A diagrammatic standardization theorem

Transcription

1 Axiomatic Rewriting Theory I A diagrammatic standardization theorem Pal-André Melliès Sbmitted Jly 2005 By extending nondeterministic transition systems with concrrency and copy mechanisms, Axiomatic Rewriting Theory proides a niform framework for a ariety of rewriting systems, ranging from higher-order rewriting systems to Petri nets and process calcli. Despite its generality, the theory is srprisingly simple, based on a mild extension of transition systems with independence: an axiomatic rewriting system is defined as a 1-dimensional transition graph G eqipped with 2-dimensional transitions describing the redex permtations of the system, and their orientation. In this article, we formlate a series of elementary axioms on axiomatic rewriting systems, and establish a diagrammatic standardization theorem. Forewords by the athor Many concepts of Rewriting Theory started in the λ-calcls which is by far the most stdied rewriting system prodced by history. A famos illstration is the conflence theorem formlated by A. Chrch and J.B. Rosser in the early years of the λ-calcls (Chrch, Rosser, 1936). The theorem was then generalized and applied extensiely to other rewriting systems, and became eentally an object of stdy in itself in a line of research pioneered by H.-B. Crry and R. Feys in their book on Combinatory Logic (1958), and clminating in a series of remarkable papers by G. Het, J. W. Klop, and J.-J. Léy pblished at the end of the 1970s and beginning of the 1980s. Today, more than half a centry after its apparition in the λ-calcls, the conflence property is niersally accepted among Compter Scientists as the theoretical principle nderlying deterministic comptations. This article is concerned with another famos and fndamental property of the λ-calcls, discoered by A. Chrch and J.B. Rosser qite at the same time as the conflence property: the standardization theorem. The main thesis of the article is that the standardization property is the theoretical principle nderlying casal comptations jst like the conflence property is the theoretical principle nderlying deterministic comptations. By casal comptation, I mean a comptation (or rewriting path) in which eery transition (or redex) is enabled by a canonical seqence (or cascade) of preios transitions. The framework in which the theory is deeloped entirely diagrammatic: the principles of casality are formlated as 3-dimensional properties satisfied by the 2-dimensional permtations on the 1-dimensional transitions of the rewriting system. A diagrammatic standardization theorem generalizes then the standardization theorem of the λ-calcls to a great ariety of rewriting systems, from higher-order calcli to Petri nets and process calcli since casal comptation is not limited to deterministic comptation. One of the most innoatie aspects of this work is certainly the obseration that the standardization theorem amonts to a 2-dimensional conflence property. The diagrammatic standardization theorem states indeed that applying 2-dimensional permtations to a rewrit- 1

2 ing path f leads eentally to a niqe rewriting path g modlo a notion of reersible permtation deeloped in the corse of the article. The standard rewriting path g is then defined as the normal form of this 2-dimensional rewriting procedre applied to the rewriting path f. For a long time, I thoght naiely that the idea was entirely new. I was ths astonished (and fascinated) to discoer dring my military stay in Amsterdam, arond 1995, that the idea appears in Jan Willem Klop s PhD thesis pblished in 1980 in his second proof of the standardization theorem for the λ-calcls. This was one occasion among seeral others to appreciate the extraordinary qality and insight of Jan Willem Klop s contribtion to Rewriting Theory. It is ths a great pleasre and honor to dedicate today this article to Jan Willem Klop, on the occasion of his 60th anniersary Standardization: from syntax to diagrams 1.1. Compting leftmost otermost is jdicios... in the λ-calcls The λ-calcls is the pre calcls of fnctionals. It has a niqe redction rle, called the β-rle, (λx.m)p M[x := P ] (1) which sbstittes eery free ariable x in the λ-term M with the λ-term P. Despite its simplicity, the β-rle enables an extraordinary range of behaiors. For instance, depending on the nmber of times the ariable x occrs in M, the β-redex (1) dplicates its argment P, or erases it... Typically, the λ-term = (λx.xx) defines a dplicator, while the λ-term K = (λx.λy.x) defines an eraser, with the following behaiors: P P P, KP Q (λy.p )Q P. Amsingly, the dplicator applied to itself defines a λ-term whose comptation loops: The λ-term Ka( ) obtained by applying the eraser K to the ariable a and to the loop is particlarly interesting, becase its behaior depends on the strategy chosen to compte it. When compted from left to right, the λ-term Ka( ) redces in two steps to its reslt a: Ka( ) (λy.a)( ) a (2) On the other hand, when compted from right to left, the same λ-term Ka( ) loops for eer on the nnecessary comptation of its sbterm : Ka( ) Ka( ) (3) To smmarize: applying the wrong strategy on the λ-term Ka( ) comptes it for eer, whereas applying the more jdicios strategy (2) transforms it into its reslt a. This raises a ery pragmatic qestion: does there exist a jdicios strategy for eery λ-term? This strategy wold aoid seless comptations, and reach the reslt of the λ-term, wheneer this reslt exists. Remarkably, sch a jdicios strategy exists, and its recipe is srpris-

3 3 ingly niform: redce at each step the leftmost otermost β-pattern of the λ-term! Note that this is precisely the strategy applied sccessflly in (2) to compte the λ-term Ka( ). We recall below the definition of the leftmost otermost strategy, formlated originally by A. Chrch and J. B. Rosser in the λi-calcls (the λ-calcls withot erasers) then adapted to the λ-calcls by H.-B. Crry and R. Feys. A β-pattern is a pattern (λx.p )Q occrring in the syntactical tree of a λ-term. The λ-terms (λx.p ) and Q are called respectiely the fnctional and the argment of the β-pattern (λx.p )Q. A λ-term which does not contain any β-pattern is called a normal form: it cannot be compted frther. Now, consider a λ-term M containing a β-pattern at least. Its leftmost otermost β-pattern is defined by indction on the size of the λ-term M: 1. as (λx.p )Q when M = λx 1...λx k.((λx.p )QR 1...R m ), 2. as the leftmost otermost β-pattern of Q when and eery P i is a normal form. M = λx 1...λx k.(xp 1...P m QR 1...R n ) Theorem 1 (Crry-Feys). Sppose that there exists a rewriting path from a λ-term M to a normal form P. The strategy consisting in rewriting at each step M i the leftmost otermost β-pattern in M i constrcts a rewriting path from M to P. M = M 0 M 1 M k 1 M k = P Theorem 1 may be stated alternatiely by defining as the least relation between λ- terms erifying the indctie steps of Figre 1.1, then by establishing that M P implies M P, for eery λ-term M and normal form P. We leae the reader check as exercise that the definition of constrcts the rewriting path (2) in the case of M = Ka( ) Compting leftmost otermost is not necessarily jdicios... in other rewriting systems This clarifies how a term shold be compted in the λ-calcls: from left to right. It appears howeer that this orientation is ery particlar to the λ-calcls. Consider for instance the (VAR) (BETA) (APP) (XI) M λx.p x x MN Q P [x := N] Q M xp 1...P k N Q MN xp 1...P k Q M P λx.m λx.p Fig. 1. An indctie definition of Crry and Feys leftmost otermost strategy.

4 4 term rewriting system defined by the rles A A B C F (x, C) D (4) Then, the rightmost otermost strategy (5) rewrites the term F (A, B) to a reslt D: F (A, B) F (A, C) D (5) whereas the leftmost otermost strategy loops for eer on the term F (A, B): F (A, B) F (A, B) (6) One mst admit here that there exists no niersal syntactic orientation in Rewriting Theory. This shold not be a srprise: after all, the syntactic orientation of a rewriting system is extremely sensitie to its notation! Think only of the λ-calcls written throgh the Looking Glass, in a reerse notation: now, the calcls is oriented right to left, instead of left to right... The general case is een worse. A rewriting system does not enjoy any niform orientation in general, and finding the jdicios strategy, een if we know that it exists, is a non decidable problem, see (Het, Léy, 1979). Despite the apparent mess, we will initiate in this article a generic theory of orientations and casality in rewriting systems. Bt on what fondations? Obiosly, we need to abstract away from syntax in order to describe niformly examples (2), (3), (5) and (6). We are ths compelled to reason diagrammatically instead of syntactically, and to deelop a syntaxfree Rewriting Theory, based on a 2-dimensional refinement of the traditional notion of Abstract Rewriting System deeloped in (Newman, 1942; Het, 1980; Klop, 1992) Forget syntax, think diagrammatically! The diagrammatic approach to Rewriting Theory which we hae in mind is jstified by a simple bt srprising obseration: despite their syntactic differences, the two terms Ka( ) and F (A, B) define exactly the same transition system, which we draw below. Ka( ) 1 Ka( ) F (A, B) A 1 F (A, B) K (λy.a)( ) 2 (λy.a)( ) K B B F (A, C) A 2 F (A, C) (7) λ a id a a λ F D id D D F Apparently, the dynamical analogy between the two terms Ka( ) and F (A, B) goes beyond the eqality of their transition systems. Obsere that in the lefthand side and the righthand side of the diagram: the steps 1 and A 1 are nnecessary becase they may be erased by the paths K λ and B F,

5 the paths K λ and B F are more jdicios than the paths 1 K λ and A 1 B F becase they aoid compting the nnecessary redexes 1 and A 1. This analogy between the two terms Ka( ) and F (A, B) is too sbtle to be reflected by the transition systems of Diagram 7. Howeer, it is possible to refine the notion of transition system, in order to captre the analogy. The refinement is based on the concept of redex permtation introdced by J.-J. Léy in his work on the λ-calcls and on term rewriting systems, see (Léy, 1978; Het, Léy, 1979; Barendregt, 1985). Permting redexes inside rewriting paths enables to express by local transformations that two different rewriting paths compte the same eents, bt in a different order. Typically, the transition system of the terms Ka( ) and F (A, B) may be eqipped with two redex permtations [1] and [2] indicated below: 5 Ka( ) 1 Ka( ) F (A, B) A 1 F (A, B) K [1] (λy.a)( ) 2 (λy.a)( ) K B [1] B F (A, C) A 2 F (A, C) (8) λ a id a [2] a λ F D [2] id D D F Consider for instance the transition system of the λ-term Ka( ) on the lefthand side of Diagram 8: the two paths 1 K λ and K 2 λ are eqialent modlo permtation [1] of the β-redexes 1 and K, and the two paths K 2 λ and K λ are eqialent modlo permtation [2] of the β-redexes 2 and λ. All pt together, the two paths f = 1 K λ and g = K λ are eqialent modlo the two permtations [1] and [2]. In particlar, they compte the same eents, bt in a different order. Note howeer that the redex 1 has disappeared in the process of reorganizing the rewriting path f into the rewriting path g. Remarkably, the same story may be told of the term F (A, B): the redex A 1 has disappeared dring the process of reorganizing the rewriting path f = A 1 B F into the rewriting path g = B F sing the two permtations [1] and [2]. The process of reorganizing a path f : P Q into the properly oriented path g : P Q is known as the standardization procedre. The rewriting path g obtained at the end of the procedre is called the standard path associated to the path f. J-J. Léy introdced the idea of an eqialence relation between rewriting paths modlo redex permtation. Here, we orient the redex permtations and ths refine Léy eqialence relation into a preorder on rewriting paths. We call this preorder the standardization preorder. This enables s to describe standardization in a prely diagrammatic way, as an extremal problem: standard paths = minimal paths wrt. the standardization order.

6 All this is explained in Sections , and illstrated by the λ-calcls in three different ways in Section 1.9. A concise and sbjectie history of the standardization theorem is proided in Section Standardization as 2-dimensional rewriting modlo Standardization is too often explained syntactically, in a qite obscring way... In order to nderstand the reorganization of redexes in a simple and diagrammatic way, we decide to orient the permtations [1] and [2], and to define standardization as the 2-dimensional process of transforming the path 1 K λ into the path K λ. Dring that transformation, each permtation [1] and [2] plays the role of a 2-dimensional rewriting step redcing a rewriting path into another more standard rewriting path: 1 K λ K 2 λ K λ. (9) The normal form of 1 K λ is the standard path K λ. In this way, we define niformly for the first time standardization for (almost) eery existing rewriting system. The 2-dimensional perspectie nifies already or two faorite examples: the rewriting path A 1 B F is rewritten as the rightmost otermost rewriting path B F by the same 2- dimensional procedre as example (9): A 1 B F B A 2 F B F. The interpretation of standardization as 2-dimensional rewriting is the athor s rediscoery of an old idea pblished fifteen years earlier by J. W. Klop in his PhD thesis. At the time of J. W. Klop s PhD thesis ( ) standardization was limited to the λ-calcls and similar leftmost-otermost standardization theorems. J. W. Klop obsered that standardization cold be expressed nicely as a plain 2-dimensional rewriting system. Qite at the same time, G. Het and J.-J. Léy reshaped the field entirely by establishing a reoltionary standardization theorem for term rewriting systems, in (Het, Léy, 1979). Unfortnately, the richer standardization mechanisms disclosed by G. Het and J.-J. Léy cannot be expressed as a plain 2-dimensional rewriting system anymore and J. W. Klop s elegant idea was simply forgotten. It is only fifteen years later, trying to abstract away from the syntactical details of (Het, Léy, 1979) that the 2-dimensional approach took shape again. This was a completely independent discoery originating from a long and obsessie reflexion on the diagrammatic presentation of (Gonthier, Léy, Melliès, 1992). Already in germ there and in the athor s PhD thesis (Melliès, 1996) the idea emerged finally that the standardization mechanism described by G. Het and J.-J. Léy redces to distingishing two classes of permtations: the reersible permtations for instance, permtation [1] in Diagram (8), the irreersible permtations for instance, permtation [2] in Diagram (8). Jan Willem Klop told Vincent an Oostrom that the idea was sggested by Martin Hyland. I mentioned that to Martin Hyland who told me that being not aware of 2-categories at the time, he was thinking of standardization as rewriting of β-redction seqences, rather than as 2-dimensional rewriting really.

7 In this way, the standardization mechanisms disclosed by G. Het and J.-J. Léy can be reformlated as a 2-dimensional rewriting system modlo reersible permtations which then specializes to a plain 2-dimensional rewriting system in the case of the leftmostotermost standardization theorems stdied by J. W. Klop in his PhD thesis. At this point, it is worth explaining briefly and informally the difference between a reersible and an irreersible permtation. Permtation [1] is called reersible becase it permtes two disjoint rewriting steps K and 1, or B and A 1 disjoint in the syntactic sense that no redex contains the other redex in the tree nesting order. The permtation is ths netral from the point of iew of standardization. 7 Ka( ) 1 Ka( ) F (A, B) A 1 F (A, B) K (λy.a)( ) [1] 2 K (λy.a)( ) B F (A, C) [1] B A 2 F (A, C) Permtation [2] is called irreersible becase it replaces the inside-ot comptation 2 λ or A 2 F by its otside-in eqialent λ or F ths strictly improing the comptation from the point of iew of standardization. (λy.a)( ) 2 (λy.a)( ) F (A, C) A 2 F (A, C) λ a id a [2] a λ F D [2] id D D F 1.5. The basic ocablary of Axiomatic Rewriting Theory It is time to introdce seeral key definitions related to or diagrammatic theory of standardization. Definition 1 (transition system). A transition system (or oriented graph) G is a qadrple (terms, redexes, sorce, target) consisting of a set terms of ertices (= terms), a set redexes of edges (= rewriting steps, or redexes), and two fnctions sorce, target : redexes terms (= the sorce and target fnctions). We write : M N when sorce() = M and target() = N. Recall that a path in a transition system G is a seqence f = (M 1, 1, M 2,..., M m, m, M m+1 ) (10) where i : M i M i+1 for eery i [1...m]. We write f : M 1 M m+1. The length of f is m and f is said to be empty when m = 0. Two paths f : M N and g : P Q are

8 coinitial (resp. cofinal) when M = P (resp. N = Q). The path f; g : M Q denotes the concatenation of two paths f : M P and g : P Q. Definition 2 (2-dimensional transition system). A 2-dimensional transition system is a pair (G, ) consisting of a transition system G and a binary relation on the paths of G. The relation is reqired to relate coinitial and cofinal paths: f : M N, g : P Q, f g (M, N) = (P, Q) The idea of Axiomatic Rewriting Theory is to replace a concrete rewriting system by its 2-dimensional transition system. This has the effect of reealing nexpected similarities: typically, the two terms Ka( ) and F (A, B) behae differently syntactically (left to right s. right to left) bt indce the same 2-dimensional transition system (drawn below) in the λ-calcls and in the term rewriting system (4). 8 X w 1 X Y w 2 Y Z id Z Z w 1 w 2 w 2 w 1 w 2 (11) It shold be obios at this point of exposition that the dynamical analogy obsered preiosly between the terms Ka( ) and F (A, B) (Section 1.3) follows from the identity of their 2-dimensional transition system. Definition 3 (permtation). A permtation (f, g) in a 2-dimensional transition system (G, ) is a pair of paths sch that f g. We often se the more explicit (and oerloaded) notation f g for a permtation (f, g). 1 Definition 4 (standardization step, ). A standardization step from a path d : M N to a coinitial and cofinal path e : M N in a 2-dimensional transition system (G, ), is a triple (d 1, f g, d 2 ) consisting of a permtation f g and two paths d 1, d 2 sch that: d = M d1 P f Q d2 N e = M d1 P We write d 1 e when there exists a standardization step from d to e. g Q d2 N Definition 5 (standardization preorder, Léy eqialence ). In eery 2-dimensional transition system (G, ) the standardization preorder is the least transitie reflexie relation containing 1. We say that a path e : M N is more standard than a path d : M N when d e. the Léy permtation eqialence is the least eqialence relation containing. Alternatiely, the eqialence relation is the least eqialence relation containing and closed nder composition.

9 To illstrate or definitions with diagram (11), the path is more standard than the path w 1 as testifies the seqence of standardization steps: w 1 1 w Reersible and irreersible permtations Permtations of (G, ) are discriminated in two classes, reersible and irreersible, according to the following definition. Definition 6 (reersible, irreersible permtation). In eery 2-dimensional transition system (G, ) 1. A permtation (f, g) is reersible when g f. A box signals reersible permtations f g in text and diagrams. 2. A permtation (f, g) is irreersible when (g f). A triangle signals irreersible permtations f g in text and diagrams. Check that the definition matches the preios qalification in Section 1.4 of permtation [1] as reersible, and permtation [2] as irreersible, in diagrams (8) and (11). We illstrate or new diagrammatic conentions on the 2-dimensional transition system (11). X w 1 X Y w 2 Y Z id Z Z w 1 w 2 w 2 (12) In the definition below, the discrimination on permtations generalizes to the obios discrimination on standardization steps. The key concept of reersible permtation eqialence is reealed, as a stronger ersion of sal Léy permtation eqialence. Definition 7 ( REV, IRR, reersible permtation eqialence ). In eery 2-dimensional transition system (G, ) A standardization step (e, f g, h) is reersible (resp. irreersible) when the permtation f g is reersible (resp. irreersible). We write d REV e d IRR e when there exists a Reersible (resp. Irreersible) standardization step from d to e. The reersible permtation eqialence is the least eqialence relation containing the relation REV.

10 Standard rewriting paths Definition 8 (standard path). A rewriting path d : M N is standard when there does not exist any seqence of standardization steps d REV d 1 REV REV d k IRR d k+1 consisting of a series of k Reersible steps followed by an Irreersible step. Remark that when the rewriting path d is standard, and d e, then d e and the rewriting path e is standard. For instance, the path X w1 X Y Z in diagram (11) is transformed in two steps in the standard path X Y Z. The rewriting path X w1 X Y is another example of standard path, becase eery standardization seqence from it to itself or to X Y w2 Y is reersible The standardization theorem One main challenge of Axiomatic Rewriting Theory is to captre the diagrammatic properties of redex permtations in syntactic rewriting systems, in order to establish the following diagrammatic standardization theorem: for eery rewriting path d : M P in the transition system G, 1. existence: there exists a standardization seqence d e transforming the rewriting path d into a standard path e, 2. niqeness: eery standardization seqence d f may be extended to a standardization seqence leading to the standard path e: d f e. The niqeness property has a series of remarkable conseqences. Sppose for instance that the rewriting path f is standard. In that case, the standardization seqence consists of Reersible steps. Ths, f e f e. From this follows that there exists a niqe standard path e sch that d e modlo reersible permtation eqialence. In fact, the niqeness property ensres that there exists a niqe standard path, modlo reersible permtation eqialence, in the Léy eqialence class of the rewriting path d.

11 In this article, we formlate a series of nine elementary axioms on the 2-dimensional transition system (G, ) and dedce from them the diagrammatic standardization theorem stated aboe. The axioms ncoer a series of simple and elegant principles of casality in comptations. They also illstrate that a prely diagrammatic and syntax-free theory of comptations is possible, and sefl, since it enscopes almost eery existing rewriting systems, from Petri nets to higher-order rewriting systems Illstration: the λ-calcls and its three standardization orders There are at least three different ways to interpret the λ-calcls as a 2-dimensional transition system, each one associated to a particlar nesting order on the β-redexes of λ-terms. The nderlying transition system G λ is the same in the three cases. It is defined in (Crry, Feys, 1958; Léy, 1978) as follows: its ertices are the λ-terms, modlo α-conersion, its edges are the β-redexes : M N. Recall that a β-redex = (M, o, N) is a triple consisting of a λ-term M, the occrrence o of a β-pattern (λx.p )Q in M and the λ-term N obtained after β-redcing (λx.p )Q P [x := Q] in the λ-term M. It is worth noting that there are two different edges I(Ia) Ia in the graph G λ : each edge corresponds to the redction of a particlar identity combinator I = (λx.x) in the λ-term I(Ia). There are at least three different ways to refine the transition system G λ as a 2-dimensional transition system, depending on the order chosen on β-redexes: the tree-order: a β-redex is smaller than a β-redex when occrs in the fnctional or argment part of ; or eqialently, when the occrrence of is a strict prefix of the occrrence of. We se the notation: tree. the left-order: a β-redex is smaller than a β-redex when occrs in the fnctional or argment part of, or when there exists an occrrence o of an application node P Q in the λ-term M, sch that occrs in P and occrs in Q. We se the notation: left. the argment-order: a β-redex is smaller than a β-redex when occrs in the argment of. We se the notation: arg. Each order indces in trn its own permtation relation tree, left and arg on the transition system G λ. Note that the order considered in the litteratre is generally the left-order, see (Crry, Feys, 1958; Léy, 1978; Klop, 1980). Here, we prefer to stdy the tree-order, becase this seems to be the most natral choice after the work by G. Het and J-J. Léy on term rewriting systems (Het, Léy, 1979). The two alternatie orders left and arg are discssed briefly in Section 9. We define the relation tree as follows. Two paths f, g are related as f tree g precisely when: 1. the paths f and g factor as f = and g = h where,, are β-redexes and h is a path,

12 2. the two β-redexes and are coinitial, and ( tree ), 3. the β-redex is the (niqe) residal of after, and the path h deelops the (possibly) seeral residals of after. [For a definition of residal and complete deelopment, see (Crry, Feys, 1958; Léy, 1978; Het, Léy, 1979; Barendregt, 1985; Klop, 1992) or Section 8.] Ths, eery permtation f tree g is of the form: M P tree Q h N f = g = h where and are different β-redexes, is a β-redex and h is a path. The three paradigmatic examples of β-redex permtation f tree g are: P Q tree P Q P Q P Q (λx.a)p a (λx.a)p tree id a a P tree P P P 1 2 P P where P P and Q Q are two β-redexes. The three permtations are respectiely reersible, irreersible and irreersible in the 2-dimensional transition system (G λ, tree ). 12 (13) A concise history of the standardization theorem Many athors hae written on the standardization theorem. Instead of drawing a comprehensie list, we delier a qick history of the sbject, in eight key steps. The list will certainly be nfair to many people, bt we want to keep it short, straight and sbjectie. [1936] A. Chrch and J.B. Rosser introdce the λi-calcls, a λ-calcls withot erasement, and proe that the nmber of β-steps from a λi-term to its normal form is bonded by the length of the leftmost otermost comptation. This reslt is the ancestor of all later standardization theorems. [1958] H.B. Crry and R. Feys formlate the first standardization theorem for the λ-calcls: the two athors proe that eery time a λ-term P β-redces to a λ-term Q, there exists also a standard way to β-redce P to Q. The theorem extends Chrch and Rosser reslt for the λi-calcls, and plays a role in Crry and Feys defense of their erasing combinator K. [1978] J.-J. Léy formlates the standardization theorem in its modern algebraic form: sing an eqialence relation on rewriting paths called today Léy permtation eqialence Léy proes that there exists a niqe standard rewriting path in each eqialence class. The niqeness reslt was so striking at the time that the theorem was called strong standardization theorem by later athors. Despite its conceptal noelty, the theorem is still limited to the λ-calcls and to its leftmost-otermost order. [1979] G. Het and J.-J. Léy formlate and establish a standardization theorem for term rewriting systems withot critical pairs. This is probably the most reoltionary step in

13 the history of standardization, the first time at least that another standardization order is considered than the leftmost otermost order of the λ-calcls. The theorem is still limited to term rewriting systems becase its proof relies heaily on syntactical notions like tree-occrrence bt the article deliers the message that standardization is a general property of rewriting systems, related to casality and domain-theoretic notions like stability and seqentiality. [1980] J. W. Klop introdces a 2-dimensional rewriting system on paths, consisting in permting anti-standard paths of length 2 into standard paths of arbitrary length. In this way, Klop dedces Léy s strong standardization theorem for leftmost-otermost λ-calcls, by establishing conflence and strong normalization of the 2-dimensional rewriting process: the standard path is obtained as normal form of the procedre. Another important contribtion of J. W. Klop is to stress the role of the finite deelopment lemma in the proof of standardization, and to extend to any left-reglar Combinatory Redction System the standardization theorem for leftmost-otermost λ-calcls. [Early 1980s] G. Bodol extends G. Het and J-J. Léy standardization theorem to term rewriting systems with critical pairs. This is another decisie step, becase it extends the principle of standardization to non deterministic rewriting systems. [1992] G. Gonthier and J-J. Léy and P-A. Melliès delier an axiomatic standardization theorem, where the syntactical proof of (Het, Léy, 1979) is replaced by diagrammatic argments on redexes, residals and the nesting relation. Sbseqently reworked by the athor in his PhD thesis (Melliès, 1996), the theorem extends G. Het and J-J. Léy s original theorem to a great ariety of rewriting systems with and withot critical pairs with the remarkable and pzzling exception (as first noted by R. Kennaway) of rewriting systems based on directed acyclic graphs. [1996] D. Clark and R. Kennaway adapt the syntactical works of G. Het, J-J. Léy and G. Bodol and establish a standardization theorem for (possibly conflicting) rewriting systems based on directed acyclic graphs (dags). It took the athor nine years to derie the crrent axiomatics from (Gonthier, Léy, Melliès, 1992). One difficlty was to find the simplest possible description of rewriting systems with critical pairs. The trinity of residal, compatibility and nesting relations operating in (Gonthier, Léy, Melliès, 1992) was certainly too complicated. Slowly, the 2-dimensional presentation emerged, leading the athor to the elementary axiomatics of this article. Twenty-fie years ago, the work of (Het, Léy, 1979; Bodol, 1985) on term rewriting systems reealed sddenly that the conflict-free left-reglar rewriting systems considered earlier was the emerged part of the mch wider and exciting world of casal comptations. This is that world and its bondaries that we are abot to explore here in a 2- dimensional diagrammatic fashion. 13 Strctre of the paper Axiomatic Rewriting Systems (AxRS) are introdced in Section 2, along with their nine standardization axioms. A less innoatie bt more traditional axiomatics based on residals, critical pairs and nesting is formlated in Section 8. Standard paths are characterized in Section 3 as the paths which do not contain a particlar anti-standard pattern, jst

14 as in (Gonthier, Léy, Melliès, 1992; Melliès, 1996). The standardization theorem is proed in Section 4, and reformlated 2-categorically in Section 5. A few additional hypothesis on axiomatic rewriting systems are discssed in Section 6, in order to relate in Section 7. this work to the companion articles (Melliès, 2000; Melliès, 1997; Melliès, 1998). Finally, we illstrate or definition of AxRS with a few examples in Section 9, like asynchronos transition systems, term rewriting systems, call-by-ale λ-calcls, λ-calcls with explicit sbstittions The 2-dimensional axiomatics A 2-dimensional transition system (G, ) is called Axiomatic Rewriting System (AxRS) when it erifies a series of nine standardization axioms presented in this section. Each axiom of the section is illstrated by the λ-calcls and its 2-dimensional transition system (G λ, tree ) defined in Section Axiom 1: shape The first axiom generalizes to eery AxRS the shape of permtations encontered in the λ-calcls see Diagram (13) in Section 1.9. Axiom 1 (Shape). We ask that in eery permtation f g, the path f is of length 2, the path g is of length at least 1, the initial redexes of f and g are different. Ths, eery permtation f g in (G, ) has the shape below: M P h Q N f = g = h where and are different redexes, is a redex and h is a path. In case of a reersible permtation f g, this shape specializes to a 2 2 sqare: (14) where,, and are redexes, and different. M P Q N f = g = 2.2. Axioms 2, 3, 4, 5: ancestor, reersibility, irreersibility and cbe The standardization theorem is sally established by a fine-grained analysis of syntactic mechanisms like erasement, dplication, etc... related to Léy theory of residals. The fragment of Léy theory necessary to the theorem, eg. the finite deelopment property, appears

15 in or axiomatics, bt transformed, since the more geometric idea of oriented permtation replaces the old concept of residal of a redex. The residal theory is particlarly isible in the for axioms ancestor, reersibility, irreersibility and cbe introdced below, as well as in axiom termination of Section 2.6. Axiom ancestor incorporates two properties of the λ-calcls, traditionally called niqeness of ancestor and finite deelopment. The existence of a permtation f tree g between two β-rewriting paths: f = M Q N g 1 = M 1 P 1 h 1 N means that the β-redex is the niqe residal of the β-redex 1 after β-redction of the redex, and that h 1 is a complete deelopment of the residals of the redex after β- redction of the redex 1. In that case, we say that the redex 1 is an ancestor of the redex before β-redction of the redex. The niqeness of ancestor property states that the redex 1 is the niqe sch ancestor of the redex. The finite deelopment property, see Section 8, states that two complete deelopments of the same set of β-redexes, are Léy eqialent. Ths, in eery permtation f tree g 2, the path g 2 factors as g = 2 h 2, where 1 = 2 and h 1 tree h 2. This leads s to formlate the Axiom 2 (Ancestor). Sppose that 1, 2 are redexes, that f, h, h are rewriting paths, forming together permtations f 1 h 1 and f 2 h 2. We ask that 1 = 2 and h 1 h Axiom reersibility indicates that eery permtation f g is either reersible, or redces to a rewriting path g for which there exists no permtation of the form g h. This mirrors the following property of the λ-calcls. Sppose that f, g, h : M N are three β-rewriting paths inoled in permtations f tree g and g tree h. The paths f and g are of length 2, the path h is of length at least 1, and the paths f, g, h decompose as f = M Q N, g = M P N, h = M O h N where the two redexes and are ancestor of the same redex, and ths = ; and where the β-redex is the niqe residal of and h is a deelopment of the residals of after, and ths h =. It follows that f = h. Axiom 3 (Reersibility). We ask that f = h when f g and g h. Axiom irreersibility completes the two preios axioms. The axiom mirrors the fact that standardization preseres complete deelopments in the λ-calcls again, for a definition of complete deelopments, see (Léy, 1978; Het, Léy, 1979) or Section 8. Starting from a complete deelopment h of a mlti-redex (M, U), sppose that the β-rewriting path h factors as h 1 h 2 h 3 where the β-rewriting path h 2 indces a permtation h 2 h 2. By definition of tree, the two β-rewriting paths h 2 and h 2 decompose as h 2 = N P O and h 2 = N Q h O. By niqeness of ancestor, the β-redex and the β-redex are residals of β-redexes in

16 U, after the β-rewriting path h 1. More precisely, the two paths h 2 and h 2 are complete deelopments of (N, {, }). The finite deelopment property of the λ-calcls ensres that the β-rewriting paths h 2 and h 2 define the same residal relation. Ths, the β-rewriting path h 1 h 2 h 3 is a complete deelopment of the mlti-redex (M, U). Now, consider an irreersible permtation f tree g between two β-rewriting paths f = M and a β-rewriting path h sch that Q N g = M P g h. h N Or preios argment shows that the β-rewriting path h is a complete deelopment of the mlti-redex (M, {, }) jst like the β-rewriting path f and g. Besides, the first β-redex redced in the path h is not the β-redex. Ths, the β-rewriting path h decomposes as where h = M P h h. h N From this follows that the β-rewriting path h is a complete deelopment of the residals of the β-redex after redction of the β-redex jst like the β-rewriting path h. Ths, f tree h. Axiom 4 (Irreersibility). We ask that f h when f g and g h. 16 Axiom cbe incorporates the cbe lemma established in (Léy, 1978; Het, Léy, 1979) as well as a carefl analysis of nesting in the λ-calcls. Sppose that C[ ] is a context, see (Barendregt, 1985) for a definition, and that a β-rewriting path g : C[M] C[N] comptes only inside M, neer inside C[ ]. Then, jst as the β-rewriting path g, eery Léy eqialent β-rewriting path f : C[M] C[N] comptes only inside M, neer inside C[ ]. So, eery β-redex w inside C[ ] has the same (niqe) residal w after the β-rewriting paths f and g. Diagrammatically speaking, the property amonts to the cbe property stated in the next axiom, when f tree g and f = and g = 1 n and w = w n+1. The axiom reqires that the property holds in eery AxRS. Axiom 5 (Cbe). We ask that eery diagram w h 1 n w 1 w n+1 h 1 h n

17 where,, and 1,..., n and w, w 1,..., w n, w n+1 are redexes and h 1,..., h n are paths forming permtations 1 n w 1 w h i w i+1 w i h i for 1 i n may be completed as a diagram: 17 w h w h h 1 n w 1 w n+1 h 1 h n h h h h 1 h n where w is a redex and h, h are paths which form permtations w n+1 w h w w h and indce the eqialence h h h h 1 h n Axiom 6: enclae Axiom enclae is based on a fndamental property of the λ-calcls, obsered for the first time in the preliminary work of (Gonthier, Léy, Melliès, 1992). Sppose that a β-redex is nested nder a β-redex that is tree and that the β-redex creates a β-redex w. By creation, we mean that the β-redex w has no ancestor before redction of the β-redex. In that case, the β-redex w is necessarily nested nder the (niqe) residal of the β- redex after redction of the β-redex. The next axiom formlates the property as its contrapose. The existence of the permtation w n+1 tree w h means that the β-redex w is not nested nder the β-redex. And from this follows that the β-redex w is not created, and ths, has an ancestor w before redction of the β-redex. The axiom reqires that this enclae property holds in eery AxRS. Axiom 6 (Enclae). We ask that eery diagram w 1 n h w n+1

18 where,, and 1,..., n and w, w n+1 are redexes, and h is path, forming the permtations (recalling or conention, the symbol means that the permtation is irreersible) may be completed as a diagram: 1 n w n+1 w h 18 w h w h 1 n w 1 w n+1 h 1 h n where w, w 1,..., w n are redexes and h, h and h 1,..., h n are paths, forming the n + 2 permtations w w h w 1 w h i w i+1 w i h i for 1 i n h 2.4. Axioms 7 and 8: stability and reersible stability Axiom stability incorporates another key property of the λ-calcls, also obsered for the first time in the preliminary work of (Gonthier, Léy, Melliès, 1992). Consider any reersible permtation M P N tree M Q N in which the β-redex creates a β-redex w 1 and the β-redex creates a β-redex w 2. It is not difficlt to establish that there exists no β-redex w 12 in the λ-term N which wold be at the same time residal of the β-redex w 1 after redction of the β-redex, and residal of the β-redex w 2 after redction of the β-redex. The property is axiomatized below as its contrapose. The axiom states that the characteristic fnction of the eent of creating the β-redex w 12 (or eqialently the β-redex w 1, or the β-redex w 2 ) is stable in the sense of G. Berry, see (Berry, 1979). Axiom reersible-stability repeats the axiom in the reersible case. Axiom 7 (Stability). We ask that eery diagram w 2 h w 1 w 12 h

19 19 where,,, and w 1, w 2, w 12 are redexes and h, h are paths, forming the permtations (recalling or conention, the symbol means that the permtation is reersible) w 12 w 2 h w 12 w 1 h may be completed as a diagram w h w 2 h w 1 w 12 where w is a redex and h, h are two paths, forming two permtations h h w 2 w h w 1 w h Axiom 8 (Reersible stability). We ask that eery diagram w w 1 w (15) where,, 1, 1 and w 1, w 2, w 12, 12, 12 are redexes forming the reersible permtations w 12 w w 12 w 1 12 may be completed as a diagram w 2 w w 1 w where w, 2, 2 are three redexes forming the reersible permtations w 2 w 2 and w 1 w 2 and

20 Remark: axiom reersible-stability may be nderstood as a conerse of the reersible ariant of axiom cbe formlated in Section 6.3. Indeed, axiom reersible-stability states that eery diagram may be completed into the diagram w w w 2 20 (16) w 1 w w 2 (17) and conersely, axiom reersible-cbe formlated in Section 6.3 states that Diagram (17) may be completed as Diagram (16). Besides, it is remarkable that the two axioms reersiblestability and reersible-cbe are dal in the sense that each axiom may be obtained from the other one by reersing the orientations of all the arrows in diagrams Drag and extraction We need to introdce a few definitions related to standardization in order to state the last axiom of the theory (axiom 9). Definition 9 (drag). A path f : M N drags a redex otgoing from N to a redex otgoing from M, when f = id M and =, or f = 1 n and there exists n + 1 redexes 1,..., n+1 and n paths h 1,..., h n sch that: 1 = and n+1 =, the rewriting paths i i+1 and i h i form a permtation i i+1 i h i for eery index 1 i n.

21 n h h 2 n h n Fig. 2. The path f = 1 n drags the redex to the redex. 1 2 i 1 h h 2 i 1 h i 1 i i+1 n Fig. 3. The redex is extractible from the path f = 1 n and the path g = h 1 h i 1 i+1 n is a projection of the rewriting path f by extraction of the redex. Notation: we write f when the rewriting path f drags the redex to the redex. See Figre 2. Lemma 10 (preseration of drag). For eery path f : M N, the relation is a partial fnction, from the redexes otgoing from N to the redexes otgoing from M. Moreoer, the relation is inariant by permtation on f: g : M N, f g f = g. f f Proof. Sppose that and. Then = by axiom ancestor, and an easy indction on the length of f. Now, by axiom cbe, the relation increases by antistandardization: if the rewriting path g drags the redex to the redex, and f g, then the rewriting path f drags the redex to the redex. By axiom enclae, the relation increases also by standardization: if the rewriting path f drags the redex to the redex, and f g, then the rewriting path g drags the redex to the redex. We conclde. Definition 11 (extraction, projection, ). A redex : M P is extractible from a path f = 1 n : M N when there exists an index 1 i n sch that the path 1 i 1 drags the redex i to the redex. In that case, we call projection of the rewriting path f by extraction of the redex : M P any rewriting path g : P N which decomposes as g = h 1 h i 1 i+1 n where there exists redexes 1,..., i with 1 = and i = i and a permtation for eery index 1 j i 1. j j+1 j h j Notation: We write f g when the redex is extractible from the path f, and g is a projection of f by extraction of the redex. See figre 3. f

22 Lemma 12 (preseration of extraction). Sppose that a redex is extractible from a path g : M N more standard than a path f : M N. Then the redex is also extractible from the path f. Moreoer, eery projection of f by extraction of is Léy eqialent to eery projection of g by extraction of. Proof. Sppose that the redex is extractible from the path f = 1 n : M N. By definition, there exists an index 1 i n sch that the path 1 i 1 drags the redex i to the redex. We show that the index i is niqe. Sppose that there exists another index 1 j n sch that 1 j 1 drags the redex j to the redex. We may sppose withot loss of generality that i < j. Let the rewriting path g be a projection of the rewriting path 1 i by extraction of the redex at position i. By definition of extraction and projection, the two rewriting paths 1 i and g are Léy eqialent. From this follows that the two paths 1 j 1 = 1 i i+1 j 1 and g i+1 j 1 are Léy eqialent. Here comes the contradiction. By Lemma 10 (preseration of drag), the path g i+1 j 1 drags the redex j to the redex. This may be decomposed in two steps: first, the path g i+1 j 1 drags the redex j to a redex, then the redex drags the redex to the redex. This ery last point means that there exists a permtation of the form h. This contradicts the axiom shape. We ths conclde that the index i is niqe. We may sppose withot loss of generality that there exists a niqe standardization step from the rewriting path f to the rewriting path g. The remainder of the lemma follows then from axioms reersibility and cbe when the standardization step between f and g is reersible, and from axioms irreersibility, ancestor and cbe when the standardization step is irreersible. Remark: the niqeness of the index i in the proof of Lemma 12 is not really necessary to establish the property, bt it is a safegard, since after all, we hae not spposed anything like the optional hypothesis descendent formlated in Section Axiom 9: termination Axiom termination mirrors in or theory the finite deelopment property of the λ-calcls, which states that eery deelopment of a set of β-redexes terminates. Jan Willem Klop ses the property in his PhD thesis to dedce that it is not possible to extract infinitely many times a β-redex from a fixed β-rewriting path, see (Klop, 1980) as well as Section 8. Axiom 9 (Termination). There exists no infinite seqence where f i are paths and i are redexes. f 1 1 f 2 2 k 1 f k k 3. A direct characterization of the standard paths In this section, we establish a key preliminary step in or proof of the standardization theorem, performed in Section 4, by characterizing standard rewriting path in a more direct

23 23 M 1 2 n N w 2 P 1 w 3 w n 2 n Q Fig. 4. The path f = 1 n : M N followed by the redex : N Q permtes reersibly to the redex : M P followed by the path g = 1 n : P Q. Alternatiely, the redex : M P followed by the path g = 1 n : P Q permtes reersibly to the path f = 1 n : M N followed by the redex : N Q. and explicit way. In Section 3.1, we introdce the notions of starts and stops of a rewriting path, and analyze their properties. From this, we dedce in Section 3.2 that eery path is epi (left cancellable) with relation to the Reersible permtation relation. In Section 3.3, we introdce the notion of anti-standard path and establish that a rewriting path is standard if and only if it does not contain any occrrence of sch anti-standard path The strctre of starts and stops Definition 13 (starts and stops). A redex : M P starts a path f : M N when there exists a path g : P N sch that f g. A redex : Q N stops a path f : M N with remainder g : M Q when f g. A redex : Q N stops a path f : M N when the redex stops the path f with some remainder g : M Q. Definition 14 (reersible permtation of path and redex). A path f : M N followed by a redex : N Q permtes reersibly to a redex : M P followed by a path g : P Q, when f = id M and g = id P and = : M P, or f = 1 n and g = 1 n and there exists a series of n + 1 redexes w 1,..., w n+1 sch that w 1 = and w n+1 =, the two paths i w i+1 and w i i form a reersible permtation i w i+1 w i i for eery index 1 i n. In that case, we say also that the redex : M P followed by the path g : P Q permtes reersibly to the path f : M N followed by the redex : N Q. See Figre 4. Remark: in Definition 14, the redex and the rewriting path g are niqely determined by the rewriting path f and the redex and conersely, the rewriting path f and the redex are niqely determined by the redex and the rewriting path g. The one-to-one relationship follows from axiom reersibility. Lemma 15 (strctre of stops). A redex : Q N stops a path f = 1 n : M N with remainder g : M Q iff there exists an index 1 i n and a path i+1 n sch that

24 24 the redex i followed by the path i+1 n permtes reersibly to the path i+1 n followed by the redex, the rewriting path ( 1 i 1 ) ( i+1 n ) is eqialent to the path g modlo. Proof. We declare that a redex : Q N sper-stops a path f = 1 n : M N at position 1 i n with remainder g : M Q when there exists a path i+1 n sch that the redex i followed by the path i+1 n permtes reersibly to the path i+1 n followed by the redex, the rewriting path ( 1 i 1 ) ( i+1 n ) is eqialent to the path g modlo. We declare that a redex sper-stops a path f with remainder g when it sper-stops the path f with remainder g at some position i. The lemma states that a redex stops a path f with remainder a path g iff the redex sper-stops f with remainder g. Right-to-left implication ( ) is immediate. The other direction ( ) redces to showing that wheneer the two assertions below holds: a redex : Q N sper-stops a path f = 1 n with remainder g, and the path f is eqialent to the path f modlo reersible permtations, then the redex sper-stops the path f with remainder the same rewriting path g. This elementary bt fndamental preseration property is established in the following way. We may sppose withot loss of generality that the two rewriting paths f = 1 n and f = 1 n are related by a niqe reersible permtation f REV f occrring at a position 1 j n 1 in the rewriting path f. We ths hae: k = k for eery index 1 k n different to j and j + 1, and j j+1 j j+1. Now, call i any position (there exists in fact only one of these positions, 1 i n, bt nobody cares abot that here) sch that the redex : Q N sper-stops the path f = 1 n at position i with remainder g. We show by case analysis on the indices i and j that there exists an index 1 k n sch that the redex : Q N sper-stops the path f = 1 k 1 k k+1 n at position k with remainder g. To that prpose, we define a rewriting path k+1 n consisting of n k redexes, sch that: a. the redex k followed by the path k+1 n permtes reersibly to the path k+1 n followed by the redex, b. the rewriting path ( 1 k 1 ) ( k+1 n) is eqialent to the path g modlo. The constrction is immediate when j + 1 i: simply take k = i and i n = i n. The constrction is also nearly immediate when j = i: simply take k = i + 1 and i+2 n = i+2 n, then apply axiom reersibility to establish the two properties a. and b.

25 The difficlt case is the remaining case when j > i. In that case, let the redex x denote the niqe redex sch that the redex i followed by the path i+1 j 1 permtes reersibly to the path i+1 j 1 followed by the redex x. Consider the diagram below, which describes in two perspecties how the redex x followed by the path j j+1 permtes reersibly to the path j j+1 followed by the redex z: 25 x j x j j j j+1 y j+1 j+1 z or j y j j+1 j+1 z j+1 By axiom reersible-stability, the diagram may be completed in the following way j x j j y j j+1 y j+1 j+1 z j+1 j+1 j j y x j j y j+1 j+1 z j+1 where y and j and j+1 denote three redexes inoled in the three reersible permtations: x j j y, and j j+1 j j+1 and y j+1 j+1 z. The completed diagram shows (in two perspecties again) that the redex x followed by the path j j+1 permtes reersibly to the path j j+1 followed by the redex z. So, by taking k = i and by defining l = l for eery index i + 1 l n different to j and j + 1, one obtains that: a. the redex i followed by the path i+1 n permtes reersibly to the path i n followed by the redex, b. the rewriting path ( 1 i 1 ) ( i+1 n) is eqialent to the path g modlo. This ery last point follows from the series of eqialence g ( 1 i 1 ) ( i+1 n ) and i+1 n i+1 n.