G\"odel's Notre Dame Course

This is a companion to a paper by the authors entitled "G\"odel's natural deduction", which presented and made comments about the natural deduction system in G\"odel's unpublished notes for the elementary logic course he gave at the University of Notre Dame in 1939. In that earlier paper, which was itself a companion to a paper that examined the links between some philosophical views ascribed to G\"odel and general proof theory, one can find a brief summary of G\"odel's notes for the Notre Dame course. In order to put the earlier paper in proper perspective, a more complete summary of these interesting notes, with comments concerning them, is given here.Comment: 18 pages. minor additions, arXiv admin note: text overlap with arXiv:1604.0307

Natural Halting Probabilities, Partial Randomness, and Zeta Functions

We introduce the zeta number, natural halting probability and natural complexity of a Turing machine and we relate them to Chaitin's Omega number, halting probability, and program-size complexity. A classification of Turing machines according to their zeta numbers is proposed: divergent, convergent and tuatara. We prove the existence of universal convergent and tuatara machines. Various results on (algorithmic) randomness and partial randomness are proved. For example, we show that the zeta number of a universal tuatara machine is c.e. and random. A new type of partial randomness, asymptotic randomness, is introduced. Finally we show that in contrast to classical (algorithmic) randomness--which cannot be naturally characterised in terms of plain complexity--asymptotic randomness admits such a characterisation.Comment: Accepted for publication in Information and Computin

Formal Languages and Compilation

This textbook describes the essential principles and methods used for defining the syntax of artificial languages, and for designing efficient parsing algorithms and syntax-directed translators with semantic attributes. A comprehensive selection of topics is presented within a rigorous, unified framework, illustrated by numerous practical examples. Features and topics: presents a novel conceptual approach to parsing algorithms that applies to extended BNF grammars, together with a parallel parsing algorithm; supplies supplementary teaching tools, including course slides and exercises with solutions, at an associated website; unifies the concepts and notations used in different approaches, enabling an extended coverage of methods with a reduced number of definitions; systematically discusses ambiguous forms, allowing readers to avoid pitfalls when designing grammars; describes all algorithms in pseudocode, so that detailed knowledge of a specific programming language is not necessary; makes extensive usage of theoretical models of automata, transducers and formal grammars; includes concise coverage of algorithms for processing regular expressions and finite automata; and introduces static program analysis based on flow equations. This clearly-written, classroom-tested textbook is an ideal guide to the fundamentals of this field for advanced undergraduate and graduate students in computer science and computer engineering. Some background in programming is required, and readers should also be familiar with basic set theory, algebra and logic