Termination and Confluence Properties of Structured Rewrite Systems Vom Fachbereich Informatik der Universit¨at Kaiserslautern zur Verleihung des akademischen Grades Doktor der Naturwissenschaften (Dr. rer. nat.) genehmigte Dissertation von Dipl.-Inform. Bernhard Gramlich Datum der wissenschaftlichen Aussprache: 19. Januar 1996 Dekan: Prof. Dr. Hans Hagen Promotionskommission: Vorsitzender: Prof. Dr. J¨urgen Nehmer Berichterstatter: Prof. Dr. J¨urgen Avenhaus Prof. Dr. Klaus E. Madlener D 386 Termination and Confluence Properties of Structured Rewrite Systems Vom Fachbereich Informatik der Universit¨at Kaiserslautern zur Verleihung des akademischen Grades Doktor der Naturwissenschaften (Dr. rer. nat.) genehmigte Dissertation von Dipl.-Inform. Bernhard Gramlich Datum der wissenschaftlichen Aussprache: 19. Januar 1996 Dekan: Prof. Dr. Hans Hagen Promotionskommission: Vorsitzender: Prof. Dr. J¨urgen Nehmer Berichterstatter: Prof. Dr. J¨urgen Avenhaus Prof. Dr. Klaus E. Madlener D 386 °c 1996 Bernhard Gramlich Abstract Termination and confluence properties of term rewriting systems are of fundamental importance, both from a theoretical point of view and also concerning many practical applications in computer science and mathematics. We study structural aspects of termination and confluence properties of unconditional and conditional term rewriting systems. Two types of structural aspects are considered. First we investigate single rewrite systems. In a systematic manner we analyze how restricted versions of termination and confluence relate to each other and to general termination and confluence. In particular we focus on innermost rewriting and its properties. Various structural syntactic and semantic conditions are isolated which guarantee equivalence of general termination (and confluence) and weakened versions of these properties. In a second major part we consider structural aspects of combining systems. Here we investigate modularity and preservation properties of rewrite systems under various types of combinations. We survey known results in a unifying framework and develop new approaches, in particular for ensuring the preservation of termination under various types of combinations. The abstract results obtained provide a deeper insight into the striking phenomena that rewriting in combined systems may exhibit, and entail various interesting positive consequences. Zusammenfassung Terminierungs- und Konfluenzeigenschaften von Termer- setzungssystemen sind von fundamentaler Bedeutung sowohl in theoretischer Hin- sicht als auch im Hinblick auf viele praktische Anwendungen in Informatik und Ma- thematik. Wir untersuchen strukturelle Aspekte von Terminierungs- und Konfluenz- eigenschaften bei unbedingten und bedingten Termersetzungssystemen. Dabei werden zweierlei Arten von strukturellen Aspekten betrachtet. Zun¨achst werden einzelne Ersetzungssysteme behandelt. In systematischer Art und Weise analysieren wir, wie abgeschw¨achte Formen von Termination und Konfluenz miteinander in Beziehung stehen, und welche Beziehungen zwischen ihnen und all- gemeiner Termination und Konfluenz bestehen. Besondere Ber¨ucksichtigung findet dabei die ‘innermost’-Reduktionsrelation, bei der minimale Teilterme, d.h. Teilterme an innnersten Stellen, ersetzt werden. Es werden verschiedene strukturelle, sowohl syn- taktische als auch semantische, Bedingungen hergeleitet, die hinreichend sind f¨ur die Aquivalenz¨ von allgemeiner Termination (und Konfluenz) und abgeschw¨achten Formen von Termination (und Konfluenz). Im zweiten Teil betrachten wir strukturelle Aspekte bei der Kombination mehrerer Sys- teme. Dabei werden Modularit¨ats- und Erhaltungseigenschaften unter verschiedenen Kombinationstypen untersucht. Bekannte Ergebnisse werden in einem einheitlichen Rahmen systematisch pr¨asentiert. Wir entwickeln wesentlich neue Ans¨atze zur Erhal- tung von Termination unter verschiedenen Kombinationstypen. Die hierbei erzielten abstrakten Resultate erm¨oglichen eine tiefere Einsicht in die erstaunlichen Ph¨anomene, die beim Termersetzen in kombinierten Systemen auftreten k¨onnen. Ferner lassen sich viele interessante positive Konsequenzen und Erhaltungss¨atze elegant daraus herleiten. Preface I am much indebted to my supervisors J¨urgen Avenhaus and Klaus Madlener who gave me the opportunity to write this thesis. Their continuous support and en- couragement as well as their critical comments, questions and hints concerning my work have always been very stimulating and helpful. Without their tenacious patience this thesis might have never been written. Furthermore I thank my colleagues for creating a pleasant and constructive atmosphere and for their interest in my work. In particular, I appreciate many valuable discussions on various (not only scientific) topics with Thomas Deiß, Roland Fettig, Ulrich K¨uhler, Norbert Kuhn, Birgit Reinert, Andrea Sattler-Klein, Joachim Steinbach and Claus- Peter Wirth. Thanks also to Mauricio Ayala Rincon, Klaus Becker, J¨org Denzinger, Matthias Fuchs, Carlos Lor´ıa-Sa´enz and Rita Kohl for their helpfulness. Special thanks to Andrea Sattler-Klein and Joachim Steinbach for many interesting discussions and feedback on topics related to this thesis, and to Claus-Peter Wirth for an intensive and fruitful cooperation on several related subjects. For their support with various LATEX problems I am grateful to Thomas Deiß and Eric Domenjoud. I am much obliged to Aart Middeldorp, for many stimulating and clarifying discussions on modularity issues as well as for his detailed and useful comments on a preliminary version of this thesis. Moreover, I wish to thank Vincent van Oostrom, Yoshihito Toyama, Nachum Dershowitz and David Plaisted for various feedback on my work and interesting discussions on related issues. And finally, I would like to apologize to all those who have suffered somehow during the last years because of all the time and energy I devoted to writing this thesis. Contents 1 Introduction and Overview 1 2 Preliminaries 9 2.1 AbstractReductionSystems . 9 2.2 TermRewritingSystems . 18 2.2.1 ConfluenceCriteria . 24 2.2.2 TerminationCriteria . 28 2.3 Conditional Term Rewriting Systems . 36 2.3.1 Confluence without Termination . 42 2.3.2 Confluence with Termination . 44 2.4 Combined Systems and Modularity Behaviour . 45 2.4.1 Introduction............................. 45 2.4.2 Basic Terminology . 49 3 Relating Termination and Confluence Properties 57 3.1 OrthogonalSystems ............................ 59 3.2 Non-OverlappingSystems . 60 3.3 LocallyConfluentOverlaySystems . 63 3.4 Extensions.................................. 69 3.5 ConfluenceofInnermostReduction . 83 3.6 ConditionalRewriteSystems. 90 3.6.1 Non-Overlapping Conditional Systems . 92 3.6.2 Conditional Overlay Systems with Joinable Critical Pairs . 95 4 Modularity of Confluence Properties 99 4.1 ConfluenceandLocalConfluence . 99 4.2 UniqueNormalFormProperties. 101 4.3 Non-DisjointUnions ............................ 104 4.4 ConditionalRewriteSystems. 108 5 Modularity of Termination Properties 113 5.1 HistoryandOverview............................ 114 5.1.1 SomeHistory ............................ 114 5.1.2 Basic Counterexamples . 115 5.1.3 Classification of Approaches . 118 5.2 Restricted Termination Properties . 119 5.2.1 Weak Termination and Weak Innermost Termination . 119 5.2.2 StrongInnermostTermination . 121 5.3 Termination................................. 121 5.3.1 The General Approach via an Abstract Structure Theorem . 121 5.3.2 The Modular Approach via Innermost Termination . 137 5.3.3 The Syntactic Approach via Left-Linearity . 141 5.4 Non-DisjointUnions ............................ 145 5.4.1 Restricted Termination Properties and Semi-Completeness . 145 5.4.2 Termination of Constructor Sharing / Composable Systems . 146 5.5 ConditionalRewriteSystems. 150 5.5.1 Termination Properties under Signature Extensions . 151 5.5.2 Restricted Termination Properties . 161 5.5.3 Termination and Completeness . 163 5.5.4 Non-Disjoint Unions . 166 6 Related Topics and Concluding Remarks 169 6.1 Hierarchical and Other Types of Combinations . 169 6.2 CombiningAbstractReductionSystems . 171 6.3 RelatedFields................................ 171 A Proofs 175 B A Parameterized Version of the Well-Founded Induction Principle 185 Bibliography 193 Index 209 Symbols 215 Chapter 1 Introduction and Overview In this introduction we shall first sketch the background of this thesis, namely the field of term rewriting, its basic idea, some history and applications. Then the main moti- vations for this work, its context and goals pursued are briefly discussed. A detailed summary of the thesis and its main contributions follow. And finally, we mention some aspects concerning the presentation philosophy. Term Rewriting: Basic Idea, History and Applications Term rewriting systems provide an elegant, abstract and simple, yet powerful, com- putation mechanism. The basic idea is very simple: replacement of equals by equals by applying symbolic equations over symbolically structured objects, terms. Apply- ing equations in one direction only immediately leads to the concept of (directed) term rewriting. Since different parts of structured objects, the subterms, can be re- placed by applying different term rewriting or rewrite rules, this obviously leads to non-deterministic computations. Moreover, after one computation (rewrite) step the same kind of computation may be possible again. Hence two basic questions naturally arise: • Do all computations eventually stop? • If there exist diverging computations (i.e., computations