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Handbook of Formal Languages: Volume 3 Beyond Words by Ferenc Gécseg, Magnus Steinby (auth.), Prof. Dr. Grzegorz

By Ferenc Gécseg, Magnus Steinby (auth.), Prof. Dr. Grzegorz Rozenberg, Prof. Dr. Arto Salomaa (eds.)

The desire for a accomplished survey-type exposition on formal languages and comparable mainstream parts of computing device technological know-how has been glaring for a few years. within the early Seventies, whilst . the publication Formal Languages by way of the second one­ pointed out editor seemed, it used to be nonetheless rather possible to put in writing a complete booklet with that name and contain additionally issues of present learn curiosity. this may no longer be attainable anymore. A standard-sized e-book on formal languages might both need to remain on a reasonably low point in any other case be really expert and limited to a few slender region of the sphere. The setup turns into vastly assorted in a suite of contributions, the place the simplest professionals on the planet sign up for forces, each one of them concentrat­ ing all alone components of specialization. the current three-volume guide constitutes one of these particular assortment. In those 3 volumes we current the present cutting-edge in formal language conception. We have been so much chuffed with the enthusiastic reaction given to our request for contributions by means of experts representing quite a few subfields. the necessity for a guide of Formal Languages was once in lots of solutions expressed in numerous methods: as an simply available his­ torical reference, a normal resource of data, an total course-aid, and a compact selection of fabric for self-study. we're confident that the ultimate consequence will fulfill such quite a few wishes. the idea of formal languages constitutes the stem or spine of the sector of technological know-how now commonly known as theoretical machine science.

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If we allow for arbitrary nodes of a production to have a label which is a nonterminal symbol, then we get a generalization of context-free grammars [Rou69, Rou70a], called context-free tree grammars, which can be viewed as symbolic nondeterministic recursive procedures with parameters. A context-free tree grammar is a system G = (N,E,X,P,ao), where is a ranked alphabet of nonterminal symbols. is a ranked alphabet of terminal symbols. is a leaf alphabet. It is assumed, that (17 U X) n N = 0. is a finite set of productions of the form a(~l' ...

Steinby It is often natural to view tree languages as subsets of term algebras the same way as one regards string languages as subsets of free monoids. We consider now morphisms of term algebras and their inverses as tree language operations. Since TE(X) is generated by X, any morphism cp : TE(X) -+ TE(Y) is completely determined by the images xcp of the letters x E X. For each EX-tree t, the EY-tree tcp is obtained by replacing each x-labeled (x E X) leaf of t by the EY-tree xcp. The mapping cp is extended to the set of EX-languages in the usual way: ifT ~ TE(X), then Tcp = {tcp I t E T}(~ TE(Y))' Similarly, if T ~ TE(Y), then Tcp-l = {t E TJJ(X) I tcp E T}.

This characterization can also be presented in terms of monoids as follows. The syntactic monoid congruence /-LT of a EX-language T is a congruence of the monoid of EX-contexts which is defined so that for any p, q E Ct( E, X), P/-LTq iff ("It E T:r;(X))(Vr E Ct(E, X))[t· p. rET {:} t· q. rET]. The syntactic monoid of T is defined in [Th084] as the quotient monoid Ct(E,X)//-LT' In [Sa183] it was shown that the syntactic monoid of any EXlanguage T is isomorphic to the translation monoid of the syntactic algebra of T.

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