By Christopher D. Godsil

ISBN-10: 0412041316

ISBN-13: 9780412041310

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**Example text**

20 (where 2 1 and x 2 may also be boxes). Intuitively the definition means that if the two productions are applied to a graph with a single edge (u1,a , u2),where ui is labeled Xi,then the same edges (21,6,22) are established between nodes of their right-hand sides, independent of the order in which the productions are applied. From this intuition the following characterization of confluence easily follows: the result of a derivation does not depend on the order in which the productions are applied.

Two edNCE grammars G and G’ are equivalent if L ( G ) - {A} = L(G’) - {A}, where A is the empty graph. 8: A street. Assumption. ,O,P,S), that i f S + ( D , C ) is in P , then C = 0. To change an arbitrary edNCE grammar into an equivalent one satisfying this assumption, take a new initial nonterminal S’ and add all productions S’ -+ ( 0 , s )with S + ( D , C ) in P for some C. 3. , [15,16,52]). It is easy to show that for each such grammar there is an equivalent edNCE grammar (for every edge (v, y,w) for which Y and w are not both terminal, add an edge (w,y,v), see [17] for details).

This means that the restriction to leftmost derivations is equivalent with the restriction t o confluent grammars. 1, respectively. Here we give a direct proof. TOsimplify the proof, we first give a natural sufficient condition for an edNCE grammar to be confluent. 10 Let G = (C, A , r, 0,P, S) be an edNCE grammar such that r = rl x for certain sets rI and and such that r2, r2 for every production X + (D, C), c, (1) i f ( ~ , ( P l l P 2 ) / ( Y I , Y 2 ) 1 ~ ,E ~~) (2) ~ f ~ ~ , ~ P 1 1 P 2 ~ l ~ Y 1E, ’ Y 2 ~ 1 ~ 1 ~ ~ ~ ~ then PI = y1 and (c’, (PI , P ~ ) / ( P,yI 2 ) , 5 , in) E C for every 0’E C and PI E rl.

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