Solving Higher-Order Equations
From Logic to Programming
Published on 18. December 1997
Book
Hardback
IX, 188 pages
978-0-8176-4032-3 (ISBN)
Description
This monograph develops techniques for equational reasoning in higher-order logic. Due to its expressiveness, higher-order logic is used for specification and verification of hardware, software, and mathematics. In these applica tions, higher-order logic provides the necessary level of abstraction for con cise and natural formulations. The main assets of higher-order logic are quan tification over functions or predicates and its abstraction mechanism. These allow one to represent quantification in formulas and other variable-binding constructs. In this book, we focus on equational logic as a fundamental and natural concept in computer science and mathematics. We present calculi for equa tional reasoning modulo higher-order equations presented as rewrite rules. This is followed by a systematic development from general equational rea soning towards effective calculi for declarative programming in higher-order logic and A-calculus. This aims at integrating and generalizing declarative programming models such as functional and logic programming. In these two prominent declarative computation models we can view a program as a logical theory and a computation as a deduction.
More details
Series
Edition
1998 ed.
Language
English
Place of publication
Boston
United States
Target group
Professional and scholarly
Research
Illustrations
IX, 188 p.
Dimensions
Height: 241 mm
Width: 160 mm
Thickness: 16 mm
Weight
477 gr
ISBN-13
978-0-8176-4032-3 (9780817640323)
DOI
10.1007/978-1-4612-1778-7
Schweitzer Classification
Other editions
Additional editions
Book
10/2012
Springer-Verlag New York Inc.
€106.99
Shipment within 15-20 days
Previous edition
Book
12/1997
Birkhäuser Verlag GmbH
€66.85
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Content
1 Introduction.- 2 Preview.- 2.1 Term Rewriting.- 2.2 Narrowing.- 2.3 Narrowing and Logic Programming.- 2.4 ?-Calculus and Higher-Order Logic.- 2.5 Higher-Order Term Rewriting.- 2.6 Higher-Order Unification.- 2.7 Decidability of Higher-Order Unification.- 2.8 Narrowing: The Higher-Order Case.- 3 Preliminaries.- 3.1 Abstract Reductions and Termination Orderings.- 3.2 Higher-Order Types and Terms.- 3.3 Positions in ?-Terms.- 3.4 Substitutions.- 3.5 Unification Theory.- 3.6 Higher-Order Patterns.- 4 Higher-Order Equational Reasoning.- 4.1 Higher-Order Unification by Transformation.- 4.2 Unification of Higher-Order Patterns.- 4.3 Higher-Order Term Rewriting.- 5 Decidability of Higher-Order Unification.- 5.1 Elimination Problems.- 5.2 Unification of Second-Order with Linear Terms.- 5.3 Relaxing the Linearity Restrictions.- 5.4 Applications and Open Problems.- 6 Higher-Order Lazy Narrowing.- 6.1 Lazy Narrowing.- 6.2 Lazy Narrowing with Terminating Rules.- 6.3 Lazy Narrowing with Left-Linear Rules.- 6.4 Narrowing with Normal Conditional Rules.- 6.5 Scope and Completeness of Narrowing.- 7 Variations of Higher-Order Narrowing.- 7.1 A General Notion of Higher-Order Narrowing.- 7.2 Narrowing on Patterns with Pattern Rules.- 7.3 Narrowing Beyond Patterns.- 7.4 Narrowing on Patterns with Constraints.- 8 Applications of Higher-Order Narrowing.- 8.1 Functional-Logic Programming.- 8.2 Equational Reasoning by Narrowing.- 9 Concluding Remarks.- 9.1 Related Work.- 9.2 Further Work.