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15
A New RecursionTheoretic Characterization Of The Polytime Functions
 COMPUTATIONAL COMPLEXITY
, 1992
"... We give a recursiontheoretic characterization of FP which describes polynomial time computation independently of any externally imposed resource bounds. In particular, this syntactic characterization avoids the explicit size bounds on recursion (and the initial function 2 xy ) of Cobham. ..."
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Cited by 234 (7 self)
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We give a recursiontheoretic characterization of FP which describes polynomial time computation independently of any externally imposed resource bounds. In particular, this syntactic characterization avoids the explicit size bounds on recursion (and the initial function 2 xy ) of Cobham.
Predicative Recursion and Computational Complexity
, 1992
"... The purpose of this thesis is to give a "foundational" characterization of some common complexity classes. Such a characterization is distinguished by the fact that no explicit resource bounds are used. For example, we characterize the polynomial time computable functions without making an ..."
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Cited by 45 (3 self)
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The purpose of this thesis is to give a "foundational" characterization of some common complexity classes. Such a characterization is distinguished by the fact that no explicit resource bounds are used. For example, we characterize the polynomial time computable functions without making any direct reference to polynomials, time, or even computation. Complexity classes characterized in this way include polynomial time, the functional polytime hierarchy, the logspace decidable problems, and NC. After developing these "resource free" definitions, we apply them to redeveloping the feasible logical system of Cook and Urquhart, and show how this firstorder system relates to the secondorder system of Leivant. The connection is an interesting one since the systems were defined independently and have what appear to be very different rules for the principle of induction. Furthermore it is interesting to see, albeit in a very specific context, how to retract a second order statement, ("inducti...
Higher Type Recursion, Ramification and Polynomial Time
 Annals of Pure and Applied Logic
, 1999
"... It is shown how to restrict recursion on notation in all finite types so as to characterize the polynomial time computable functions. The restrictions are obtained by enriching the type structure with the formation of types !oe, and by adding linear concepts to the lambda calculus. 1 Introduction ..."
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It is shown how to restrict recursion on notation in all finite types so as to characterize the polynomial time computable functions. The restrictions are obtained by enriching the type structure with the formation of types !oe, and by adding linear concepts to the lambda calculus. 1 Introduction Recursion in all finite types was introduced by Hilbert [9] and later became known as the essential part of Godel's system T [8]. This system has long been viewed as a powerful scheme unsuitable for describing small complexity classes such as polynomial time. Simmons [16] showed that ramification can be used to characterize the primitive recursive functions by higher type recursion, and Leivant and Marion [14] showed that another form of ramification can be used to restrict higher type recursion to PSPACE. However, to characterize the much smaller class of polynomialtime computable functions by higher type recursion, it seems that an additional principle is required. By introducing linear...
Functionalgebraic characterizations of log and polylog parallel time
 Computational Complexity
, 1994
"... Abstract. The main results of this paper are recursiontheoretic characterizations of two parallel complexity classes: the functions computable by uniform bounded fanin circuit families of log and polylog depth (or equivalently, the functions bitwise computable by alternating Turing machines in log ..."
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Abstract. The main results of this paper are recursiontheoretic characterizations of two parallel complexity classes: the functions computable by uniform bounded fanin circuit families of log and polylog depth (or equivalently, the functions bitwise computable by alternating Turing machines in log and polylog time). The present characterizations avoid the complex base functions, function constructors, and a priori size or depth bounds typical of previous work on these classes. This simplicity is achieved by extending the \tiered recursion &quot; techniques of Leivant and Bellantoni&Cook. Key words. Circuit complexity � subrecursion. Subject classi cations. 68Q15, 03D20, 94C99. 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 computab ..."
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Cited by 3 (3 self)
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We de ne the notion of adding \small amounts &quot; 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 &quot; 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 &quot; characterizations to the functions computable in (quasi)linear time with small amounts of nondeterminism, and discuss implications for both sequential and parallel complexity.
Safe Weak Minimization Revisited
 SIAM Journal on Computing
, 2002
"... Minimization operators of di#erent strength have been studied in the framework of "predicative (safe) recursion". In this paper a modification of these operators is presented. By adding the new operator to those used by BellantoniCook and Leivant to characterize the polynomialtime co ..."
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Minimization operators of di#erent strength have been studied in the framework of "predicative (safe) recursion". In this paper a modification of these operators is presented. By adding the new operator to those used by BellantoniCook and Leivant to characterize the polynomialtime computable functions one obtains a characterization of the nondeterministic polynomialtime computable multifunctions.
Safe Recursive Set Functions
, 2012
"... We introduce the safe recursive set functions based on a BellantoniCook style subclass of the primitive recursive set functions. We show that under a natural encoding of finite strings by hereditarily finite sets, the functions computed by safe recursive set functions are exactly the functions comp ..."
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We introduce the safe recursive set functions based on a BellantoniCook style subclass of the primitive recursive set functions. We show that under a natural encoding of finite strings by hereditarily finite sets, the functions computed by safe recursive set functions are exactly the functions computed by alternating exponential time Turing machines with polynomially many alternations. We characterize the safe recursive set functions on arbitrary sets in definabilitytheoretic terms. In its strongest form, we show that a function on arbitrary sets is safe recursive if, and only if, it is uniformly definable in some polynomial level of a refinement of Jensen’s Jhierarchy, relativised to the transitive closure of the function’s arguments. We observe that saferecursive functions on infinite binary strings are equivalent to functions computed by socalled infinitetime Turing machines in time less than ω ω. We also give a machine model for safe recursion which is based on setindexed parallel processors and the natural bound on running times.