Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page

From Nonstandard Analysis to various flavours of Computability Theory

As suggested by the title, it has recently become clear that theorems of Nonstandard Analysis (NSA) give rise to theorems in computability theory (no longer involving NSA). Now, the aforementioned discipline divides into classical and higher-order computability theory, where the former (resp. the latter) sub-discipline deals with objects of type zero and one (resp. of all types). The aforementioned results regarding NSA deal exclusively with the higher-order case; we show in this paper that theorems of NSA also give rise to theorems in classical computability theory by considering so-called textbook proofs.Comment: To appear in the proceedings of TAMC2017 (http://tamc2017.unibe.ch/

Ramon Llull:

Abstract not availabl

The logic of the reverse mathematics zoo

Building on previous work by Mummert, Saadaoui and Sovine, we study the logic underlying the web of implications and nonimplications which constitute the so called reverse mathematics zoo. We introduce a tableaux system for this logic and natural deduction systems for important fragments of the language

Computability and Complexity Properties of Automatic Structures and their Applications

Finite state automata are Turing machines with fixed finite bounds on resource use. Automata lend themselves well to real-time computations and efficient algorithms. Continuing a tradition of studying computability in mathematics, we examine automatic structures, mathematical objects which can be represented by automata, and apply resulting observations to computer science. We measure the complexity of automatic structures via well-established concepts from model theory, topology, and set theory. We prove the following results. The ordinal height of any automatic well-founded partial order is bounded by \omega^\omega. The ordinal heights of automatic well-founded relations are unbounded below the first uncomputable ordinal. For any computable ordinal, there is an automatic structure of Scott rank at least that ordinal. Moreover, there are automatic structures of Scott rank the first uncomputable ordinal and the successor of the first uncomputable ordinal. For any computable ordinal, there is an automatic successor tree of Cantor-Bendixson rank that ordinal. Next, we study infinite graphs produced from a natural unfolding operation applied to finite graphs. Graphs produced via such operations have finite degree and can be described by finite automata over a one-letter alphabet. We investigate algorithmic properties of such graphs in terms of their finite presentations. In particular, we ask how hard it is to check whether a given node belongs to an infinite component, whether two given nodes in the graph are reachable from one another, and whether the graph is connected. We give polynomial-time algorithms answering each of these questions. For a fixed input graph, the algorithm for infinite component membership works in constant time and reachability is decided uniformly by a single automaton. Hence, we improve on previous work, in which nonelementary or nonuniform algorithms were found. We turn our attention to automata techniques for deciding first-order logical theories. These techniques are useful in Integer Linear Programming and Mixed Integer Linear Programming, which in turn have applications in diverse areas of computer science and engineering. We extend known work to address the enumeration problem for linear programming solutions. Then, we apply a similar paradigm to give an automata theoretic decision procedure for the p-adic valued ring under addition and for formal Laurent series over a finite field with valuation and addition

Designing Dependencies

Given a binary recursively enumerable relation R, one or more logic programs over a language L can be constructed and interconnected to produce a dependency relation D on selected predicates within the Herbrand base BL of L isomorphic to R. D can be, optionally, a positive, negative or mixed dependency relation. The construction is applied to representing any effective game of the type introduced by Gurevich and Harrington, which they used to prove Rabin\u27s decision method for S2S, as the dependency relation of a logic program. We allow games over an infinite alphabet of possible moves. We use this representation to reveal a common underlying reason, having to do with the shape of a program\u27s dependency relation, for the complexity of several logic program properties