Combinatory Logic : Pure, Applied and Typed book cover
1st Edition

Combinatory Logic
Pure, Applied and Typed

ISBN 9781439800003
Published July 27, 2011 by Chapman & Hall
357 Pages 10 B/W Illustrations

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Book Description

Combinatory logic is one of the most versatile areas within logic that is tied to parts of philosophical, mathematical, and computational logic. Functioning as a comprehensive source for current developments of combinatory logic, this book is the only one of its kind to cover results of the last four decades. Using a reader-friendly style, the author presents the most up-to-date research studies. She includes an introduction to combinatory logic before progressing to its central theorems and proofs. The text makes intelligent and well-researched connections between combinatory logic and lambda calculi and presents models and applications to illustrate these connections.

Table of Contents

Elements of combinatory logic
Objects, combinators and terms
Various kinds of combinators
Reductions and combinatory bases
Main theorems
Church–Rosser property
Normal forms and consistency
Fixed points
Second fixed point theorem and undecidability
Recursive functions and arithmetic
Primitive and partial recursive functions
First modeling of partial recursive functions in CL
Second modeling of partial recursive functions in CL
Undecidability of weak equality
Connections to l-calculi
l-calculi: L
Combinators in L
Back and forth between CL and L
(In)equational combinatory logic
Inequational calculi
Equational calculi
Term models
Operational models
Encoding functions by numbers
Models for typed CL
Relational models
Dual and symmetric combinatory logics
Dual combinators
Symmetric combinators
Structurally free logics
Applied combinatory logic
Illative combinatory logic
Elimination of bound variables
Typed combinatory logic
Simply typed combinatory logic
Intersection types for combinators
Elements of combinatory logic
Main theorems
Recursive functions and arithmetic
Connections to l-calculi
(In)equational combinatory logic
Dual and symmetric combinatory logic
Applied combinatory logic
Typed combinatory logic
List of Symbols

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Katalin Bimbo is an assistant professor in the Department of Philosophy at the University of Alberta in Edmonton, Canada.


For beginners, it is a compact introduction, including exercises, to the classical syntactic theory of combinators with some pointers to their models and their relation with λ-calculus. More advanced readers may find in the book much information on the connections between combinators and non-classical and substructural logics that are now a prominent topic in several areas, from philosophical logic to theoretical computer science, information that is mostly scattered through the research literature.

One of the commendable aspects of the book is its extensive and up-to-date bibliography, which deals with CL and other relevant topics in logic; it will surely aid many readers who may need to brush up on background information in the course of their study.
—Computing Reviews, 2012