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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...
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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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.
On Tiered Small Jump Operators
"... Predicative analysis of recursion schema is a method to characterize complexity classes like the class FPTIME of polynomial time computable functions. This analysis comes from the works of Bellantoni and Cook, and Leivant by data tiering. Here, we refine predicative analysis by using a ramified Ack ..."
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Predicative analysis of recursion schema is a method to characterize complexity classes like the class FPTIME of polynomial time computable functions. This analysis comes from the works of Bellantoni and Cook, and Leivant by data tiering. Here, we refine predicative analysis by using a ramified Ackermann’s construction of a nonprimitive recursive function. We obtain a hierarchy of functions which characterizes exactly functions, which are computed in O(n k) time over register machine model of computation. For this, we introduce a strict ramification principle. Then, we show how to diagonalize in order to obtain an exponential function and to jump outside ∪kDTIME(n k). Lastly, we suggest a dependent typed lambdacalculus to represent this construction.
Both can be found at the ENTCS Macro Home Page. The Garland Measure and Computational Complexity of Stack Programs
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Certifying
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Characterizing the Grzegorczyk hierarchy by safe recursion
, 1999
"... We show how the charaterization of the polytime functions by Bellantoni and Cook [1] can be extended to characterize any stage of the Grzegorzyk hierarchy above the second, thus proposing an answer to a problem posted by Clote [3]. This is done by allowing an arbitrary fixed number of distinct posit ..."
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We show how the charaterization of the polytime functions by Bellantoni and Cook [1] can be extended to characterize any stage of the Grzegorzyk hierarchy above the second, thus proposing an answer to a problem posted by Clote [3]. This is done by allowing an arbitrary fixed number of distinct positions for variables instead of only two as in the original work of Bellantoni and Cook. As turned out after writing down this paper, comparable results were also proved by Bellantoni and Niggl [2]. Keywords: Recursion theory, Complexity theory, Grzegorzcyk. 1 Introduction Bellantoni and Cook [1] characterized the polytime functions by distinguishing between two sorts of arguments of functions, called normal and safe arguments. Recursion is only allowed over normal arguments, whereas the recursively computed values must be inserted in a safe position, and function composition is defined accordingly. Related tiering notions also appeared elsewhere, e.g. in Leivant [5] or Simmons [10]. Clote [...
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"... Draft not for distribution, Comments are welcome! Abstract. Predicative analysis of recursion schema is a method to characterize complexity classes like the class of polynomial time functions. This analysis comes from the works of Bellantoni and Cook, and Leivant. Here, we refine predicative analysi ..."
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Draft not for distribution, Comments are welcome! Abstract. Predicative analysis of recursion schema is a method to characterize complexity classes like the class of polynomial time functions. This analysis comes from the works of Bellantoni and Cook, and Leivant. Here, we refine predicative analysis by using a ramified Ackermann’s construction of a nonprimitive recursive function. We obtain an hierarchy of functions which characterizes exactly functions, which are computed in O(n k) over register machine model of computation. Then, we are able to diagonalize using dependent types in order to obtain an exponential function. 1
Journal of Automata, Languages and Combinatorics u (v) w, x–y c ○ OttovonGuerickeUniversität Magdeburg NP PREDICATES COMPUTABLE IN THE WEAKEST LEVEL OF THE GRZEGORCZYCK HIERARCHY
"... Let (Er)r∈N be the hierarchy of Grzegorczyk. Its weakest level, E0 is indeed quite weak, as it doesn’t even contains functions such as max(x, y) or x + y. In this paper we show that SAT ∈ E0 by developing a technique which can be used to show the same result holds for other NP problems. Using this t ..."
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Let (Er)r∈N be the hierarchy of Grzegorczyk. Its weakest level, E0 is indeed quite weak, as it doesn’t even contains functions such as max(x, y) or x + y. In this paper we show that SAT ∈ E0 by developing a technique which can be used to show the same result holds for other NP problems. Using this technique, we are able to show that also the Hamiltonian Cycle Problem is solvable in E0.