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23
Linear Types and NonSizeIncreasing Polynomial Time Computation
 Information and Computation
, 1998
"... this paper we present a typetheoretic approach to this problem. We will develop a fairly natural linear type system which has the property that all definable functions are nonsize increasing and which boasts higherorder recursion on datatypes without any predicativity restriction. We will show th ..."
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Cited by 69 (12 self)
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this paper we present a typetheoretic approach to this problem. We will develop a fairly natural linear type system which has the property that all definable functions are nonsize increasing and which boasts higherorder recursion on datatypes without any predicativity restriction. We will show that nevertheless all definable firstorder functions are polynomial time computable even if they contain higherorder functions as subexpressions
Optimizing Object Queries Using an Effective Calculus
 ACM Transactions on Database Systems
, 1998
"... This paper concentrates on query unnesting (also known as query decorrelation), an optimization that, even though improves performance considerably, is not treated properly (if at all) by most OODB systems. Our framework generalizes many unnesting techniques proposed recently in the literature and i ..."
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Cited by 45 (2 self)
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This paper concentrates on query unnesting (also known as query decorrelation), an optimization that, even though improves performance considerably, is not treated properly (if at all) by most OODB systems. Our framework generalizes many unnesting techniques proposed recently in the literature and is capable of removing any form of query nesting using a very simple and efficient algorithm. The simplicity of our method is due to the use of the monoid comprehension calculus as an intermediate form for OODB queries. The monoid comprehension calculus treats operations over multiple collection types, aggregates, and quantifiers in a similar way, resulting in a uniform way of unnesting queries, regardless of their type of nesting.
Computational Complexity and Induction for Partial Computable Functions in Type Theory
 In Preprint
, 1999
"... An adequate theory of partial computable functions should provide a basis for defining computational complexity measures and should justify the principle of computational induction for reasoning about programs on the basis of their recursive calls. There is no practical account of these notions in ..."
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Cited by 11 (7 self)
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An adequate theory of partial computable functions should provide a basis for defining computational complexity measures and should justify the principle of computational induction for reasoning about programs on the basis of their recursive calls. There is no practical account of these notions in type theory, and consequently such concepts are not available in applications of type theory where they are greatly needed. It is also not clear how to provide a practical and adequate account in programming logics based on set theory. This paper provides a practical theory supporting all these concepts in the setting of constructive type theories. We first introduce an extensional theory of partial computable functions in type theory. We then add support for intensional reasoning about programs by explicitly reflecting the essential properties of the underlying computation system. We use the resulting intensional reasoning tools to justify computational induction and to define computationa...
The strength of nonsize increasing computation
, 2002
"... We study the expressive power of nonsize increasing recursive definitions over lists. This notion of computation is such that the size of all intermediate results will automatically be bounded by the size of the input so that the interpretation in a finite model is sound with respect to the standar ..."
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Cited by 11 (0 self)
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We study the expressive power of nonsize increasing recursive definitions over lists. This notion of computation is such that the size of all intermediate results will automatically be bounded by the size of the input so that the interpretation in a finite model is sound with respect to the standard semantics. Many wellknown algorithms with this property such as the usual sorting algorithms are definable in the system in the natural way. The main result is that a characteristic function is definable if and only if it is computable in time O(2 p(n)) for some polynomial p. The method used to establish the lower bound on the expressive power also shows that the complexity becomes polynomial time if we allow primitive recursion only. This settles an open question posed in [1, 7]. The key
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.
The geometry of linear higherorder recursion
 In Logic in Computer Science, 20th International Symposium, Proceedings
, 2005
"... Linearity and ramification constraints have been widely used to weaken higherorder (primitive) recursion in such a way that the class of representable functions equals the class of polytime functions, as the works by Leivant, Hofmann and others show. This paper shows that finetuning these two cons ..."
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Cited by 9 (4 self)
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Linearity and ramification constraints have been widely used to weaken higherorder (primitive) recursion in such a way that the class of representable functions equals the class of polytime functions, as the works by Leivant, Hofmann and others show. This paper shows that finetuning these two constraints leads to different expressive strengths, some of them lying well beyond polynomial time. This is done by introducing a new semantics, called algebraic context semantics. The framework stems from Gonthier’s original work and turns out to be a versatile and powerful tool for the quantitative analysis of normalization in the lambdacalculus with constants and higherorder recursion. 1
Realizability Models for BLLlike languages
, 2000
"... We give a realizability model of GirardScedrovScott's Bounded Linear Logic (BLL). This gives a new proof that all numerical functions representable in that system are polytime. Our analysis naturally justifies the design of the BLL syntax and suggests further extensions. 1 Introduction Bounded L ..."
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Cited by 6 (1 self)
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We give a realizability model of GirardScedrovScott's Bounded Linear Logic (BLL). This gives a new proof that all numerical functions representable in that system are polytime. Our analysis naturally justifies the design of the BLL syntax and suggests further extensions. 1 Introduction Bounded Linear Logic (BLL) [3] was an early attempt to provide an intrinsic notion of polynomial time computation within a logical system. That is, the aim was not merely to express polynomial time computability in terms of provability of certain restricted formulas, but rather to provide a typed logical system in which computation via cutelimination or proof normalization is inherently polytime. Since the appearance of this paper, several di#erent typed functional systems for analyzing ptime computability have appeared in the literature [5, 4, 10, 11, 6, 7]. For deeper foundational purposes, we should mention Girard's Light Linear Logic (LLL) [4] as a major improvement of the syntax of BLL, in that...
An Equational Characterization of the Polytime Functions on any Constructor Data Structure
, 1996
"... We give a purely syntactical, equational characterization of the polytime functions on any constructor data structure (free algebra). The equations defining a function f have the shape of simple patterns: (f (c y 1 . . . y m ) x 2 . . . xn ) = r, where c is a constructor, y 1 ,. . . ,y m , x 2 ,. . ..."
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Cited by 4 (0 self)
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We give a purely syntactical, equational characterization of the polytime functions on any constructor data structure (free algebra). The equations defining a function f have the shape of simple patterns: (f (c y 1 . . . y m ) x 2 . . . xn ) = r, where c is a constructor, y 1 ,. . . ,y m , x 2 ,. . . , xn are di#erent variables. There are restrictions on the righthand sides (rhs) r. The first restrictions concern the general shape of calls to mutually recursive functions, and they imply that we recur on first argument. To express the two main restrictions on rhs we use a concept of "critical position" which is closely related to the notion "safe" of Bellantoni and Cook, and to the "tiers" of Leivant. A function f's i'th argument position is critical i# in this position f may have access to the result of a recursive call. Then the two main restrictions are (there will be some exceptions for ifthenelse, projections and unary addition): 1. The first position of every recursive funct...