On the computational properties of the Baire category theorem

Computability is one of the fundamental notions of mathematics and computer science, trying to capture the effective content of mathematics and its applications. Computability Theory explores the frontiers and limits of effectiveness and algorithmic methods. It has its origins in Godel's Incompleteness Theorems and the formalization of computability by Turing and others, which later led to the emergence of computer science as we know it today. Computability Theory is strongly connected to other areas of mathematics and theoretical computer science. The core of this theory is the analysis of relative computability and the induced degrees of unsolvability; its applications are mainly to Kolmogorov complexity and randomness as well as mathematical logic, analysis and algebra. Current research in computability theory stresses these applications and focuses on algorithmic randomness, computable analysis, computable model theory, and reverse mathematics (proof theory). Recent advances in these research directions have revealed some deep interactions not only among these areas but also with the core parts of computability theory. The goal of this Dagstuhl Seminar is to bring together researchers from all parts of computability theory and related areas in order to discuss advances in the individual areas and the interactions among those.

To enable the study of open sets in computational approaches to mathematics, lots of extra data and structure on these sets is assumed. For both foundational and mathematical reasons, it is then a natural question, and the subject of this paper, what the influence of this extra data and structure is on the logical and computational properties of basic theorems pertaining to open sets. To answer this question, we study various basic theorems of analysis, like the Baire category, Heine, Heine–Borel, Urysohn and Tietze theorems, all for open sets given by their (third-order) characteristic functions. Regarding computability theory, the objects claimed to exist by the aforementioned theorems undergo a shift from ‘computable’ to ‘not computable in any type 2 functional’, following Kleene’s S1–S9. Regarding reverse mathematics, the latter’s main question, namely which set existence axioms are necessary for proving a given theorem, does not have a unique or unambiguous answer for the aforementioned theorems, working in Kohlenbach’s higher-order framework. A finer study of representations of open sets leads to the new ‘$\varDelta$-functional’ that has unique (computational) properties.

In this paper we discuss two logic techniques of constructing nonstandard models of real number system. The first method is the ultraproduct construction of hyperreals in nonstandard analysis. This method provides infinitesimals as mathematical objects, which can not be formally expressed in the classical account of mathematical analysis. Since every equivalence class of hyperreals modulo infinitesimals can be seen as a granule, such a nonstandard model can be seen as a kind of Granular Mathematics. The second method is the standard compactness argument in mathematical logic. However, both methods cause some troubles: (1) There are some infinitesimals in unexpected forms. (2) The compactness construction of nonstandard models also shows that the standard initial segments of the nonstandard model for infinite sets (which look just like the set of natural numbers or the set of real numbers) are not always legitimate sets, from the set-theoretic point of view, as we usually expect in classical mathematics. We conclude that infinitesimals and nonstandard reals may exist, and we can not claim their nonexistence even using set theory.

1. INTRODUCTION AND EARLY DEVELOPMENTS. Logic and foundations are a domain of mathematics concerned with basic mathematical structures (in terms of which one can define all other mathematical structures), with the correctness and significance of mathematical reasoning, and with the effectiveness of mathematical computations. In the twentieth century, these areas have crystallized into three large chapters of mathematics: set theory, mathematical logic (including model theory), and computability theory, which are intertwined in significant ways. In this paper we describe the evolution and present state of each of them. In modern times the study of logic and foundations has attracted eminent mathematicians and philosophers such as Cantor, Cohen, Frege, Godel, Hilbert, Kleene, Martin, Russell, Solovay, Shelah, Skolem, Tarski, Turing, Zermelo, and others, and has given rise to a large body of knowledge. Although our paper is only a brief sketch of this development, we discuss essential results such as G6del's theorem on the completeness of first-order logic and his theorems on the incompleteness of most mathematical theories, some independence theorems in set theory, the role of axioms of existence of large cardinal numbers, Turing's work on computability, and some recent developments. There are still many interesting unsolved problems in logic and foundations. For example, logic does not explain what mathematics is. We know that mathematics has a very precise structure: axioms, definitions, theorems, proofs. Thus we know what is correct mathematics but not why the works of certain mathematicians delight us while others strike us as downright boring. Nor do foundations tell us how mathematicians construct proofs of their conjectures. Since we have no good theoretical model of the process for constructing proofs, we are far from having truly effective procedures for automatically proving theorems, although spectacular successes have been achieved in this area. We mention other unsolved problems at the end of this paper. We now give a brief sketch of the history of logic and foundations prior to 1900. The ancient Greeks asked: what are correct arguments? and: what are the real numbers? As partial answers they created the theory of syllogisms and a theory of commensurable and incommensurable magnitudes. These questions resurfaced in the 18th century, due to the development of analysis and to the lack of sufficiently clear concepts of sets, functions, continuity, convergence, etc. In a series of papers (1878-1897) Georg Cantor created set theory. In 1879 Gottlob Frege described a formal system of logic that explained precisely the logical structure of all mathematical proofs. In 1858 Richard Dedekind gave a definitive answer to the question: what are the real numbers? by defining them in terms of sets of rational numbers. He proved the axiom of continuity of the real line, an axiom accepted hitherto (beginning with the Greeks) without proof.

Computational type theory provides answers to questions such as: What is a type? What is a natural number? How do we compute with types? How are types related to sets? Can types be elements of types? How are data types for numbers, lists, trees, graphs, etc. related to the corresponding notions in mathematics? What is a real number? Are the integers a subtype of the reals? Can we form the type of all possible data types? Do paradoxes arise in formulating a theory of types as they do in formulating a theory of sets, such as the circular idea of the set of all sets or the idea of all sets that do not contain themselves as members? Is there a type of all types? What is the underlying logic of type theory? Why isn’t it the same logic in which standard set theories are axiomatized? What is the origin of the notion of a type? What distinguishes computational type theory from other type theories? In computational type theory, is there a type of all computable functions from the integers to the integers? If so, is it the same as the set of Turing computable functions from integers to integers? Is there a type of computable functions from any type to any type? Is there a type of the partial computable functions from a type to a type? Are there computable functions whose values are types? Do the notations of an implemented computational type theory include programs in the usual sense? What does it mean that type theory is a foundational theory for both mathematics and computer science? There have been controversies about the foundations of mathematics, does computational type theory resolve any of them, do these controversies impact a foundation for computing theory? This article answers some of these questions and points to literature answering all of them.

The present paper introduces a novel notion of ‘(effective) computability,’ called viability, of strategies in game semantics in an intrinsic (i.e., without recourse to the standard Church–Turing computability), non-inductive, non-axiomatic manner and shows, as a main technical achievement, that viable strategies are Turing complete. Consequently, we have given a mathematical foundation of computation in the same sense as Turing machines but beyond computation on natural numbers, e.g., higher-order computation, in a more abstract fashion. As immediate corollaries, some of the well-known theorems in computability theory such as the smn theorem and the first recursion theorem are generalized. Notably, our game-semantic framework distinguishes high-level computational processes that operate directly on mathematical objects such as natural numbers (not on their symbolic representations) and their symbolic implementations that define their ‘computability,’ which sheds new light on the very concept of computation. This work is intended to be a stepping stone toward a new mathematical foundation of computation, intuitionistic logic and constructive mathematics.

The real numbers as a wreath product

The Real Numbers as a Wreath Product

The theory of the computable functions is a mathematical theory of total and partial functions of the form f : Nn →N, and sets of the form. . . SÍ Nn. . .that can be defined by means of algorithms on the set . . . N = {0,1,2, . . . } . . . of natural numbers. The theory establishes what can and cannot be computed in an explicit way using finitely many simple operations on numbers. The set of naturals and a selection of these simple operations together form an algebra. A mathematical objective of the theory is to develop, analyse and compare a variety of models of computation and formal systems for defining functions over a range of algebras of natural numbers. Computability theory on N is of importance in science because it establishes the scope and limits of digital computation. The numbers are realised as concrete symbolic objects and the operations on the numbers can be carried out explicitly, in finitely many concrete symbolic steps. More generally, the numbers can be used to represent or code any form of discrete data. However, the question arises: . . . Can we develop theories of functions that can be defined by means of algorithms on other sets of data?. . . The obvious examples of numerical data are the integer, rational, real and complex numbers; and associated with these numbers there are data such as matrices, polynomials, power series and various types of functions. In addition, there are geometric objects that are represented using the real and complex numbers, including algebraic curves and manifolds. Examples of syntactic data are finite and infinite strings, terms, formulae, trees and graphs. For each set of data there are many choices for a collection of operations from which to build algorithms. . .How specific to the set of data and chosen operations are these computability theories? What properties do the computability theories over different sets of data have in common? . . . The theory of the computable functions on N is stable, rich and useful; will the theory of computable functions on the sets of real and complex numbers, and the other data sets also be so? The theory of computable functions on arbitrary many-sorted algebras will answer these questions.

Algorithmic information theory studies description complexity and randomness and is now a well known field of theoretical computer science and mathematical logic. There are several textbooks and monographs devoted to this theory where one can find the detailed exposition of many difficult results as well as historical references. However, it seems that a short survey of its basic notions and main results relating these notions to each other, is missing. This report attempts to fill this gap and covers the basic notions of algorithmic information theory: Kolmogorov complexity (plain, conditional, prefix), Solomonoff universal a priori probability, notions of randomness (Martin-L\"of randomness, Mises--Church randomness), effective Hausdorff dimension. We prove their basic properties (symmetry of information, connection between a priori probability and prefix complexity, criterion of randomness in terms of complexity, complexity characterization for effective dimension) and show some applications (incompressibility method in computational complexity theory, incompleteness theorems). It is based on the lecture notes of a course at Uppsala University given by the author.

We give a nontechnical account of the mathematical theory of randomness. The theory of randomness is founded on computability theory, and it is nowadays often referred to as algorithmic randomness. It comes in two varieties: A theory of finite objects, that emerged in the 1960s through the work of Solomonoff, Kolmogorov, Chaitin and others, and a theory of infinite objects (starting with von Mises in the early 20th century, culminating in the notions introduced by Martin-Löf and Schnorr in the 1960s and 1970s) and there are many deep and beautiful connections between the two. Research in algorithmic randomness connects computability and complexity theory with mathematical logic, proof theory, probability and measure theory, analysis, computer science, and philosophy. It also has surprising applications in a variety of fields, including biology, physics, and linguistics. Founded on the theory of computation, the study of randomness has itself profoundly influenced computability theory in recent years.

Signifiable computability aims to separate what is theoretically computable from what is computable through performable processes on computers with finite amounts of memory. Mathematical objects are signifiable in a formalism L on an alphabet A if they can be written as spatiotemporally finite texts in L on A. In a previous article, we formalized the signification and reference of real numbers and showed that data structures representable as multidimensional matrices of discretely finite real numbers are signifiable. In this investigation, we continue to formulate our theory of signifiable computability by offering an axiomatization of signifiable computation on discretely finite real numbers. The axiomatization implies an ontology of functions on discretely finite real numbers that classifies them as signifiable, signifiably computable, and signifiably partially computable. Relative to L and A, signification is performed with two formal systems: the Former F¨^A,L that forms texts in L on A and the Transformer T¨^A,L that transforms texts formed by F¨^A,L into other texts in L on A. Singifiable computation is defined relative to L on A as a finite sequence of signifiable program states, the first of which is generated by F¨^A,L and each subsequent state is deterministically obtained from the previous one by T¨^A,L. We define a debugger function to investigate signifiable computation on finite-memory devices and to prove two theorems, which we call the Debugger Theorems. The first theorem shows that, for a singifiably partially computable function signified by a program on a finite-memory device D, the memory capacity of D is exceeded when running the program on signifiable discretely finite real numbers outside of the function’s domain. The second theorem shows that there are functions signifiably computable in general that become partially signifiably computable when signified by programs on D insomuch as the memory capacity of D can be exceeded even when the programs are executed on some signifiable discretely finite real numbers in the domains of these functions.

Generic approach is one of the approaches to the study of algorithmic problems for almost all inputs, born at the intersection of computational algebra and computer science. Within the framework of this approach, algorithms are studied that solve a problem for almost all inputs, and for the remaining rare inputs give an undefined answer. This review reflects two areas of research of generic complexity of algorithmic problems in algebra, mathematical logic, number theory, and theoretical computer science. The first direction is devoted to the construction of generic algorithms for problems that are unsolvable and hard in the classical sense. In the second direction, algorithmic problems are sought that remain unsolvable or hard even in the generic sense. Such problems are important in cryptography.

The main purpose of the book is a detailed exposition of methods used in semantical and deductive analysis of ordinary mathematical reasoning by means of classical mathematical logic. The book contains both a statement of modern mathematical logic and many its applications in various fields of mathematics. Introduction gives a brief review of the trends in mathematics in the 19th century which have resulted in a renewed interest in formal logic and its rapid development in the 20th century. Chapters I–XI contain a rather detailed exposition of basic facts in modern mathematical logic including classical propositional and predicate logics, their semantics and axiomatizations. Chapters XII–XVII present applications of mathematical logic to mathematics. In this connection formalizations of group theory, linear orderings, arithmetic of natural numbers, integers, rationals, and real numbers are considered. First-order and second-order arithmetic theories are discussed. Chapters XVIII and XIX written in collaboration with Lesław Szczerba give...

مستخلص البحث: هدف البحث إلى تدريس برنامج قائم على المنطق الرياضي من اجل تنمية مهارات الاثبات الجبري لدى تلاميذ الصف الثاني الاعدادي، وتکونت عينة البحث من (40) تلميذا تم تقسيمهم الى مجموعتين : مجموعة تجريبية وبلغ عددهم (21) تلميذا درسوا البرنامج القائم على المنطق الرياضي بالتوازي مع دراستهم لوحدتي الجبر(الاعداد الحقيقية والعلاقة بين متغيرين ) المقررتين على الصف الثاني الاعدادي، ومجموعة ضابطة بلغ عددهم (19) تلميذا درسوا وحدتي الجبر بالطريقة المعتادة، وتم اختبار المجموعتين قبليا وبعديا في اختبار قياس مهارات الاثبات الجبري، واظهرت النتائج فعالية البرنامج القائم على المنطق الرياضي في تنمية تلک المهارات، وفي ضوء ما اسفر عنه البحث من نتائج، قدم البحث مجموعة من التوصيات، کان من ابرزها الاهتمام بإبراز بعض موضوعات المنطق الرياضي في منهج الرياضيات للمرحلة الاعدادية. الکلمات الدالة: المنطق الرياضي – مهارات الاثبات الجبري. Abstract: The aim of the research is to teach a program based on mathematical logic in order to develop the skills of algebraic proof among second-grade prep school pupils. The research sample consisted of (40) pupils divided into two groups: an experimental group consisted of (21) pupils who studied the program in parallel with their study of the two units of algebra (real numbers and the relationship between two variables), and a control group consisted of (19) pupils who studied the two units of algebra in the usual way. The two groups were tested before and after the experiment. Results showed the effectiveness of the program in developing these skills, and in light of these results, the research presented a set of recommendations, the most prominent of which was the interest in adding some issues of mathematical logic in the mathematics curriculum for the prep school. Key words: mathematical logic- algebraic proof skills.