A Geometric Proof of Confluence by Decreasing Diagrams

Transcription

1 A Geometric Proof of Confluence by Decreasing Diagrams Jan Willem Klop Department of Software Technology, CWI, Kruislaan 43, 098 SJ Amsterdam; Department of Computer Science, Vrije Universiteit, De Boelelaan 08a, 08 HV Amsterdam Vincent van Oostrom Department of Philosophy, University of Utrecht, Heidelberglaan 8, 3584 CS Utrecht Roel de Vrijer Department of Computer Science, Vrije Universiteit, De Boelelaan 08a, 08 HV Amsterdam Abstract Recently a new confluence criterion for confluence was found using decreasing diagrams, as a generalization of several wellknown confluence criteria in abstract rewriting such as the strong confluence lemma. We give a new proof of the decreasing diagram theorem based on a geometric study of infinite reduction diagrams, arising from unsuccesful attempts to obtain a confluent diagram by tiling with elementary diagrams. Contents. Introduction. Abstract Reduction Systems 3. Finite reduction diagrams 4. Infinite reduction diagrams and towers 5. Tree coverings of reduction diagrams 6. Impossible tree coverings 7. Confluence by decreasing diagrams 8. References

2 . Introduction Abstract rewriting is the initial part of the theory of rewriting where objects have no structure and the rewrite relation is just a binary relation on the set of objects. Usually there is not one but an indexed family of rewrite relations present. There are several useful and well-known lemma s for such abstract rewrite systems that give conditions for confluence: Newman s Lemma (Newman (94)), Huet s strong confluence lemma (Huet (980)), Staples request lemma s (Staples 975), the lemma of Hindley-Rosen (Hindley (964)). A common generalization of all these lemma s has been obtained in van Oostrom (994, 994a), elaborating an unpublished note of De Bruijn (De Bruijn 978). De Bruijn s original proof was a complicated nested induction, while van Oostrom used a certain invariant for the diagram construction called decreasing diagrams. A slightly different invariant called trace-decreasing diagrams was used in Bezem et al. (996); this invariant will be used in the present paper. The theorem of De Bruijn and van Oostrom is concerned with labeled reductions. For a version of the theorem where points instead of edges are labeled, see Bognar [97], with a proof checked by the Coq proof checker. In this paper we give a proof of this confluence by decreasing diagrams theorem that is totally different from the two mentioned above. The proof is by an analysis of the geometry of, possibly infinite, reduction diagrams, resulting from two co-initial diverging finite reduction sequences, by tiling with elementary reduction diagrams. Infinite diagrams arise this way, when we have a failure of confluence. Such infinite reduction diagrams are interesting geometric objects themselves; the simplest one is the diagram in Figure.. that we will call the Escher-diagram. In the sequel we will give several more examples of infinite reduction diagrams, some of them exhibiting an interesting fractal-like boundary, some of them reminiscent to the pictures of M.C. Escher, with a repetition of the same pattern, receding in infinity. Actually, we consider an enrichment of mere reduction diagrams, namely diagrams with a tree covering. A tree covering of a diagram determines a ancestor-descendant relation between the edges appearing in a reduction diagram. By means of a tree covering an edge can be traced back to its ancestor edge on one of the original divergent reduction sequences. The theorem proved in this paper states the impossibility of certain infinite diagrams with a treecovering. Since the notion of (trace-)decreasing diagram gives rise in a natural

3 way to a tree covering of the impossible kind we have as an immediate corollary then the theorem of confluence by decreasing diagrams. The method of proof of our theorem is purely geometric. It employs topological notions such as condensation points of point sets in the real plane. Figure.. Escher-diagram. Astract Reduction Systems An Abstract Reduction System (ARS) A is a set A equipped with a collection of rewrite or reduction relations α, indexed by some set I of indexes: A = ŸA, ( α ) α I ±. The index set I is a well-founded partial order. In examples, we will use the set of natural numbers with the usual ordering as index set. The union of the rewrite relations α will be. We use the notation ÿ for the transitive-reflexive closure of. The ARS A is called confluent (see Figure 3.0(a) if a,b,c A d A (a ÿ b & a ÿ c b ÿ d & c ÿ d). It is called locally confluent or WCR (weak Church-Rosser) see Figure 3.0(b) if a,b,c A d A (a b & a c b ÿ d & c ÿ d). a b a b c d c d Figure 3.0

4 For more notions about ARSs we refer to Klop [9]. 3. Finite reduction diagrams An important ingredient in finding a common reduct of the end points of two diverging reduction sequences consists of the elementary diagrams, as given by local confluence (Figure 3.0(b)); see the examples in Figure 3.. They are the 'atomic' or basic building blocks for constructing reduction diagrams, arising from the local confluence property. A non-trivial elementary diagram consists of two diverging steps (arrows), joined by two sequences of steps of arbitrary length. Note that in the e.d.'s we may use empty sides (the dashed sides), to keep matters orthogonal (in the usual geometric sense, not in the technical sense used for first-order term rewriting systems, see e.g. Klop (99)). This gives rise to some trivial e.d.'s as in the lower part of Figure 3.. The e.d.'s are used as 'tiles' with the intention to obtain a completed reduction diagram as in Figure 3. Usually we will forget the direction of the arrows (second picture in Figure 3.): they always are from left to right, or downwards (except the empty 'steps' that have no direction). elementary diagrams Figure 3.

5 Figure Infinite reduction diagrams and towers 4.. Infinite reduction diagrams We will consider infinite reduction diagrams as they arise from unsuccesfully tiling with elementary reduction diagrams. The simplest infinite reduction diagram is the Escher diagram in Figure., repeated in Figure 4.. Some more examples are given in Figure 4. and 4.. Note the fractal-like boundary that arises in Figure 4..

6 Figure 4.

7 Figure 4.

8 Figure REMARK. Since we admit also empty steps, it is not immediately clear that an infinite diagram contains infinitely many non-empty edges. However, this is indeed the case; Bezem et al. (996) proves the stronger fact that an infinite diagram possesses an infinite reduction containing infinitely many splitting steps. (An elementary diagram is splitting if one of the converging sides contains two or more steps which then are called splitting steps. Clearly, splitting steps are non-empty.) 4.. Towers in reduction diagrams A tower in an infinite reduction diagram is the result of adjoining elementary reduction diagrams in a linear way, as suggested in Figure 4.5. We will always be interested in infinite towers. Towers can be either horizontal or vertical. Figure 4.6 displays two towers in the fractallike diagram of Figure 4.; Figure 4.7 displays (shaded) one of the two towers constituting the Escher diagram.

9 infinite towers Figure 4.5 Figure 4.6

10 Figure PROPOSITION. Every infinite diagram contains an infinite horizontal tower and an infinite vertical tower. PROOF. Consider the infinite diagram, and draw in each tile arrows from the left side to the steps in the right side (see Figure 4.8). In this way finitely many trees arise. By the pigeon-hole principle and König's Lemma, one of these trees must have an infinite branch. This branch determines an infinite horizontal tower. Dually we find an infinite vertical tower. ø Figure 4.8

11 Consider again the left-to-right trees in the preceding proof. Their branches are linearly ordered according to whether the one is 'above' the other. A branch σ is above branch τ, when after running together for some (possibly 0) steps, σ branches off to above compared to τ. Furthermore it is clear that there is a highest infinite branch in the left-to-right trees of an infinite diagram. It is constructed in the obvious way: to start, choose the highest root of the leftright trees that has an infinite branch, then choose the highest successor with the same property, and so on. Since branches in the left-right trees correspond with horizontal towers, there also exists a highest horizontal infinite tower. This will play an important role later on REMARK. In fact, the towers of a reduction diagram are linearly ordered by the relation above. There may be continuum many towers (see Figure 4.9). continuum many towers Figure REMARK. As apparent from the example figures, there is quite a variety in the shape of a border of a reduction diagram and one may ask for a characterization of such borders. Let us adopt coordinates (x,y) such that horizontal edges are parallel to the x-axis and vertical edges parallel to the y-axis, with the origin (0,0) in the upper left-corner of the diagram. Now define the strict partial order on points (x,y) of the xy-plane by: (x,y) (x,y ) iff x x and y y and for subsets X,Y of the xy-plane: X Y iff for all P in X, P in Y we have P < P. Then, we claim: (i) any border is a dense linear order with respect to. (Proof sketch: the border δ(t) of a tower T is either a point or a line segment. The border of the diagram is the union of all these borders of towers (since the diagram is the union of all its towers) So let P be on the border of tower T and P on the border of tower T. Now consider the relative positions of δ (T) and δ(t ). The case of intersecting T, T is easy; we clearly then

12 have δt < δt. Otherwise T, T are both horizontal towers. If T = T : easy. Otherwise one of the towers is above the other, say T is above T. Then the result is obvious. (ii) Any border has the intersection property : every horizontal line that intersects with the diagram intersects with the border; likewise every vertical line. (iii) every border is continuous (i.e. is the image of a continuous map with domain a line segment in the xy-plane). Now one can prove that any curve C arising in the following way can be a border of an infinite diagram: C is the image of a continuous map from the unit line segment [0,] on the x- axis to the unit square [0,] in the xy-plane with f(0) = (0,0), f() = (,) such that C is linearly ordered with respect to as defined above. We will not pursue this matter in the sequel. 5. Tree coverings of reduction diagrams Next, we will define the concept of a tree covering of a reduction diagram. Elementary reduction diagrams will be equipped with arrows leading from the initial (diverging) edges of the elementary reduction diagram to the opposite (converging) edges. Each converging edge is traced back via an arrow to one of the two initial edges (if the elementary diagram is not trivial; empty sides are not traced back). Figure 5. shows an example of a finite, completed reduction diagram with a tree covering. In this example the branches of the trees do not intersect, in general they may however. tree covering of reduction diagram Figure 5.

13 Figure 6. contains a number of periodic tree coverings of the Escher diagram. The upper part of Figure 6. gives some of the tree coverings (not exhaustive) of the elementary diagram of which the Escher diagram is built. (Note that the Escher diagram is indeed built from elementary diagrams of a single shape). These tree covered elementary diagrams are then used to build the Escher diagram in various combinations,,... E.g. 3 means that the tree covered elementary diagram is used, next the elementary diagram 3 (after mirroring); then the 3 configuration is recursively repeated. 5.. REMARK. Note that an infinite horizontal tower does not always contain an infinite straight branch; see e.g. in Figure 6. the tree covering DEFINITION. (i) A step in a branch is straight if it leads from an initial edge to an opposing edge. Thus all steps in the canonical tree covering are straight. (ii) A branch changes orientation if it goes from vertical to horizontal or dually. (iii) An infinite branch is meandering if it changes orientation infinitely often. (iv) Let τ be a horizontal branch. We say that τ branches off downward to branch σ, if τ, σ are concurrent for some steps, after which σ branches off to a lower opposing edge, or changes orientation. Likewise dually: a vertical branch may branch off to the right. (v) There is exactly one tree covering all of whose steps are straight. We call it the canonical tree covering. infinite branch of tree covering, contained in tower, branching off only in downward direction Figure 5. Consider an infinite diagram, and an infinite horizontal tower in it. Consider of each elementary diagram in the tower, its upper edge (see the heavy edges in Figure 3.0). Trace back each of these upper edges all the way to the initial diverging reductions of the diagram.

14 Figure 5.3 Then, by a simple argument using the fact that the covering trees in the diagram are finitely branching, at least one infinite branch arises that we will call an upper boundary branch of the tower under consideration. It has the property that from any point on it infinitely many upper edges of the tower are reachable (by some branch of the tree covering). Figure 5.4 gives an example of an infinite horizontal tower with upper boundary branch, unique in this case; note that it is not eventually straight. infinite tower (shaded) with upper boundary branch

15 Figure 5.4 Figure 5.5 gives an example of an infinite horizontal tower with exactly two upper boundary branches, one of them straight, the other one not. More precisely, the construction of an upper boundary branch is as follows: Consider an arbitrary infinite horizontal tower. Consider of each elementary diagram in the tower its upper edge.trace each of these edges all the way back to the initial diverging reductions of the diagram. A path from an upper edge back to one of the initial edges is called an `upper edge branch'. Since there only finitely many initial steps, by pigeon holing at least one initial edge will be the origin of infinitely many upper edge branches. Choose such an initial edge and consider the infinite tree formed by all the upper branches originating from that edge. Since this is a finitely branching infinite tree, by Konig's Lemma it must have an infinite branch. Such a branch will be called an `upper boundary branch'. Note that an upper boundary branch is itself not an upper edge branch, since the latter are all finite. However, each initial segment of an upper boundary branch is also the initial segment of at least one upper edge branch (by definition). Even of infinitely many, according the following proposition PROPOSITION. Each initial segment of an upper boundary branch is also the initial segment of infinitely many upper edge branches. PROOF. Consider an initial segment s of the upper boundary branch and extend it to an upper edge branch e.since the upper edge branch is finite, there must be a (first) further point on the upper boundary branch that is not on the upper edge branch. Consider the initial segment s corresponding to that further point, and repeat the construction, resulting in a second upper edge branch e and a still further point on the upper boundary branch. Continuing this process indefinitely yields infinitely many upper edge branches e, e,... that all extend the original initial segment s. ø 5.5. COROLLARY. From any point on a upper boundary branch infinitely many upper edges of the tower are reachable. PROOF. Consider the initial segment corresponing to the point P on the ubb. According to the lemma there are infinitely many upper edge branches that extend it. All endpoints of these upper edge branches are reachable from P and are on an upper edge of the tower. ø

16 infinite tower (shaded) with two upper boundary branches Figure Impossible tree coverings We will now prove that it is impossible to cover an infinite reduction diagram with a tree covering such that: (i) all infinite branches are eventually straight, while (ii) eventually horizontal branches only may split off, eventually, in downward direction (iii) and dually, eventually vertical branches only may split off, eventually, to the right.

17 Figure 6. It is instructive to consider the ten cases of Figure 6.. These cases have the satisfy the properties (i)-(iii) as summed up in Table 6.. Indeed, no case has all three properties.

18 (i) (ii) (iii) Table 6. Let us elaborate on the underlying intuition. Condition (i) says that there are no infinitely meandering branches. Let us simplify the situation by forbidding any meandering, so assume all branches are straight. This means that we are dealing with the canonical tree covering (Def. 5.). Now consider the lowest horizontal branch σ and the rightmost vertical branch τ. Now say an edge in D is accounted for if a branch of the tree covering under consideration passes through it. the branches σ and τ account for infinitely many edges, as they are infinite. But there remain infinitely many edges not touched by σ and τ. Some experiments make this clear; e.g. in the Escher diagram we find that the steps in bold are not accounted for (see Figure 6.). In Figure 6.3 this is the grey area, containing infinitely many edges. Now if σ and τ are not allowed to branch off towards this infinitely large area, the tree covering can never cover all these edges. σ τ Figure 6.

19 τ σ Figure THEOREM. An infinite reduction diagram does not possess a tree covering such that (i) all infinite branches are eventually contained in towers, (ii) infinite branches contained in horizontal towers split, eventually, only downwards, (iii) infinite branches contained in vertical towers split, eventually, only to the right. PROOF. Now consider in the infinite diagram under consideration the highest infinite horizontal tower T, which exists according to Proposition 4... Consider any of its upper boundary branches; by () we know that there must exist one. Let us call it σ. By hypothesis (i) of the statement in the theorem σ must be contained in a tower T', which may be horizontal or vertical. CASE. T' is horizontal. Since T is the highest horizontal tower, T' must be T or be lower than T. Both cases are contradictory, since by hypothesis (ii) σ can branch off (after some steps) only in downward direction, hence can never be a boundary branch of T. CASE. T' is vertical. This requires more argument to show its impossibility. Consider the relative position of the towers T, T' as in Figure 6.6. Case (a), where the vertical tower T' intersects the horizontal T, is impossible: the boundary branch σ contained in T' can never reach the upper edges of T. Case (b), where the horizontal T proceeds beyond the vertical line starting at a condensation point of T', is equally impossible. This follows from a consideration of Figures 6.4 and 6.5: above an elementary diagram (the shaded rectangular zone in Figure 6.4) there can not occur a condensation point of the diagram. This follows since that elementary diagram together with the zone above, must be part of some finite stage of the diagram; and as Figure 6.5 makes clear, no point in a finite stage of a diagram construction can turn into a condensation point.

20 So only case (c) of Figure 6.6 remains as possibility. But in this case the branch σ contained in tower T', can not reach the upper edges of T, since eventually σ branches off only to the right. ø Figure 6.4 finite stage of diagram does not contain a condensation point Figure 6.5

21 (a) T' T (b) T' condensation point T (c) T' T relative position of horizontal and vertical tower Figure 6.6

22 7. Confluence by decreasing diagrams De Bruijn (978) gave a very strong confluence criterion for abstract reduction systems with indexed reduction relations. It consists of a combinatorial property of the distribution of indexes in the elementary diagrams. The original formulation in De Bruijn (978) was asymmetrical; Van Oostrom (94, 94a) gave a symmetrical version, as follows. We define an elementary diagram to be decreasing, if it has the following form (see Figure 7.): a n b <n m m <n or <m c <m n <n or <m d Figure 7. Elementary decreasing diagram This means that given two diverging steps a n b and a m c with indices n, m there is a common reduct d such that b <n. m... Â <n or <m d and c <m. n... Â <n or <m d So from b we take some steps with indices < n, followed by 0 or step with index m, followed by some steps with index < n or < m, with result d. Dually, from c we have a reduction to d as indicated. In Figure 7.(a) some non-decreasing elementary diagrams are given; in (b) some de-

23 creasing elementary diagrams. (The labels are subject to the usual ordering < on natural numbers.) (a) not decreasing (b) 3 3 decreasing Figure 7. Examples of (non-) decreasing elementary diagrams We will now connect the present definition with the tree coverings of above. In a decreasing elementary diagram we will trace back the converging steps to the two diverging steps. In doing so, it will be helpful to use a heavy arrow in case the index remains the same, and a light arrow in case the index decreases. The heavy and light arrows are determined as follows. Consider the vertical reduction b <n. m... Â <n or <m d. Now we let the first part of this reduction, consisting of steps with index less than the index n of the horizontal step a n b, trace back lightly to that step. If the second part consists of step with label m, it is traced back heavily to the vertical step a c. If it consists of 0 steps, we do nothing. The part consisting of steps with label less than n or m is treated as follows. If the step label is less than n we trace back lightly to a b, if less than m then lightly to a c, if both then we choose one. Likewise dually. So a decreasing elementary diagram with the tracing arrows has one of the shapes of Figure 7.3: containing two heavy arrows, or one, or none. It is important that heavy arrows (along which the indices remain the same) are straight, while the light arrows (along which the indices decrease) may involve a change of orientation.

24 decreasing elementary diagram Figure 7.3 Elementary diagrams with tree covering See Figure 7.4, consisting of the decreasing elementary diagrams of Figure 7. but now enriched with the tracing arrows (with the convention for heavy and light just mentioned). 3 3 Figure 7.4 Note that the tracing pattern (the tree covering) is not uniquely determined by the decreasing elementary diagram; e.g. Figure 7.5 contains two tracings for the same elementary diagram. Figure 7.5

25 We now have 7.. PROPOSITION. Every diagram construction using decreasing elementary diagrams will terminate eventually in a finite confluent diagram. PROOF. Equip the decreasing elementary diagrams with heavy and light arrows as explained above. Note that heavy arrows preserve indices and are straight, while light ones decrease indices and may change orientation. Note furthermore that a horizontal heavy arrow cannot split off in upward direction (see Figure 7.6) and like wise dually. (a) (b) 3 (c) not allowed (d) allowed Figure 7.6 Now consider an infinite branch in the diagram enriched with heavy and light arrows. Because the partial order I is well-founded, eventually only heavy (index-preserving) arrows can occur in this branch. But these are straight. So, every infinite branch must be eventually straight (and thus contained in a tower). Furthermore, from infinite horizontal branches we can only have split offs in downward direction (either by straight arrows as in Figure 7.6(c) or by a change in orientation as in 7.6(d). Likewise dually. That is, the three hypotheses of Theorem 6. are fulfilled. According to this theorem the diagram cannot be infinite. ø

26 7.. COROLLARY (Confluence by decreasing diagrams) Every ARS with reduction relations indexed by a well-founded partial order I, and satisfying the decreasing criterion for its elementary diagram s, is confluent. Acknowledgement. We thank Stefan Blom for providing us with the proof of a statement in Remark References BEZEM, M., KLOP, J.W. & VAN OOSTROM, V.,(996) Diagram Techniques for Confluence. Information and Computation, Vol.4, No., p.7-04, 998. BOGNAR, M. (997). The point version of decreasing diagrams. In: Proc. of Accolade 96, Dutch Graduate School in Logic, (eds. J. Engelfriet, Tigran Spaan), pp.-4, 997. DE BRUIJN, N.G. (978). A note on weak diamond properties. Memorandum 78-08, Eindhoven University of Technology, August 978. DERSHOWITZ, N. & JOUANNAUD, J.-P. (990). Rewrite systems. In: Formal models and semantics, Handbook of Theoretical Computer Science, Vol.B (J. van Leeuwen, editor), Elsevier - The MIT Press, Chapter 6, p HINDLEY, J.R. (964). The Church-Rosser property and a result in combinatory logic. Ph.D. Thesis, Univ. Newcastle-upon-Tyne, 964. HUET, G. (980). Confluent reductions: Abstract properties and applications to term rewriting systems. JACM, Vol.7, No.4 (980), KLOP, J.W. (980). Combinatory Reduction Systems. Mathematical Centre Tracts 7, Amsterdam 980. KLOP, J.W. (99). Term rewriting systems. In Vol. of Handbook of Logic in Computer Science (eds. S. Abramsky, D. Gabbay & T. Maibaum), Oxford University Press 99, p.-6. NEWMAN, M.H.A. (94). On theories with a combinatorial definition of "equivalence". Annals of Mathematics, 43():3-43, April 94. VAN OOSTROM, V. (994). Confluence for Abstract and Higher-Order Rewriting. Ph.-D. thesis, Vrije Universiteit, Amsterdam, March 994. VAN OOSTROM, V. (994a). Confluence by decreasing diagrams. Theoretical Computer Science 6. p STAPLES, J. (975). Church-Rosser theorems for replacement systems. In 'Algebra and Logic', Springer Lecture Notes in Mathematics, Vol. 450, 975,