We consider a scenario where (functional) programs in pre-compiled form are exchanged among untrusted parties. Our contribution is a system of annotations for the code that can be verified at load time so as to ensure bounds on the time and space resources required for its execution, as well as to guarantee the usual integrity properties.

We demonstrate that the class of rst order functional programs over lists which terminate by multiset path ordering and admit a polynomial quasi-interpretation, is exactly the class of function computable in polynomial time. The interest of this result lies (i) on the simplicity of the conditions on programs to certify their complexity, (ii) on the fact that an important class of natural programs is captured, (iii) and on potential applications on program optimizations. 1 Introduction This paper is part of a general investigation on the implicit complexity of a specication. To illustrate what we mean, we write below the recursive rules that computes the longest common subsequences of two words. More precisely, given two strings u = u1 um and v = v1 vn of f0; 1g , a common subsequence of length k is dened by two sequences of indices i 1 < < i k and j1 < < jk satisfying u i q = v j q . lcs(; y) ! 0 lcs(x; ) ! 0 lcs(i(x); i(y)) ! lcs(x; y) + 1 lcs(i(...

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

Abstract. The main results of this paper are recursion-theoretic characterizations of two parallel complexity classes: the functions computable by uniform bounded fan-in circuit families of log and polylog depth (or equivalently, the functions bitwise computable by alternating Turing machines in log and polylog time). The present characterizations avoid the complex base functions, function constructors, and a priori size or depth bounds typical of previous work on these classes. This simplicity is achieved by extending the \tiered recursion &quot; techniques of Leivant and Bellantoni&Cook. Key words. Circuit complexity � subrecursion. Subject classi cations. 68Q15, 03D20, 94C99. 1.

. This paper introduces a simply-typed lambda calculus with both modal and linear function types. Through the use of subtyping extra term formers associated with modality and linearity are avoided. We study the basic metatheory of this system including existence and inference of principal types. The system serves as a platform for certain higher-order generalisations of Bellantoni-Cook's function algebra capturing polynomial time using a separation of the variables into "safe" and "normal" ones. The distinction between and the syntactic restrictions involved with the safe and normal variables in the Bellantoni-Cook framework are captured by the modal function space and the associated typing rules. The linear function spaces on the other hand are used to enable a certain form of primitive recursion with functional result type which is conservative over polynomial time. The proofs associated with these applications are based on an interpretation of the lambda calculus in a category-theor...

This paper presents a new query algebra based on fold iterations that facilitates database implementation. An algebraic normalization algorithm is introduced that reduces any program expressed in this algebra to a canonical form that generates no intermediate data structures and has no more nested iterations than the initial program. Given any inductive data type, our system can automatically synthesize the definition of the fold operator that traverses instances of this type, and, more importantly, it can produce the necessary transformations for optimizing expressions involving this fold operator. Database implementation in our framework is controlled by userdefined mappings from abstract types to physical structures. The optimizer uses this information to translate abstract programs and queries into concrete algorithms that conform to the type transformation. Database query optimization can be viewed as a search over the reduced space of all canonical forms which are equivalent to t...

We present a functional framework for database query languages, which is analogous to the conventional logical framework of first-order and fixpoint formulas over finite structures. We use atomic constants of order 0, equality among these constants, variables, application, lambda abstraction, and let abstraction; all typed using fixed order ( 5) functionalities. In this framework, proposed in [21] for arbitrary order functionalities, queries and databases are both typed lambda terms, evaluation is by reduction, and the main programming technique is list iteration. We define two families of languages: TLI = i or simply-typed list iteration of order i +3 with equality, and MLI = i or ML-typed list iteration of order i+3 with equality; we use i+3 since our list representation of databases requires at least order 3. We show that: FO-queries ` TLI = 0 ` MLI = 0 ` LOGSPACE-queries ` TLI = 1 = MLI = 1 = PTIME-queries ` TLI = 2 , where equality is no longer a primitive in TLI = 2 . We also show that ML type inference, restricted to fixed order, is polynomial in the size of the program typed. Since programming by using low order functionalities and type inference is common in functional languages, our results indicate that such programs suffice for expressing efficient computations and that their ML-types can be efficiently inferred.

Abstract. Soft linear logic ([Lafont02]) is a subsystem of linear logic characterizing the class PTIME. We introduce Soft lambda-calculus as a calculus typable in the intuitionistic and affine variant of this logic. We prove that the (untyped) terms of this calculus are reducible in polynomial time. We then extend the type system of Soft logic with recursive types. This allows us to consider non-standard types for representing lists. Using these datatypes we examine the concrete expressiveness of Soft lambda-calculus with the example of the insertion sort algorithm. 1

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...

We use the category of presheaves over PTIME-functions in order to show that Cook and Urquhart's higher-order function algebra PV ! defines exactly the PTIME-functions. As a byproduct we obtain a syntax-free generalisation of PTIME-computability 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 category-theoretic approach to a new higher-order extension of Bellantoni-Cook's system BC of safe recursion. 1 Introduction Cook and Urquhart's system PV ! [3] is a simply-typed lambda calculus providing constants to denote natural numbers and an operator for bounded recursion on notation like in Cobham's characterisation of polynomial-time computability. 1 Although functionals of arbitrary type can be defined in this system one can show that thei...