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32
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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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...
Unifying equivalencebased definitions of protocol security
 In WITS 2004
, 2004
"... , and Vitaly Shmatikov 2 1 Stanford University ..."
Semantics of Linear/modal Lambda Calculus
 Journal of Functional Programming
, 1998
"... In previous work the author has introduced a lambda calculus SLR with modal and linear types which serves as an extension of BellantoniCook's function algebra BC to higher types. It is a step towards a functional programming language in which all programs run in polynomial time. While this previous ..."
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In previous work the author has introduced a lambda calculus SLR with modal and linear types which serves as an extension of BellantoniCook's function algebra BC to higher types. It is a step towards a functional programming language in which all programs run in polynomial time. While this previous work was concerned with the syntactic metatheory of SLR in this paper we develop a semantics of SLR in terms of Chu spaces over a certain category of sheaves from which it follows that all expressible functions are indeed in PTIME. We notice a similarity between the Chu space interpretation and CPS translation which as we hope will have further applications in functional programming. 1 Introduction In [10] we have introduced a lambda calculus SLR which generalises the BellantoniCook characterisation of PTIME [4] to higherorder functions. The separation between normal and safe variables which is crucial to the BellantoniCook system has been achieved by way of an S 4 modality on types. ...
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).
A Functional Language for Logarithmic Space
 In APLAS
, 2004
"... More than being just a tool for expressing algorithms, a welldesigned programming language allows the user to express her ideas efficiently. The design choices however effect the efficiency of the algorithms written in the languages. It is therefore of importance to understand how such choices effe ..."
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More than being just a tool for expressing algorithms, a welldesigned programming language allows the user to express her ideas efficiently. The design choices however effect the efficiency of the algorithms written in the languages. It is therefore of importance to understand how such choices effect the expressibility of programming languages. The paper pursues the very low complexity programs by presenting a firstorder function algebra BC # that captures exactly LF, the functions computable in logarithmic space. This gives insights into the expressiveness of recursion. Moreover, it can be useful for the automatic analysis of programs' resource usage and the separation of complexity classes. The important technical features of BC # are (1) a separation of variables into safe and normal variables where recursion can only be done over the latter; (2) linearity of the recursive call; and (3) recursion with a variable step length (courseofvalue recursion). Unlike formulations LF via Turin Machines, BC # makes no references to outside resource measures, e.g., the size of the memory used. This appears to be the first such characterization of LFcomputable functions (not just predicates). The proof that all BC #programs can be evaluated in LF is of separate interest to programmers: it trades space for time and evaluates recursion with at most one recursive call without a call stack.
Ranking Arithmetic Proofs by Implicit Ramification
 in Proof Complexity and Feasible Arithmetics, P. Beame and S. Buss, eds., DIMACS Series in Discrete Mathematics
, 1996
"... Proofs in an arithmetic system are ranked according to a ramification hierarchy based on occurrences of induction. It is shown that this ranking of proofs corresponds exactly to a natural ranking of the primitive recursive functions based on occurrences of recursion. A function is provably convergen ..."
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Cited by 4 (3 self)
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Proofs in an arithmetic system are ranked according to a ramification hierarchy based on occurrences of induction. It is shown that this ranking of proofs corresponds exactly to a natural ranking of the primitive recursive functions based on occurrences of recursion. A function is provably convergent using a rank r proof, if and only if it is a rank r function. The result is of interest to complexity theorists, since rank one corresponds to polynomial time. Remarkably, this characterization of polynomialtime provability admits induction over formulas having arbitrary quantifier complexity. 1 Introduction The primitive recursive functions can be assigned ranks, based on an examination of the structure of their derivations as built up from the initial functions by the rules of composition and recursion. One of the hierarchies defined using such a ranking consists of the polynomialtime computable functions at level 1, and at higher levels consists of certain of the Grzegorczyk classe...
Divide and Conquer in Parallel Complexity and Proof Theory
, 1992
"... Copyright Stephen Austin Bloch, 1992 All rights reserved. The dissertation of Stephen Bloch is approved, and it is acceptable in quality and form for publication on micro ..."
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Cited by 4 (2 self)
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Copyright Stephen Austin Bloch, 1992 All rights reserved. The dissertation of Stephen Bloch is approved, and it is acceptable in quality and form for publication on micro
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.