Zur Erlangung des akademischen Grades eines. Doktors der Naturwissenschaften. der Technischen Fakultat. der Universitat Bielefeld. vorgelegte.

Transcription

1 Modular Properties of Composable Term Rewriting Systems Zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften der Technischen Fakultat der Universitat Bielefeld vorgelegte Dissertation von Enno Ohlebusch Bielefeld, im Mai 1994

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3 Acknowledgements I am much obliged to my supervisor Robert Giegerich who made the writing of this thesis possible. His attitude towards teaching and research and his ability to break with traditional thought patterns has always been an example to me. I am much indebted to Aart Middeldorp. His excellent thesis called my attention to modularity issues in term rewriting and his willingness to referee this thesis completes the circle. I have benetted from many discussions with him and a good many of his suggestions have improved previous papers. I am grateful to my former and present colleagues in Bielefeld for creating a pleasant atmosphere. Special thanks go to Horst Hogenkamp for the LaT E X support, to Anke Bodzin for typesetting parts of previous papers, and to Stefan Kurtz and Bernd Butow. Last, but not least I thank my wife Silvia for her continuous encouragement.

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5 Contents 1 Introduction 1 2 Preliminaries Abstract Reduction Systems : : : : : : : : : : : : : : : : : : : : : : : : : : : Partial Orderings : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Term Rewriting Systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Decidability of Properties : : : : : : : : : : : : : : : : : : : : : : : : 22 3 Modular Properties Various Combinations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Results: An Overview : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Composable Systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Basic Notions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Testing for Pairwise Composibility : : : : : : : : : : : : : : : : : : : Dependency Components : : : : : : : : : : : : : : : : : : : : : : : : : Local Conuence and Normalization : : : : : : : : : : : : : : : : : : : : : : Semi-Completeness : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 44 4 Conuence A Counterexample : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : A Sucient Criterion : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Collapsing Reduction : : : : : : : : : : : : : : : : : : : : : : : : : : : Conuence of White Preserved Terms : : : : : : : : : : : : : : : : : : Conuence is Modular if! c is Normalizing : : : : : : : : : : : : : : Consequences of the Sucient Criterion : : : : : : : : : : : : : : : : : : : : : Toyama's Theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : Modularity of Semi-Completeness for Constructor-Sharing Systems : Left-Linear Systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 59 i

6 ii CONTENTS 5 Termination Simple Termination : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Disjoint Systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : The Famous Counterexample : : : : : : : : : : : : : : : : : : : : : : Non-Collapsing and Non-Duplicating Systems : : : : : : : : : : : : : Sucient Conditions for the Modularity of Termination : : : : : : : : Composable Systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Layer-Preservation and Non-Duplication : : : : : : : : : : : : : : : : The Simplifying Property : : : : : : : : : : : : : : : : : : : : : : : : 84 6 Completeness Disjoint Systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Counterexamples : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Left-Linear Systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : Variable-Preserving Systems : : : : : : : : : : : : : : : : : : : : : : : Composable Systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Overlay Systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Combining Reduction Strategies Disjoint Systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Constructor-Sharing Systems : : : : : : : : : : : : : : : : : : : : : : : : : : Conditional Term Rewriting Systems Basic Notions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Results: A Brief Summary : : : : : : : : : : : : : : : : : : : : : : : : : : : : Constructor-Sharing Systems : : : : : : : : : : : : : : : : : : : : : : : : : : Local Conuence and Normalization : : : : : : : : : : : : : : : : : : Conuence and Semi-Completeness : : : : : : : : : : : : : : : : : : : Termination and Completeness : : : : : : : : : : : : : : : : : : : : : Combining Reduction Strategies : : : : : : : : : : : : : : : : : : : : : Composable Systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : The Simplifying Property : : : : : : : : : : : : : : : : : : : : : : : : Conditional Constructor Systems : : : : : : : : : : : : : : : : : : : : Related Subjects Unique Normal Forms and Related Properties : : : : : : : : : : : : : : : : : 153

7 CONTENTS iii 9.2 Hierarchical Combinations : : : : : : : : : : : : : : : : : : : : : : : : : : : : Commutation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Term Graph Rewriting : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Many-Sorted Term Rewriting : : : : : : : : : : : : : : : : : : : : : : : : : : Applicative Term Rewriting : : : : : : : : : : : : : : : : : : : : : : : : : : : Open Problems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 161 A Kruskal's Tree Theorem 165 A.1 Well-Quasi-Orderings : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 165 A.2 Kruskal's Tree Theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 166 Bibliography 169 Index 179 List of Notations 183

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9 Chapter 1 Introduction and Overview Equations belong to the rst mathematical achievements of mankind. For instance, they appear in old Babylonian texts written in cuneiform characters dating back as far as to the third millennium B.C. This is not so surprising because equational reasoning (replacing equals with equals, that is) occurs frequently in day-to-day life, for example in calculating simple arithmetic expressions like (1 + 2) 3. Using the equation = 3, the expression or the term (1 + 2) 3 is replaced with 3 3 which further rewrites to 9. In calculating the intuitively simplest term equal to (1 + 2) 3, no one would use the equation = 3 to replace the term (1+2)3 with the term (1+2)(1+2). So the equation 1+2 = 3 is used only in one direction, and this one-directional replacement of equals with equals is the basis of term rewriting. Thus a term rewriting system consists of a set of directed equations, so-called rewrite rules. Representing natural numbers by the terms 0; S(0); S(S(0)); S(S(S(0))); : : : the simplest form of every simple arithmetical expression of the above form can be computed using the term rewriting system R = 8 >< >: 0 + x! x S(x) + y! S(x + y) 0 x! 0 S(x) y! (x y) + y For instance, 2 1 is represented by the term S(S(0)) S(0) which, by the above rewrite rules, rewrites to the term S(S(0)) as follows: S(S(0)) S(0)! (S(0) S(0)) + S(0)! ((0 S(0)) + S(0)) + S(0)! (0 + S(0)) + S(0)! S(0) + S(0)! S(0 + S(0))! S(S(0)) The term S(S(0)) cannot be further rewritten, this so-called normal form is the result of the computation. So a term rewriting system is a kind of program that computes by rewriting or reducing terms to other terms until a normal form is obtained. Hence the theory of rewriting is concerned with the computation of normal forms. The above term rewriting system has two fundamental properties, it is conuent and terminating. If a term rewriting system is conuent, then the normal form of a given term is 1

10 2 CHAPTER 1. INTRODUCTION unique. A terminating term rewriting system does not permit innite computations, that is, every computation starting at a term ends nally in a normal form. A term rewriting system that is both conuent and terminating is said to be complete. So if a term rewriting system is complete, then every computation leads to a result and this result is unique { no matter in which order the rewrite rules are applied. Another property weaker than termination is normalization. A term rewriting system is normalizing if every term t has a normal form; in other words, there exists a terminating computation starting from t. The combination of conuence and normalization is called semi-completeness. Thus, if a term rewriting system is semi-complete, then every term has a unique normal form and all that is further needed is a good normalizing reduction strategy { a strategy which eciently computes that unique normal form. Term rewriting has applications in various elds of computer science such as symbolic computation, functional programming, abstract data type specications, program verication, program synthesis, and automated theorem proving. We refer to the surveys of Dershowitz & Jouannaud [DJ90] and Klop [Klo92] for details. Term rewriting has its roots in -calculi (Church [Chu41]) and combinatory logic (Curry & Feys [CF58]). In their outstanding paper [KB70], Knuth & Bendix describe a completion procedure which can often be successfully used to transform a given set of equations into a complete term rewriting system which denes the same equational theory (see also Bachmair [Bac89]). Thus term rewriting systems provide an operational model of algebraic specications of abstract data types (cf. [GTW78]). Large specications, however, must be written in a modular way according to the one page principle of Mark Ardis: \A specication that will not t on one page of 8:5 11 inch paper cannot be understood". Modularity is a well-known programming paradigm in computer science. Programmers should design their programs in a modular way, that is, as a combination of small programs. These so-called modules are implemented separately and are then integrated to form the whole program. Since term rewriting systems have important applications in computer science, it is { not only from a theoretical viewpoint but also from a practical point of view { of utmost importance to know under which conditions a combined system inherits desirable properties from its constituent systems. For this reason modular aspects of term rewriting have been receiving increasing attention. A property P of term rewriting systems (like conuence, termination etc.) is called modular if whenever R 1 and R 2 are term rewriting systems both satisfying P, then their combined system R 1 [ R 2 also satises P. Since all interesting properties are in general not modular, the starting-point of research were disjoint unions, combinations of term rewriting systems having no function symbols in common (see [Toy87b, Toy87a, Rus87a, TKB89, Mid90b]). Kurihara and Ohuchi [KO91, KO92] investigated constructor-sharing systems; constructors are function symbols that do not occur at the root position of the left-hand side of any rewrite rule, the others are called dened symbols. Middeldorp and Toyama [MT93] introduced composable systems which have to contain all rewrite rules that dene a dened symbol whenever that symbol is shared. The authors, however, restricted their investigations to constructor systems (where no proper subterm of a left-hand side of a rewrite rule is allowed to contain dened symbols). We drop the latter requirement, so the composable systems we consider are a proper generalization of constructor-sharing systems. The title of this thesis reects that the combination of composable systems is the most general kind of combination which will be investigated here. Nevertheless, many results only apply to constructor-sharing or disjoint systems. The

11 3 remainder of this chapter consists of a brief overview of the thesis. In Chapter 2, we follow Klop's approach to term rewriting (see [Klo92]). This introduction to term rewriting is more or less self-contained. Section 2.1 is concerned with abstract reduction systems which are just pairs consisting of a set and a binary relation thereon. The fundamental properties of term rewriting systems are described on this abstract level. Section 2.2 deals with the basic notions of terms and term rewriting systems which are just special abstract reduction systems. In Chapter 3, we start the investigation of modular properties of term rewriting systems. Section 3.1 contains the exact denitions of the dierent kinds of combinations including hierarchical combinations which are also a proper generalization of constructor-sharing systems (cf. [Der93, KR93a, KR94]). Composable systems, however, are neither a generalization of hierarchical systems nor vice versa. We focus on composable systems rather than hierarchical systems. In Section 3.2, the principal new results of this thesis (concerning unconditional term rewriting) are summarized and related to known modularity results. Section 3.3 is dedicated to composable systems: At rst, all the technical denitions needed for the subsequent analysis of combinations of composable systems are supplied. Then it is shown how a nite number of term rewriting systems can eciently be tested for pairwise composibility. Finally, we prove that under certain conditions a term rewriting system has a property modular for composable systems if and only if each of its dependency components has that property. Subsections and will hopefully help the reader to acquaint herself/himself with combinations of composable systems notwithstanding the fact that the results of these subsections will not be used in subsequent sections. In the remainder of Chapter 3, the modularity of local conuence, normalization, innermost termination, and semi-completeness for composable term rewriting systems is proved. The most remarkable result is certainly the modularity of semi-completeness. Chapter 4 deals exclusively with the conuence property. By now, it is well-known that conuence is modular for disjoint systems. This was rst proved by Toyama in his pioneering paper [Toy87b]. This result is thus sometimes referred to as Toyama's Theorem. In contrast to this encouraging result, conuence is not modular for constructor-sharing systems (cf. [KO92]). So Kurihara and Krishna Rao posed the problem whether there are some interesting sucient conditions (independent of termination) which ensure the modularity of conuence for constructor-sharing systems. This question occurred as Problem 59 in the list of open problems in term rewriting, see [DJK93]. A positive solution to Problem 59 has appeared in [Ohl94b]. In Chapter 4, the results of [Ohl94b] are partially generalized. By an extension of the techniques used by Klop et al. [KMTV94] for disjoint unions, it will indeed be proved that conuence is a modular property for composable systems provided that a certain collapsing reduction relation is normalizing. Toyama's Theorem is a corollary thereof and furthermore an alternative proof for the modularity of semi-completeness for constructor-sharing systems can be given. The chapter is concluded with some remarks on the special behavior of leftlinear systems with respect to the preservation of conuence. Chapter 5 is devoted to the termination property. First, the closely related properties simple termination, C E -termination, and the simplifying property are introduced and the relationships between them are analysed. Section 5.2 starts with the famous counterexample of Toyama [Toy87a] to the modularity of termination for disjoint systems. The rst sucient criteria ensuring a modular behavior of termination were obtained by Rusinowitch [Rus87a].

12 4 CHAPTER 1. INTRODUCTION He showed that termination is modular for non-collapsing and for non-duplicating disjoint term rewriting systems. Moreover, he conjectured that the disjoint union of two terminating term rewriting systems is also terminating provided that one of them contains neither collapsing nor duplicating rewrite rules. This conjecture has been proven true by Middeldorp [Mid89a]. Subsection contains one simple proof for the three results mentioned above. This proof has already appeared in [Ohl93a]. In Subsection more sucient conditions for the modularity of termination for disjoint systems are supplied (see also [Ohl93c]). We generalize results of [Gra93a] from nitely branching systems to arbitrary systems, thereby solving a conjecture of Gramlich [Gra93a]. The most important consequences are the modularity of simple termination and C E -termination, respectively. Unfortunately, these results do not extend to constructor-sharing systems as is shown at the beginning of Section 5.3. To the contrary, the results of Rusinowitch and Middeldorp do extend, mutatis mutandis, to constructor-sharing systems (see [Ohl93c]), and even to composable systems (Subsection 5.3.1). In an inuential paper, Kurihara and Ohuchi [KO92] proved that the simplifying property (simple termination in their terminology) is modular for constructor-sharing systems. Chapter 5 is concluded with a proof showing that this holds true even for composable systems. In Chapter 6, we consider completeness, the conjunction of conuence and termination. To begin with, we recall the interesting history of conjectures and counterexamples concerning the modularity of completeness for disjoint systems. Then we provide an alternative proof for a special case of the deep theorem of Toyama et al. [TKB89] stating that completeness is modular for left-linear disjoint term rewriting systems. Their theorem suggests investigating whether completeness is also modular for variable-preserving disjoint systems. Surprisingly, this is not the case. Owing to a suggestion of Hans Zantema, the presentation of the complex counterexample in Subsection diers from the one in [Ohl93d]. A counterexample at the beginning of Section 6.2 demonstrates that the result of Toyama et al. [TKB89] does not extend to constructor-sharing systems. Furthermore, we proer a new simple proof for an interesting result of Gramlich [Gra92b, Gra93b] saying that an overlay system is complete if and only if it is locally conuent and innermost terminating. Since local conuence and innermost termination are modular for composable systems, the modularity of completeness for composable overlay systems is an immediate consequence. This slightly generalizes modularity results of [Gra92b, Gra93b] and [MT93]. In Chapter 7, we introduce the so-called modular reduction relation which is due to Kurihara and Kaji [KK88]. It has been shown by the aforesaid authors that this modular reduction relation is terminating whenever the constituent term rewriting systems are pairwise disjoint (cf. [KK88, KK90]). In [KO91], Kurihara and Ohuchi remarked that the proof idea of [KK90] also applies to pairwise constructor-sharing systems. We provide a rigorous proof for this fact under the premise that the constituent systems are normalizing. Moreover, the modular reduction relation proves to be complete whenever the constituent term rewriting systems are pairwise constructor-sharing and semi-complete (this was already known for pairwise disjoint systems, see [KK88, Mid90b]). Chapter 8 deals with conditional term rewriting systems. The rewrite rules of those systems may possess conditions, and such a conditional rewrite rule is only applicable if its conditions are fullled. We focus on the most prominent kind of conditional term rewriting systems, the so-called join or standard systems. Section 8.1 provides a summary of the basic concepts of conditional term rewriting which is inherently more complicated than

13 5 unconditional term rewriting. A brief overview of our new results concerning conditional term rewriting is contained in Section 8.2. Most of our results only apply to constructorsharing systems { as a matter of fact, up until now no positive modularity result is known for the combination of composable conditional term rewriting systems which may have extra variables in their conditions. Middeldorp [Mid90b, Mid93b] was the rst to investigate modular properties of (disjoint) conditional term rewriting systems. Among other things, he showed that for disjoint conditional term rewriting systems conuence and semi-completeness are modular whereas local conuence and normalization lack a modular behavior. So the best one can hope for when considering constructor-sharing conditional term rewriting systems is the modularity of semi-completeness (all other above-mentioned properties cannot be modular for those systems since they already fail to be modular for more restricted systems). We prove that semi-completeness is indeed modular for constructor-sharing conditional term rewriting systems. Middeldorp [Mid90b, Mid93b] has also shown that termination is modular for non-collapsing, and completeness is modular for non-duplicating disjoint conditional term rewriting systems. Furthermore, he conjectured (see [Mid90b, Mid93b]) that the disjoint union of two terminating join conditional term rewriting systems is terminating if one of them contains neither collapsing nor duplicating rules and the other is conuent. We will refute this conjecture by a simple counterexample. Moreover, it will be shown that his results also hold, mutatis mutandis, in the presence of shared constructors. We point out that our proof (though based on the ideas of [Mid93b]) is considerably simpler than that of [Mid93b]. At the end of Section 8.3, we extend the results of Chapter 7 to pairwise constructor-sharing conditional term rewriting systems. In this context, nite and decreasing conditional term rewriting systems play a special r^ole. Finally, it is shown in Section 8.4 that the simplifying property is modular, even for composable conditional term rewriting systems. Parts of the results of Chapter 8 are stated in [Ohl94a] and previously obtained special cases of some of these results have appeared already in [Ohl93b]. Various results from related subjects are collected in Chapter 9. In Section 9.1, simple counterexamples show that none of the known modularity results for disjoint term rewriting systems about unique normal forms and related properties extend to constructor-sharing systems. Several recent modularity results for hierarchical systems are gathered in Section 9.2. Some commutation results proved to be useful in establishing the preservation of properties like conuence and termination. The fundamental ones are recalled in Section 9.3. Since term rewriting is often implemented by means of the more ecient graph rewriting, we also summarize the modularity results concerning term graph rewriting systems. In Section 9.5, modular aspects of many-sorted term rewriting are recapitulated. Finally, in Section 9.6, we briey summarize related work on applicative term rewriting and -calculus extensions. Chapter 9 is concluded with a list of interesting open problems. Some of the results of Chapter 5 depend on a well-known theorem, viz. Kruskal's Tree Theorem. For the sake of completeness, Appendix A contains a proof of the nite version of that theorem in the setting of term rewriting.

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15 Chapter 2 Preliminaries We will introduce term rewriting systems by rst abstracting from the term structure. That is to say, at rst we shall concentrate on the so-called abstract reduction systems. This approach to term rewriting has been propagated by Klop [Klo87, Klo92] and it has two main advantages. Firstly, it is instructive to see which denitions and properties depend on the term structure and which are more fundamental. Secondly, abstract reduction systems comprise both unconditional and conditional term rewriting systems. So a repetition of similar denitions and concepts is avoided by stating them once and for all on an abstract level. 2.1 Abstract Reduction Systems Denition An abstract reduction system (ARS) is a pair A = (A;!) consisting of a set A and a binary relation! AA. We write a! b for (a; b) 2! and say that a reduces to b in one step and that b is a one step reduct of a. A reduction sequence or derivation is a sequence a 1 ; a 2 ; a 3 ; : : : of elements of A such that a 1! a 2! a 3! : : :. Denition Let A = (A;!) be an ARS. 1. The transitive closure of! is denoted by! + and! = denotes the reexive closure. 2. The reexive transitive closure of! is denoted by!. If a! b, we say that a reduces or rewrites to b and we call b a reduct of a. We use the notation a a b if b! a and b! a, respectively. b and 3. The symmetric closure of! is denoted by $ (i.e., $ =! [ ). The reexive transitive closure of $ is called conversion or convertibility and denoted by $. 4. Two elements a; b 2 A are joinable, written a # b, if there is a c 2 A such that a! c b. We call c a common reduct of a and b. The following denitions describe basic properties of ARSs. Dierent terminology also used in the literature is supplied in parenthesis. In the sequel, the phrases \an ARS A = (A;!) has a certain property" and \! has a certain property" are used synonymously whenever the underlying set A is clear from the context. 7

16 8 CHAPTER 2. PRELIMINARIES Denition Let A = (A;!) be an ARS. 1. A is locally conuent (weakly Church-Rosser) if, for all a; b; c 2 A with b a! c, the elements b and c have a common reduct, that is, there exists a d 2 A such that b! d c. 2. A is conuent (Church-Rosser) if, for all a; b; c 2 A with b a! c, the elements b and c have a common reduct d. a a b local confluence c b * * confluence c d * * * * d Figure 2.1 It is easy to see that conuence implies local conuence. The next example given by Hindley [Hin74] shows that the converse is not true. Example The following ARS is locally conuent but not conuent: a b c d Figure 2.2 The next well-known proposition will be tacitly used in the sequel. Proposition An ARS A = (A;!) is conuent if and only if every pair of convertible elements of A is joinable. Proof: Let A = (A;!) be conuent and let a; b 2 A such that a $ b. It will be shown that a and b have a common reduct. We proceed by induction on the number of reduction steps in a $ b. The case of zero steps holds trivially. Consider a $ c $ b. By the induction hypothesis, there is a d 2 A such that a! d c. Now c b immediately implies a # b. If c! b, then conuence of! yields a common reduct e of d and b which is also a common reduct of a and b. The converse direction (if-case) is trivial. 2 Denition Let A = (A;!) be an ARS. 1. An element a 2 A is in normal form w.r.t! or irreducible w.r.t! if there is no element b 2 A with a! b. An element a 2 A has a normal form w.r.t.! if a! b for some normal form b. The phrase \w.r.t.!" will be suppressed whenever! can be inferred from the context. The set of normal forms of A is denoted by NF(A) or NF(!) if there is no ambiguity about the set A.

17 2.1. ABSTRACT REDUCTION SYSTEMS 9 2. A is normalizing (weakly normalizing) if every term has a normal form. 3. A is terminating (strongly normalizing, noetherian) if there is no innite reduction sequence a 1! a 2! a 3! : : :. Termination obviously implies normalization. Example refutes the converse. Denition A conuent and normalizing ARS is said to be semi-complete (uniquely normalizing). 2. A conuent and terminating ARS is said to be complete (convergent). It is clear that completeness implies semi-completeness. The removal of the arrow from b to a in Example yields an ARS which is semi-complete but not complete. Denition Let A = (A;!) be an ARS. 1. A has unique normal forms with respect to reduction if no element of A reduces to dierent normal forms (i.e., 8a; b; c 2 A if a! b, a! c and b; c 2 NF (!), then b = c). 2. A has unique normal forms if dierent normal forms are not convertible (i.e., 8a; b 2 A if a $ b and a; b 2 NF (!), then a = b). 3. A has the normal form property if every element convertible to a normal form reduces to that normal form (i.e., 8a; b 2 A if a $ b and b 2 NF (!), then a! b). Proposition and Example illustrate the connections between the properties dened in Denition The simple proofs are omitted. Proposition Conuence implies the normal form property. 2. The normal form property implies unique normal forms. 3. An ARS with unique normal forms has unique normal forms w.r.t. reduction. 2 Example Consider the ARSs A 1, A 2 and A 3 dened by: (1) a (2) a (3) a d b c b c b c e It is easy to see that A j disproves the converse of the j-th statement in Proposition Another important fact is that an ARS is semi-complete if and only if it is normalizing and has unique normal forms w.r.t. reduction. This can be concluded from Proposition in combination with the following proposition.

18 10 CHAPTER 2. PRELIMINARIES Proposition A normalizing ARS A = (A;!) with unique normal forms w.r.t. reduction is conuent. Proof: Let a; b, and c be elements of A such that b a! c. It has to be shown that b # c. Since A is normalizing, b and c reduce to normal forms b 0 and c 0, respectively. Hence b 0 b a! c! c 0, where b 0 ; c 0 2 NF (A). It follows that b 0 = c 0 because A has unique normal forms w.r.t. reduction. b 0 = c 0 is evidently a common reduct of b and c. 2 The next lemma was rst proved by Newman [New42] and is thus referred to as Newman's Lemma. A simple proof of this very famous lemma was given by Huet [Hue80] but instead we will follow the elegant indirect proof of Barendregt [Bar84]. Lemma A terminating ARS A = (A;!) is conuent if and only if it is locally conuent. Proof: We have to prove that for a terminating ARS, local conuence implies conuence. According to Proposition it suces to show that A has unique normal forms w.r.t. reduction. For a proof by contradiction, suppose that A does not have unique normal forms w.r.t. reduction. Consequently, the set S = fa 2 A j a 1 a! a 2 ; a 1 ; a 2 2 NF (A); a 1 6= a 2 g is not empty. It is shown that for every a 2 S, there is an a 0 2 S such that a! a 0. This implies the existence of an innite reduction sequence a! a 0! a 00! : : : and contradicts the termination of A. So let a 2 S, i.e., a reduces to two distinct normal forms a 1 and a 2. It is clear that there are b 1 ; b 2 2 A such that a 1 b 1 a! b 2! a 2. Since A is locally conuent, b 1 and b 2 have a common reduct b. b reduces to some a 3 2 NF (A) because A is terminating (see Figure 2.3). Now set a 0 = b 1 if a 3 = a 2 6= a 1 and a 0 = b 2 otherwise. Evidently, a 0 2 S and a! a a X X X b XXXz 1 X X X z b 9 b 2??? a 1 a 3 a 2 Figure 2.3 Denition An ARS A = (A;!) is called nitely branching if for all a 2 A the set fb 2 A j a! bg of one step reducts of a is nite. Moreover, the set G(a) = fb 2 A j a! bg is called the reduction graph of a 2 A. Structuring the set G(a) by the relation! yields an ARS (cf. Klop [Klo92]). The next lemma is a reformulation of Konig's Lemma. Lemma Let A = (A;!) be a nitely branching and terminating ARS. Then G(a) is nite for any a 2 A. 2 Figure 2.4 summarizes the relationships between the various properties.

19 2.1. ABSTRACT REDUCTION SYSTEMS 11 termination & local completeness confluence normalization confluence normal form property semi- & completeness Partial Orderings unique normal forms w.r.t. -> Figure 2.4 unique normal forms Partial orderings can be regarded as abstract reduction systems with special properties. In the following, the phrase \partial ordering" will always mean \irreexive partial ordering". Denition A (irreexive) partial ordering (A; >) is a pair consisting of a set A and a binary irreexive and transitive relation > on A. 2. A partial ordering (A; >) is said to be well-founded if there are no innite reduction sequences a 1 > a 2 > a 3 > : : : of elements from A. Note that every partial ordering (A; >) is asymmetric (i.e., there are no elements a; b 2 A such that a > b and b > a). Denition A quasi-ordering (A; ) is a pair consisting of a set A and a binary reexive and transitive relation on A. 2. A reexive partial ordering (A; ) is a quasi-ordering which is also anti-symmetric. Note that a quasi-ordering may fail to be a reexive partial ordering because it is not anti-symmetric, i.e., there may be two distinct elements a; b 2 A such that a b and b a. Due to the next lemma, we can alternate between reexive and irreexive partial orderings.

20 12 CHAPTER 2. PRELIMINARIES Lemma Given a partial ordering (A; >), the denition a b if a > b or a = b yields a reexive partial ordering (A; ). 2. Given a reexive partial ordering (A; ), the denition a > b if a b and a 6= b yields a partial ordering (A; >). 3. Given a quasi-ordering (A; ), the denition a > b if a b but b 6 a yields a partial ordering (A; >). Proof: Routine. 2 The reexive or irreexive partial ordering obtained by one of the above denitions is called the associated partial ordering. It should be pointed out that given a quasi-ordering the denition a > b if a b and a 6= b does not in general yield an irreexive partial ordering. If, however, the quasi-ordering is actually a reexive partial ordering (i.e., it is also anti-symmetric), then the two above denitions of the associated irreexive partial ordering coincide. New partial orderings evolve by means of the denitions below. Denition The lexicographic product of n partial orderings (A j ; > j ), j 2 f1; : : : ; ng, is the partial ordering (A 1 : : : A n ; > lex ), where > lex is dened by (a 1 ; : : : ; a n ) > lex (b 1 ; : : : ; b n ) if a j > j b j at the least j 2 f1; : : : ; ng for which a j 6= b j. 2. For a set A, let S(A) denote the set of all nite subsets of A. The set extension of a partial ordering (A; >) is the partial ordering (S(A); > set ) dened as follows: S 1 > set S 2 if there exist sets X; Y 2 S(A) such that ; 6= X S 1, S 2 = (S 1 n X) [ Y for all y 2 Y there exists an x 2 X such that x > y. 3. A multiset is a collection in which elements are allowed to occur more than once. 1 In order to distinguish multisets from sets, we use brackets instead of braces for the former. If A is a set, then the set of all nite multisets over A is denoted by M(A). 4. The multiset extension of a partial ordering (A; >) is the partial ordering (M(A); > mul ) dened by: M 1 > mul M 2 if there exist multisets X; Y 2 M(A) such that ; 6= X M 1, M 2 = (M 1 n X) [ Y, where [ denotes the sum of multisets, for all y 2 Y there exists an x 2 X such that x > y. 1 Formal denitions concerning multisets can for example be found in [MW85].

21 2.1. ABSTRACT REDUCTION SYSTEMS 13 Example Let > be the usual well-founded partial ordering on the natural numbers IN. In its set extension (S(IN); > set ), there is the following reduction sequence: f3; 4; 0g > set f2; 1; 4; 0g > set f4; 0g > set f3; 1; 0g > set fg Analogously, in its multiset extension (M(IN); > mul ), we have a reduction sequence: [3; 3; 4; 0] > mul [3; 2; 2; 1; 1; 1; 4; 0] > mul [3; 4] > mul [3; 3; 3; 3; 2; 2] > mul [ ] Proposition A lexicographic product of well-founded partial orderings is wellfounded. Proof: Straightforward. 2 Proposition The multiset extension (M(IN); > mul ) of the usual well-founded ordering > on natural numbers is well-founded. Proof: It will be shown that the well-foundedness of (M(IN); > mul ) follows from the wellfoundedness of (IN n+1 ; > lex ), where IN n+1 denotes the (n + 1)-fold cartesian product of IN. Suppose there is an innite sequence M 1 > mul M 2 > mul M 3 > mul : : : of nite multisets M j 2 M(IN). Let n be the largest number occurring in M 1. Moreover, let c : IN M(IN)! IN be the function dened by the property that c(i; M j ) counts the multiplicity (i.e., the number of occurrences) of the number i in the multiset M j. Then (c(n; M 1 ); : : : ; c(0; M 1 )) > lex (c(n; M 2 ); : : : ; c(0; M 2 )) > lex (c(n; M 3 ); : : : ; c(0; M 3 )) > lex : : : is an innite reduction sequence in (IN n+1 ; > lex ) contrary to Proposition Hans Zantema (personal communication) called the author's attention to the simple proof of the preceding proposition. Dershowitz and Manna [DM79] proved that the preceding proposition extends to arbitrary partial orderings. Theorem The multiset extension of a partial ordering is well-founded if and only if the partial ordering is well-founded. 2 Although the proof of Theorem is not very dicult, we would like to stress that the proof of the preceding proposition is much simpler. Proposition The set extension of a partial ordering is well-founded if and only if the partial ordering is well-founded. Proof: Let (A; >) be a well-founded partial ordering. Suppose that the partial ordering (S(A); > set ) is not well-founded, that is to say, there is an innite sequence S 1 > set S 2 > set S 3 > set : : : of elements from S(A). Obviously, every S j can be considered as an element of M(A). Moreover, S j > set S j+1 implies S j > mul S j+1. Hence there is an innite sequence S 1 > mul S 2 > mul S 3 > mul : : : of elements from M(A). This contradicts Theorem The other implication is trivial. 2

22 14 CHAPTER 2. PRELIMINARIES 2.2 Term Rewriting Systems Now we present the basic concepts of term rewriting which are required in this thesis. More details on term rewriting, its applications, and related subjects can be found in the surveys of Dershowitz & Jouannaud [DJ90] and Klop [Klo92]. An early forerunner which can also be consulted, is the survey of Huet & Oppen [HO80]. Denition A signature is a countable set F of function symbols or operators, where every f 2 F is associated with a natural number denoting its arity (the number of \arguments" it is supposed to have). F n denotes the set of all function symbols of arity n, hence F = S n0 F n. Elements of F 0 are called constants. Denition Let F be a signature and let V be a countable set of variables with F \ V = ;. The set of terms T (F; V) is dened to be the smallest set such that V T (F; V) if f 2 F n and t 1 ; : : : ; t n 2 T (F; V), then f(t 1 ; : : : ; t n ) 2 T (F; V). We write f instead of f() for every constant f. The set of function symbols appearing in a term t 2 T (F; V) is denoted by Fun(t), and the set of variables occurring in t is denoted by Var(t). Terms without variables are called ground terms. The set of all ground terms is denoted by T (F). If a term t does not contain multiple occurrences of the same variable, it is said to be linear. Denition Let T (F; V) be given, and let t 2 T (F; V). symbol of t and is dened by: root(t) denotes the root root(t) = t if t 2 V, and root(t) = f if t = f(t 1 ; : : : ; t n ). Let sym 2 F [ V. jtj sym stands for the number of occurrences of the symbol sym in t. More precisely: jtj sym = 1 if t 2 V and t = sym, jtj sym = 1 + jt 1 j + : : : + jt n j if t = f(t 1 ; : : : ; t n ) and f = sym, where f 2 F 0 if n = 0 jtj sym = 0 if t 2 V and t 6= sym, jtj sym = 0 + jt 1 j + : : : + jt n j if t = f(t 1 ; : : : ; t n ) and f 6= sym, where f 2 F 0 if n = 0 jtj = S sym2f[v jtj sym denotes the overall number of symbols in t and is called the size of t. Denition Let 2 be a special constant symbol. A context is a term in T (F]f2g; V) 2. From the very beginning we distinguish between three kinds of contexts. This will be very convenient for later purposes. The context 2 The symbol ] denotes the union of disjoint sets.

23 2.2. TERM REWRITING SYSTEMS 15 C[; : : :; ] contains at least one occurrence of 2 and may be equal to 2. Ch; : : : ; i contains zero or more occurrences of 2 and may be equal to 2. Cf; : : : ; g contains zero or more occurrences of 2 and is dierent from 2. If C[; : : : ; ] is a context with n occurrences of 2 and t 1 ; : : : ; t n are terms, then C[t 1 ; : : : ; t n ] is the result of replacing from left to right the occurrences of 2 with t 1 ; : : : ; t n. The meaning of Cht 1 ; : : : ; t n i and Cft 1 ; : : : ; t n g is dened analogously. A context containing precisely one occurrence of 2 is denoted by C[ ]. A term t is a subterm of a term s if there exists a context C[ ] such that s = C[t]. A subterm t of s is proper, denoted by s > t, if s 6= t. By abuse of notation we write T (F; V) for T (F ] f2g; V), interpreting 2 as a special constant which is always available but used only for the aforementioned purpose. In those cases where the above notion of subterm is not precise enough, we will distinguish occurrences of subterms by means of positions. Denition Let t 2 T (F; V). 1. The set Pos(t) of positions in t is dened inductively: - Pos(t) = fg if t 2 V, - Pos(t) = fg [ fi:p j p 2 Pos(t i ) and 1 i ng if t = f(t 1 ; : : : ; t n ). A position p within a term t is thus a sequence of natural numbers (where denotes the empty sequence and the numbers are separated by dots) describing the path from the root of t to the root of the subterm occurrence tj p, where ( tj p = t if p = t i j q if p = i:q; 1 i n; and t = f(t 1 ; : : : ; t n ): 2. Positions are partially ordered by the so-called prex ordering, i.e., p q if there is an o such that p = q:o. In this case we say that p is below q or q is above p. If moreover p 6= q, then we say that p is strictly below q or q is strictly above p. Two positions are disjoint or independent if neither one is below the other. 3. Let p 2 Pos(t) and s 2 T (F; V). The term t[p s] is dened as follows: - t[p s] = s if p =, - t[p s] = f(t 1 ; : : : ; t i [q s]; : : : ; t n ) if p = i:q, 1 i n, and t = f(t 1 ; : : : ; t n ). If p 1 ; : : : ; p n 2 Pos(t) are pairwise disjoint, then t[p i s i j 1 i n] stands for t[p 1 s 1 ] : : : [p n s n ] (the order of the p i is irrelevant because the positions are pairwise disjoint). Example Let the signature F = fadd; Mult; 0; 1; 2; 3; : : :g be given. Consider the ground term t = Add(Mult(2; 3); Add(1; Mult(2; 3))). Its root symbol is Add, jtj M ult = 2, and its size is 9. The positions in t are depicted in Figure 2.5. There are two occurrences of the subterm Mult(2; 3) in t, viz. at the positions 1 and 2:2. Moreover, we have for instance t[2 Mult(4; 3)] = Add(Mult(2; 3); Mult(4; 3)).

24 16 CHAPTER 2. PRELIMINARIES Add λ Mult Add Mult Figure 2.5 Denition A substitution is a mapping from V to T (F; V) such that the domain Dom() = fx 2 V j (x) 6= xg of is nite. Occasionally we present a substitution as fx 7! (x) j x 2 Dom()g. The substitution with empty domain will be denoted by ". Every substitution extends uniquely to a morphism : T (F; V)! T (F; V), where (f(t 1 ; : : : ; t n )) = f((t 1 ); : : : ; (t n )) for each n-ary function symbol f and terms t 1 ; : : : ; t n. (t) is said to be an instance of t. We also use the notation t for (t). The composition of two substitutions and is dened by ( )(x) = ((x)). A substitution is a variable renaming if there is a substitution such that is the identity function on V. Denition Let s; t 2 T (F; V). s matches t if t is an instance of s. s and t are uniable if there is a substitution such that s = t. If s = t for some substitution, then is called a unier of s and t. A most general unier of s and t is a unier such that for every unier there exists a substitution with =. Matching and uniability of two terms s and t is decidable. Furthermore, if two terms are uniable, then they have a most general unier which is unique up to variable renamings. For instance, the algorithms of Robinson [Rob65] and Martelli & Montanari [MM82] construct a most general unier of two terms. Example Let F = ff; g; ag. If s = x and t = f(x), then s matches t via the substitution = fx 7! f(x)g but the terms s and t are not uniable. On the other hand, if s = g(f(x); y) and t = g(y; f(x)), then s does not match t but s and t are uniable. A unier is for instance the substitution = fy 7! f(a); x 7! ag. However, is not most general. A most general unier of s and t is = fy 7! f(x)g. Denition A term rewriting system (TRS for short) is a pair (F; R) consisting of a signature F and a set R T (F; V) T (F; V) of rewrite rules or reduction rules. Every rewrite rule (l; r) must satisfy the following two constraints:

25 2.2. TERM REWRITING SYSTEMS 17 l 62 V, i.e., the left-hand side l is not a variable, and Var(r) Var(l), i.e., variables occurring in the right-hand side r also occur in l. Rewrite rules (l; r) will be denoted by l! r. The rewrite rules of a TRS (F; R) dene a rewrite relation! R on T (F; V) as follows: s! R t if there exists a rewrite rule l! r in R, a substitution and a context C[ ] such that s = C[l] and t = C[r]. We call s! R t a rewrite step or reduction step. We also say that s rewrites to t by contracting redex l, where an instance of a left-hand side of a rewrite rule is called a redex (reducible expression). We often simply write R instead of (F; R) if the signature is clear from the context (by default we will assume F to consist of the function symbols occurring in R). All notions dened in Section 2.1 for abstract reduction systems carry over to term rewriting systems by associating the ARS (T (F; V);! R ) with the TRS (F; R). We will also use \localized" versions of these notions. The phrase \a term t is terminating" means for instance that there is no innite reduction sequence starting from t. We illustrate the above denition by a small example. The TRS we are going to present translates simple arithmetic expressions into code for a stack-machine. Example Let F = fcons, Nil, app, Load, trans, Add, Mult, ADD, MULT, 0; 1; 2; 3; : : :g and R = 8 >< >: app(n il; z)! z app(cons(x; y); z)! Cons(x; app(y; z)) app(x; app(y; z))! app(app(x; y); z) trans(add(x; y))! app(trans(x); app(trans(y); Cons(ADD; N il))) trans(m ult(x; y))! app(trans(x); app(trans(y); Cons(M U LT; N il))) trans(0)! Cons(Load(0); N il) trans(1)! Cons(Load(1); N il) trans(2)! Cons(Load(2); N il). In the following examplary reduction sequence every contracted redex is written in bold-face. trans(add(mult(2; 3); 4))! R app(trans(mult(2; 3)); app(trans(4); Cons(ADD; Nil)))! R app(trans(mult(2; 3)); app(cons(load(4); Nil); Cons(ADD; Nil)))! R app(trans(mult(2; 3)); Cons(Load(4); app(nil; Cons(ADD; Nil))))! R app(trans(mult(2; 3)); Cons(Load(4); Cons(ADD; Nil)))! + R Cons(Load(2); Cons(Load(3); Cons(MULT; Cons(Load(4); Cons(ADD; Nil))))) Denition Let (F; R) be a TRS. A reduction step s! R t is innermost, denoted by s! im R t if no proper subterm of the contracted redex is itself a redex. An innermost reduction sequence consists only of innermost reduction steps. R is innermost normalizing if, for every term s, there is an innermost reduction sequence s im! R t so that t 2 NF (! R ). R is innermost terminating if there is no innite innermost reduction sequence. A reduction step s! R t is outermost if the contracted redex is not a proper subterm of another redex. The notions outermost normalization and outermost termination are dened in analogy to the above notions.

26 18 CHAPTER 2. PRELIMINARIES Lemma We have the following implications: termination ) innermost termination ) innermost normalization ) normalization None of the converse implications is true in general. Of course the lemma remains true when \innermost" is replaced with \outermost". Proof: Routine. 2 Note that the beginning of the reduction sequence of Example is innermost. Like Dershowitz and Jouannaud [DJ90], we distinguish canonical TRSs from complete TRSs (although, in the literature, canonical is usually synonymous with complete). Denition A TRS (F; R) is called irreducible or (inter-)reduced, if every rewrite rule l! r of R satises: 1. the left-hand side l is irreducible w.r.t. R n fl! rg. 2. the right-hand side r is irreducible w.r.t. R. Denition A TRS (F; R) is called canonical if it is complete and irreducible. The TRS of Example cannot be canonical because the right-hand side of the fourth rewrite rule can for instance be reduced by the third rule. In this context, the following theorem of Metivier [Met83] is important. Theorem For every complete TRS (F; R), there is a canonical TRS (F; R 0 ) such that the relations $ R and $ R0 coincide on T (F; V). 2 The following classication of operators will be of utmost importance. It separates functions from data. Denition The rules of a TRS (F; R) partition F into two disjoint sets D = froot(l) j l! r 2 Rg of dened symbols, and C = F n D of constructors. The phrase dened function symbol is justied by the existence of dening rules for that symbol. The same holds true for constructors which only construct terms. Denition A TRSs (F; R) is a constructor system if every left-hand side F (t 1 ; : : : ; t n ) of a rewrite rule of R satises t 1 ; : : : ; t n 2 T (C; V). For the TRS (F; R) of Example , the set of dened symbols is D = fapp; transg and the set of constructors is C = fcons; Nil; Load; Add; Mult; ADD; MULT; 0; 1; 2; 3; : : :g. The system is not a constructor system but the removal of the third rule from R yields a constructor system. We shall next concern ourselves with a syntactic classication of reduction rules.

27 2.2. TERM REWRITING SYSTEMS 19 Denition Let (F; R) be a TRS. A rewrite rule l! r 2 R is called 1. left-linear if l is linear, 2. right-linear if r is linear, 3. variable-preserving (or non-erasing) if Var(l) = Var(r). (F; R) is said to have one of the properties (1){(3) if all of its rewrite rules have the respective property. The reduction rule l! r is called 4. collapsing if r 2 V, 5. duplicating if there is a variable x 2 V with jrj x > jlj x. (F; R) has one of the properties (4){(5) if one of its rewrite rules has the respective property. A glance at Example reveals that that system is left-linear, right-linear, variablepreserving, collapsing, and non-duplicating. In Section 5.1 we will acquaint ourselves with methods for proving termination of a given TRS. For terminating systems, there is a renowned sucient criterion which ensures conuence (Theorem ). Let us state some prerequisites rst. Denition Let R be a TRS. 1. Let l 1! r 1 and l 2! r 2 be (variable) renamed versions of rewrite rules of R such that they have no variables in common. Suppose l 1 = C[t] with t 62 V such that t and l 2 are uniable. Let be a most general unier of t and l 2. We call (C[r 2 ]; r 1 ) a critical pair of R. If the two rules are renamed versions of the same rewrite rule of R, we do not consider the case C[ ] = CP (R) denotes the set of all critical pairs between rules of R. 3. A critical pair (s; t) 2 CP (R) is called convergent if s and t are joinable w.r.t. R. Once again, consider the TRS R of Example The left-hand sides of the renamed rules app(cons(x; y); z)! Cons(x; app(y; z)) and app(u; app(v; w))! app(app(u; v); w) are unied via the most general unier = fu 7! Cons(x; y); z 7! app(v; w)g. Thus (app(app(cons(x; y); v); w); Cons(x; app(y; app(v; w)))) is a critical pair of R. The reader is invited to check that this critical pair is convergent and that CP (R) = f(app(app(n il; y); z); app(y; z)); (app(app(cons(x; y); v); w); Cons(x; app(y; app(v; w))))g: The next lemma is due to Huet [Hue80]. It is called the Critical Pair Lemma. Lemma A term rewriting system is locally conuent if and only if all its critical pairs are convergent. 2