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16
A Linguistic Characterization of Bounded Oracle Computation and Probabilistic Polynomial Time
, 1998
"... We present a higherorder functional notation for polynomialtime computation with an arbitrary 0, 1valued oracle. This formulation provides a linguistic characterization for classes such as NP and BPP, as well as a notation for probabilistic polynomialtime functions. The language is derived from H ..."
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Cited by 26 (9 self)
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We present a higherorder functional notation for polynomialtime computation with an arbitrary 0, 1valued oracle. This formulation provides a linguistic characterization for classes such as NP and BPP, as well as a notation for probabilistic polynomialtime functions. The language is derived from Hofmann 's adaptation of BellantoniCook safe recursion, extended to oracle computation via work derived from that of Kapron and Cook. Like Hofmann's language, ours is an applied typed lambda calculus with complexity bounds enforced by a type system. The type system uses a modal operator to distinguish between two sorts of numerical expressions. Recursion can take place on only one of these sorts. The proof that the language captures precisely oracle polynomial time is modeltheoretic, using adaptations of various techniques from category theory.
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.
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).
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...
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).
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
Tiering as a Recursion Technique
 Bulletin of Symbolic Logic
"... I survey the syntactic technique of tiering which can be used to restrict the power of a recursion scheme. I show how various results can be obtained entirely proof theoretically without the use of a model of computation. The essence of the method is to move between explicit numerals and simulated ( ..."
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Cited by 3 (0 self)
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I survey the syntactic technique of tiering which can be used to restrict the power of a recursion scheme. I show how various results can be obtained entirely proof theoretically without the use of a model of computation. The essence of the method is to move between explicit numerals and simulated (Church) numerals.
Minimization and NP multifunctions
 Theoretical Computer Science
, 2000
"... . The implicit characterizations of the polynomialtime computable functions FP given by BellantoniCook and Leivant suggest that this class is the complexitytheoretic analog of the primitive recursive functions. Hence it is natural to add minimization operators to these characterizations and in ..."
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Cited by 2 (0 self)
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. The implicit characterizations of the polynomialtime computable functions FP given by BellantoniCook and Leivant suggest that this class is the complexitytheoretic analog of the primitive recursive functions. Hence it is natural to add minimization operators to these characterizations and investigate the resulting class of partial functions as a candidate for the analog of the partial recursive functions. We do so in this paper for Cobham's denition of FP by bounded recursion and for BellantoniCook's safe recursion and prove that the resulting classes capture exactly NPMV, the nondeterministic polynomialtime computable partial multifunctions. We also consider the relationship between our schemes and a notion of nondeterministic recursion dened by Leivant and show that the latter characterizes the total functions of NPMV. We view these results as giving evidence that NPMV is the appropriate analog of partial recursive. We reinforce this view by showing that for many ...