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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.
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.