W. Thomas, Ehrenfeucht games, the composition method, and the monadic theory of ordinal words, in Structures in Logic and Computer Science. A Selection of Essays in Honor of Andrzej Ehrenfeucht (G. Rozenberg, A. Salomaa, and J. Mycielski, eds.), Lecture Notes in Computer Science, vol.1261, Springer-Verlag, Berlin, 1997, pp. 118--143.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....with tree automatic presentation strictly includes the class of automatic structures (see [5] 2) The structure (N, stands for having no common divisor is automatic. 7 Composition of structures The composition method developed by Feferman and Vaught [21] and by Shelah [39] see also [26, 41]) considers compositions (products and sums) of structures according to some index structure and allows one to compute depending on the type of composition the first order or monadic second order theory of the whole structure from the respective theories of its components and the monadic ....

W. Thomas, Ehrenfeucht games, the composition method, and the monadic theory of ordinal words, in Structures in Logic and Computer Science. A Selection of Essays in Honor of Andrzej Ehrenfeucht, G. Rozenberg, A. Salomaa, and J. Mycielski, eds., no. 1261 in LNCS, Springer, 1997, pp. 118--143.

....a dag t = V; E; de nes a ( Q) labeled dag by t 0 = V; E; r) Then r is a run i for any y 2 V , the restriction of t 0 to y and its lower neighbors is a transition, i.e. if t 0 can be tiled by the transitions. Di erently from nondeterministic graph acceptors, considered e.g. in [Tho97a,Tho97b], here we take only into account the lower neighbors and not all neighbors of a node y. The reason for this restriction is that we understand a dag as a process that continues in time. Having this in mind, the state reached by performing a node y can depend only on its history but not on the ....

W. Thomas. Ehrenfeucht games, the composition method, and the monadic theory of ordinal words. In J. Mycielski et al., editor, Structures in Logic and Computer Science, A Selection of Essays in Honor of A. Ehrenfeucht, pages 25{ 40. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 1997.

....Th k ( P D t ) 5 We already mentioned that the D sum of n graphs is a special case of the generalized sum as considered by Shelah. The following theorem follows from his result [18, Thm. 2. 4] A condensed proof of the full result can be found in [11] for lexicographic sums of linear orders, [21] contains the full proof. Shelah is also interested in the e ective computability of the combined theory from the argument theories. If one is only interested in the result as stated above, an alternative and much simpler proof of both statements can be given using Ehrenfeucht Fra ss e games for ....

W. Thomas. Ehrenfeucht games, the composition method, and the monadic theory of ordinal words. In J. Mycielski et al., editor, Structures in Logic and Computer Science, A Selection of Essays in Honor of A. Ehrenfeucht, Lecture Notes in Comp. Science vol. 1261, pages 118-143. Springer, 1997.

....parallel product k, and infinite sequential product Q are special cases of the generalized sum as considered by Shelah [She75] The preceding theorem follows from his result [She75, Thm. 2. 4] A condensed proof of Shelah s result can be found in [Gur79] for lexicographic sums of linear orders, Tho97] contains the full proof. In the terminology of [CM00] the first statements say that the sequential and the parallel product are Hintikka operations. These results can also be derived from [CM00] on quantifier free definable operations. Shelah is also interested in the effective computability of ....

W. Thomas. Ehrenfeucht games, the composition method, and the monadic theory of ordinal words. In J. Mycielski et al., editor, Structures in Logic and Computer Science, A Selection of Essays in Honor of A. Ehrenfeucht, Lecture Notes in Comp. Science vol. 1261, pages 118--143. Springer, 1997. 36

....second order theory of the whole structure from the respective theories of its components and the monadic theory of the index structure. The characterisation given in the previous section can be used to prove closure of automatic structures under such compositions of finitely many structures (see [23, 13, 16]) A generalised product as it is defined below is a generalisation of a direct product, a disjoint union, and an ordered sum. We will prove that given a finite sequence (A i ) i of structures which belong to some class K containing a complete structure, all their generalised products are ....

W. Thomas. Ehrenfeucht games, the composition method, and the monadic theory of ordinal words. LNCS, 1261:118-- 143, 1997.

....second order theory of the whole structure from the respective theories of its components and the monadic theory of the index structure. The characterisation given in the previous section can be used to prove closure of automatic structures under such compositions of finitely many structures (see [23, 13, 16]) A generalised product as it is defined below is a generalisation of a direct product, a disjoint union, and an ordered sum. We will prove that given a finite sequence (A i ) i of structures which belong to some class K containing a complete structure, all their generalised products are ....

W. Thomas. Ehrenfeucht games, the composition method, and the monadic theory of ordinal words. LNCS, 1261:118-- 143, 1997.

....of theories MThhN; P i for recursive P ; it turns out, for example, that for recursive P the theory MThhN; P i is truth table reducible to a complete 2 set, and that this bound cannot be improved. The situation changes when recursive predicates over countable ordinals are considered (see [22]) In [20] it was shown that for each predicate P , the full monadic theory MThhN; P i is decidable i the weak monadic theory WMThhN; P i is (where all set quanti ers are assumed to range only over nite sets) However, there are examples P such that the rst order theory FThhN; P i is ....

Thomas, W. Ehrenfeucht games, the composition method, and the monadic theory of ordinal words. In Structures in Logic and Computer Science, A Selection of Essays in Honor of A. Ehrenfeucht, J. Mycielski et al., Eds., Lect. Notes in Comput. Sci. Springer-Verlag, 1997, pp. 118-143. 22

....of theories MThhN; P i for recursive P ; it turns out, for example, that for recursive P the theory MThhN; P i is truth table reducible to a complete 2 set, and that this bound cannot be improved. The situation changes when recursive predicates over countable ordinals are considered (see [19]) In [18] it was shown that for each predicate P , the full monadic theory MThhN; P i is decidable i the weak monadic theory WMThhN; P i is (where all set quanti ers are assumed to range only over nite sets) However, there are examples P such that the rst order theory FThhN; P i is ....

Thomas, W. Ehrenfeucht games, the composition method, and the monadic theory of ordinal words. In Structures in Logic and Computer Science, A Selection of Essays in Honor of A. Ehrenfeucht, J. Mycielski et al., Eds., Lect. Notes in Comput. Sci. Springer-Verlag, 1997, pp. 118-143. 21

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W. Thomas, Ehrenfeucht games, the composition method, and the monadic theory of ordinal words, in Structures in Logic and Computer Science. A Selection of Essays in Honor of Andrzej Ehrenfeucht (G. Rozenberg, A. Salomaa, and J. Mycielski, eds.), Lecture Notes in Computer Science, vol.1261, Springer-Verlag, Berlin, 1997, pp. 118--143.

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W. Thomas. Ehrenfeucht Games, the Composition Method, and the Monadic Theory of Ordinal Words. In Structures in Logic and Computer Science, A Selection of Essays in Honor of Andrzej Ehrenfeucht. Pages 118--143,Springer, 1997.

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W. Thomas, Ehrenfeucht games, the composition method, and the monadic theory of ordinal words, in: Structures in Logic and Computer Science: A Selection of Essays in Honor of A. Ehrenfeucht, Lecture Notes in Computer Science, vol. 1261, Springer-Verlag, Berlin, 1997, pp. 118--143.

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