15 Hyp ergraph Rewriting Critical Pairs and Undecidabili ty of Conuence Detlef Plump INTRODUCTION In their pioneering pap er KB Knuth and Bendix showed that conuence or equivalently the ChurchRosser prop erty is decidable for terminating term rewriting systems It suces to compute all critical pairs t s u of rewrite steps in which s is the sup erp osition of the lefthand sides of two rules and to check whether t and u reduce to a common term This pro cedure is justied by the socalled Critical Pair Lemma Hue which states that a term rewriting system is lo cally conuent if and only if all critical pairs have a common reduct For hypergraph rewriting systems however no such simple characterization of lo cal conuence is p ossible The reason is that the embedding of derivations into context is more complicated than for tree rewriting It is shown b elow that in the graph case conuence of all critical pairs need not imply general lo cal conuence This phenomenon refutes a critical pair lemma published by Raoult Rao p ersonal communication Okada and Hayashi OH avoid the problem by giving a critical pair lemma under the strong restriction that distinct no des in a graph must not have the same lab el In this chapter a critical pair lemma for general hypergraph rewriting is presented which provides a sucient condition for lo cal conuence It requires that all critical pairs are conuent by derivations that satisfy certain conditions The second part of Term Graph Rewriting Theory and Practice eds MRSleep MJPlasmeijer and MC van Eekelen c John Wiley Sons Ltd PLUMP this chapter reveals that a simple characterization of lo cal conuence is indeed im p ossible conuence is shown to b e undecidable for terminating hypergraph rewriting systems HYPERGRAPH REWRITING In this section the Berlin approach to graph rewriting is briey reviewed see Ehr for a comprehensive survey but all notions are lifted to the hypergraph case which is more exible in applications In particular three theorems of the Berlin approach are recalled concerning the commutation restriction and extension of derivations These results are essential to ols in the pro of of the Critical Pair Lemma Hyp ergraphs and hypergraph morphisms Let h i b e a signature that is and are sets of no de and edge V E V E lab els and each comes with a pair ty pe h i of strings E V A hypergraph over is a system G hV E l m s t i where V and E are G G G G G G G G nite sets of nodes and hyperedges or edges for short l V and m E G G V G G are functions that assign strings are labeling functions and s t E V E G G G G s e t e of source and target nodes to each hyperedge e such that ty pem e G G G hl s e l t ei The extension f A B of a function f A B maps the G G G G empty string to itself and a a to f a f a n n G is said to b e discrete if E G In pictures of hypergraphs no des are drawn as circles and hyperedges as b oxes b oth with inscrib ed lab els Lines without arrowheads connect a hyperedge with its source no des while arrows p oint to the target no des For example the graphical structure n n a a m m n S Sw n n b b n represents a hyperedge together with its source and target no des where ty pe ha a b b i Ordinary edges with one source and one target no de are fre m n quently depicted as arrows with lab els written aside Let GH b e hypergraphs Then G is a subhypergraph of H denoted by G H if V V E E and l m s t are restrictions of the corresp onding G H G H G G G G functions of H A hypergraph morphism f G H consists of two functions f V V and V G H f E E that preserve lab els and assignments of source and target no des that E G H is l f l m f m s f f s and t f f t f is H V G H E G H E G H E G V V injective surjective if f and f are injective surjective f is an isomorphism if it V E is injective and surjective in this case G and H are isomorphic denoted by G H HYPERGRAPH REWRITING The subhypergraph of H with no de set f V and edge set f E is denoted by V G E G f G If G H then G H denotes the inclusion morphism Rules and derivations A rule r L K R consists of three hypergraphs LK R and a morphism K R where K L A hypergraph rewriting system G R consists of a signature and a set R of rules with hypergraphs over For the rest of this section and the following section G denotes an arbitrary hypergraph rewriting system Let G H b e hypergraphs Given a rule r L K R from G and a morphism g L G G directly derives H through r and g denoted by G H if there are rg two hypergraph pushouts of the following form L K R g c G D H See Ehr for the denition and construction of graph pushouts the extension to hypergraphs is straightforward Intuitively D is obtained from G by removing the no des and edges in g L g K and H is constructed from D by identifying items in g K as sp ecied by K R and by adding the items in R K The relations and are dened in the obvious way G H means G H r or G H G derives H denoted by G H if G H or there are hypergraphs G G n such that G G G G H n n Proposition Let G be a hypergraph r L K R be a rule and g L G be a morphism Then there exists a direct derivation G H if and only rg if the fol lowing two conditions are satised Contact Condition No edge in G g L is incident to any node in g L g K Identication Condition For al l items x y in L g x g y implies x y or x y K The following track function allows to follow no des through derivations For a direct derivation G H tr ack V V is the partial function dened by G)H G H c v if v D V tr ack v G)H undened otherwise For a derivation G H tr ack i if G H by an isomorphism i G H G) H V if G H by a sequence G tr ack and tr ack tr ack G )G G) H G )G 0 1 n1 n G G G H n G is conuent if for all hypergraphs G H H with H G H there is a hypergraph M such that H M H G is locally conuent if for all direct derivations of the form H G H there is an M such that H M H Finally G is terminating if it do es not admit an innite sequence G G G of direct derivations PLUMP Commutation restriction and extension of derivations The following three theorems were originally formulated for graphs rather than for hypergraphs But insp ecting their pro ofs shows that they can b e extended to the hypergraph case without further ado H G Theorem Commutation theorem EK Let H r g r g 2 2 1 1 be direct derivations through rules r L K R for i If g L g L i i i i H M g K g K then there is a hypergraph M such that H r r 1 2 The following variant of the socalled Clip Theorem applies only to direct deriva tions which suces for the purp oses of the present chapter Theorem Kre Let G H be a direct derivation through a rule rg 0 U r L K R If S is a subhypergraph of G such that g L S then S rg 0 where g is the restriction of g to S and U H Moreover tr ack is the restriction S )U of tr ack G)H The next theorem allows a derivation to extend by arbitrary context provided that context edges are not attached to no des that are removed by the derivation The present form of the theorem is tailored to the pro of of the Critical Pair Lemma Theorem Ehr Kre Let S T U be a derivation and G be a rg hypergraph with S G Let Boundary be the discrete subhypergraph of S that consists of al l nodes that are touched by any edge in G S If tr ack is dened for al l S )T ) U H M such that T H and nodes in Boundary then there is a derivation G g r g is the extension of g to G Moreover M is dened by the pushout tr Boundary U Context M where Context G S Boundary is a subhypergraph of G Boundary Context is the inclusion of Boundary in Context and tr is the restriction of tr ack to S )T ) U Boundary considered as a morphism THE CRITICAL PAIR LEMMA The quest for a critical pair lemma is motivated by the problem of testing hypergraph rewriting systems for lo cal conuence The idea is to infer the conuence of arbi H from the conuence of those steps where G G trary divergent steps H r r 2 1 represents a critical overlap of the lefthand sides of r and r By the Commutation Theorem such an overlap is critical only if it comprises no des or edges that are removed by r or r This suggests the following denition of a critical pair Definition Critical pair Let r L K R be rules for i i i i i U is a critical pair if S A pair of direct derivations of the form T r g r g 2 2 1 1 S g L g L and g L g L g K g K Moreover g g is required for the case r r HYPERGRAPH REWRITING In the sequel two critical pairs are not distinguished if they dier only by renaming of no des and edges The critical pairs arising from r and r can b e computed by U where L L L L constructing all pairs of direct derivations T r r 2 1 is a quotient of the disjoint union L L that identies at least one item in L K resp L K with some item in L L By the Commutation Theorem a strong conuence prop erty can b e estab lished for the case that G has no critical pairs at all This is substantially dierent from term rewriting where only lo cal conuence holds see for example Hue Theorem Hypergraph rewriting systems without critical pairs are strongly conuent that is whenever H G H then there is a hypergraph X such that H X H H If g L g L g K g K then there are direct G Proof Let H r g r g 2 2 1 1 H by Theorem Assume therefore the contrary The M