Stephen A. Cook and Bruce M. Kapron Department of Computer Science University of Toronto Toronto, Canada M5S 1A4 1 Introduction Functionals are functions which take natural numbers and other functionals as arguments and return natural numbers as values. The class of "feasible" functionals of finite type was introduced in [6] via the typed lambda calculus, and used to interpret certain formal systems of arithmetic: systems capturing the notion of "feasibly constructive proof" (we equate feasibility with polynomial time computability) . Here we name the functionals of [6] the basic feasible functionals and justify the designation by presenting results which include two programming style characterizations of the class. We also give examples of both feasible and infeasible functionals, and argue that the notion plays a natural role in complexity theory. Type 2 functionals take numbers and ordinary numerical functions as arguments. When these argument functions are 0-1 valued (i.e. sets) ...

We investigate queries in the presence of external functions with arbitrary inputs and outputs (atomic values, sets, nested sets etc). We propose a new notion of domain independence for queries with external functions which, in contrast to previous work, can also be applied to query languages with fixpoints or other kinds of iterators. Next, we define two new notions of computable queries with external functions, and prove that they are equivalent, under the assumption that the external functions are total. Thus, our definition of computable queries with external functions is robust. Finally, based on the equivalence result, we give examples of complete query languages with external functions. A byproduct of the equivalence result is the fact that Relational Machines are complete for complex objects: it was known that they are not complete over flat relations. 1 Introduction Database functionalities are important both for practical and for theoretical purposes. E.g. the system O 2 of ...

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...

A type 1 function is a total mapping from N to N. We will denote the set of all such functions by N N. A type 2 functional is a total mapping from ( N N) k \Theta N l to N, for some k; l. More specifically, we will call a mapping of this sort a functional with rank (k; l). For type 1 functions, there is a well established notion of computational feasibility. Namely a function is feasible if it is computable in polynomial time on a Turing machine. More specifically, a function f is poly time if there is a TM M and a polynomial p such that for all x, M with input x computes f(x) and runs in time p(n), where n = jxj, and for x 2 N, jxj denotes the length of the binary notation of x, that is dlog(x + 1)e. This notion of f...

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Abstract. In [12] we defined a class of functions called Type-2 Time Bounds (henceforth T2TB) for clocking the Oracle Turing Machine (OTM) in order to capture the long missing notion of complexity classes at type-2. In the present paper we further advance the type-2 complexity theory under our notion of type-2 complexity classes. We have learned that the theory is highly sensitive to how the oracle answers are handled. We present a reasonable alternative called unit cost model, and examine how this model shapes the outlook of the type-2 complexity theory. Under the unit cost model we prove two theorems opposite to the classical union theorem and gap theorem. We also investigate some properties of T2TB including a very useful theorem stating that there is an effective operator to convert any β ∈ T2TB into an equivalent one that is locking-detectable. The existence of such operator allows us to simplify many proofs without loss of generality. 1 1

There are now a number of things called "higher-type complexity classes." The most promenade of these is the class of basic feasible functionals [CU93, CK90], a fairly conservative higher-type analogue the (type-1) polynomial-time computable functions. There is however currently no satisfactory general notion of what a higher-type complexity class should be. In this paper we propose one such notion for type-2 functionals and begin an investigation of its properties. The most striking di#erence between our type-2 complexity classes and their type-1 counterparts is that, because of topological constrains, the type-2 classes have a much more ridged structure. Example: It follows from McCreight and Meyer's Union Theorem [MM69] that the (type-1) polynomial-time computable functions form a complexity class (in the strict sense of Definition 1 below). The analogous result fails for the class of type-2 basic feasible functionals. 1.

Abstract. We study polynomial time complexity of type 2 functionals. For that purpose, we introduce a first order functional stream language. We give criteria, named well-founded, on such programs relying on second order interpretation that characterize two variants of type 2 polynomial complexity including the Basic Feasible Functions (BFF). These characterizations provide a new insight on the complexity of stream programs. Finally, we adapt these results to functions over the reals, a particular case of type 2 functions, and we provide a characterization of polynomial time complexity in Recursive Analysis. 1