Daniel Leivant. Ramified recurrence and computational complexity I: Word recurrence and poly-time. In Peter Clote and Je#rey Remmel, editors, Feasible Mathematics II, Perspectives in Computer Science, pages 320--343. BirkhauserBoston, New York, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....directly carries over to PS, PTLS, and LS and the corresponding classes of type two functionals. Last but not least, let us mention the important activities in the program of so called implicit computational complexity and tiered formalisms in the sense of Bellantoni, Cook, and Leivant (cf. e.g. [4, 20, 22]) There questions regarding higher types have recently been of interest, see for example Leivant [21] Bellantoni, Niggl, Schwichtenberg [5] and Hofmann [17] For applicative theories based on safe induction, see Cantini [8] Recently and independently, Leivant has given a proof theoretic ....

Leivant, D. Ramified recurrence and computational complexity I: Word recurrence and poly-time. In Feasible Mathematics II, P. Clote and J. Remmel, Eds. Birkhauser, 1994, pp. 320--343.

....between the prior two in order to better explain the machine based model. Our type2 inflationary tiered loop programs (abbreviated, ITLP 2 ) system is a typed programming formalism inspired by type theoretic characterizations of PF due to Bellantoni and Cook [BC92] and Leivant and Marion [Lei95, LM93]. ITLP 2 is nonetheless very close to the polynomially clocked OTMs. ITLP 2 di#ers from BTLP 2 in that ITLP 2 types can incorporate certain complexity theoretic information. It also di#ers in that certain types and iteration bounds are inflationary in the sense that in the course of a computation ....

....to higher type polynomial time. A large part of the motivation of this paper is to provide appropriate settings for the E = BFF Theorem in order to extract full analogues of the Kapron Cook Theorem. Implicit Computational Complexity. Bellantoni and Cook s [BC92] and Leivant and Marion s [Lei95, LM93] type theoretic characterizations of PF inspired Bellantoni, Girard, Hofmann, Leivant, Marion, Niggl, Schwichtenberg [BNS00, Gir98, Lei94, LM02, Hof97, Hof99b] and others to investigate the use of type systems in restricted programming languages to characterize various complexity classes. Hofmann ....

D. Leivant, Ramified recurrence and computational complexity I: Word recurrence and poly-time, Feasible Mathematics II (P. Clote and J. Remmel, eds.), Birkhauser, 1995, pp. 320--343.

....in the applicative theory PT for the polynomial time computable functions. Apart from the world of bounded recursion schemas, bounded arithmetic and bounded applicative theories there is the world of so called tiered systems in the sense of Cook and Bellantoni (cf. e.g. 5] and Leivant (cf. e.g. [49, 51]) Crucial for this approach to characterizing complexities is a strictly predicative regime which distinguishes between di#erent uses of variables in induction and recursion schemas, thus severely restricting the definable or provably total functions in various unbounded formalisms. In our ....

Leivant, D. Ramified recurrence and computational complexity I: Word recurrence and poly-time. In Feasible Mathematics II, P. Clote and J. Remmel, Eds. Birkhauser, 1994, pp. 320--343. 48

....characterization of the class is possible. The answer is yes, as was proved in a fundamental paper [1] by Stephen Bellantoni and Stephen Cook. A complete account, together with related developments, is in Bellantoni s thesis [2] Here I shall follow the beautiful treatment by Daniel Leivant [3], with some changes in terminology derived from nonstandard analysis. Let W be the free word algebra with the constant ffl (denoting the empty string) and the unary functions 0 and 1. Thus W consists of all strings of zeroes and ones. We call certain strings standard, with the assumptions that ....

Daniel Leivant, Ramified recurrence and computational complexity I: Word recurrence and poly-time, in Peter Cole and Jeffrey Remmel, editors, Feasible Mathematics II, Perspectives in Computer Science, pages 320-343, Birkhauser-Boston,New York, 1994.

.... of Hartmanis and Stearns, On the Computational Complexity of Algorithms [60] Many textbooks cover this material, for example [75] The extension of this theory to higher order objects is an active field [99] and the study of feasible computation is another active area related to this article [12, 69, 72, 73, 74]. These topics are covered also in Schwichtenberg [13] and in the articles of Jones [70] Schwichtenberg [98] and Wainer [87] in this book. The work reported here is new and based largely on Constable and Crary [38] and Benzinger [14] as well as examples from Kreitz and Pientka [90] One ....

Daniel Leivant. Ramified recurrence and computational complexity I: Word recurrence and polynomial time. In P. Clote and J. Remmel, editors, Feasible Mathematics II, Perspectives in Computer Science. Birkhauser, 1995.

....applicative theory PT for the polynomial time computable functions. 41 Apart from the world of bounded recursion schemas, bounded arithmetic and bounded applicative theories there is the world of so called tiered systems in the sense of Cook and Bellantoni (cf. e.g. 4] and Leivant (cf. e.g. [38, 40]) Crucial for this approach to characterizing complexities is a strictly predicative regime which distinguishes between di#erent uses of variables in induction and recursion schemas, thus severely restricting the definable or provably total functions in various unbounded formalisms. In our ....

Leivant, D. Ramified recurrence and computational complexity I: Word recurrence and poly-time. In Feasible Mathematics II, P. Clote and J. Remmel, Eds. Birkhauser, 1994, pp. 320--343.

....between the prior two in order to better explain the machine based model. Our type2 inflationary tiered loop programs (abbreviated, ITLP 2 ) system is a typed programming formalism inspired by type theoretic characterizations of PF due to Bellantoni and Cook [BC92] and Leivant and Marion [Lei95, LM93]. ITLP 2 2 As evidence for this, the hard direction of the proof of the Kapron Cook Theorem is the translation of polynomial time bounded OTMs into equivalent BTLP2 procedures. In contrast, it is relatively simple to translate a BTLP2 procedure into an equivalent polynomialtime bounded OTM. 25 ....

....to higher type polynomial time. A large part of the motivation of this paper is to provide appropriate settings for the E = BFF Theorem in order to extract full analogues of the Kapron Cook Theorem. Implicit Computational Complexity. Bellantoni and Cook s [BC92] and Leivant and Marion s [Lei95, LM93] type theoretic characterizations of PF inspired Bellantoni, Girard, Goerdt, Hofmann, Leivant, Marion, Niggl, Schwichtenberg [BNS00, GSS92, Gir98, Goe92, Lei94, LM01, Hof97, Hof99b] and others to investigate the use of type systems in restrict programming languages to characterize various ....

D. Leivant, Ramified recurrence and computational complexity I: Word recurrence and poly-time, Feasible Mathematics II (P. Clote and J. Remmel, eds.), Birkhauser, 1995, pp. 320--343.

....J. Irwin # Bruce M. Kapron James S. Royer # 25 September 2000 Abstract We introduce a typed programming formalism, type 2 inflationary tiered loop programs or ITLP 2 , that characterizes the type 2 basic feasible functionals. ITLP 2 is based on Bellantoni and Cook s [BC92] and Leivant s [Lei95] type theoretic characterization of polynomial time and turns out to be closely related to Kapron and Cook s [KC91, KC96] machine based characterization of the type 2 basic feasible functionals. # Dept. of Elec. Eng. and Computer Science; Syracuse University; Syracuse, NY 13244 USA. Email: ....

....better explain this machine based model, we introduce our type 2 inflationary tiered loop programs (abbreviated ITLP 2 ) formalism in Section 8, with Section 7 laying the groundwork for its definition. ITLP 2 is a typed programming formalism inspired by Bellantoni and Cook s [BC92] and Leivant s [Lei95] type theoretic characterizations of PF. ITLP 2 is nonetheless very close to the polynomially clocked oracle Turing machine scheme. The price for this closeness is that certain types and iteration bounds are inflationary in the sense that in the course of a computation they can grow (inflate) with ....

[Article contains additional citation context not shown here]

D. Leivant, Ramified recurrence and computational complexity I: Word recurrence and poly-time, Feasible Mathematics II (P. Clote and J. Remmel, eds.), Birkhauser, 1995, pp. 320--343.

....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 polynomial time computable functions by higher type recursion, it seems that an additional principle is required. By introducing linearity ....

....0 = g R oe g h (s i n) h(s i n) R oe ghn) Now a single recursion in type can define a function of exponential growth: e : R s 1 (u V y :V (V y) satisfies je(m) n)j = 2 jmj jnj. Note that the function e can be assigned a ramified type under the scheme of Leivant [14], in which m is tier 1 and n is tier 0. What this shows is that another requirement, in addition to ramification of the recursion variable, is required to restrict higher type recursion to polytime computability. The culprit seems to lie in the nested, nonlinear use of the Research supported by: ....

D. Leivant and J.Y. Marion. Ramified Recurrence and Computational Complexity IV: Predicative Functionals and Poly-space. Information and Computation, to appear.

....system would be very desirable, but must at present await further research. Some preliminary work together with Radha Jagadeesan has shown that at least Bellantoni Cook s original system [ admits a compositional translation into LLL. The systems of tiered recursion studied by Leivant and Marion [15, 16, 17] also use restrictions of primitive recursion in order to achieve complexity effects. One difference is that the use of modality is replaced by the use of several copies of the base types ( tiers ) Another difference is that linearity and the ensuing recursion patterns with higher result type ....

Daniel Leivant and Jean-Yves Marion. Ramified Recurrence and Computational Complexity IV: Predicative functionals and Poly-space. Manuscript, 1997.

....system would be very desirable, but must at present await further research. Some preliminary work together with Radha Jagadeesan has shown that at least Bellantoni Cook s original system [ admits a compositional translation into LLL. The systems of tiered recursion studied by Leivant and Marion [15, 16, 17] also use restrictions of primitive recursion in order to achieve complexity effects. One difference is that the use of modality is replaced by the use of several copies of the base types ( tiers ) Another difference is that linearity and the ensuing recursion patterns with higher result type ....

Daniel Leivant and Jean-Yves Marion. Ramified Recurrence and Computational Complexity II: Substitution and Poly-space. In Jerzy Tiuryn and Leszek Pacholski, editors, Proc. CSL '94, Kazimierz, Poland, Springer LNCS, Vol. 933, pages 4486--500, 1995.

....Robert J. Irwin # Bruce M. Kapron James S. Royer # 21 July 1999 Abstract We introduce a typed programming formalism, type 2 inflationary tiered loop programs or ITLP 2 , that characterizes the type 2 basic feasible functionals. ITLP 2 is based on Bellantoni and Cook s [BC92] and Leivant s [Lei95] type theoretic characterization of polynomial time and turns out to be closely related to Kapron and Cook s [KC91, KC96] machine based characterization of the type 2 basic feasible functionals. # Dept. of Elec. Eng. and Computer Science; Syracuse University; Syracuse, NY 13244 USA. Email: ....

....better explain this machine based model, we introduce our type 2 inflationary tiered loop programs (abbreviated ITLP 2 ) formalism in Section 8, with Section 7 laying the groundwork for its definition. ITLP 2 is a typed programming formalism inspired by Bellantoni and Cook s [BC92] and Leivant s [Lei95] type theoretic characterizations of PF. ITLP 2 is nonetheless very close to the polynomially clocked oracle Turing machine scheme. The price for this closeness is that certain types and iteration bounds are inflationary in the sense that in the course of a computation they can grow (inflate) with ....

[Article contains additional citation context not shown here]

D. Leivant, Ramified recurrence and computational complexity I: Word recurrence and poly-time, Feasible Mathematics II (P. Clote and J. Remmel, eds.), Birkhauser, 1995, pp. 320--343.

....while still obtaining polytime provability . However, the actual system defined there was inadequate as a working logic e.g. it was awkwardly defined and not closed under modus ponens. 2 For background in ramified recursion, see the systems of [3] 5] 4] 16] 17] 18] and Leivant [23] [24] and further references cited there. Although it is carried out more in the tradition of ramified recursion than linear logic, this work has obvious and important connections to linear logic. In linear logics, one introduces special operators to control and track the usage of formulas. See ....

D. Leivant and J.Y. Marion, "Ramified Recurrence and Computational Complexity IV: Predicative Functionals and Poly-space", in Information and Computation, to appear.

....gratefully acknowledged. Laboratory for Foundations of Computer Science, University of Edinburgh. 1 At the same time, researchers in recursion theory have developed systems in which computational complexity is controlled by type information rather than by explicit resource bounds [29] 3] [23], 17] 4] Each of the various types #, #, #, in a ramified system is a di#erent intension for the same extensional values. Typically, one may recurse on a value that is comprehended through a type # reference, while one may only access a few low order bits from a value referred to by a ....

....while still obtaining polytime provability . However, the actual system defined there was inadequate as a working logic e.g. it was awkwardly defined and not closed under modus ponens. 2 For background in ramified recursion, see the systems of [3] 5] 4] 16] 17] 18] and Leivant [23], 24] and further references cited there. Although it is carried out more in the tradition of ramified recursion than linear logic, this work has obvious and important connections to linear logic. In linear logics, one introduces special operators to control and track the usage of formulas. See ....

D. Leivant, "Ramified Recurrence and Computational Complexity I: Word Recurrence and Poly-time", in Feasible Mathematics II, P. Clote and J. Remmel, eds., p. 320-343, series Perspectives in Computer Science, Birkhauser, 1994.

....as unary coded numbers if we don t care about their elements. In other words, as unary numbers can be embedded into many commonly used data structures, counterexamples with unary numbers also work with almost any of the interesting data structures. Approaches based on predicative recursion [1, 8] argue that this is due to the way double is called: the previous value of an outer recursion (i.e. exp(x) is used as recursive argument of an inner recursion (i.e. is used as argument for double) However, although all functions computable in polynomial time can be defined in such a restricted ....

D. Leivant. Ramified recurrence and computational complexity I: Word recurrence and poly--time. In P. Clote and J. Remmel, editors, Feasible Mathematics II, pages 320--343. Birkhauser, Boston, 1995.

.... Intuitionistic Polyspace Affine Logic (IPAL) see [Asp98] for the simplifying effect of adding unrestricted weakening and the necessity of adopting an intuitionistic formalism) There have already been a number of machine independent characterizations of polyspace, for instance, Bus86] LM94b] [LM98], Lei99] and characterizations in finite model theory (see [LM94b] for a survey) We emphasize, however, that our characterization inherits from LLL its purely logical nature, particularly (i) and (ii) above, and thus is distinguished from others. Splitness and Polyspace Functions. A natural ....

....the reduction rules. Section 3 shows that every term (representing a proof) of strictly lazy type can be normalized within a polynomial size bound if we perform computation by slices. Section 4 gives an encoding of polytime alternating Turing machine, borrowing the essential idea from [LM94b] and [LM98], and thus shows that every polyspace function is representable in IPAL. Section 5 concludes the paper. Due to lack of space, most lemmas and theorems are stated without proofs. Appendix A collects proofs for some of them. Appendix B includes several figures and derivations, which may help the ....

[Article contains additional citation context not shown here]

D. Leivant and J.-Y. Marion. Ramified recurrence and computational complexity IV: predicative functionals and polyspace. Information and Computation, 1998.

.... is called Intuitionistic Polyspace Affine Logic (IPAL) see [Asp98] for the simplifying effect of adding unrestricted weakening and the necessity of adopting an intuitionistic formalism) There have already been a number of machine independent characterizations of polyspace, for instance, Bus86] [LM94b], LM98] Lei99] and characterizations in finite model theory (see [LM94b] for a survey) We emphasize, however, that our characterization inherits from LLL its purely logical nature, particularly (i) and (ii) above, and thus is distinguished from others. Splitness and Polyspace Functions. A ....

.... simplifying effect of adding unrestricted weakening and the necessity of adopting an intuitionistic formalism) There have already been a number of machine independent characterizations of polyspace, for instance, Bus86] LM94b] LM98] Lei99] and characterizations in finite model theory (see [LM94b] for a survey) We emphasize, however, that our characterization inherits from LLL its purely logical nature, particularly (i) and (ii) above, and thus is distinguished from others. Splitness and Polyspace Functions. A natural way of delimiting the expressive power of a computational system is ....

[Article contains additional citation context not shown here]

D. Leivant and J.-Y. Marion. Ramified recurrence and computational complexity II: substitution and polyspace. In J. Tiuryn and L. Pacholsky, editors, Computer Science Logic. Springer, 1994.

....to forbid iterating f(x; x) to limit the representable functions to sub exponential ones. While there are several approaches to achieve this by directly putting some constraint on the recursion schema itself (such as bounded recursion of [Cob65] and safe recursion (ramified recurrence) of [BC92] [LM94a], etc. see also [Hof98] BNS99] ILAL (LLL) 1 y x f(x,y) y x h(x,y) c) d) e) b) a) f) A 1 ; An B A 1 ; A n B (g) A; A A A; A A A A Figure 1: Functions and Iterations takes a purely logical approach, namely, achieves this through a control of ....

D. Leivant and J.-Y. Marion. Ramified recurrence and computational complexity I: Word recurrence and poly-time. In P. Clote and J. Remmel, editors, Feasible Mathematics II, pages 320 -- 343. Birkhauser, 1994.

....N j with j i. 3 Let us confess that the restrictions on For loops are a bit ugly. However, they serve a valuable role in keeping our type system for ITLP simple. In place For loops, we could have built our system around a variant of limited recursion on notation as was done, for example, in [Lei95, Hof99, BNS00]. The di#culty with this is that the restrictions on For loops would reappear as complications in the type system. Since an analysis of feasible higher type recursion was not among our goals in this work, we are happier with our simpler type system and inelegant For loop. 21 April 2000 ....

D. Leivant, Ramified recurrence and computational complexity I: Word recurrence and poly-time, Feasible Mathematics II (P. Clote and J. Remmel, eds.), Birkhauser, 1995, pp. 320--343.

....arguments of functions, called normal and safe arguments. Recursion is only allowed over normal arguments, whereas the recursively computed values must be inserted in a safe position, and function composition is defined accordingly. Related tiering notions also appeared elsewhere, e.g. in Leivant [5] or Simmons [10] Clote [3] gives a short review of some recent results and rises the problem of relating these concepts to the Grzegorczyk hierarchy (Problem 3.102) This paper proposes an answer to that question. Many characterizations of the Grzegorczyk hierarchy are based on controlling the ....

Leivant, D. Ramified recurrence and computational complexity I: Word recurrence and poly-time. In Feasible Mathemathics II, P. Clote and J. Remmel, Eds. Birkhauser-Boston, New York, 1994.

.... restrictions of the terms in Godel s T [1, 2, 5] Consider for example the following functions on unary coded natural numbers 1 , which lead to exponential growth: double(0) 0 double(Sx) SSdouble(x) exp(0) S0 exp(Sx) double(exp(x) Approaches based on predicative recursion [1, 8] argue that this is due to the way double is called: the previous value of an outer recursion (i.e. exp(x) is used as recursive argument of an inner recursion (i.e. is used as argument for double) 1 Of course, no one actually uses unary coded numbers when dealing with questions of complexity. ....

Daniel Leivant. Ramified recurrence and computational complexity I: Word recurrence and poly--time. In P. Clote and J. Remmel, editors, Feasible Mathematics II, pages 320--343. Birkhauser, Boston, 1995.

....concepts (Bellantoni, Niggl, Schwichtenberg, 1998; Hofmann, 1999) Typed lambda calculi. A series of papers by Leivant (some with Marion) characterize complexity clases in terms of simply typed lambda calculi with recurrence constants, including (Leivant, 1989; Leivant and Marion, 1993; Leivant and Marion, 1999). In particular they study calculi extended by functions and operations on an algebra of words over f0; 1g, and characterize ptime, pspace and several other classes. The earlier works had rather complex formulations mostly related to ramified recurrence, but (Leivant, 1999) characterizes these ....

....Marion, 1993; Leivant and Marion, 1999) In particular they study calculi extended by functions and operations on an algebra of words over f0; 1g, and characterize ptime, pspace and several other classes. The earlier works had rather complex formulations mostly related to ramified recurrence, but (Leivant, 1999) characterizes these classes by much simpler syntactic restrictions on the form of program control. Hillebrand, Kanellakis, 1996) studies expressivity of lambda calculus variants that abstract an ML subset, characterizing the regular sets, and k Exptime and The Expressive Power of Higher order ....

Leivant, D. and Marion, J.-Y. Ramified recurrence and computational complexity IV: predicative functionals and poly-space. Information and Computation (1999).

No context found.

Daniel Leivant. Ramified recurrence and computational complexity I: Word recurrence and poly-time. In Peter Clote and Je#rey Remmel, editors, Feasible Mathematics II, Perspectives in Computer Science, pages 320--343. BirkhauserBoston, New York, 1994.

No context found.

Daniel Leivant and Jean-Yves Marion. Ramified Recurrence and Computational Complexity IV: Predicative functionals and Poly-space. Manuscript, 1997.

No context found.

D. Leivant, Ramified recurrence and computational complexity I: Word recurrence and polytime, in: P. Clote and J. Remmel, eds., Feasible Mathematics II , Birkhauser, Boston, 1995, 320--343.

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