An introduction to computable model theory on groups and fields

We introduce infinite time computable model theory, the com- putable model theory arising with infinite time Turing machines, which provide infinitary notions of computability for structures built on the reals R. Much of the finite time theory generalizes to the infinite time context, but several fundamental questions, including the infinite time computable analogue of the Completeness Theorem, turn out to be independent of zfc. Computable model theory is model theory with a view to the computability of the structures and theories that arise (for a standard reference, see (EGNR98)). Infinite time computable model theory, which we introduce here, carries out this program with the infinitary notions of computability provided by infinite time Turing ma- chines. The motivation for a broader context is that, while finite time computable model theory is necessarily limited to countable models and theories, the infinitary context naturally allows for uncountable models and theories, while retaining the computational nature of the undertaking. Many constructions generalize from finite time computable model theory, with structures built on N, to the infinitary theory, with structures built on R. In this article, we introduce the basic theory and con- sider the infinitary analogues of the completeness theorem, the Lowenheim-Skolem Theorem, Myhill's theorem and others. It turns out that, when stated in their fully general infinitary forms, several of these fundamental questions are independent of zfc. The analysis makes use of techniques both from computability theory and set theory. This article follows up (Ham05). 1.1. Infinite time Turing machines. The definitive introduction to infinite time Turing machines appears in (HL00), but let us quickly describe how they work. The

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.

Chapter 1 Pure computable model theory

One of the roots of evolutionary computation was the idea of Turing about unorganized machines. The goal of this paper is the development of foundations for evolutionary computations, connecting Turing’s ideas and the contemporary state of art in evolutionary computations. The theory of computation is based on mathematical models of computing automata, such as Turing machines or finite automata. In a similar way, the theory of evolutionary computation is based on mathematical models of evolutionary computing automata, such as evolutionary Turing machines or evolutionary finite automata. The goal of the chapter is to study computability in the context of the theory of evolutionary computation and genetic algorithms. We use basic models of evolutionary computation, such as different types of evolutionary machines, evolutionary automata and evolutionary algorithms, for exploration of the computing and accepting power of various kinds of evolutionary automata. However, we consider not only how evolutionary automata compute but also how they are generated because a rigorous study of construction techniques for computational systems is an urgent demand of information processing technology. Generation schemas for evolutionary automata are studied and applied to computability problems.

Computable model theory, also called effective or recursive model theory, studies algorithmic properties of mathematical structures, their relations, and isomorphisms. These properties can be described syntactically or semantically. One of the major tasks of computable model theory is to obtain, whenever possible, computability-theoretic versions of various classical model-theoretic notions and results. For example, in the 1950's, Fröhlich and Shepherdson realized that the concept of a computable function can make van der Waerden's intuitive notion of an explicit field precise. This led to the notion of a computable structure. In 1960, Rabin proved that every computable field has a computable algebraic closure. However, not every classical result “effectivizes”. Unlike Vaught's theorem that no complete theory has exactly two nonisomorphic countable models, Millar's and Kudaibergenov's result establishes that there is a complete decidable theory that has exactly two nonisomorphic countable models with computable elementary diagrams. In the 1970's, Metakides and Nerode [58], [59] and Remmel [71], [72], [73] used more advanced methods of computability theory to investigate algorithmic properties of fields, vector spaces, and other mathematical structures.

Classical computable model theory is most naturally concerned with countable domains. There are, however, several methods – some old, some new – that have extended its basic concepts to uncountable structures. Unlike in the classical case, however, no single dominant approach has emerged, and different methods reveal different aspects of the computable content of uncountable mathematics. This book contains introductions to eight major approaches to computable uncountable mathematics: descriptive set theory; infinite time Turing machines; Blum-Shub-Smale computability; Sigma-definability; computability theory on admissible ordinals; E-recursion theory; local computability; and uncountable reverse mathematics. This book provides an authoritative and multifaceted introduction to this exciting new area of research that is still in its early stages. It is ideal as both an introductory text for graduate and advanced undergraduate students and a source of interesting new approaches for researchers in computability theory and related areas.

The computational complexity of a problem is the amount of resources, such as time or space, required by a machine that solves the problem. The descriptive complexity of problems is the complexity of describing problems in some logical formalism over finite structures. One of the exciting developments in complexity theory is the discovery of a very intimate connection between computational and descriptive complexity. It is this connection between complexity theory and finite-model theory that we term computational model theory. In this overview paper we offer one perspective on computational model theory. Two important observations underly our perspective: (1) while computational devices work on encodings of problems, logic is applied directly to the underlying mathematical structures, and this “mismatch” complicates the relationship between logic and complexity significantly, and (2) first-order logic has severely limited expressive power on finite structures, and one way to increase the limited expressive power of first-order logic is by adding fixpoint constructs. These observations motivated the introduction of two absract formalisms: that of finite-variable infinitary logic and that of relational machines. This paper is the story of how these formalisms are used in computational model theory and their interrelationship.

Infinite time Turing machines extend the operation of ordinary Turing machines into transfinite ordinal time, thereby providing a natural model of infinitary computability, with robust notions of computability and decidability on the reals, while remaining close to classical concepts of computability. Here, I survey the theory of infinite time Turing machines and recent developments. These include the rise of infinite time complexity theory, the introduction of infinite time computable model theory, the study of the infinite time analogue of Borel equivalence relation theory, and the introduction of new ordinal computational models. The study of infinite time Turing machines increasingly relies on the interaction of methods from set theory, descriptive set theory and computability theory.

Preface

Since A. M. Turing introduced the notion of computability in 1936, various theories of real number computation have been studied [1][10][13]. Some are of interest in nonlinear and statistical physics while others are extensions of the mathematical theory of computation. In this review paper, we introduce a recently developed computability theory for Julia sets in complex dynamical systems by Braverman and Yampolsky [3]. 1 Computability and complexity Chaos and fractals have been studied from the viewpoint of computability in physics [6][12][1][2]. Investigation has focused on the nature of complexity arising from simple nonlinear equations. Unpredictability in chaotic attractors and final state sensitivity in fractal basins are discussed in terms of computability and complexity in the theory of computation. We summarize the leading results of a recently developed computability theory for Julia sets by Braverman and Yampolsky [3]. First, we introduce the classical notions of computability introduced by Turing [14], and computable real functions introduced by Pour-El [13]. Turing computability is defined by rather a physical model of human computation, called a Turing machine, which is an automaton which consisting of finite internal states and a head to read/write symbols on an external tape. The length of the tape is not restricted, but the number of alphabets is finite. It can manipulate individual symbols on a tape, according to a transition diagram, which tells the machine what action to undertake depending on the current internal state ∗s133038@math.sci.hokudai.ac.jp †ysato@math.sci.hokudai.ac.jp ‡zin@math.sci.hokudai.ac.jp

Inspired by the study of generic and coarse computability in computability theory, we extend such investigation to the context of computable model theory. In this paper, we continue our study initiated in the previous paper (Journal of Logic and Computation 32 (2022) 581–607) , where we introduced and studied the notions of generically and coarsely computable structures and their generalizations. In this paper, we introduce the notions of generically and coarsely computable isomorphisms, and their weaker variants. We sometimes also require that the isomorphisms preserve the density structure. For example, for any coarsely computable structure A, there is a density preserving coarsely computable isomorphism from A to a computable structure. We demonstrate that each notion of generically and coarsely computable isomorphisms, density preserving or not, gives interesting insights into the structures we consider, focusing on various equivalence structures and injection structures.

Turing's legacy: developments from Turing's ideas in logic Rod Downey 1. Computability and analysis: the legacy of Alan Turing Jeremy Avigad and Vasco Brattka 2. Alan Turing and the other theory of computation (expanded) Lenore Blum 3. Turing in Quantumland Harry Buhrman 4. Computability theory, algorithmic randomness and Turing's anticipation Rod Downey 5. Computable model theory Ekaterina B. Fokina, Valentina Harizanov and Alexander Melnikov 6. Towards common-sense reasoning via conditional simulation: legacies of Turing in artificial intelligence Cameron E. Freer, Daniel M. Roy and Joshua B. Tenenbaum 7. Mathematics in the age of the Turing machine Thomas C. Hales 8. Turing and the development of computational complexity Steven Homer and Alan L. Selman 9. Turing machines to word problems Charles F. Miller, III 10. Musings on Turing's thesis Anil Nerode 11. Higher generalizations of the Turing model Dag Normann 12. Step by recursive step: Church's analysis of effective calculability Wilfried Sieg 13. Turing and the discovery of computability Robert Irving Soare 14. Transfinite machine models P. D. Welch.

Computational models are mostly based on the interaction of two strategies, one aimed at constructing data types and one aimed at collecting statistical data. The construction strategy type is grounded on the theory of computation while data collection is supported by statistical practices. The theory of information is deeply rooted in mathematical statistics and has been developed by Claude Shannon into a data transmission model. I submit that an explicit interaction of information theory with computation theory can bring balance to the epistemic status of computational models.

The standard course in theory of computation introduces students to Turing machines and computability theory. This model prescribes what can be computed, and what cannot be computed, but the negative results have far more consequences. To take the common example, suppose an operating systems designer wants to determine whether or not a program will halt given enough memory or other resources. Even a Turing machine program cannot be designed to solve this problem---and Turing machines have far more memory than any physical computer. The negative results of computability theory are also robust (a principle enshrined as Church's thesis): since many other models of computation, including λ-calculus, Post systems, and µ-recursive functions, compute the same class of functions on the natural numbers, negative results from one description apply to all other descriptions.But the discipline of programming and the architecture of modern computers impose other constraints on what can be computed. The constraints are ubiquitous. For example, a combination of hardware and software in operating systems prevents programs from manipulating protected data structures except through the system interface. In programming languages, there are programs that "cannot be written," e.g., a sort procedure in Pascal that works on arrays of any size. In databases, there is no Datalog program to calculate the parity of a relation (see [1]). Each of these settings involves a uniprocessor machine, but the constraints become even more pronounced in distributed systems: for instance, there is no mutual exclusion protocol for n processors using fewer than n atomic read/write registers [5]. All of these problems are computable in Turing's sense: one can encode each of these problems as computation over the natural numbers, and one can write programs to solve the problems. So in what sense is Church's thesis applicable? It is important to remember that computability theory only describes properties of the set of computable functions on the natural numbers (although there have been attempts to extend computability theory and complexity theory to higher-order functions; see, e.g., [13, 12, 20].) If one adopts computability theory as the only theory of computation, one is naturally forced to encode other forms of computation as functions on the natural numbers. Alan Perlis' phrase "Turing tarpit" highlights this potential misuse of computability theory: the encoding of computation into one framework forces many relevant distinctions to become lost.Any attempt to explain other computing constraints must necessarily look for theories beyond computability theory. Semantics aims to fill this niche: it is the mathematical analysis and synthesis of programming structures. The definition is admittedly broad and not historically based: semantics was originally a means of describing programming languages, and the definition covers areas not usually called "semantics." This essay attempts to flesh out this definition of semantics with examples, comparisons, and sources of theories. While most of the ideas will be familiar to the practicing semanticist, the perspective may be helpful to those in and out of the field.

[Church 1931] and [Turing 1936] developed equivalent models of computation based on the concept of an algorithm, which by definition is provided an input from which it is to compute a value without external interaction. After physical computers were constructed, they soon diverged from computing only algorithms meaning that the Church/Turing theory of computation no longer applied to computation in practice because computer systems are highly interactive as they compute, which inspired the development of the Actor Model in 1972 to characterize all digital computation. Indeterminate computation can be thousands of times faster than the parallel λ-calculus (cf. [Takahashi 1989]) or pure logic programs [Hewitt 1969]. This article presents metatheory for a universal theory of digital computation that is more general and powerful than Church/Turing computation [Church 1935, Turing 1936]. Meta[Actors] (meta theory of Actors) proves that theory Actors has the following properties: • The theory Actors is uniquely categorical, i.e., it characterizes Actors (including types) up to an unique isomorphism. • The theory Actors is model sound, i.e., every theorem of Actors of is true in the unique up to isomorphism model. • The unique up to isomorphism model has partial predicates and propositions because of indeterminacy. • Proof checking in the theory Actors is computationally decidable. • There are no “monsters” [Lakatos 1976] in models of the theory Actors such as the ones in 1st-order theories of computation in which there can be Zeno-like computations, which have an infinite number of computational steps between two steps. Consequently unlike 1st-order theories, the theory Actors is not subject to cyberattacks using “monsters” in models. • The theory Actors is algorithmically inexhaustible, i.e., it is impossible to computationally enumerate theorems of the theory thereby reinforcing the intuition behind [Franzen, 2004]. Contrary to [Church 1934], the conclusion in this article is to abandon the assumption that theorems of a theory must be computationally enumerable while retaining the requirement that proof checking must be computationally decidable. Thesis: Any digital system can be directly modeled and implemented using Actors.