D. Giammarresi. Two-dimensional languages and recognizable functions. In G. Rozenberg and A. Salomaa, editors, Developments in Language Theory, Proceedings of the conference, Turku (Finnland) '93, pages 290--301. world scientific, Singapore, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....function f exists such that L f is Sigma k definable. For the first claim (which proves the desired lower bound on expressiveness of Sigma k formulas) we give two proofs. The first uses an automata theoretic view of formulas as developed in [GRST96] which is then applied in the spirit of [Gia94] (Section 3 below) In [GRST96] it was shown that a Sigma 1 formula over grids can be transformed into a nondeterministic acceptor, called tiling system , which roughly consists of a list of state quadruples. A run of such a tiling system associates a state with each grid vertex such that any ....

D. Giammarresi. Two-dimensional languages and recognizable functions. In G. Rozenberg and A. Salomaa, editors, Developments in Language Theory, Proceedings of the conference, Turku (Finnland) '93, pages 290--301. world scientific, Singapore, 1994.

....of growth of a function realizable by a Sigma n formula can be at most n fold exponential in m. For the latter we use an automata theoretic view of formulas as developed in [GRST96] this in turn allows to apply pumping arguments. The approach is a generalization of a method due to Giammarresi ([Gia94]) which argues against Sigma 1 formulas as follows: Such a formula, say with a single leading existential quantifier, will be equivalent to a nondeterministic finite automaton which works through a given grid, column by column from left to right. A state of the automaton is a column vector of ....

....of a shortest word accepted by an NFA A is at most the number of states of A, so the following is an easy consequence of Theorem 12. Corollary 13 ( third claim of Theorem 4) Any Sigma n definable function m 7 R(m) is s n (O(m) and hence f n (O(m) 2 The preceding fact has been shown in [Gia94, GR96] for the special case n = 1 using essentially the same argument but without explicit usage of the close connection of Sigma 1 definable picture languages to finite word automata. 4 Fast Growing MSO Definable Functions In this section we will develop formulas that define n fold exponential ....

D. Giammarresi. Two-dimensional languages and recognizable functions. In G. Rozenberg and A. Salomaa, editors, Developments in Language Theory, Proceedings of the conference, Turku (Finnland) '93, pages 290--301. world scientific, Singapore, 1994.

....that a picture language is recognizable if and only if it can be defined by a formula of monadic second order logic. From this, one derives a new proof for Fagin s theorem on descriptive complexity of finite structures. Numerical functions that can be defined by these languages are studied in [271]. A topic closely related to that of two dimensional languages considers cellular automata on general graphs. An extension of the Moore and Myhill theorem on the existence of garden of Eden configuration to cellular automata on Cayley graphs is given in [317] 4 Area 4: Parallel and Distributed ....

D. Giammarresi. Two-dimensional languages and recognizable functions. In G. Rozenberg and A. Salomaa, editors, Proc. Developments in Language Theory. World Scientific Publishing Co., July 1993.

....is at most k fold exponential; F is at least k fold exponential iff some F definable function is at least k fold exponential; F is k fold exponential iff F is both at least and and most k fold exponential. Example 12. Example 4 shows that Delta 1 is at least 0 fold exponential. In [Gia94], the author shows that Sigma 1 is 1 fold exponential. This means that every Sigma 1 definable function is at most singly exponential, and there is one singly exponential Sigma 1 definable function (namely m 7 2 m ) Note that, of course, there are other Sigma 1 definable functions ....

D. Giammarresi. Two-dimensional languages and recognizable functions. In G. Rozenberg and A. Salomaa, editors, Developments in Language Theory, Proceedings of the conference, Turku (Finnland) '93, pages 290--301. world scientific, Singapore, 1994.

....of growth of a function re alizable by a Sigma n formula can be at most n fold exponential in m. For the latter we use an automata theoretic view of formulas as developed in [11] this in turn allows to apply pumping arguments. The approach is a generalization of a method due to Giammarresi [9] which argues against Sigma 1 formulas as follows: Consider grids of a fixed height m; then such a formula, say with a single leading existential quantifier, will be equivalent to a nondeterministic finite automaton (NFA) which works through the given grid, column by column from left to right. ....

....f : N1 N1 . By Theorem 9 there is a c such that for all m 1 there is an NFA with s n Gamma1 (c m ) states and singleton alphabet that accepts only the word of length f(m) This implies that f(m) s n Gamma1 (c m ) s n (m log 2 c) for all m. 2 The preceding fact has been shown in [9, 10] for the special case n = 1 using essentially the same argument. In the remainder of this section we strengthen this result to cover the functions which are definable by boolean combinations of Sigma n sentences, rather than just Sigma n sentences. This becomes relevant in the presence of the ....

D. Giammarresi. Two-dimensional languages and recognizable functions. In G. Rozenberg and A. Salomaa, editors, Developments in LanguageTheory, Proceedings of the Conference, Turku (Finnland) '93, pages 290--301. world scientific, Singapore, 1994.

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