Algebraic Approaches to Graph Transformationpart I : Basic Concepts and Double Pushout

derivation is consistent with the notion of abstract graph, i.e., any abstract derivation should start from an abstract graph and should end in an abstract graph. De nition 27 (well-de ned equivalences). For each abstract derivation [ ], de ne ([ ]) = f ( 0) j 0 2 [ ]g, and similarly ([ ]) = f ( 0) j 0 2 [ ]g. Then equivalence is well-de ned if for each derivation : G ) H we have that ([ ]) = [ ( )], and ([ ]) = [ ( )]. ut If is well-de ned, then for each derivation : G ) H we can write [ ] : [G] ) [H ]. Since our goal is to de ne abstract models of computation as quotient categories of concrete ones, the second requirement is that the equivalence is a congruence with respect to sequential composition. De nition 28 (equivalence allowing for sequential composition). Given two abstract derivations [ ] and [ 0] such that ([ ]) = ([ 0]), their sequential composition [ ] ; [ 0] is de ned i for all 1; 2 2 [ ] and 01; 02 2 [ 0] such that ( 1) = ( 01) and ( 2) = ( 02), one has that 1 ; 01 2 ; 02. In this case [ ] ; [ 0] is, by de nition, the abstract derivation [ 1 ; 01]. An equivalence allows for sequential composition i [ ] ; [ 0] is de ned for all [ ] and [ 0] such that ([ ]) = ([ 0]). ut The third requirement guarantees that we reobtain the result of uniqueness of canonical derivations in the abstract framework, i.e., that each abstract derivation has a unique shift-equivalent abstract canonical derivation. De nition 29 (uniqueness of canonical derivations). Equivalence enjoys uniqueness of canonical derivations i for each pair of equivalent derivations 0 and for each pair h c; 0ci of canonical derivations, sh c and 0 sh 0c implies that c 0c. ut 5.3 Towards an equivalence for representation independence. In this section we introduce two di erent equivalences on derivations, and analyze their properties with respect to the requirements established in Section 5.2. Actually, since by the de nitions all the equivalences are clearly well-de ned in the sense of De nition 27, we will focus only on the other requirements. The rst equivalence we consider is obtained from equivalence 0 of De nition 26 by requiring that the productions applied at each (parallel) direct derivation are isomorphic and not just span-isomorphic. De nition 30 (equivalence 1). Let : G0 ) Gn and 0 : G00 ) G0n be two derivations such that 0 0 as in Figure 14. Then they are 1-equivalent (written 1 0) if 46 1. there exists a family of permutations h 1; : : : ; ni such that for each i 2 f1; : : : ; ng, productions qi and q0 i are isomorphic via i; 2. For each i 2 f1; : : : ; ng, the family of isomorphisms f Li : Li ! L0i; Ki : Ki ! K 0 i; Ri : Ri ! R0 ig that exists by De nition 26 is exactly the family of isomorphisms between corresponding graphs of qi and q0 i which is induced by permutation i, according to Fact 16. We say that and 0 are 1-equivalent via h iii n if h iii n is the family of permutations of point 1 above. Given a derivation , its equivalence class with respect to 1 is denoted by [ ]1, and is called a 1-abstract derivation. ut Let us consider now the algebraic properties of equivalence 1. We prove below that 1 enjoys uniqueness of canonical derivations. This result is based on an important lemma which states that 1 behaves well with respect to analysis, synthesis, and shift of derivations. This is the main technical result of this section, and the whole Appendix B is devoted to its proof. Lemma31 (analysis, synthesis and shift preserve equivalence 1). Let 1 0 via h iii n, and let G0 and Gn be the isomorphisms between their starting and ending graphs (see Figure 14). 1. If 1 2 ANALij( ) then there is at least one derivation 2 2 ANALi i(j)( 0). Moreover, for each 2 2 ANALi i(j)( 0), it holds that 2 1 1. 2. If 1 2 SYNT i( ) then there is at least one derivation 2 2 SYNT i( 0). Moreover, for each 2 2 SYNT i( 0), it holds that 2 1 1. 3. If 1 2 SHIFT i j ( ) then there is at least one derivation 2 2 SHIFT i i(j)( 0). Moreover, for each 2 2 SHIFT i i(j)( 0), it holds that 2 1 1. Furthermore, in all these three cases, the isomorphisms relating the starting and ending graphs of the 1-equivalent derivations 2 and 1 are exactly isomorphisms G0 and Gn . ut The observation that analysis, synthesis and shift preserve not only equivalence 1, but also the isomorphisms between the starting and ending graphs of equivalent derivations will be used later in Proposition 36. Theorem32 ( 1 enjoys uniqueness of canonical derivations). Let , 0 be two derivations such that 1 0, and let c, 0c be two canonical derivations such that sh c and 0 sh 0c. Then c 1 0c. 47 5:S 1:idle 4:C 2: job 3: req G1 4:C 5:S 1:idle 3: req 5:S 4:C 6:busy 4: job 5: req G3 1:S 1:idle 0:C 0: req 1:S 0:C 1:S 0:C 2: busy 0:C 0: job 0:C 0:C 2: job 1:req 0 4 2 4 1 5 0 2 0 4 1 5 2 6 5:S 1:idle 4:C 4: job 3: req 5: req 0 3 1 1 5:S 4:C 4: job 5: req