F. Drewes, J. Engelfriet; Decidability of finiteness of ranges of tree transductions, Inform. and Comput. 145 (1998), 1-50  Home/Search   Document Details and Download   Summary   Related Articles   Check

....transformation system (in the double pushout approach ) terminates for all input graphs is undecidable. This is shown in [Plu98] by a reduction of the Post Correspondence Problem. As a consequence, total correctness of programs in graph transformation languages is in general undecidable, too. In [DE98] the finiteness of ranges of a large class of tree transductions is proved to be decidable. It is shown that this yields a generalization of known decidability results concerning boundedness problems for functions on graphs and other domains. The functions that can be dealt with are definable by ....

Frank Drewes and Joost Engelfriet. Decidability of the finiteness of ranges of tree transductions. Information and Computation, 1998. To appear.

....provides the use of already existing analysis techniques in the field of temporal logic. Furthermore, the construction of a minimal model is introduced allowing for the use of model checking. The contents of the following papers have been described in the First Annual Progress Report: Dre98a, DE98, Plu98] 4.1.4 Language Design Why not combining two successful principles in one formalism: rules and graphs This is the common vision of the graph transformation community and the main motivation behind the development of PROGRES, a visual language that supports PROgramming with Graph ....

Frank Drewes and Joost Engelfriet. Decidability of the finiteness of ranges of tree transductions. Information and Computation, 145:1--50, 1998.

....composition with T (Theorem 4.12 of [EV85] this means that rub b1 ; b n (L) is in yMTT (L) 2 The proof of Lemma 8 in fact shows that yMTT (L) is closed under deterministic generalized sequential machine (GSM) mappings. For the case of nondeterministic MTTs it is shown in Theorem 6. 3 of [DE98] that the class of string languages generated by them is closed under nondeterministic GSM mappings. We are now ready to prove that there is a string language which can be generated by a nondeterministic top down tree transducer with monadic input but not by the composition closure of MTTs. ....

Frank Drewes and Joost Engelfriet. Decidability of finiteness of ranges of tree transductions. Inform. and Comput., 145:1--50, 1998.

....x 1 i(nil; hq Ln ; x 1 i(nil; hq L ; x 2 i(nil; y 2 ) Finally, since the first son of a state is always nil, we need rules hq L ; nili(y 1 ; y 2 ) y 2 . The formal construction is straightforward. Then decidability of emptiness and finiteness follows from Lemma 3.14 and Theorem 4. 5 of [7], respectively. 7.3 Safe Transformations We now define a dynamic restriction on DTL mso d programs that allows copying but nevertheless induces transformations of only linear size increase. We bound the number of times that any node u of an input tree s may be selected during the transformation ....

Frank Drewes and Joost Engelfriet. Decidability of finiteness of ranges of tree transductions. Inform. and Comput., 145:1--50, 1998.

....by 7( x,x2,e) i, b,x,x2, where has rank 4. Hence, since T r REGT, peb( peb( Tr) yDtMTT(REGT) Since M is a DPT0 transduction, Lemma 2 now implies that out(M) yDtMTT(REGT) Since all these results are effective, and the finiteheSS problem for languages in yDtMTTk(REGT) is decidable [6], Theorem 4 implies that the finiteheSS problem for output languages of deterministic pebble transducers is decidable. Hence, since OUT(DPT) is closed under intersection with regular languages (by an obvious product construction) it is decidable for a deterministic pebble transducer M and a ....

F. Drewes, J. Engelfriet; Decidability of finiteness of ranges of tree transductions, Inform. and Comput. 145 (1998), 1-50

....# M # (s[u # p] # M ## (t)equals sts M (s, u) seen as a monadic tree. Hence, for K = s # T# (P ) #P (s) 1 , M is fci i# # M ## (# M # (K) is finite. This is decidable, because finiteness of the range of a composition of MTT R s, restricted to a regular tree language K,is decidable [DE98]. ii) It is straightforward to construct an MTT R M # with input alphabet # # q (1) q # Q and output alphabet # # Ym (for an appropriate m) such that Characterizing and Deciding MSO Definability of Macro Tree Transductions 547 # M # (q(s) M q (s) for every q # Q and s # T# ....

....M # (q(s) M q (s) for every q # Q and s # T# , and to construct an MTT R M ## such that for every t # T# (Ym ) size(# M ## (t) 1 # # y j (t) j # [m] Hence, for K = q(s) q # Q, s # T# , M is fcp i# # M ## (# M # (K) is finite. As above this is decidable by [DE98]. # Pumping Lemmas We now present two pumping lemmas for non fci T R s and for non fcp MTT R gfci s, respectively. They are the core of the proof, in Section 5, that linear size increase implies gfci and fcp. The first lemma is similar to Lemma 4.2 of [AU71] We use the following notation (to ....

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F. Drewes and J. Engelfriet. Decidability of finiteness of ranges of tree transductions. Inform. and Comput., 145:1--50, 1998.

....Group APPLIGRAPH through the University of Bremen. 1 The idea to associate a tree grammar or tree transducer with an algebra that maps trees into a semantic domain was already mentioned in [14] For graphs generated by node or hyperedge replacement this has been worked out in [15] see also [3, 11, 9]) For the field of picture generation the idea seems to be new, however. In the main part of the paper four of the well known classes of picture generating devices found in the literature, namely collage grammars [22] mutually recursive function systems (a generalised type of iterated function ....

....research. A proof technical advantage is that constructions can often be formulated as tree transductions, as it was done in [8] see also [7] In these cases one can make use of known closure properties and other results from the theory of tree transductions in order to get concise proofs (cf. [11]) Another point is that a construction or a proof idea may not only apply to a single type of picture generating systems, but may be applicable to related devices as well. In such cases it is convenient to use a unified framework in order to avoid having to write proofs twice (see [7] again) ....

Frank Drewes and Joost Engelfriet. Decidability of the finiteness of ranges of tree transductions. Information and Computation, 145:1--50, 1998.

.... p] M 00 (t) equals sts M (s; u) seen as a monadic tree. Hence, for K = fs 2 T Sigma (P ) j #P (s) 1g, M is fci iff M 00 ( M 0 (K) is finite. This is decidable, because finiteness of the range of a composition of MTT R s, restricted to a regular tree language K, is decidable [DE98]. ii) It is straightforward to construct an MTT R M 0 with input alphabet Sigma [ fq (1) j q 2 Qg and output alphabet Delta [ Ym (for an appropriate m) such that M 0 (q(s) M q (s) for every q 2 Q and s 2 T Sigma , and to construct an MTT R M 00 such that for every t 2 T ....

....= M q (s) for every q 2 Q and s 2 T Sigma , and to construct an MTT R M 00 such that for every t 2 T Delta (Ym ) size( M 00 (t) 1 P f# y j (t) j j 2 [m]g. Hence, for K = fq(s) j q 2 Q; s 2 T Sigma g, M is fcp iff M 00 ( M 0 (K) is finite. As above this is decidable by [DE98]. Pumping Lemmas We now present two pumping lemmas for non fci T R s and for non fcp MTT R gfci s, respectively. They are the core of the proof, in Section 5, that linear size increase implies gfci and fcp. The first lemma is similar to Lemma 4.2 of [AU71] We use the following notation (to ....

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F. Drewes and J. Engelfriet. Decidability of finiteness of ranges of tree transductions. Inform. and Comput., 145:1--50, 1998.

....and of the induced transformations turn out to be decidable, provided that the chosen grammars, Partially supported by the EC TMR Network GETGRATS (General Theory of Graph Transformation Systems) through the University of Bremen. transducers, and algebras are taken from suitable classes [Eng94, Dre96a, DE98]. In this paper, a software system called Treebag (Tree Based Generator) is presented in which these facts are exploited in order to allow for the generation, manipulation, and visualization of objects of various kinds. From the user s point of view (which in many ways just reflects the internal ....

....turtle algebras) and for collages. As the user may interactively compose regular tree grammars with arbitrary sequences of top down tree transductions and YIELD transductions one gets in fact a very large class of tree languages and tree transductions the one which is denoted by R TBY in [DE98] and which equals the class of output languages of the composition closure of macro tree transductions (see also, e.g. EV85, EV88] In particular, the IO hierarchy of tree languages (whose first levels are the regular and the IO context free tree languages) is obtained in this way. Furthermore, ....

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Frank Drewes and Joost Engelfriet. Decidability of the finiteness of ranges of tree transductions. Information and Computation, 1998. To appear.

....with every derivation tree the corresponding picture. The idea to associate a tree grammar or tree transducer with an algebra that maps trees into a semantic domain was already mentioned in [Eng80] For graphs generated by node or hyperedge replacement this has been worked out in [Eng94] see also [CE95, DE96, Dre97]) For the field of picture generation the idea seems to be new, however. In the main part of the paper four of the well known classes of picture generating devices found in the literature, namely collage grammars [HK91] mutually recursive function systems (or hierarchical iterated function ....

.... A proof technical advantage is that constructions can often be formulated as tree transductions, as it was done in [Dre96b] see also [Dre96a] In these cases one can make use of known closure properties and other results from the theory of tree transductions in order to get concise proofs (cf. [DE96]) Another point is that a construction or a proof idea may not only apply to a single type of picture generating systems, but may be applicable to related devices as well. In such cases it is convenient to use a unified framework in order to avoid having to write proofs twice (see [Dre96a] ....

Frank Drewes and Joost Engelfriet. Decidability of the finiteness of ranges of tree transductions. Report 9/96, Univ. Bremen, 1996. Revised version to appear in Information and Computation.

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