# Solving Higher-Order Equations

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Dezember 1997

## Beschreibung

### Beschreibung

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.### Inhaltsverzeichnis

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.

2.8.1 Functional-Logic Programming.

2.8.2 Conditional Narrowing.

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.

4.3.1 Equational Logic.

4.3.2 Confluence.

4.3.3 Termination.

5 Decidability of Higher-Order Unification.

5.1 Elimination Problems.

5.2 Unification of Second-Order with Linear Terms.

5.2.1 Unifying Linear Patterns with Second-Order Terms.

5.2.2 Extensions.

5.3 Relaxing the Linearity Restrictions.

5.3.1 Extending Patterns by Linear Second-Order Terms.

5.3.2 Repeated Second-Order Variables.

5.4 Applications and Open Problems.

5.4.1 Open Problems.

6 Higher-Order Lazy Narrowing.

6.1 Lazy Narrowing.

6.2 Lazy Narrowing with Terminating Rules.

6.2.1 Avoiding Lazy Narrowing at Variables.

6.2.2 Lazy Narrowing with Simplification.

6.2.3 Deterministic Eager Variable Elimination.

6.2.4 Avoiding Reducible Substitutions by Constraints.

6.3 Lazy Narrowing with Left-Linear Rules.

6.3.1 An Invariant for Goal Systems: Simple Systems.

6.3.2 A Strategy for Call-by-Need Narrowing.

6.3.3 An Implementational Model.

6.4 Narrowing with Normal Conditional Rules.

6.4.1 Conditional Rewriting.

6.4.2 Conditional Lazy Narrowing with Terminating Rules.

6.4.3 Conditional Lazy Narrowing with Left-Linear Rules.

6.5 Scope and Completeness of Narrowing.

6.5.1 Oriented versus Unoriented Goals.

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.

8.1.1 Hardware Synthesis.

8.1.2 Symbolic Computation: Differentiation.

8.1.3 A Functional-Logic Parser.

8.1.4 A Simple Encryption Problem.

8.1.5 "Infinite" Data-Structures and Eager Evaluation.

8.1.6 Functional Difference Lists.

8.1.7 The Alternating Bit Protocol.

8.2 Equational Reasoning by Narrowing.

8.2.1 Program Transformation.

8.2.2 Higher-Order Abstract Syntax: Type Inference.

9 Concluding Remarks.

9.1 Related Work.

9.1.1 First-Order Narrowing.

9.1.2 Other Work on Higher-Order Narrowing.

9.1.3 Functional-Logic Programming.

9.1.4 Functional Programming.

9.1.5 Higher-Order Logic Programming.

9.2 Further Work.

9.2.1 Implementation Issues.

9.2.2 Other Extensions.

EAN: 9780817640323

ISBN: 0817640320

Untertitel: From Logic to Programming.
1998. Auflage.
Book.
Sprache: Englisch.

Verlag: Birkhäuser

Erscheinungsdatum: Dezember 1997

Seitenanzahl: 204 Seiten

Format: gebunden

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