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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 ..."
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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...
A Functional Language for Logarithmic Space
 In APLAS
, 2004
"... More than being just a tool for expressing algorithms, a welldesigned programming language allows the user to express her ideas efficiently. The design choices however effect the efficiency of the algorithms written in the languages. It is therefore of importance to understand how such choices effe ..."
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More than being just a tool for expressing algorithms, a welldesigned programming language allows the user to express her ideas efficiently. The design choices however effect the efficiency of the algorithms written in the languages. It is therefore of importance to understand how such choices effect the expressibility of programming languages. The paper pursues the very low complexity programs by presenting a firstorder function algebra BC # that captures exactly LF, the functions computable in logarithmic space. This gives insights into the expressiveness of recursion. Moreover, it can be useful for the automatic analysis of programs' resource usage and the separation of complexity classes. The important technical features of BC # are (1) a separation of variables into safe and normal variables where recursion can only be done over the latter; (2) linearity of the recursive call; and (3) recursion with a variable step length (courseofvalue recursion). Unlike formulations LF via Turin Machines, BC # makes no references to outside resource measures, e.g., the size of the memory used. This appears to be the first such characterization of LFcomputable functions (not just predicates). The proof that all BC #programs can be evaluated in LF is of separate interest to programmers: it trades space for time and evaluates recursion with at most one recursive call without a call stack.
HigherOrder Linear Ramified Recurrence
, 2004
"... HigherOrder Linear Ramified Recurrence (HOLRR) is a linear (affine) calculus  every variable occurs at most once  extended with a recursive scheme on free algebras. Two simple conditions on type derivations enforce both polytime completeness and a strong notion of polytime soundness on typ ..."
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HigherOrder Linear Ramified Recurrence (HOLRR) is a linear (affine) calculus  every variable occurs at most once  extended with a recursive scheme on free algebras. Two simple conditions on type derivations enforce both polytime completeness and a strong notion of polytime soundness on typeable terms. Completeness for PTIME holds by embedding Leivant's ramified recurrence on words into HOLRR. Soundness is established at all types  and not only for first order terms. Type connectives are limited to tensor and linear implication.
Linear Logical Characterization of Polyspace Functions (Extended Abstract)
 Presented at the Workshop on Implicit Computational Complexity ICC’00, Santa Barbara, 2000  Unpublished
, 2000
"... ) Kazushige Terui 3 Abstract Light Linear Logic (LLL) of [Gir95] characterizes the polytime functions through a careful handling of structural inference rules of logic. Based on this purely logical approach, we give a characterization of the polyspace functions. Our logical system is an extension ..."
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) Kazushige Terui 3 Abstract Light Linear Logic (LLL) of [Gir95] characterizes the polytime functions through a careful handling of structural inference rules of logic. Based on this purely logical approach, we give a characterization of the polyspace functions. Our logical system is an extension of Intuitionistic Light Affine Logic (ILAL) of [Asp98], a variant of LLL with full (unrestricted) weakening. We introduce the notion of split terms and enrich ILAL by allowing a ! box to be formed from a split term. The resulting system, called Intuitionistic Polyspace Affine Logic, precisely characterizes the polyspace functions. 1 Introduction Linear Logical Approach to Computational Complexity. Over the past few decades, a considerable number of studies have been made on logical characterizations of computational complexity classes, aiming at giving an inherent (machineindependent) account on computational complexity from the logical point of view. Among those, Light Linear Logic (LL...
Weak Affine Light Typing is complete with respect to Safe Recursion on Notation
, 2008
"... Weak affine light typing (WALT) assigns light affine linear formulae as types to a subset ofλterms of System F. WALT is polytime sound: if aλterm M has type in WALT, M can be evaluated with a polynomial cost in the dimension of the derivation that gives it a type. The evaluation proceeds under an ..."
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Weak affine light typing (WALT) assigns light affine linear formulae as types to a subset ofλterms of System F. WALT is polytime sound: if aλterm M has type in WALT, M can be evaluated with a polynomial cost in the dimension of the derivation that gives it a type. The evaluation proceeds under any strategy of a rewriting relation which is a mix of both callbyname and callbyvalueβreductions. WALT weakens, namely generalizes, the notion of “stratification of deductions”, common to some Light Systems — those logical systems, derived from Linear logic, to characterize the set of Polynomial functions —. A weaker stratification allows to define a compositional embedding of Safe recursion on notation (SRN) into WALT. It turns out that the expressivity of WALT is strictly stronger than the one of the known Light Systems. The embedding passes through the representation of a subsystem of SRN. It is obtained by restricting the composition scheme of SRN to one that can only use its safe variables linearly. On one side, this suggests that SRN, in fact, can be redefined in terms of more primitive constructs. On the other, the embedding of SRN into WALT enjoys the two following remarkable aspects. Every datatype, required by the embedding, is represented from scratch, showing the strong structural prooftheoretical roots of WALT. Moreover, the embedding highlights a stratification structure of the normal and safe arguments, normally hidden inside
Implicit complexity in recursive analysis
 TENTH INTERNATIONAL WORKSHOP ON LOGIC AND COMPUTATIONAL COMPLEXITY LCC'09
, 2009
"... Recursive analysis is a model of analog computation which is based on type 2 Turing machines. Various classes of functions computable in recursive analysis have recently been characterized in a machine independent and algebraical context. In particular nice connections between the class of computabl ..."
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Recursive analysis is a model of analog computation which is based on type 2 Turing machines. Various classes of functions computable in recursive analysis have recently been characterized in a machine independent and algebraical context. In particular nice connections between the class of computable functions (and some of its sub and supclasses) over the reals and algebraically defined (sub and sup) classes of Rrecursive functions à la Moore have been obtained. We provide in this paper a framework that allows to dive into complexity for functions over the reals. It indeed relates classical computability and complexity classes with the corresponding classes in recursive analysis. This framework opens the field of implicit complexity of functions over the reals. While our setting provides a new reading of some of the existing characterizations, it also provides new results: inspired by Bellantoni and Cook’s characterization of polynomial time computable functions, we provide the first algebraic characterization of polynomial time computable functions over the reals.
Algebraic Characterizations of ComplexityTheoretic Classes of Real Functions, Laboratoire d’informatique de l’école polytechnique
 Alexandria University  Alexandria University
, 2009
"... Recursive analysis is the most classical approach to model and discuss computations over the real numbers. Recently, it has been shown that computability classes of functions in the senseof recursive analysis can bedefined (or characterized) in an algebraic machine independent way, without resorting ..."
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Recursive analysis is the most classical approach to model and discuss computations over the real numbers. Recently, it has been shown that computability classes of functions in the senseof recursive analysis can bedefined (or characterized) in an algebraic machine independent way, without resorting to Turing machines. In particular nice connections between the class of computable functions (and some of its sub and supclasses) over the reals and algebraically defined (sub and sup) classes of Rrecursive functions à la Moore 96 have been obtained. However, until now, this has been done only at the computability level, and not at the complexity level. In this paper we provide a framework that allows us to dive into the complexity level of real functions. In particular we provide the first algebraic characterization of polynomialtime computable functions over the reals. This framework opens the field of implicit complexity of analog functions, and also provides a new reading of some of the existing characterizations at the computability level. 1
Safe Recursion Over an Arbitrary Structure: PAR, PH and DPH
"... Considering the Blum, Shub, and Smale computational model for real numbers, extended by Poizat to general structures, classical complexity can be considered as the restriction to nite structures of a more general notion of computability and complexity working over arbitrary structures. In a previous ..."
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Considering the Blum, Shub, and Smale computational model for real numbers, extended by Poizat to general structures, classical complexity can be considered as the restriction to nite structures of a more general notion of computability and complexity working over arbitrary structures. In a previous paper, we showed that the machineindependent characterization of Bellantoni and Cook of sequential polynomial time for classical complexity is actually the restriction to nite structures of a characterization of sequential polynomial time over arbitrary structures. In this paper, we prove that the same phenomenon happens for several other complexity classes: over arbitrary structures, parallel polynomial time corresponds to safe recursion with substitutions, and the polynomial hierarchy corresponds to safe recursion with predicative minimization. Our results yield machineindependent characterizations of several complexity classes subsuming previous ones when restricted to nite structures. 1 Partially supported by City University of Hong Kong SRG grant 7001290. Preprint submitted to Elsevier Preprint 1 April 2005