Recursion Theory Research Articles

The Slaman-Wehner theorem in higher recursion theory

Slaman and Wehner have independently shown that there is a countable structure whose degree spectrum consists of the nonzero Turing degrees. We show that the analogue fails in the degrees of constructibility. While we do not settle the problem for the hyperdegrees, we show that every almost computable structure, in the sense of Kalimullin, has a copy computable from Kleene's O.

Levels of theory‐of‐mind reasoning in competitive games

AbstractThe literature on recursive theory of mind (TOM) reasoning in interactive decision making (reasoning of the type “I think that you think that I think…”) has been pessimistic, suggesting that adults attribute to others levels of reasoning that are low and slow to increase with learning. In four experiments with college‐age adults playing sequential games, we examined whether choices and predictions were consistent with believing that others pursue their immediate self‐interest, or with believing that others reason through their own decision making, with fixed‐sum games that were simpler and more competitive. This manipulation led to higher‐level default TOM reasoning; indeed, reasoning against a lower‐level opponent was frequently consistent with assuming the opponent's reasoning to be higher‐level, leading to sub‐optimal choices. We conclude that TOM reasoning is not of a low level in all game settings; rather, individuals may display effective TOM reasoning, reflecting realistic assumptions about their opponents, in competitive and relatively simple games. Copyright © 2010 John Wiley & Sons, Ltd.

A lambda calculus for real analysis

Abstract Stone Duality is a new paradigm for general topology in which computable continuous functions are described directly, without using set theory, infinitary lattice theory or a prior theory of discrete computation. Every expression in the calculus denotes both a continuous function and a program, and the reasoning looks remarkably like a sanitised form of that in classical topology. This is an introduction to ASD for the general mathematician, with application to elementary real analysis. This language is applied to the Intermediate Value Theorem: the solution of equations for continuous functions on the real line. As is well known from both numerical and constructive considerations, the equation cannot be solved if the function hovers near 0, whilst tangential solutions will never be found. In ASD, both of these failures, and the general method of finding solutions of the equation when they exist, are explained by the new concept of overtness. The zeroes are captured, not as a set, but by higher-type modal operators. Unlike the Brouwer degree of a mapping, these are naturally defined and (Scott) continuous across singularities of a parametric equation. Expressing topology in terms of continuous functions rather than using sets of points leads to treatments of open and closed concepts that are very closely lattice- (or de Morgan-) dual, without the double negations that are found in intuitionistic approaches. In this, the dual of compactness is overtness. Whereas meets and joins in locale theory are asymmetrically finite and infinite, they have overt and compact indices in ASD. Overtness replaces metrical properties such as total boundedness, and cardinality conditions such as having a countable dense subset. It is also related to locatedness in constructive analysis and recursive enumerability in recursion theory.

The Third Man Argument, Parmenides 132a1-b2

Over the last half-century, Plato's Third Man Argument [TMA] has received a surge of attention. The challenge which numerous critics have undertaken is to provide a viable interpretation of Plato's puzzling passage at Parm. 132a1-b2. The exegetical part of this paper attempts to bring together some of the most plausible interpretations offered to date, distilling the good moves from the bad. The not-so-exegetical part of this paper draws out the consequences of these most plausible interpretations. In pa1ticular, it considers the possibility, inspired in the first instance by a stage-functional interpretation of Plato's one-over-many principle, that Plato held a recursive theory of knowledge. In Part I, I give a textual and logical analysis of the TMA I try to formulate the TMA such that it validly generates a regress from consistent premises, while remaining faithful to the text. In Part 11, I ask whether Plato is vulnerable to the TMA so conceived. I argue that some textual evidence suggests that he is not. In Part III, I assume for the sake of argument that Plato is vulnerable, and ask: (1) ls the conclusion of the TMA vicious - does it pose a problem for Plato? And (2), what are the consequences of the IMA (if it goes through) for Plato's claim that knowledge is possible?

The diagonalization method in quantum recursion theory

As quantum parallelism allows the effective co-representation of classical mutually exclusive states, the diagonalization method of classical recursion theory has to be modified. Quantum diagonalization involves unitary operators whose eigenvalues are different from one.

Uncomputability and undecidability in economic theory

Economic theory, game theory and mathematical statistics have all increasingly become algorithmic sciences. Computable Economics, Algorithmic Game Theory [Noam Nisan, Tim Roiughgarden, Éva Tardos, Vijay V. Vazirani (Eds.), Algorithmic Game Theory, Cambridge University Press, Cambridge, 2007] and Algorithmic Statistics [Péter Gács, John T. Tromp, Paul M.B. Vitányi, Algorithmic statistics, IEEE Transactions on Information Theory 47 (6) (2001) 2443–2463] are frontier research subjects. All of them, each in its own way, are underpinned by (classical) recursion theory – and its applied branches, say computational complexity theory or algorithmic information theory – and, occasionally, proof theory. These research paradigms have posed new mathematical and metamathematical questions and, inadvertently, undermined the traditional mathematical foundations of economic theory. A concise, but partial, pathway into these new frontiers is the subject matter of this paper. Interpreting the core of mathematical economic theory to be defined by General Equilibrium Theory and Game Theory, a general – but concise – analysis of the computable and decidable content of the implications of these two areas are discussed. Issues at the frontiers of macroeconomics, now dominated by Recursive Macroeconomic Theory (The qualification ‘recursive’ here has nothing to do with ‘recursion theory’. Instead, this is a reference to the mathematical formalizations of the rational economic agent’s intertemporal optimization problems, in terms of Markov Decision Processes, (Kalman) Filtering and Dynamic Programming, where a kind of ‘recursion’ is invoked in the solution methods. The metaphor of the rational economic agent as a ‘signal processor’ underpins the recursive macroeconomic paradigm.), are also tackled, albeit ultra briefly. The point of view adopted is that of classical recursion theory and varieties of constructive mathematics.

Operational set theory and small large cardinals

A new axiomatic system OST of operational set theory is introduced in which the usual language of set theory is expanded to allow us to talk about (possibly partial) operations applicable both to sets and to operations. OST is equivalent in strength to admissible set theory, and a natural extension of OST is equivalent in strength to ZFC. The language of OST provides a framework in which to express “small” large cardinal notions—such as those of being an inaccessible cardinal, a Mahlo cardinal, and a weakly compact cardinal—in terms of operational closure conditions that specialize to the analogue notions on admissible sets. This illustrates a wider program whose aim is to provide a common framework for analogues of large cardinal notions that have appeared in admissible set theory, admissible recursion theory, constructive set theory, constructive type theory, explicit mathematics, and systems of recursive ordinal notations that have been used in proof theory.

Das kündlich große Geheimnis: Wort und Sakrament in der evangelischen Kirche

Abstract Drawing on insights from Peter Brunner and Friedrich Schleiermacher, the author develops a liturgical theory of recursivity of word and sacrament. Consequences for the question of who should be admitted to Holy Communion are suggested.

Ordinal machines and admissible recursion theory

We generalize standard Turing machines, which work in time ω on a tape of length ω , to α -machines with time α and tape length α , for α some limit ordinal. We show that this provides a simple machine model adequate for classical admissible recursion theory as developed by G. Sacks and his school. For α an admissible ordinal, the basic notions of α -recursive or α -recursively enumerable are equivalent to being computable or computably enumerable by an α -machine, respectively. We emphasize the algorithmic approach to admissible recursion theory by indicating how the proof of the Sacks–Simpson theorem, i.e., the solution of Post’s problem in α -recursion theory, could be based on α -machines, without involving constructibility theory.

Kolmogorov Complexity Theory over the Reals

Kolmogorov Complexity constitutes an integral part of computability theory, information theory, and computational complexity theory—in the discrete setting of bits and Turing machines. Over real numbers, on the other hand, the BSS-machine (aka real-RAM) has been established as a major model of computation. This real realm has turned out to exhibit natural counterparts to many notions and results in classical complexity and recursion theory; although usually with considerably different proofs. The present work investigates similarities and differences between discrete and real Kolmogorov Complexity as introduced by Montaña and Pardo (1998).

Dually nondeterministic functions

Nondeterminacy is a fundamental notion in computing. We show that it can be described by a general theory that accounts for it in the form in which it occurs in many programming contexts, among them specifications, competing agents, data refinement, abstract interpretation, imperative programming, process algebras, and recursion theory. Underpinning these applications is a theory of nondeterministic functions; we construct such a theory. The theory consists of an algebra with which practitioners can reason about nondeterministic functions, and a denotational model to establish the soundness of the theory. The model is based on the idea of free completely distributive lattices over partially ordered sets. We deduce the important properties of nondeterministic functions.

Lowness of higher randomness notions

We study randomness notions given by higher recursion theory, establishing the relationships Π 1 1 -randomness ⊂ Π 1 1 -Martin-Lof randomness ⊂ Δ 1 1 -randomness = Δ 1 1 -Martin-Lof randomness. We characterize the set of reals that are low for Δ 1 1 randomness as precisely those that are Δ 1 1 -traceable. We prove that there is a perfect set of such reals.

Introduction to Turing categories

We give an introduction to Turing categories, which are a convenient setting for the categorical study of abstract notions of computability. The concept of a Turing category first appeared (albeit not under that name or at the level of generality we present it here) in the work of Longo and Moggi; later, Di Paolo and Heller introduced the closely related recursion categories. One of the purposes of Turing categories is that they may be used to develop categorical formulations of recursion theory, but they also include other notions of computation, such as models of (partial) combinatory logic and of the (partial) lambda calculus. In this paper our aim is to give an introduction to the basic structural theory, while at the same time illustrating how the notion is a meeting point for various other areas of logic and computation. We also provide a detailed exposition of the connection between Turing categories and partial combinatory algebras and show the sense in which the study of Turing categories is equivalent to the study of PCAs.

Extended recursion-based formalization of virus mutation

Computer viruses are programs that can replicate themselves by infecting other programs in a system. Bonfante, Kaczmarek and Marion have recently proposed a classification of viruses which relies on the recursion theory and its recursion theorems. We propose an extension of their formalism to consider in a more practical way the mutation of viruses. In particular, we are interested in modelling any depth of mutation, not just the first two levels. We show that this formalism still relies on recursion theorems, whatever the depth of mutation, even in the case of infinite depth. We also extend furthermore this formalism to model the viability of viral replication, which ensures that an infected program still can propagate the virus. An application of the proposed formalism to the class of combined viruses (multi-part viruses) is studied. Finally, given that metamorphic viruses can be modelled by grammars operating on grammars, we study a recursion-based approach of formal grammars and show that the recursion theorems of the recursion theory can be ported to the formal grammars theory.

Recursion Theoretic Operators for Function Complexity Classes

We characterize the gap between time and space complexity of functions by operators and completeness. First, we introduce a new notion of operators for function complexity classes based on recursive function theory and construct an operator which generates FP SPACE from FP. Then, we introduce new function classes composed of functions whose output lengths are bounded by the input length plus some constant. We characterize FP and FP SPACE by using these classes and operators. Finally, we define a new notion of completeness for FP SPACE and show a FP SPACE-complete function.

The Turing degrees for some computation model with the real parameter

L. Blum, M. Shub and S. Smale defined a kind of computation model having parameters that take real values [2]. In this paper we will extend their computation theory on the basis of the conventional recursion theory which is constructed on the domain of integers. In particular, we will define the Turing degree for the set of reals similarly to the Turing degree for the set of integers, so as to derive some related results.

Learning indexed families of recursive languages from positive data: A survey

In the past 40 years, research on inductive inference has developed along different lines, e.g., in the formalizations used, and in the classes of target concepts considered. One common root of many of these formalizations is Gold’s model of identification in the limit. This model has been studied for learning recursive functions, recursively enumerable languages, and recursive languages, reflecting different aspects of machine learning, artificial intelligence, complexity theory, and recursion theory. One line of research focuses on indexed families of recursive languages — classes of recursive languages described in a representation scheme for which the question of membership for any string in any of the given languages is effectively decidable with a uniform procedure. Such language classes are of interest because of their naturalness. The survey at hand picks out important studies on learning indexed families (including basic as well as recent research), summarizes and illustrates the corresponding results, and points out links to related fields such as grammatical inference, machine learning, and artificial intelligence in general.

On the Relationship between Filter Spaces and Weak Limit Spaces

Countably based filter spaces have been suggested in the 1970's as a model for recursion theory on higher types. Weak limit spaces with a countable limit base are known to be the class of spaces which can be handled by the Type-2 Model of Effectivity (TTE). We prove that the category of countably based filter spaces is equivalent to the category of weak limit spaces with a countable limit base. As a consequence we obtain that filter spaces form yet another category from which the category QCB of quotients of countably based topological spaces inherits its cartesian closed structure.

Preface to Special Issue devoted to the memory of Sauro Tulipani

2006 was a special year for both mathematical logic and computer science, as it celebrated Gödel's centenary. Although Gödel's work was mainly concerned with mathematics and metamathematics, the crucial role it had in the foundation of modern theoretical computer science is undeniable: for instance, one only has to remember Gödel's contributions to the birth of recursion theory as well as his part in the debate in the nineteen thirties on the subject of the Church Thesis.

Mitotic Classes in Inductive Inference

For the natural notion of splitting classes into two disjoint subclasses via a recursive classifier working on texts, the question of how these splittings can look in the case of learnable classes is addressed. Here the strength of the classes is compared using the strong and weak reducibility from intrinsic complexity. It is shown that, for explanatorily learnable classes, the complete classes are also mitotic with respect to weak and strong reducibility, respectively. But there is a weakly complete class that cannot be split into two classes which are of the same complexity with respect to strong reducibility. It is shown that, for complete classes for behaviorally correct learning, one-half of each splitting is complete for this learning notion as well. Furthermore, it is shown that explanatorily learnable and recursively enumerable classes always have a splitting into two incomparable classes; this gives an inductive inference counterpart of the Sacks splitting theorem from recursion theory.