This almost self-contained introduction to higher recursion theory is essential reading for all researchers in the field.

Volume II of Classical Recursion Theory describes the universe from a local (bottom-up or synthetical) point of view, and covers the whole spectrum, from the recursive to the arithmetical sets. The first half of the book provides a detailed picture of the computable sets from the perspective of Theoretical Computer Science. Besides giving a detailed description of the theories of abstract Complexity Theory and of Inductive Inference, it contributes a uniform picture of the most basic complexity classes, ranging from (...) small time and space bounds to the elementary functions, with a particular attention to polynomial time and space computability. It also deals with primitive recursive functions and larger classes, which are of interest to the proof theorist. The second half of the book starts with the classical theory of recursively enumerable sets and degrees, which constitutes the core of Recursion or Computability Theory. Unlike other texts, usually confined to the Turing degrees, the book covers a variety of other strong reducibilities, studying both their individual structures and their mutual relationships. The last chapters extend the theory to limit sets and arithmetical sets. The volume ends with the first textbook treatment of the enumeration degrees, which admit a number of applications from algebra to the Lambda Calculus. The book is a valuable source of information for anyone interested in Complexity and Computability Theory. The student will appreciate the detailed but informal account of a wide variety of basic topics, while the specialist will find a wealth of material sketched in exercises and asides. A massive bibliography of more than a thousand titles completes the treatment on the historical side. (shrink)

Traces the development of recursive functions from their origins in the late nineteenth century to the mid-1930s, with particular emphasis on the work and influence of Kurt Gödel.

This volume contains papers from the recursion theory session of the Kazan Workshop on Recursion and Complexity Theory. Recursion theory, the study of computability, is an area of mathematical logic that has traditionally been particularly strong in the United States and the former Soviet Union. This was the first workshop ever to bring together about 50 international experts in recursion theory from the United States, the former Soviet Union and Western Europe.

The book examines one of the most contested topics in linguistics and cognitive science: the role of recursion in language. It offers a precise account of what recursion is, what role it should play in cognitive theories of human knowledge, and how it manifests itself in the mental representations of language and other cognitive domains.

iterations of REA operators, as well as extensions, generalizations and other applications are given in [6] while those for the ...

This graduate-level_text by a master in the field builds a function theory of the rational field that combines aspects of classical and intuitionist analysis. Topics include recursive convergence, recursive and relative continuity, recursive and relative differentiability, the relative integral, elementary functions, and transfinite ordinals. 1961 edition.

In this paper we show that there are various pseudo-jump operators definable over inadmissible $J_{\beta}$ that relate to the failure of admissiblity and to non-regularity. We will use these ideas to construct some intermediate degrees.

It has been claimed that recursion is one of the properties that distinguishes human language from any other form of animal communication. Contrary to this claim, a recent study purports to demonstrate center‐embedded recursion in starlings. I show that the performance of the birds in this study can be explained by a counting strategy, without any appreciation of center‐embedding. To demonstrate that birds understand center‐embedding of sequences of the form AnBn (such as A1A2B2B1, or A3A4A5B5B4B3) would require not (...) only that they discriminate such patterns from other patterns, but that they appreciate that elements must be bound from the outside in (thus, in the above examples, A1B1, A2B2, A3B3, A4B4, A5B5 are bound pairs). This has not been shown in nonhuman species, and sentences with this structure are difficult even for humans to parse. There appears to be no evidence to date that nonhuman species understand recursion. (shrink)

It has been suggested that hierarchically structured symbols, a remarkable feature of human language, are produced via the operation of recursive combination. Recursive combination is frequently observed in human behavior, not only in language but also in action sequences, mind-reading, technology, et cetera.; in contrast, it is rarely observed in animals. Why is it that only humans use this operation? What is the adaptability of recursive combination? We aim (1) to identify the environmental feature(s) in which recursive combination is effective (...) for survival and reproduction, and that has facilitated the evolution of this ability, and (2) to demonstrate the possible evolutionary processes of recursive combination. To achieve this, we constructed an evolutionary simulation of agents that generated products using recursive combination and used the results to explore the types of fitness functions (that reflect the kinds of adaptive environments) that give rise to this ability. We identified two types of adaptability of the recursive combination: (1) diversifiability of production and (2) diversifiability of products. Through the former, recursive combination promotes robustness against failure of production caused by inaccurate manipulations or irreversible changes. In an environment in which diversified products are preferable, sharing a portion of the production process for these products entails producing multiple products in which recursive combination plays a key role. We suppose that recursive combination works as a driving force of material culture. Finally, we discuss the possible evolutionary scenarios of recursive combination that is later generalized to encompass many aspects of human cognition, including human language. (shrink)

GENERALIZED RECUBION THEORY II © North-Holland Publishing Company (1978) MONOTONE QUANTIFIERS AND ADMISSIBLE SETS Ion Barwise University of Wisconsin ...

Provability, Computability and Reflection.

Let the chain antichain principle (CAC) be the statement that each partial order on $\mathbb{N}$ possesses an infinite chain or an infinite antichain. Chong, Slaman, and Yang recently proved using forcing over nonstandard models of arithmetic that CAC is $\Pi^1_1$-conservative over $\text{RCA}_0+\Pi^0_1\text{-CP}$ and so in particular that CAC does not imply $\Sigma^0_2$-induction. We provide here a different purely syntactical and constructive proof of the statement that CAC (even together with WKL) does not imply $\Sigma^0_2$-induction. In detail we show using a (...) refinement of Howard's ordinal analysis of bar recursion that $\text{WKL}_0^\omega+\text{CAC}$ is $\Pi^0_2$-conservative over PRA and that one can extract primitive recursive realizers for such statements. Moreover, our proof is finitary in the sense of Hilbert's program. CAC implies that every sequence of $\mathbb{R}$ has a monotone subsequence. This Bolzano-Weierstraß}-like principle is commonly used in proofs. Our result makes it possible to extract primitive recursive terms from such proofs. We also discuss the Erdős-Moser principle, which—taken together with CAC—is equivalent to $\text{RT}^2_2$. (shrink)

What can computers do in principle? What are their inherent theoretical limitations? These are questions to which computer scientists must address themselves. The theoretical framework which enables such questions to be answered has been developed over the last fifty years from the idea of a computable function: intuitively a function whose values can be calculated in an effective or automatic way. This book is an introduction to computability theory (or recursion theory as it is traditionally known to mathematicians). Dr (...) Cutland begins with a mathematical characterisation of computable functions using a simple idealised computer (a register machine); after some comparison with other characterisations, he develops the mathematical theory, including a full discussion of non-computability and undecidability, and the theory of recursive and recursively enumerable sets. The later chapters provide an introduction to more advanced topics such as Gildel's incompleteness theorem, degrees of unsolvability, the Recursion theorems and the theory of complexity of computation. Computability is thus a branch of mathematics which is of relevance also to computer scientists and philosophers. Mathematics students with no prior knowledge of the subject and computer science students who wish to supplement their practical expertise with some theoretical background will find this book of use and interest. (shrink)

A real number is recursively approximable if there is a computable sequence of rational numbers converging to it. If some extra condition to the convergence is added, then the limit real number might have more effectivity. In this note we summarize some recent attempts to classify the recursively approximable real numbers by the convergence rates of the corresponding computable sequences ofr ational numbers.

A class of problems is called decidable if there is an algorithm which will give the answer to any problem of the class after a finite length of time. The purpose of this paper is to discuss the classes of problems that can be solved by infinitely long decision procedures in the following sense: An algorithm is given which, for any problem of the class, generates an infinitely long sequence of guesses. The problem will be said to be solved in (...) the limit if, after some finite point in the sequence, all the guesses are correct and the same. Functions, sets, and functionals which are decidable by such infinite algorithms will be calledlimiting recursive. These, together with classes of objects which can beidentified in the limit, are the subjects of this report.Without qualification,setwill mean set of numbers;functionwill mean number-theoretic function of 1 variable, possibly partial;functionalswill take numerical values and have any number of numerical and/or function variables, the latter ranging solely over total functions of 1 variable. Thus a function is a special case of a functional,xwill invariably stand for a numerical variable;φfor a function variable;gfor a guess ;nfor the numerical variable which indexes the guesses, referred to as thetime. (shrink)

We give recursion-theoretic characterizations of the counting class \(\textsf {\#P} \), the class of those functions which count the number of accepting computations of non-deterministic Turing machines working in polynomial time. Moreover, we characterize in a recursion-theoretic manner all the levels \(\{\textsf {\#P} _k\}_{k\in {\mathbb {N}}}\) of the counting hierarchy of functions \(\textsf {FCH} \), which result from allowing queries to functions of the previous level, and \(\textsf {FCH} \) itself as a whole. This is done in the (...) style of Bellantoni and Cook’s safe recursion, and it places \(\textsf {\#P} \) in the context of implicit computational complexity. Namely, it relates \(\textsf {\#P} \) with the implicit characterizations of \(\textsf {FPTIME} \) (Bellantoni and Cook, Comput Complex 2:97–110, 1992) and \(\textsf {FPSPACE} \) (Oitavem, Math Log Q 54(3):317–323, 2008), by exploiting the features of the tree-recursion scheme of \(\textsf {FPSPACE} \). (shrink)

Although Peano arithmetic is necessarily incomplete, Isaacson argued that it is in a sense conceptually complete: proving a statement of the language of PA that is independent of PA will require conceptual resources beyond those needed to understand PA. This paper gives a test of Isaacon’s thesis. Understanding PA requires understanding the functions of addition and multiplication. It is argued that grasping these primitive recursive functions involves grasping the double ancestral, a generalized version of the ancestral operator. Thus, we can (...) test Isaacon’s thesis by seeing whether when we phrase arithmetic in a context with the double ancestral operator, the result is conservative over PA. This is a stronger version of the test given by Smith, who argued that understanding the predicate “natural number” requires understanding the ancestral operator, but did not investigate what is required to understand the arithmetic functions. (shrink)

Language sciences have long maintained a close and supposedly necessary coupling between the infinite productivity of the human language faculty and recursive grammars. Because of the formal equivalence between recursion and non-recursive iteration; recursion, in the technical sense, is never a necessary component of a generative grammar. Contrary to some assertions this equivalence extends to both center-embedded relative clauses and hierarchical parse trees. Inspection of language usage suggests that recursive rule components in fact contribute very little, and likely (...) nothing significant, to linguistic creativity. Further than this, if the productivity of human language is considered as not rigidly bound, but not infinite, then the need for any sort of iteration in generative grammars vanishes and can be replaced with a Non-iterative Explicit Alternatives Rule grammar. The knock-on effects of dispensing with recursive grammar components are: that language diversity can simply be based on rule and lexicon combinatorics with no potentially infinite dimensions derived from recursive or iterative components of rules; the oddity of a vast, multiply-infinite competence set of ‘grammatical but unacceptable’ productions is gone; and the development of a language faculty based on rules that eschew iterative rule components avoids any need for explaining ‘special’ mechanisms. On the broader front of searching for the mechanisms of mind, our analysis can be similarly applied to the proposals for a recursive basis for mind as an explanation for humanity’s great leap forward. (shrink)

The current literature standardly conceives of voting power in terms of decisiveness: the ability to change the voting outcome by unilaterally changing one’s vote. I argue that this classic conception of voting power, which fails to account for partial decisiveness or efficacy, produces erroneous results because it saddles the concept of voting power with implausible microfoundations. This failure in the measure of voting power in turn reflects a philosophical mistake about the concept of social power in general: a failure to (...) recognize that an agent can exercise individual social power with others’ assistance, in virtue of the group’s collective power, sometimes even when she could not unilaterally scuttle the group’s collective power. I therefore develop a conception of efficacy that admits of degrees and defend a Recursive Measure of voting power that takes partial efficacy into account. (shrink)

Humans grasp discrete infinities within several cognitive domains, such as in language, thought, social cognition and tool-making. It is sometimes suggested that any such generative ability is based on a computational system processing hierarchical and recursive mental representations. One view concerning such generativity has been that each of the mind’s modules defining a cognitive domain implements its own recursive computational system. In this paper recent evidence to the contrary is reviewed and it is proposed that there is only one supramodal (...) computational system with recursion in the human mind. A recursion thesis is defined, according to which the hominin cognitive evolution is constituted by a recent punctuated genetic mutation that installed the general, supramodal capacity for recursion into the human nervous system on top of the existing, evolutionarily older cognitive structures, and it is argued on the basis of empirical evidence and theoretical considerations that the recursion thesis constitutes a plausible research program for cognitive science. (shrink)

This work is a self-contained elementary exposition of the theory of recursive functionals, that also includes a number of advanced results.