion (concatenation recursion and logarithmic recursion), a ramified type structure and imposing of a linearity constraint.
On parallel hierarchies and R i k
 Submitted Annals of Pure and Applied Logic
, 1996
"... This paper de nes natural hierarchies of function and relation classes, constructed from parallel complexity classes in a manner analogous to the polynomialtime hierarchy. A number of structural results about these classes are proven: relationships between them and the levels of PH � relationships ..."
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This paper de nes natural hierarchies of function and relation classes, constructed from parallel complexity classes in a manner analogous to the polynomialtime hierarchy. A number of structural results about these classes are proven: relationships between them and the levels of PH � relationships between these classes and de nability in the bounded arithmetic theories Ri k � and several results relating conservation among theories of bounded arithmetic to the collapse of complexity classes. 1
Recursion Schemata For Slow Growing Depth Circuit Classes
"... . In this note we characterize iterated log depth circuit classes LD i and ND i by Cobhamlike bounded recursion schemata. We also give alternative characterizations which utilizes the safe recursion method developed by Bellantoni and Cook. 1. Introduction The search for recursion theoretic ..."
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. In this note we characterize iterated log depth circuit classes LD i and ND i by Cobhamlike bounded recursion schemata. We also give alternative characterizations which utilizes the safe recursion method developed by Bellantoni and Cook. 1. Introduction The search for recursion theoretic characterizations of various complexity classes was began by A. Cobham [Cob], who characterized the class of polynomial time computable functions by a scheme now called bounded recursion on notation. (See also [Ro] for the proof.) The essence of this recursion scheme is two fold: firstly, on input x the recursive call is made for jxj O(1) times where jxj is the length of x, and . secondly, the growth rate is bounded by a previously defined polynomial time function. The second condition is crucial for the characterization of resource bounded computations since the computation on each recursive call takes the value of the function as an argument, so the number of steps that each recursive ...