Showing papers on "Chomsky hierarchy published in 1984"

An application of the Ehrenfeucht-Fraisse game in formal language theory

Abstract: A version of the Ehrenfreucht-Fraisse game is used to obtain a new proof of a hierarchy result in formal language theory: It is shown that the concatenation hierarchy ("dot-depth hierarchy") of star-free languages is strict. Resume Une version du jeu de Ehrenfeucht-Fraisse est appliquee pour obtenir une nouvelle preuve d'un theoreme dans la theorie des-langages formels: On montre que la hierarchic de concatenation ("dot-depth hierarchy") deslangages sans etoile est stricte. 1 . Introduction. The present paper is concerned with a connection between formal language theory and model theory. We study a hierarchy of formal languages {namely, the dot-depth hierarchy of star-free regular languages) using logical notions such as quantifier complexity of first-order sentences. In this context we apply a form of the Ehrenfeucht-Fraisse game which serves to establish the elementary equivalence between structures with respect to sentences of certain prefix types. The class of star-free regular languages is of a very basic nature: It consists of all languages (= word-sets) over a given alphabet A which can be obtained from the finite languages by finitely many applications of boolean operations and the concatenation product. (For technical reasons we consider only nonempty words over A , i . e . 0037-9484/8403 11 11/S 3.10/ © Gauthier-Villars 11 + languages L c A ; in particular, the complement operation is applied w . r . t . A^) General references on the star-free regular languages are McNaughton-Papert ( 1 9 7 1 ) , Chapter IX of Eilenberg ( 1 9 7 6 ) , or Pin ( 1 9 8 4 b ) . A natural classification of the star-free regular languages is obtained by counting the "levels of concatenation" which are necessary to build up such a language: For a fixed alphabet A , let B.. = {LcA'^lL finite or cofinite), B = { L c A |L is a boolean combination of languages of the form L • . . . L (n > 1 ) with L ^ , . . . , L ^ € B^} . The language classes B , B , , . « . form the so-called dot-depth hierarchy (or: Brzozowski hierarchy), introduced by Cohen/Brzozowski ( 1 9 7 1 ) . In the framework of semigroup theory, Brzozowski/Knast ( 1 9 7 8 ) showed that the hierarchy is infinite ( i . e . that B^ B^_^ for k > 1 ) . The aim of the present paper is to give a new proof of this result, based on a logical characterization of the hierarchy that was obtained in Thomas ( 1 9 8 2 ) . The present proof does not rely on semigroup-theory; instead, an intuitively appealing model-theoretic technique is applied: the Ehrenfeucht-Fraisse game. Let us first state the mentioned characterization.result, taking A = { a , b } . One identifies any word W G A 4 ' , say of length n , with a "word model" w = ( { 1 , . . . , n } , < , m i n , m a x , S , P , Q ^ , Q ^ ) where the domain { 1 , . . . , n } represents the set of positions of letters in the word w , ordered by < , where min and max are the first and the last position, i . e . min = 1 and max=n, S and P are the successor and predecessor function on { 1 , . . . , n } with the convention that S(max) =max and P(min) =min, and Q^Q^ are unary predicates over { 1 , . . . , n } containing the positions with letter a , b respectively. (Sometimes it is convenient to assume that the position-sets of two words u, v are disjoint; then one takes any two nonoverlapping segments of the integers as the position-sets of u and v . ) Let L be the first-order language with equality and nonlogical symbols <,min, m a x , S , P , Q , Q . . Then the satisfaction of an Lsentence tp in a word w a D * • 12 EHRENFEUCHT-FRAISSE GAME (written: w t= ip) can be defined in a natural way, and we say that L c: A is defined by the L-sentence tp if L == {w € A Iw^ ip} . For example, the language L= (ab) is defined by Q min A Q,max A Vy (y < max -• (Q y ̂ Q,S ( y ) ) ) . As usual, a I,-formula is a formula in pr.enex normal form with a prefix consisting of k alternating blocks of quantifiers, beginning with a block of existential quantifiers. A B (£,)-formula is a boolean combination of £,-formulas. 1 .1 Theorem. (Thomas ( 1 9 8 2 ) ) . Let k>0. A language L <= A belongs to B iff L is defined by a B(£)-sentence of L. For the formalization of properties of words the symbols min,max,S,P are convenient. But of course they are definable in the restricted first-order language L,. with the nonlogical constants <,Q ,Q, alone. u a JD Indeed, we have: 1 .2 Lemma. Let k>0 . If L <= A is defined by a B (I,)-sentence of L, then L is defined by a B (Z,^ )-sentence of L . Proof. The quantifier-free kernel of a Z,-formula tp of L can be expressed both by a £..and a n,--formula of L/.. For example, Q S(min) is expressible in the following two ways: (+) 3y(y=S(min) A Q^y) , Vy(y=S(min) -» Q^y) where y = S (min) is rewritten as a II.-formula of L.. using x = min —^ V z ( x = z v x < z ) , x = max -* V z ( z = x v z < x ) S (x) = y -^ (x = max A x = y ) v ( x < y A Vz~l (x < z A z < y) ) . Hence we obtain a £, .-sentence of Lwhich is equivalent (in all word-models) to tp by applying one of the two definitions in (+), depending on the case whether the innermost quantifier-block of cp is existential or universal. We mention without proof that (for k >0) the B(Z,)-sentences of L.. define exactly those languages L c: A which occur on the k-th level of another hierarchy of star-free regular languages, introduced by

Outline of an Algebraic Language Theory

Abstract: The algebraic theory we present here continues the early work of Chomsky-Schutzenberger [Ch,Sch], Shamir [Sh] and Nivat [N]. The leading idea is to develop a machine and production free language theory. The interest in such a theory is support by the hope that the proofs in such a theory don't need so much case discussions which often lead to errors and that a view which is free from non-essentials of language theory will lead to a progress in the direction of our problems. Even if the theory is in an early stage the attempt pays out in a machine free definition of LL(k) and LR(k) languages which leads easily to generalisations of non-deterministic LL(k) and LR(k) languages with the same space and time complexity behaviour. Too, we are able to show that this theory is not restricted to the context free languages but also concerns the whole Chomsky hierarchy. Our theory is in a sense a dual to the theory of formal power series as introduced by M. Schutzenberger.