Yiannis N. Moschovakis

What Is an Algorithm

When algorithms are defined rigorously in Computer Science literature (which only happens rarely), they are generally identified with abstract machines, mathematical models of computers, sometimes idealized by allowing access to “unbounded memory”.1 My aims here are to argue that this does not square with our intuitions about algorithms and the way we interpret and apply results about them; to promote the problem of defining algorithms correctly; and to describe briefly a plausible solution, by which algorithms are recursive definitions while machines model implementations, a special kind of algorithms.

Determinacy and Prewellorderings of the Continuum

Publisher Summary This chapter discusses determinacy and prewellorderings of the continuum. The chapter establishes the notation and summarizes the properties of recursive functions, projective sets and determinacy. The new constructions, the consequences when Λ = R x R, and the consequences when Λ is assumed to satisfy various structure properties are discussed. The results about the projective classes are also presented. A different approach to the projective classes is through definability in the language of second order number theory or analysis. Classes of sets that are definable in languages richer than second order number theory may be studied. Infinite games were introduced where it was shown that every closed set is determined and that there exist non-determined sets. The chapter summarizes fundamental structure properties and highlights the results about them.

The formal language of recursion

This is the first of a sequence of papers in which we will develop a foundation for the theory of computation based on a precise, mathematical notion of abstract algorithm . To understand the aim of this program, one should keep in mind clearly the distinction between an algorithm and the object (typically a function) computed by that algorithm. The theory of computable functions (on the integers and on abstract structures) is obviously relevant to this work, but we will focus on making rigorous and identifying the mathematical properties of the finer (intensional) notion of algorithm. It is characteristic of this approach that we take recursion to be a fundamental (primitive) process for constructing algorithms, not a derived notion which must be reduced to others—e.g. iteration or application and abstraction, as in the classical λ -calculus. We will model algorithms by recursors , the set-theoretic objects one would naturally choose to represent (syntactically described) recursive definitions. Explicit and iterative algorithms are modelled by (appropriately degenerate) recursors. The main technical tool we will use is the formal language of recursion , FLR, a language of terms with two kinds of semantics: on each suitable structure, the denotation of a term t of FLR is a function, while the intension of t is a recursor (i.e. an algorithm) which computes the denotation of t . FLR is meant to be intensionally complete , in the sense that every (intuitively understood) “algorithm” should “be” (faithfully modelled, in all its essential properties by) the intension of some term of FLR on a suitably chosen structure.

Recursion in Higher Types

Publisher Summary Recursion in higher types is an extension of the theory of recursive functions on the integers. This chapter presents an exposition of the basic notions and facts of this theory. It develops higher-type recursion in the context of the general theory of inductive definability. The study is based on Moschovakis approach. The chapter is divided into two parts—functional induction and recursion in higher types. The functional inductions includes monotone operators on partial functions, recursion in type 2 objects and quantifiers, the stage comparison theorem, semirecursive relations, and the enumeration theorem. The topics discussed under recursion in higher types are normality and enumeration in higher type recursion, the original definition of Kleene, substitution theorems of Kleene, sections and envelopes, inductive analysis of semirecursive sets, and closure under higher existential quantification.

Axioms for Computation Theories-First Draft

Publisher Summary This chapter presents the first draft of the axioms for computation theories. A definition of computation theories is proposed and these structures are studied to get the first step in classifying the known theories. One of the difficulties in trying to compare and classify these theories has been the lack of a definition of recursion theory. The chapter outlines the basic properties of computation theories. This includes proving some of the basic results of ordinary recursion theory, and identifying some of the known theories as computation theories. The important concept of finiteness relative to a theory is introduced and studied. The approach in the chapter differs from the recursive and bounded definition of metarecursion theory. A set is finite relative to a theory if the functional representing quantification over that set is computable in the theory. This is a natural approach and allows for direct generalization of the fundamental properties of finite sets.

Abstract Computability and Invariant Definability

By language we understand a lower predicate calculus with identity and (perhaps) relation and function symbols. It is convenient to allow for more than one sort of variable. Now each individual constant (if there are any) is of a specified sort, the formal expressions R(t 1 , … t n ), f(t 1 ,…, t n ) are well formed only if the terms t 1 , …, t n are of specified sorts determined by the relation symbol R and the function symbol f, and the term f(t 1 , …, t n ) (if well formed) is of a sort determined by f. We admit s = t as well formed, no matter what the sorts of s and t.

Two theorems about projective sets

AbstractIn this paper we prove two (rather unrelated) theorems about projective sets. The first one asserts that subsets of ℵ1 which are

On the Basic Notions in the Theory of Induction

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