Results 1  10
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12
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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Cited by 22 (3 self)
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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...
Algorithms With Polynomial Interpretation Termination Proof
 Journal of Functional Programming
, 1999
"... We study the effect of polynomial interpretation termination proofs of deterministic (resp. nondeterministic) algorithms defined by confluent (resp. nonconfluent) rewrite systems over data structures which include strings, lists and trees, and we classify them according to the interpretations of t ..."
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Cited by 14 (3 self)
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We study the effect of polynomial interpretation termination proofs of deterministic (resp. nondeterministic) algorithms defined by confluent (resp. nonconfluent) rewrite systems over data structures which include strings, lists and trees, and we classify them according to the interpretations of the constructors. This leads to the definition of six classes which turn out to be exactly the deterministic (resp. nondeterministic) polytime, linear exponentialtime and doubly linear exponential time computable functions when the class is based on conuent (resp. nonconfluent) rewrite systems. We also obtain a characterisation of the linear space computable functions. Finally, we demonstrate that functions with exponential interpretation termination proofs are superelementary.
A Flow Calculus of mwpBounds for Complexity Analysis
"... We present a method for certifying that the values computed by an imperative program will be bounded by polynomials in the program’s inputs. To this end, we introduce mwpmatrices and define a semantic relation  = C: M where C is a program and M is an mwpmatrix. It follows straightforwardly from o ..."
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Cited by 4 (3 self)
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We present a method for certifying that the values computed by an imperative program will be bounded by polynomials in the program’s inputs. To this end, we introduce mwpmatrices and define a semantic relation  = C: M where C is a program and M is an mwpmatrix. It follows straightforwardly from our definitions that there exists M such that  = C:M holds iff every value computed by C is bounded by a polynomial in the inputs. Furthermore, we provide a syntactical proof calculus and define the relation ⊢ C:M to hold iff there exists a derivation in the calculus where C:M is the bottom line. We prove that ⊢ C:M implies  = C:M. By means of exhaustive proof search, an algorithm can decide if there exists M such that the relation ⊢ C:M holds, and thus, our results yield a computational method. Categories and Subject Descriptors: D.2.4 [Software engineering]: Software/Program Verification; F.2.0 [Analysis of algorithms and problem complexity]: General; F.3.1 [Logics and meanings of programs]: Specifying and Verifying and Reasoning about Programs
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.
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.
An Hierarchy of Terminating Algorithms With Semantic Interpretation Termination Proofs
, 1998
"... We study deterministic (nondeterministic) algorithms define by mean of confluent (resp. nonconuent) rewrite system admitting polynomial interpretation termination proofs. Data structures of the algorithms include strings, lists and trees. We classify them according to the interpretations of constr ..."
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Cited by 1 (1 self)
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We study deterministic (nondeterministic) algorithms define by mean of confluent (resp. nonconuent) rewrite system admitting polynomial interpretation termination proofs. Data structures of the algorithms include strings, lists and trees. We classify them according to the interpretations of constructors This leads to the definition of six classes, which turn out to be exactly the deterministic (nondeterministic) polytime, linear exponentialtime and doubly linear exponential time computable functions when the class is based on confluent (resp. nonconfluent) systems. Next, we demonstrate that functions with exponential interpretation termination proofs are superelementary.
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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