A Formalization of Polytime Functions

We present a deep embedding of Bellantoni and Cook's syntactic characterization of polytime functions. We prove formally that it is correct and complete with respect to the original characterization by Cobham that required a bound to be proved manually. Compared to the paper proof by Bellantoni and Cook, we have been careful in making our proof fully contructive so that we obtain more precise bounding polynomials and more efficient translations between the two characterizations. Another difference is that we consider functions on bitstrings instead of functions on positive integers. This latter change is motivated by the application of our formalization in the context of formal security proofs in cryptography. Based on our core formalization, we have started developing a library of polytime functions that can be reused to build more complex ones.Comment: 13 page

An Embedding of the BSS Model of Computation in Light Affine Lambda-Calculus

This paper brings together two lines of research: implicit characterization of complexity classes by Linear Logic (LL) on the one hand, and computation over an arbitrary ring in the Blum-Shub-Smale (BSS) model on the other. Given a fixed ring structure K we define an extension of Terui's light affine lambda-calculus typed in LAL (Light Affine Logic) with a basic type for K. We show that this calculus captures the polynomial time function class FP(K): every typed term can be evaluated in polynomial time and conversely every polynomial time BSS machine over K can be simulated in this calculus.Comment: 11 pages. A preliminary version appeared as Research Report IAC CNR Roma, N.57 (11/2004), november 200

Polynomial Path Orders: A Maximal Model

This paper is concerned with the automated complexity analysis of term rewrite systems (TRSs for short) and the ramification of these in implicit computational complexity theory (ICC for short). We introduce a novel path order with multiset status, the polynomial path order POP*. Essentially relying on the principle of predicative recursion as proposed by Bellantoni and Cook, its distinct feature is the tight control of resources on compatible TRSs: The (innermost) runtime complexity of compatible TRSs is polynomially bounded. We have implemented the technique, as underpinned by our experimental evidence our approach to the automated runtime complexity analysis is not only feasible, but compared to existing methods incredibly fast. As an application in the context of ICC we provide an order-theoretic characterisation of the polytime computable functions. To be precise, the polytime computable functions are exactly the functions computable by an orthogonal constructor TRS compatible with POP*

Algebraic Characterizations of Complexity-Theoretic Classes of Real Functions

Recursive analysis is the most classical approach to model and discuss computations over the reals. It is usually presented using Type 2 or higher order Turing machines. Recently, it has been shown that computability classes of functions computable in recursive analysis can also be defined (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 sup-classes) over the reals and algebraically defined (sub- and sup-) classes of $\R$-recursive 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 functions over the reals. In particular we provide the first algebraic characterization of polynomial time computable functions over the reals. This framework opens the field of implicit complexity of functions over the reals, and also provide a new reading of some of the existing characterizations at the computability level

Polynomial time operations in explicit mathematics

In this paper we study (self-)applicative theories of operations and binary words in the context of polynomial time computability. We propose a first order theory PTO which allows full self-application and whose provably total functions on = {0, 1}* are exactly the polynomial time computable functions. Our treatment of PTO is proof-theoretic and very much in the spirit of reductive proof theor