Learning from Different Teachers

A model for computation over an arbitrary (ordered) ring R is presented. In this general setting, universal machines, partial recursive functions, and NP-complete problems are obtained. While the theory reflects of classical over Z (e.g. the computable functions are the recursive functions), it also reflects the special mathematical character of the underlying ring R (e.g. complements of Julia sets provide natural examples of recursively enumerable undecidable sets over the reals) and provides a natural setting for studying foundational issues concerning algorithms in numerical analysis.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Recursive function theory studies functions that can be evaluated by an algorithm based on some computing model such as a Turing machine or computer program. These functions are called partial recursive (p.r.) and are defined on subsets of N, where N = {0, 1,... }, with values in N. (A good introduction to the theory may be found in [2].) This paper explains the Parameter Theorem, illustrates some unusual computer programs, and explains their connection to recursive function theory. Basing our computing model on a modern high level language, we can construct programs whose existence is known through more formal means. Our functions will operate on strings rather than numbers, however. There is no loss in generality since strings can be viewed as members of N, and vice-versa. From a practical aspect, this approach is somewhat more desirable since conventional programming languages often allow making use of long strings, but place more restrictive limits on the size of integers. We first pose a problem: Does there exist a program (written in some conventional programming language) which halts if and only if its input is identical to the program's text (i.e., source code)? For the remainder of this note X is the set of ASCII characters 'a,' 'b, +,' space, line feed, and so forth, needed to write programs in most programming languages. We denote by X * the set of all strings (finite sequences) over X. Let P1 be the set of all one variable partial recursive functions on X *. A function I belongs to P1 if and only if there is a subset of S of X* and some algorithm which computes 't by associating with each element of S a well-defined element of X *. Similarly, 't belongs to P2 if and only if there is a set S c X * X X *, and an algorithm for 't which associates with each element of S a well-defined element of X *. A partial recursive function I in P1 (or in P2) is recursive if the algorithm is defined on all of X * (or X * X X *). This is equivalent to saying that the algorithm which evaluates I always halts. Our computing model will be based on Adat; its high-level string-handling properties simplify later examples. For readability, however, we will express algorithms in a more familiar Pascal-like style. Readers familiar with either programming language should find forthcoming algorithms sufficiently clear. A formal presentation in Ada is given in the Appendix. A string of ASCII characters, such as in FIGURE 1, represents a function in our language which takes strings to strings. Of course, strings of ASCII characters such as The quick brown fox and %*!! represent nothing at all in our programming language. Let A1 be the set of all strings in X * representing functions in our language of one input STRING which outputs a STRING. Similarly let A2 be the set of all strings in

We discuss a scheme for defining and reasoning about partial recursive functions within a classical two-valued logic in which all terms denote. We show how a total extension of the partial function introduced by a recursive declaration may be axiomatized within a classical logic, and illustrate by an example the kind of reasoning that our scheme supports. By presenting a naive set-theoretic semantics, we show that the system we propose is logically consistent. Our work is motivated largely by the pragmatic issues arising from mechanical theorem proving – we discuss some of the practical benefits and limitations of our scheme for mechanical verification of software and hardware systems.

Minimization and NP multifunctions

We highlight the limitations of formal methods by exhibiting two results in recursive function theory: that there is no effective means of finding a program that satisfies a given formal specification; or checking that a program meets a specification. We exhibit a simple MAS which has all the power of a Turing machine. We argue that any pure design methodology will face insurmountable difficulties in today's open and complex MAS. We recommend instead a methodology based on experimental method - scientific foundations for MAS construction and control.

Our goal is to define a type of partial recursive functions in constructive type theory. In a series of previous articles, we studied two different formulations of partial functions and general recursion. We could obtain a type only by extending the theory with either an impredicative universe or with coinductive definitions. Here we present a new type constructor that eludes such entities of dubious constructive credentials. We start by showing how to break down a recursive function definition into three components: the first component generates the arguments of the recursive calls, the second evaluates them, and the last computes the output from the results of the recursive calls. We use this dissection as the basis for the introduction rule of the new type constructor. Every partial recursive function is associated with an inductive domain predicate; evaluation of the function requires a proof that the input values satisfy the predicate. We give a constructive justification for the new construct by interpreting it into the base type theory. This shows that the extended theory is consistent and constructive.

There are essentially two results described in this paper. First, it is shown that for any random access machine (RAM) time-constructable function $T(n) \geqq n$, there are languages L such that L can be recognized in time $O(T(n))$ by a RAM (using the unit cost measure), but L cannot be recognized by any deterministic multitape Turing machine in time $O(T(n))$. Secondly, a family of random access stored program machines (RASP’S) are considered. For these RASP’S it is shown that there is an arbitrarily complex (infinitely often) partial recursive function $f(n)$. which has only 0–1 values (whenever defined) such that $f(n)$ can be computed in time $F(n)$ by some RASP, but cannot be computed in time $(1 - \varepsilon )F(n)$, for any $\varepsilon > 0$, by any RASP in this family.

The modern theory of computability is based on the works of Church, Markov and Turing who, starting from quite different models of computation, arrived at the same class of computable functions. The purpose of this paper is the show how the main results of the Church-Markov-Turing theory of computable functions may quickly be derived and understood without recourse to the largely irrelevant theories of recursive functions, Markov algorithms, or Turing machines. We do this by ignoring the problem of what constitutes a computable function and concentrating on the central feature of the Church-Markov-Turing theory: that the set of computable partial functions can be effectively enumerated. In this manner we are led directly to the heart of the theory of computability without having to fuss about what a computable function is.The spirit of this approach is similar to that of [RGRS]. A major difference is that we operate in the context of constructive mathematics in the sense of Bishop [BSH1], so all functions are computable by definition, and the phrase “you can find” implies “by a finite calculation.” In particular ifPis some property, then the statement “for eachmthere isnsuch thatP(m, n)” means that we can construct a (computable) functionθsuch thatP(m, θ(m))for allm. Church's thesis has a different flavor in an environment like this where the notion of a computable function is primitive.One point of such a treatment of Church's thesis is to make available to Bishopstyle constructivists the Markovian counterexamples of Russian constructivism and recursive function theory. The lack of serious candidates for computable functions other than recursive functions makes it quite implausible that a Bishopstyle constructivist could refute Church's thesis, or any consequence of Church's thesis. Hence counterexamples such as Specker's bounded increasing sequence of rational numbers that is eventually bounded away from any given real number [SPEC] may be used, as Brouwerian counterexamples are, as evidence of the unprovability of certain assertions.

4 - Recursive Enumerability

With the recent trend of analysing the process of computation through the linear logic looking glass, it is well understood that the ability to copy and erase data is essential in order to obtain a Turingcomplete computation model. However, erasing and copying don't need to be explicitly included in Turing-complete computation models: in this paper we show that the class of partial recursive functions that are syntactically linear (that is, partial recursive functions where no argument is erased or copied) is Turing-complete

Complexity classes of partial recursive functions

Computation of recursive functionals using minimal initial segments

In this paper we proved that the function class CFRF and its proper subclass CFPRF are respectively the partial recursive functions and primitive recursive functions of context free languages (CFLs). Also we discussed the relation between them and recursive functions defined on other domains. It is indicated that the functions of natural numbers and/or symbol strings (words) are functions of CFLs. Several frequently used primitive recursive functions on words were given, including logical connectives, conditional expressions. Also the powerful operators (bounded maximization and minimization operators) for constructing primitive recursive functions were defined. Two important nontrivial algorithms, the characteristic function of arbitrary CFL and the parse function of CFL sentences were constructed. Based on them, the method for extending or restricting function domain was described.

In what follows, by retracing function we will always mean special retracing function: a partial recursive function f whose range is contained in its domain and which is associated with a unique point b such that for each x in the domain of f, f(x) x and for some n, fn (x) = b. Suppose R were a recursively enumerable set which was the union of n immune retraceable sets with n as small as possible. Then R must be infinite by definition of immunity. Let R=S1U .. USn where each Si is retraced by partial recursive function fi to basepoint bi. We observe that if T is any infinite recursive subset of R, for each i, Tn'Si is immune retraceable. For let B= {bi, b2, * * , bn} and T* = TUB. Then T* = S* U ... US* wrhere St = T*flSi. Now St is retraceable for it is retraced by the function f* where domain ft = T n Dom f, and xEDom f =>ft (x) is the largest element y of T* such that for some m f7(x) =y. Hence St could only be immune or finite, but the minimality of n insures that it must be immune. But B is finite so our conclusion holds without adjunction of B. Next, we use the previous observation to show that the intersection of the domains of the fi is infinite. For if this were not the case we could choose a largest subcollection of the fi whose domains had infinite intersection and enumerate the intersection of their domains. Tlhis infinite recursively enumerable subset of R contains only finitely imany members of the domain of some fj not in the subcollection hence only finitely many members of the corresponding Sj. But this set contains an infinite recursive subset with finite intersection with Sj contradicting our observation. Since the intersection of the domains of thefi is infinite recursively enumerable we may choose an infinite recursive subset of this intersection which (by our observation) is the union of n immune retraceable sets; or equivalently, assume that R is recursive and each fi has R as domain. If, for some n, fn(x) =y we will write i: x->y. If this happens for no n, we write i: x-f-y. We notice that if yE Sj then {z|j:zy} is finite. For if it were infinite, since if it is disjoint from Sj by the defi-

Smullyan in [Smu61] identified the recursion theoretic essence of incompleteness results such as Gödel's first incompleteness theorem and Rosser's theorem. Smullyan (improving upon [Kle50] and [Kle52]) showed that, for sufficiently complex theories, the collection of provable formulae and the collection of refutable formulae are effectively inseparable—where formulae and their Gödel numbers are identified. This paper gives a similar treatment for proof speed-up. We say that a formal system S1is speedable over another system S0on a set of formulaeAiff, for each recursive functionh, there is a formulaαinAsuch that the length of the shortest proof ofαin S0is larger thanhof the shortest proof ofαin S1. (Here we equate the length of a proof with something like the number of characters making it up,notits number of lines.) We characterize speedability in terms of the inseparability by r.e. sets of the collection of formulae which are provable in S1but unprovable in S0from the collectionA–where again formulae and their Gödel numbers are identified. We provide precise definitions of proof length, speedability and r.e. inseparability below.We follow the terminology and notation of [Rog87] with borrowings from [Soa87]. Below,ϕis an acceptable numbering of the partial recursive functions [Rog87] andΦa (Blum) complexity measure associated withϕ[Blu67], [DW83].