Combinators : a centennial view

Combinators have inspired ideas about computation ever since they were first invented in 1920, and in this innovative book, Stephen Wolfram provides a modern view of combinators and their significance. Informed by his work on the computational universe of possible programs and on computational language design, Wolfram explains new and existing ideas about combinators with unique clarity and stunning visualizations, as well as provides insights on their historical connections and the curious story of Moses Schönfinkel, inventor of combinators. Though invented well before Turing machines, combinators have often been viewed as an inaccessibly abstract approach to computation. This book brings them to life as never before in a thought-provoking and broadly accessible exposition of interest across mathematics and computer science, as well as to those concerned with the foundations of formal and computational thinking, and with the history of ideas. Contents Preface Combinators: A Centennial View: Ultimate Symbolic Abstraction • Computing with Combinators • A Hundred Years Later... • Combinators in the Wild: Some Zoology • Visualizing Combinators • Updating Schemes and Multiway Systems • The Question of Evaluation Order • The World of the S Combinator • Causal Graphs and the Physicalization of Combinators • Combinator Expressions as Dynamical Systems • Equality and Theorem Proving for Combinators • Lemmas and the Structure of Combinator Space • Empirical Computation Theory with Combinators • The Future of Combinators • Historical & Other Notes Combinators and the Story of Computation The Abstract Representation of Things • What Is Mathematics―and Logic―Made Of? • Combinators Arrive • What Is Their Mathematics? • Gödel's Theorem and Computability • Lambda Calculus • Practical Computation • Combinators in Culture • Designing Symbolic Language • Combinators in the Computational Universe • Combinators All the Way Down? Where Did Combinators Come From? Hunting the Story of Moses Schönfinkel December 7, 1920 • Who Was Moses Schönfinkel? • The Beginning of the Story • Going to College in Odessa • Göttingen, Center of the Mathematical Universe • Problems Are Brewing • The 1924 Paper • The "1927" Paper • To Moscow and Beyond... • Other Schönfinkels... • Haskell Curry • Schönfinkel Rediscovered • What Should We Make of Schönfinkel? A Little Closer to Finding What Became of Moses Schönfinkel, Inventor of Combinators 1920, 2020 and a $20,000 Prize: Announcing the S Combinator Challenge Hiding in Plain Sight for a Century? • The Basic Setup • The Operation of the S Combinator Challenge Excerpts from A New Kind of Science (2002) A Bibliography of Combinators Foundational Documents • Books • Surveys & Summaries • Combinators as Symbolic Expressions • Combinators as Mathematical Constructs • Combinator Computation • Extensions & Applications • Confusing Issues Index Cover Title Page Copyright Page Table of Contents Preface 1 Combinators: A Centennial View Ultimate Symbolic Abstraction Computing with Combinators A Hundred Years Later... Combinators in the Wild: Some Zoology Visualizing Combinators Updating Schemes and Multiway Systems The Question of Evaluation Order The World of the S Combinator Causal Graphs and the Physicalization of Combinators Combinator Expressions as Dynamical Systems Equality and Theorem Proving for Combinators Lemmas and the Structure of Combinator Space Empirical Computation Theory with Combinators The Future of Combinators Historical & Other Notes 2 Combinators and the Story of Computation The Abstract Representation of Things What Is Mathematics—and Logic—Made Of? Combinators Arrive What Is Their Mathematics? Gödel’s Theorem and Computability Lambda Calculus Practical Computation Combinators in Culture Designing Symbolic Language Combinators in the Computational Universe Combinators All the Way Down? 3 Where Did Combinators Come From? Hunting the Story of Moses Schönfinkel December 7, 1920 Who Was Moses Schönfinkel? The Beginning of the Story Going to College in Odessa Göttingen, Center of the Mathematical Universe Problems Are Brewing The 1924 Paper The “1927” Paper To Moscow and Beyond... Other Schönfinkels... Haskell Curry Schönfinkel Rediscovered What Should We Make of Schönfinkel? 4 A Little Closer to Finding What Became of Moses Schönfinkel, Inventor of Combinators One More Thing 5 1920, 2020 and a $20,000 Prize: Announcing the S Combinator Challenge Hiding in Plain Sight for a Century? The Basic Setup The Operation of the S Combinator Challenge 6 Excerpts from A New Kind of Science (2002) Section 10: Symbolic Systems Notes Section 12: Universality in Turing Machines and Other Systems Notes A Bibliography of Combinators Foundational Documents Books Surveys & Summaries Combinators as Symbolic Expressions Combinators as Mathematical Constructs Combinator Computation Extensions & Applications Confusing Issues Index "Machine learning - a computer's ability to learn - is transforming our world: it is used to understand images, process text, make predictions by analyzing large amounts of data, and much more. It can be used in nearly every industry to improve efficiency and help stakeholders make better decisions. Whatever your industry or hobby, chances are that these modern artificial intelligence methods will be useful to you as well. "Introduction to Machine Learning" weaves reproducible coding examples into explanatory text to show what machine learning is, how it can be applied, and how it works. Perfect for anyone new to the world of AI or those looking to further their understanding, the text begins with a brief introduction to the Wolfram Language, the programming language used for the examples throughout the book. From there, readers are introduced to key concepts before exploring common methods and paradigms such as classification, regression, clustering, and deep learning. The math content is kept to a minimum to focus on what matters-applying the concepts in useful contexts. This book is sure to benefit anyone curious about the fascinating field of machine learning" "Combinators have inspired ideas about computation ever since they were first invented in 1920, and in this innovative book, Stephen Wolfram provides a modern view of combinators and their significance. Informed by his work on the computational universe of possible programs and on computational language design, Wolfram explains new and existing ideas about combinators with unique clarity and stunning visualizations, as well as provides insights on their historical connections and the curious story of Moses Schèonfinkel, inventor of combinators. Though invented well before Turing machines, combinators have often been viewed as an inaccessibly abstract approach to computation. This book brings them to life as never before in a thought-provoking and broadly accessible exposition of interest across mathematics and computer science, as well as to those concerned with the foundations of formal and computational thinking, and with the history of ideas"--ISBN : 9781579550431