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14
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
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...
Safe recursion with higher types and BCKalgebra
 Annals of Pure and Applied Logic
, 2000
"... In previous work the author has introduced a lambda calculus SLR with modal and linear types which serves as an extension of BellantoniCook's function algebra BC to higher types. It is a step towards a functional programming language in which all programs run in polynomial time. In this paper we de ..."
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Cited by 21 (3 self)
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In previous work the author has introduced a lambda calculus SLR with modal and linear types which serves as an extension of BellantoniCook's function algebra BC to higher types. It is a step towards a functional programming language in which all programs run in polynomial time. In this paper we develop a semantics of SLR using BCKalgebras consisting of certain polynomialtime algorithms. It will follow from this semantics that safe recursion with arbitrary result type built up from N and ( as well as recursion over trees and other data structures remains within polynomial time. In its original formulation SLR supported only natural numbers and recursion on notation with first order functional result type. 1 Introduction In [10] and [11] we have introduced a lambda calculus SLR which generalises the BellantoniCook characterisation of PTIME [2] to higherorder functions. The separation between normal and safe variables which is crucial to the BellantoniCook system has been achieved...
An Application of CategoryTheoretic Semantics to the Characterisation of Complexity Classes Using HigherOrder Function Algebras
, 1997
"... We use the category of presheaves over PTIMEfunctions in order to show that Cook and Urquhart's higherorder function algebra PV ! defines exactly the PTIMEfunctions. As a byproduct we obtain a syntaxfree generalisation of PTIMEcomputability to higher types. By restricting to sheaves for a sui ..."
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Cited by 11 (6 self)
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We use the category of presheaves over PTIMEfunctions in order to show that Cook and Urquhart's higherorder function algebra PV ! defines exactly the PTIMEfunctions. As a byproduct we obtain a syntaxfree generalisation of PTIMEcomputability to higher types. By restricting to sheaves for a suitable topology we obtain a model for intuitionistic predicate logic with \Sigma b 1 induction over PV ! and use this to reestablish that the provably total functions in this system are in polynomial time computable. Finally, we apply the categorytheoretic approach to a new higherorder extension of BellantoniCook's system BC of safe recursion. 1 Introduction Cook and Urquhart's system PV ! [3] is a simplytyped lambda calculus providing constants to denote natural numbers and an operator for bounded recursion on notation like in Cobham's characterisation of polynomialtime computability. 1 Although functionals of arbitrary type can be defined in this system one can show that thei...
Semantics of Linear/modal Lambda Calculus
 Journal of Functional Programming
, 1998
"... In previous work the author has introduced a lambda calculus SLR with modal and linear types which serves as an extension of BellantoniCook's function algebra BC to higher types. It is a step towards a functional programming language in which all programs run in polynomial time. While this previous ..."
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Cited by 5 (2 self)
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In previous work the author has introduced a lambda calculus SLR with modal and linear types which serves as an extension of BellantoniCook's function algebra BC to higher types. It is a step towards a functional programming language in which all programs run in polynomial time. While this previous work was concerned with the syntactic metatheory of SLR in this paper we develop a semantics of SLR in terms of Chu spaces over a certain category of sheaves from which it follows that all expressible functions are indeed in PTIME. We notice a similarity between the Chu space interpretation and CPS translation which as we hope will have further applications in functional programming. 1 Introduction In [10] we have introduced a lambda calculus SLR which generalises the BellantoniCook characterisation of PTIME [4] to higherorder functions. The separation between normal and safe variables which is crucial to the BellantoniCook system has been achieved by way of an S 4 modality on types. ...
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.
Structural Recursion on Ordered Trees and Listbased Complex Objects Expressiveness and PTIME Restrictions
"... Abstract. XML query languages need to provide some mechanism to inspect and manipulate nodes at all levels of an input tree. In this paper we investigate the expressive power provided in this regard by structural recursion. We show that the combination of vertical recursion down a tree combined with ..."
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Cited by 2 (0 self)
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Abstract. XML query languages need to provide some mechanism to inspect and manipulate nodes at all levels of an input tree. In this paper we investigate the expressive power provided in this regard by structural recursion. We show that the combination of vertical recursion down a tree combined with horizontal recursion across a list of trees gives rise to a robust class of transformations: it captures the class of all primitive recursive queries. Since queries are expected to be computable in at most polynomial time for all practical purposes, we next identify a restriction of structural recursion that captures the polynomial time queries. Although this restriction is semantical in nature, and therefore undecidable, we provide an effective syntax. We also give corresponding results for listbased complex objects. 1
The computational SLR: a logic for reasoning about computational indistinguishability
"... Abstract. Computational indistinguishability is a notion in complexitytheoretic cryptography and is used to define many security criteria. However, in traditional cryptography, proving computational indistinguishability is usually informal and becomes errorprone when cryptographic constructions ar ..."
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Abstract. Computational indistinguishability is a notion in complexitytheoretic cryptography and is used to define many security criteria. However, in traditional cryptography, proving computational indistinguishability is usually informal and becomes errorprone when cryptographic constructions are complex. This paper presents a formal proof system based on an extension of Hofmann’s SLR language, which can capture probabilistic polynomialtime computations through typing and is sufficient for expressing cryptographic constructions. We in particular define rules that justify directly the computational indistinguishability between programs and prove that these rules are sound with respect to the settheoretic semantics, hence the standard definition of security. We also show that it is applicable in cryptography by verifying, in our proof system, Goldreich and Micali’s construction of pseudorandom generator, and the equivalence between nextbit unpredictability and pseudorandomness. 1
Type Systems for ResourceBounded Programming and Compilation: Case for Support
"... Introduction Recent decades have seen a gradual move from lowlevel programming languages such as assembler, Basic, and COBOL to highlevel languages such as C++, Haskell, Java, and ML. Highlevel languages provide abstraction layers and powerful programming idioms, making it much easier to impleme ..."
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Cited by 1 (0 self)
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Introduction Recent decades have seen a gradual move from lowlevel programming languages such as assembler, Basic, and COBOL to highlevel languages such as C++, Haskell, Java, and ML. Highlevel languages provide abstraction layers and powerful programming idioms, making it much easier to implement and analyse complicated algorithms. At the beginning, lack of efficiency prevented the use of these languages in reallife applications, but nowadays powerful hardware and efficient compilers make highlevel programming languages the method of choice for most applications. Nonetheless, efficiency concerns are still paramount when computing resources are limited, such as in embedded and realtime systems, or for applications to be run across the internet. For embedded and realtime systems, programmers use assembler code (or assemblerclose fragments of `C') to ensure a close control over resource consumption. For internet applications, highlevel languages such as Java ar