Myhill's work in recursion theory

AbstractIn this paper we discuss the following contributions to recursion theory made by John Myhill: (1) two sets are recursively isomorphic iff they are one-one equivalent; (2) two sets are recursively isomorphic iff they are recursively equivalent and their complements are also recursively equivalent; (3) every two creative sets are recursively isomorphic; (4) the recursive analogue of the Cantor–Bernstein theorem; (5) the notion of a combinatorial function and its use in the theory of recursive equivalence types

Realizability and recursive mathematics

Section 1: Philosophy, logic and constructivityPhilosophy, formal logic and the theory of computation all bear on problems in the foundations of constructive mathematics. There are few places where these, often competing, disciplines converge more neatly than in the theory of realizability structures. Uealizability applies recursion-theoretic concepts to give interpretations of constructivism along lines suggested originally by Heyting and Kleene. The research reported in the dissertation revives the original insights of Kleene—by which realizability structures are viewed as models rather than proof-theoretic interpretations—to solve a major problem of classification and to draw mathematical consequences from its solution.Section 2: Intuitionism and recursion: the problem of classificationThe internal structure of constructivism presents an interesting problem. Mathematically, it is a problem of classification; for philosophy, it is one of conceptual organization. Within the past seventy years, constructive mathematics has grown into a jungle of fullydeveloped "constructivities," approaches to the mathematics of the calculable which range from strict finitism through hyperarithmetic model theory. The problem we address is taxonomic: to sort through the jungle, set standards for classification and determine those features which run through everything that is properly "constructive."There are two notable approaches to constructivity; these must appear prominently in any proposed classification. The most famous is Brouwer's intuitioniam. Intuitionism relies on a complete constructivization of the basic mathematical objects and logical operations. The other is classical recursive mathematics, as represented by the work of Dekker, Myhill, and Nerode. Classical constructivists use standard logic in a mathematical universe restricted to coded objects and recursive operations.The theorems of the dissertation give a precise answer to the classification problem for intuitionism and classical constructivism. Between these realms arc connected semantically through a model of intuitionistic set theory. The intuitionistic set theory IZF encompasses all of the intuitionistic mathematics that does not involve choice sequences. (This includes all the work of the Bishop school.) IZF has as a model a recursion-theoretic structure, V(A7), based on Kleene realizability. Since realizability takes set variables to range over "effective" objects, large parts of classical constructivism appear over the model as inter¬ preted subsystems of intuitionistic set theory. For example, the entire first-order classical theory of recursive cardinals and ordinals comes out as an intuitionistic theory of cardinals and ordinals under realizability. In brief, we prove that a satisfactory partial solution to the classification problem exists; theories in classical recursive constructivism are identical, under a natural interpretation, to intuitionistic theories. The interpretation is especially satisfactory because it is not a Godel-style translation; the interpretation can be developed so that it leaves the classical logical forms unchanged.Section 3: Mathematical applications of the translation:The solution to the classification problem is a bridge capable of carrying two-way mathematical traffic. In one direction, an identification of classical constructivism with intuitionism yields a certain elimination of recursion theory from the standard mathematical theory of effective structures, leaving pure set theory and a bit of model theory. Not only are the theorems of classical effective mathematics faithfully represented in intuitionistic set theory, but also the arguments that provide proofs of those theorems. Via realizability, one can find set-theoretic proofs of many effective results, and the set-theoretic proofs are often more straightforward than their recursion-theoretic counterparts. The new proofs are also more transparent, because they involve, rather than recursion theory plus set theory, at most the set-theoretic "axioms" of effective mathematics.Working the other way, many of the negative ("cannot be obtained recursively") results of classical constructivism carry over immediately into strong independence results from intuitionism. The theorems of Kalantari and Retzlaff on effective topology, for instance, turn into independence proofs concerning the structure of the usual topology on the intuitionistic reals.The realizability methods that shed so much light over recursive set theory can be applied to "recursive theories" generally. We devote a chapter to verifying that the realizability techniques can be used to good effect in the semantical foundations of computer science. The classical theory of effectively given computational domains a la Scott can be subsumed into the Kleene realizability universe as a species of countable noneffective domains. In this way, the theory of effective domains becomes a chapter (under interpre¬ tation) in an intuitionistic study of denotational semantics. We then show how the "extra information" captured in the logical signs under realizability can be used to give proofs of classical theorems about effective domains.Section 4: Solutions to metamathematical problems:The realizability model for set theory is very tractible; in many ways, it resembles a Boolean-valued universe. The tractibility is apparent in the solutions it offers to a number of open problems in the metamathematics of constructivity. First, there is the perennial problem of finding and delimiting in the wide constructive universe those features that correspond to structures familiar from classical mathematics. In the realizability model, it is easy to locate the collection of classical ordinals and to show that they form, intuitionistically, a set rather than a proper class. Also, one interprets an argument of Dekker and Myhill to prove that the classical powerset of the natural numbers contains at least continuum-many distinct cardinals.Second, a major tenet of Bishop's program for constructivity has been that constructive mathematics is "numerical:" all the properties of constructive objects, including the real numbers, can be represented as properties of the natural numbers. The realizability model shows that Bishop's numericalization of mathematics can, in principle, be accomplished. Every set over the model with decidable equality and every metric space is enumerated by a collection of natural numbers

Reducibilities in recursive function theory.

Massachusetts Institute of Technology. Dept. of Mathematics. Thesis. 1966. Ph.D.Bibliography: leaves 102-103.Ph.D

Proceedings of the 22nd Conference on Formal Methods in Computer-Aided Design – FMCAD 2022

The Conference on Formal Methods in Computer-Aided Design (FMCAD) is an annual conference on the theory and applications of formal methods in hardware and system verification. FMCAD provides a leading forum to researchers in academia and industry for presenting and discussing groundbreaking methods, technologies, theoretical results, and tools for reasoning formally about computing systems. FMCAD covers formal aspects of computer-aided system design including verification, specification, synthesis, and testing

Proceedings of the 22nd Conference on Formal Methods in Computer-Aided Design – FMCAD 2022

The Conference on Formal Methods in Computer-Aided Design (FMCAD) is an annual conference on the theory and applications of formal methods in hardware and system verification. FMCAD provides a leading forum to researchers in academia and industry for presenting and discussing groundbreaking methods, technologies, theoretical results, and tools for reasoning formally about computing systems. FMCAD covers formal aspects of computer-aided system design including verification, specification, synthesis, and testing

Environnement humain de l'érosion

Au Niger, en culture traditionnelle, sur terrain à fort coefficient de ruissellement, la banquette de diversion diminue le rendement des récoltes et le temps de concentration de l'écoulement, elle augmente le ruissellement et l'érosion. Le calcul hydraulique justifiant ces résultats, permet de chiffrer l'augmentation de vitesse de circulation de l'eau et la réduction des volumes infiltrés. Le calcul permet également de démontrer que les éléments de banquette ont une contenance insuffisante pour absorber le ruissellement des pluies érosives, et que la banquette isohypse déborde, les eaux s'écoulant trop lentement. Sur terrain à fort coefficient d'infiltration, et à faible pouvoir de rétention, les éléments de banquette augmentent les rendements, mais de façon insuffisante pour rentabiliser les investissements. L'aménagement en banquette (CES) semble provoquer une érosion technocratique et une réduction du niveau de vie des exploitants. Cette technique, préconisée par de nombreux manuels et exécutée, à grand frais, par de multiples organismes, devrait faire l'objet de recherches approfondies pour en vérifier le bien fondé. La méthode de recherche proposée est simple, économique et rapide. Elle utilise des asperseurs du commerce et des paires de parcelles. La création routinière de périmètres CES dont on attribue l'échec à l'inexpérience des exécutants est une illustration des blocages psychologiques et de l'autocensure engendrée par les lacunes de la formation. (Résumé d'auteur

Mediterranean Soils with Particular Reference to Archaeology

Following a review ofthe Mediterranean environment and its development during the late Quaternary, eleven published papers, one paper in preparation, a book (which accompanies the thesis volume) and as yet unpublished data are presented on seven areas of the Mediterranean (in s Portugal, Sicily and Greece). The aims of the research are to consider methodologies in the study of .soils in relation to archaeology (including the relationships between soils, geomorphology and archaeology), and to consider questions of relevance to such research, including a number which have been raised, but not satisfactorily answered, in the wider literature on the Mediterranean. The research comes within the area of geoarchaeology. The studies presented cover a range of spatial scales: single soil profile and archaeological excavation context - the hillslope - river catchmentand broader region. General conclusions in respect of the questions raised include the following. 1. Significant differences in environment and soils occur between the Mediterranean and neighbouring regions arising particularly from climate, but also from other aspects of the environment and human history. Distinctively Mediterranean soils began forming in the Pleistocene or earlier; Holocene soils tend to be weakly de~eloped and similar to soils of cool temperate regions. 2. Geomorphological changes in the physical landscape during the Holocene are generally well defined, if not always well dated. 3. Evidence from much of the Mediterranean points to environmental resilience (an ability to recover from disturbance) rather than 'degradation', though some 20th and 21 st century land use pressures have caused changes that are probably significantly faster and possibly more severe than any during the Holocene. 4. The 'paradox' ofthe Mediterranean - much diversity within elements of strong regional conformity - may result in local factors in environmental change overriding major regional. In many cases, data are not available to permit more than speculation as to the relative importance of anthropogenic versus natural triggers of change during the last five millennia. Intensive, local studies are required to test assertions about major regional effects. 5. For its impact on the archaeological record, 'erosion' must be defined in terms of precise processes and their potential effects in the context of detailed conditions on the hillslope; analysis of valley alluvial sediments, though invaluable in the study of erosion history, cannot fully address these questions. 6. Geoarchaeological analysis requires close integration of archaeological, geomorphological and pedological analyses. Research applying soil information to archaeological diagnosis of excavation contexts also requires a closely integrated, multidisciplinary approach to sampling and analysis, and to intensive computer processing and advanced statistical methods of data analysis

Evaluation and mapping of tropical African rangelands

Proceedings of the seminar on evaluation and mapping of tropical African rangelands, particular categories of rangeland survey and evaluation, review of experiences; site development, parameters and methods; sampling and data processing, cartography, with recommendations and guidelines for the future