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Substitution in families of languages
Abstract
The effect of substitution in families of languages, especially an AFL (i.e., abstract family of languages), is considered. Among the main results shown are the following: The substitution of one AFL into another is an AFL. Under suitable hypotheses, the AFL generated by the family obtained from the substitution of one family into another is the family obtained from the substitution of the corresponding AFL. A condition is given for the AFL generated by the substitution closure of a family to be the substitution closure of the AFL generated by the family.
... [14] for a monograph on this approach. Similar as in ordinary algebra -where one went from groups to semigroups, rings, and fields-full AFL's gave rise to weaker structures (full trios, full semi-AFL's [14]) and to more powerful ones: full substitution-closed AFL's [16], full super-AFL's [18] and full hyper-AFL's [1]. ...
... Since we assumed that each family of fuzzy languages is closed under isomorphism, the Sûb-operator is associative, i.e., Sûb(K 1 , Sûb(K 2 , K 3 )) = Sûb(Sûb(K 1 , K 2 ), K 3 ); cf. [16,14]. ...
... [14]. Full substitution-closed AFL's have been investigated in [16]. ...
We study operations on fuzzy languages such as union, concatenation, Kleene ★, intersection with regular fuzzy languages, and several kinds of (iterated) fuzzy substitution. Then we consider families of fuzzy languages, closed under a fixed collection of these operations, which results in the concept of full abstract family of fuzzy languages or full AFFL. This algebraic structure is the fuzzy counterpart of the notion of full abstract family of languages that has been encountered frequently in investigating families of crisp (i.e., non-fuzzy) languages. Some simpler and more complicated algebraic structures (such as full substitution-closed AFFL, full super-AFFL, full hyper-AFFL) will be considered as well.In the second part of the paper we focus our attention to full AFFLs closed under iterated parallel fuzzy substitution, where the iterating process is prescribed by given crisp control languages. Proceeding inductively over the family of these control languages, yields an infinite sequence of full AFFL-structures with increasingly stronger closure properties.
... Thus AFL and full AFL are unifying concepts in language theory as it relates to machines. Since [4], a series of papers have been written [5, 7, 9, 10, 11, 12], dealing either with AFL theory per se, or with problems about languages rendered prominent by the discovery of AFL. The present article deals further with AFL theory. ...
... L~a~ for each a in 27 L . If ~ and .W 2 are AFL, then so are ~ cr .LP 2 ([11], Corollary 1 of Theorem 4.2) and 5 .Wa [7]. If ~ and .W~ are full AFL, then ~ a .La 2 = ~a x 5 .W~ is a full AFL by Remark 4 after Theorem 4.1 of [11]. ...
... Finally, note that ~,~(L1) o ~-(L~) = ~-(L1) o(~ 0 a.A[(L~)), Hence o~-(L1) ~ ~-(L2) _C ~-(~-tLzc,)+((LlCa)+)), whence equality. 1~ It was shown in [7] that ~ is associative on families of languages closed under isomorphism, i.e., (.o~10.~2) ~ = "~t~ if ..9'1, .~, ...
A (full) principal AFL is a (full) AFL generated by a single language, i.e., it is thesmallest (full) AFL containing the given language. In the present paper, a study is made of such AFL. First, an AFA (abstract family of acceptors) characterization of (full) principal AFL is given. From this result, many well-known families of AFL can be shown to be (full) principal AFL. Next, two representation theorems for each language in a (full) principal AFL are given. The first involves the generator and one application each of concatenation, star, intersection with a regular set, inverse homomorphism, and a special type of homomorphism. The second involves an a-transducer, the generator, and one application of concatenation and star. Finally, it is shown that if ℒ1 and ℒ2 are (full) principal AFL, then so are (a) the smallest (full) AFL containing {L1∩L2/L1 in ℒ1, L2 in ℒ2 and (b) the family obtained by substituting ε-free languages of ℒ2 into languages of ℒ1.
... An important step in this algebraic approach to families of languages has been the introduction of the notion of full Abstract Family of Languages (full AFL), being a nontrivial family of languages closed under the operations: union, concatenation, Kleene ⋆, homomorphism, inverse homomorphism, and intersection with regular sets [9]. Similar as in ordinary algebra -where one went from groups to semigroups, rings, and fields-full AFL's gave rise to weaker structures (full trios, full semi-AFL's [9]) and more powerful ones: full substitution-closed AFL's [10], full super-AFL's [11] and full hyper-AFL's [1]. ...
... Proposition 2.4.[10,9,2] A family K of languages is a full AFL if and only if K is a prequasoid closed under regular substitution (i.e., Sûb(K, REG) ⊆ K), and under substitution in the regular languages (i.e., Sûb(REG, K) ⊆ K). ...
We investigate different sets of operations on languages which results in corresponding algebraic structures, viz.\ in different types of full AFL's (full Abstract Family of Languages). By iterating control on ETOL-systems we show that there exists an infinite sequence () of classes of such algebraic structures (full AFL-structures): each class is a proper superset of the next class . In turn each class contains a countably infinite hierarchy, i.e., a countably infinite chain of language families () such that (i) each is closed under the operations that determine , and (ii) each is properly included in the next one: .
... The result follows from Lemma 2.1 and the fact (Theorem 2.1 of [11]) that sub(~a~, ~LP2) is an AFL if ~ and ~ are AFL. Corollary 1 is of interest in its own right. ...
... By Corollary 1, Sub(~, ~) is a slip AFL. By Corollary 2 of Theorem 2.1 of [11], sub(~, ~,e) ~-~.(~a). Suppose that ~ is a slip family. ...
A slip language is a language whose Parikh mapping is a semilinear set. A slip family is a family containing only slip languages. The purpose of this paper is to study slip AFL. A sufficiency condition is given on a slip family which ensures that the family generates a slip AFL. Using this condition, it is proved that (i) there exists a largest slip AFL and (ii) if ℒ is a slip family, then the smallest AFL containing the commutative closure of ℒ is a slip AFL. A new operation called “homomorphic replication” is then introduced. It is shown that the smallest AFL containing a homomorphic replication of a slip AFL is also a slip AFL. Furthermore, the resulting AFL is principal if the original AFL is principal. It is then proved that the smallest AFL containing all homomorphic replications of the regular sets is not principal. Finally, abstract families of acceptors are presented which, respectively, define the smallest AFL containing a particular homomorphic replication of the regular sets and all homomorphic replications of the regular sets.
... If ~ and ~ are full semiAFLs, so is ~0~ [18]. The operator 8-is associated with the operation a defined in Definition 1.7. ...
... Since concatenation is associative, we have a simple way of expressing ~o 9 Thus an insertion is a substitution such that at most one symbol in a word is replaced at any time. By the same arguments used in [18] for substitution we have, although insertion is not associative: We shall first show that, is the simple operator associated with ~(L x , L2). ...
The class of syntactic operators is defined. If a full AFL ℒ is not closed undera syntactic operator ⊗, then repeated application of ⊗ to ℒ produces an infinite hierarchy of full AFLs and the closure of ℒ under ⊗ is not full principal. If ℒ1 and ℒ2 are incomparable full AFLs, then the least full AFL containing ℒ1 and ℒ2 is not closed under any syntactic operator. If L is any generator of a full AFL ℒ closed under any syntactic operator, then all of ℒ may be expressed as finite state translations of L (without applying concatenation or star). It is shown that substitution, insertion, intercalation and homomorphic replication are all syntactic operators.
... The closure of CF under nested iterated substitution was proved by Král [75] in 1970. Following some further results (for example, [52]), an abstract basis for substitution was developed by Ginsburg and Spanier [53]; one particular important notion developed there was treating the (nested) substitution-closure of a full AFL as a form of 'algebraic closure'. In particular, it is proved that the substitution-closure of a full AFL is a full AFL [53, Theorem 2.1]. ...
This article studies the properties of word-hyperbolic semigroups and monoids, that is, those having context-free multiplication tables with respect to a regular combing, as defined by Duncan and Gilman [‘Word hyperbolic semigroups’, Math. Proc. Cambridge Philos. Soc. 136 (3) (2004), 513–524]. In particular, the preservation of word-hyperbolicity under taking free products is considered. Under mild conditions on the semigroups involved, satisfied, for example, by monoids or regular semigroups, we prove that the semigroup free product of two word-hyperbolic semigroups is again word-hyperbolic. Analogously, with a mild condition on the uniqueness of representation for the identity element, satisfied, for example, by groups, we prove that the monoid free product of two word-hyperbolic monoids is word-hyperbolic. The methods are language-theoretically general, and apply equally well to semigroups, monoids or groups with a -multiplication table, where is any reversal-closed super- . In particular, we deduce that the free product of two groups with with respect to indexed multiplication tables again has an with respect to an indexed multiplication table.
... Following some further results (e.g. [55]), an abstract basis for substitution was developed by Ginsburg & Spanier [56]; one particular important notion developed there was treating the (nested) substitution-closure of a full AFL as a form of "algebraic closure". In particular, it is proved that the substitutionclosure of a full AFL is a full AFL [56, Theorem 2.1]. ...
This article studies the properties of word-hyperbolic semigroups and monoids, i.e. those having context-free multiplication tables with respect to a regular combing, as defined by Duncan & Gilman. In particular, the preservation of word-hyperbolicity under taking free products is considered. Under mild conditions on the semigroups involved, satisfied e.g. by monoids or regular semigroups, we prove that the semigroup free product of two word-hyperbolic semigroups is again word-hyperbolic. Analogously, with a mild condition on the uniqueness of representation for the identity element, satisfied e.g. by groups, we prove that the monoid free product of two word-hyperbolic monoids is word-hyperbolic. The methods are language-theoretically general, and apply equally well to semigroups, monoids, or groups with a -multiplication table, where is any reversal-closed super-, in the sense of Greibach. In particular, we deduce that the free product of two groups with resp. indexed multiplication tables again has an resp. indexed multiplication table.
... Let K ∞ denote the smallest family closed under fuzzy substitution that includes a given family K. In [14] Ginsburg and Spanier proved for crisp language families that closure under substitution can be obtained by iterating the Sûb-operator. A proof of the following fuzzy counterpart of their result can be found in [8]. ...
A fuzzy context-free K-grammar is a fuzzy context-free grammar with a countable rather than a finite number of rules satisfying the following condition: for each symbol α, the set containing all right-hand sides of rules with left-hand side equal to α forms a fuzzy language that belongs to a given family K of fuzzy languages. In this paper we study the effect of the non-self-embedding restriction on the generating power of fuzzy context-free K-grammars. Our main result shows that under weak assumptions on the family K, a fuzzy language is generated by a non-self-embedding fuzzy context-free K-grammar if and only if either it is a fuzzy regular language or it belongs to the substitution closure K∞ of the family K. The proof heavily relies on the closure properties of the families K and K∞.
... Let ~0 = ~176 and ~ = ~~ 1 for n/> 1. Then each ~o is a full AFL and ~(~) = U~>0~ [3]. Let k =min{n[Le~~ By Lemma 2.1, k = 0. ThusL is in ~0 = ~&j/~(~o) _C .~2(~~176 ...
A set of languages is independent if no language in the set can be obtained from other languages in the set by a sequence of full AFL operations. It is proved that the set {Pk|k≥2} is independent, where Pk={akn|n≥1}Pk={ank|n≥1}. For J⊆{2, 3, 4,…}, it is proved that {tEk|k∈J} is independent if and only if no two numbers in J are powers of the same integer, where Ek={akn|n≥1}Ek={akn|n≥1}.
... Thus 7%(h(L~)) is in Sfib(~). Let q/1 ----*LP2 and Sfib(Xe z , q/k) = Y/k+1 for each k >/ 1. oo It is known that Sfib(~) = Uk=l q/k and that each q/k is a full AFL [5]. Let n be the smallest integer such that "cL~(h(L2) ) is in q/,. ...
Given an abstract family of languages (AFL) ℒ the question is considered if there exists an AFL incomparable with ℒ. In case there is an AFL ℒ′ incomparable with ℒ the paper considers if there exists a largest AFL incomparable with ℒ, and if there is a maximal AFL containing ℒ′ incomparable with ℒ. The main results characterize those full AFL ℒ′ having a largest full incomparable AFL ℒ′ and relate properties of ℒ to properties of ℒ′.
... In this case, if v(w)=ciq-rlpl+ '..q-rkpT~, Pl,...,PT~ePi, ri>/ We shall need to use various facts about a-transducers, semi-AFLs and substitutions. Greibach, 1969;Ginsburg and Spanier, 1970;Goldstine, 1972;Lewis, 1971). ...
A homomorphism acting on a context-free language L is either e-limited on L, linearly bounded on L, or of unbounded erasing on L. There are context-free AFLs not closed under linear erasing; there are context-free AFLs closed under linear erasing but not under arbitrary erasing. It is undecidable whether a context-free language is erasable, linearly erasable, or uniformly erasable.
In Teil 1 dieses Buches führten wir einige spezielle Sprachfamilien ein und studierten sie von verschiedenen Gesichtspunkten her. Wenn wir zeigten, daß diese Familien einige spezielle Eigenschaften besitzen, benötigten wir nur einige ihrer definierenden Eigenschaften. Damit kommen diese speziellen Eigenschaften auch jeder Sprachfamilie zu, die die benötigten Eigenschaften besitzt. Dies liefert die Motivation zu folgender Abstraktion: Wir betrachten nicht irgendeine spezielle SprachfamiÜe, sondern eine Menge von Sprachfamilien, von denen jede gewisse Grundeigenschaften besitzt. Unter Verwendung dieser Grundeigenschaften weisen wir dann andere Eigenschaften nach, die damit allen Familien in dieser Menge zukommen. Alle vier Famüien ℒi
,i = 0, 1, 2, 3, in unserer fundamentalen Hierarchie besitzen diese Grundeigenschaften. Folglich gelten die gewonnenen Ergebnisse auch für die Famüien ℒi
. Man nennt die Elemente der obengenannten Menge von Sprachfamilien abstrakte Familien von Sprachen. Dieser Begriff ermöglicht es uns, einige Teüe der Theorie zu vereinheitlichen. Denn es wird unnötig, Beweise für verschiedene Familien zu wiederholen, die alle die Grundeigenschaften besitzen und somit eine abstrakte Familie von Sprachen sind.
We extend some results of Ginsburg and Spanier [21] on incomparable full AFL’s (full Abstract Families of Languages) to similar structures like full hyper-AFL’s, full hyper(l)-AFL’s and prequasoids. A general approach based on universal algebra and lattice theory enables us to establish Ginsburg and Spanier-like theorems for several types of language families. It turns out that a considerable part of the proofs is algebraic or latticetheoretic in nature. On the other hand formal language theory provides language families that may serve as (counter)examples in algebra.
A linguistic model to generate matrices (arrays of terminals) is presented. The sixclosure axioms which define abstract families of languages (AFL) are extended to matrices to obtain abstract families of matrices (AFM). Corresponding automata to recognize matrices are defined and equivalence established. Immediate application is shown in picture processing by generating a wide class of interesting pictures. Illustrations include generation of kolam patterns, wall paper designs and geometric patterns replicated several times. Simple transformations of a picture such as reflection, translation, halftum, and turning upside down are very naturally obtained.
OL systems and TOL systems are the simplest mathematical models for the study of the development of biological organisms with or without a variable environment, respectively. This paper contributes to the study of the properties of the languages generated by these systems and by their generalizations. Macro OL (TOL) languages are languages obtained by substituting languages of a given type in OL (TOL) languages. We study properties of certain families of macro OL (TOL) languages in particular we show that they are full AFL's. We observe that OL, TOL systems and many of their generalizations can be viewed as special classes of index grammars.
The parallelism inherent in a parallel context free grammar is incorporated into OL-systems. A parallel OL-language (POL-language) is obtained in the same manner as an OL-language except that when there is more than one occurrence of a symbol in a particular line of derivation, the same production is applied simultaneously to all the occurrences of the same symbol. This class combines the peculiar features of OL languages and parallel context-free languages.
We study context free closed families of languages, that is families L verifying the equation CF º L = Lwhere CF denotes the family of context free languages and º is the substitution operation. Call a family L a context free AFL if it is both context free closed and a rational cone. The families CF and RE (of recursively enumerable languages) have this property. We show that the class of context free AFL's is properly contained into that of full AFL's. Moreover the context free AFL generated by a family L, is given by the formulaL Δ = CF º LΓwhere LΓ denotes the rational cone generated by L. Using this result we establish the remarkable inclusion(L º M)Δ ⊃ LΔ º MΔholding for all families L and M as well as the equalityLΔ = CF º LΓLΓ being the full AFL generated by L.
Motivated by aspects of robustness in parsing a context-free language, we study generalized fuzzy context-free grammars. These fuzzy context-free K-grammars provide a general framework to describe correctly as well as erroneously derived sentences by a single generating mechanism. They model the situation of making a finite choice out of an infinity of possible grammatical errors during each context-free derivation step. Formally, a fuzzy context-free K-grammar is a fuzzy context-free grammar with a countable rather than a finite number of rules satisfying the following condition: for each symbol α, the set containing all right-hand sides of rules with left-hand side equal to α forms a fuzzy language that belongs to a given family K of fuzzy languages. We investigate the generating power of fuzzy context-free K-grammars, and we show that under minor assumptions on the parameter K, the family of languages generated by fuzzy context-free K-grammars possesses closure properties very similar to those of the family of ordinary context-free languages.
We investigate derivation-controlled K-iteration grammars, called (Γ, K)-iteration grammars, where Γ can be any family of control languages. We prove that already under very weak restrictions on Γ and K the following hold: (i) Regular control does not increase the generating power of K-iteration grammars, (ii) for each (Γ, K)-iteration grammar there exists an equivalent propagating (Γ, K)-iteration grammar, (iii) the family of (Γ, K)-iteration languages is a full hyper-AFL, (iv) for each (Γ, K)-iteration grammar there exists an equivalent (Γ, K)-iteration grammar with exactly two substitutions. We also discuss some additional properties and applications of (uncontrolled) K-iteration grammars and controlled (deterministic) ETOL systems and their languages in a wider context.
Interest in the study of sets of trees, tree languages, has led to the definition of finite automata which accept trees [2,11] and transducers which map trees into other trees [7,9,10]. These generalized machines may read treesfinite automata which accept trees [2,11] and of transducers which map trees into other trees [7,9,10]. These generalized machines may read trees either “top-down” (from the root toward the leaves) or “bottom-up” (from the leaves toward the root). Here it is shown that both the class of top-down transductions and the class of bottom-up transductions can be characterized in terms of two restricted classes of tree transductions. From these ductions and the class of bottom-up transductions can be characterized in terms of two restricted classes of tree transductions. From these characterizations, it is shown that the composition of any n bottom-up transductions can be realized by the composition of n+1 top-down transductions, and similarly, the composition of any n top-down transductions can be realized by the composition of n+1 bottom-up transductions. Next, we study the families of tree languages which can be obtained from the recognizable sets (sets accepted by finite tree automata) by the composition of n top-down or bottom-up transductions, n>0. The yield operation, which concatenates the leaves of a tree from left to right to form a of string, languages from the hierarchy of families of tree languages. It is shown that each family of string languages in this hierarchy is properly contained in the family of context-sensitive languages.
This paper treats the closure of families of formal languages under operators which may be viewed as substitution into a particular language. A language L over alphabet {a1,...,an} induces an n-place operator on languages by substitution of the n arguments Li for the symbols ai. For example, if L is regular, it induces an operator under which any full AFL is closed. In section two we find a large class of full AFL's which are closed under no other such operators than those induced by regular languages. Also, for any full AFL @@@@', let @@@@ be the class of languages which @@@@' is closed under substitution into. Then @@@@ is itself a full AFL and is closed under substitution. Finally we show that any substitution-closed full AFL @@@@ is obtained in this manner from some non-substitution-closed full AFL @@@@ (except when @@@@ is the universal family). The study is based on the concept of a full AFL, and in section one considerable effort is devoted to a novel approach to the subject. A full AFL is taken to be a family of languages closed under finite-state transducer mappings and substitution into regular sets. By replacing the family of regular sets with more general families we obtain some broad results about canonical forms for derivations of languages from other languages using transducers and substitution. These forms yield several known forms as special cases, and provide tools needed for the study of full AFL's in section two.
For each (abstract family of languages) AFL ℒ, two families of languages, the family ℐ(ℒ) of nondeterministic and the family
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Device representations, via multitape AFA (abstract families of acceptors), are given for the families of languages which result from applying the wedge (A) m)d the substitu- tion operations to AFL (abstract families of languages). In particular, if ~ and ~2 are multi- tape AFA (i.e. certain families of multistorage tape acceptors), then 5)~ A ~2 is defined as the family of multitape acceptors which results when the tapes of ~ and ~2 are coalesced, with the ~-tapes preceding those in ~2 . It is shown that the smallest full AFL containing 2(~) A 2 (~D2) = {L~ N L2 ILi in2 (~Di) } is ,12 (~ h~:). For each multitape AFAr, aset ~.v of "nested" multitape acceptors is defined. It is shown that if ~ and ~2 are single-tape AFA, then the family of languages obtained from (~ A ~2) N is the family of langnages obtained by substitut- ing the AFL defined by ~2 into the AFL defined by ~D~ .
Tree transducers (automata which read finite labeled trees and output finite labeled trees) are used to define a hierarchy of families of “tree languages” (sets of trees). In this hierarchy, families generated by “top-down” tree transducers (which read trees from the root toward the leaves) alternate with families generated by “bottom-up” tree transducers (which read trees from the leaves toward the root). A hierarchy of families of string languages is obtained from the first hierarchy by the “yield” operation (concatenating the labels of the leaves of the trees). Both hierarchies are conjectured to be infinite, and some results are presented concerning this conjecture. A study is made of the closure properties of the top-down and bottom-up families in the hierarchies under various tree and string operations. The families are shown to be closed under certain operations if and only if the hierarchies are finite.
By iterating iterated substitution not all regular languages can be copied. Hence the smallest full hyper (1)-AFL is properly contained in ETOL, the smallest full hyper-AFL. The number of iterations of iterated substitution gives rise to a proper hierarchy. Consequently the smallest full hyper (1)-AFL is not a full principal AFL.
A superAFL is a family of languages closed under union with unitary sets, intersection with regular sets, and nested iterated substitution and containing at least one nonunitary set. Every superAFL is a full AFL containing all context-free languages. If L is a full principal AFL, then S∞(L, the least superAFL containing L, is full principal. If L is not substitution closed, the substitution closure of L is properly contained in S∞ (L). The index languages form a superAFL which is not the least superAFL containing the one way stack languages. If L has a decidable emptiness problem, so does S∞ (L). If Ds is an AFA, L=L (Ds) and Dw is the family of machines whose data structure is a pushdown store of tapes of Ds, then L (Dw) = S∞(L) if and only if Ds is nontrivial. If Ds is uniformly erasable and L(Ds) has a decidable emptiness problem, then it is decidable if a member of Dw is finitely nested.
Finite automata are considered in this paper as instruments for classifying finite tapes. Each one-tape automaton defines a set of tapes, a two-tape automaton defines a set of pairs of tapes, et cetera. The structure of the defined sets is studied. Various generalizations of the notion of an automaton are introduced and their relation to the classical automata is determined. Some decision problems concerning automata are shown to be solvable by effective algorithms; others turn out to be unsolvable by algorithms.
The axiomatic properties used to define abstract families of languages are considered. The properties are shown not to be independent. All independent subsets that imply the remaining properties are found. Finally alternative operations are considered. (Author)
A derivation in a phrase-structure grammar is said to be k-bounded if each word in the derivation contains at most k occurrences of nonterminals. A set L is said to be derivation bounded if there exists a phrase-structure grammar G and a positive integer k such that L is the set of words in the language generated by G which have some k-bounded derivation. The main result is that every derivation-bounded set is a context-free language. Various characterizations of the derivation-bounded languages are then given. For example, the derivation-bounded languages coincide with the standard matching-choice sets discussed by Yntema. They also coincide with the smallest family of sets containing the linear context-free languages and closed under arbitrary substitution, a family discussed by Nivat.
The notion of an abstract family of languages (AFL) as a family of sets of words satisfying certain properties common to many types of formal languages is introduced. Operations preserving AFL are then considered. The concept of an abstract family of acceptors (AFA) is also introduced and shown to give rise to an AFL. A necessary and sufficient condition on an AFL is presented in order that the AFL come from some AFA. Finally, abstract families of transducers (AFA with output) are discussed.
A slip language is a language whose Parikh mapping is a semilinear set. A slip family is a family containing only slip languages. The purpose of this paper is to study slip AFL. A sufficiency condition is given on a slip family which ensures that the family generates a slip AFL. Using this condition, it is proved that (i) there exists a largest slip AFL and (ii) if ℒ is a slip family, then the smallest AFL containing the commutative closure of ℒ is a slip AFL. A new operation called “homomorphic replication” is then introduced. It is shown that the smallest AFL containing a homomorphic replication of a slip AFL is also a slip AFL. Furthermore, the resulting AFL is principal if the original AFL is principal. It is then proved that the smallest AFL containing all homomorphic replications of the regular sets is not principal. Finally, abstract families of acceptors are presented which, respectively, define the smallest AFL containing a particular homomorphic replication of the regular sets and all homomorphic replications of the regular sets.
To better understand the syntax of problem-oriented languages, the equa- tional form used in defining ALGOL (1) was recently abstracted (6). On varying the coeffi- cients in the abstract equational form two families of languages arose. (A language is viewed as a set of strings, called words, of symbols from a finite, fixed alphabet.) The two types of languages, called definable and sequentially definable, were then investigated and a num- ber of properties discovered. In particular, it was shown that the definable languages were identical to the type-two phrase structure languages introduced by Chomsky (4). The present paper deals with operations T (on languages) which preserve definability ~nd frequently sequential definability. All operations considered have the additional prop- erty of preserving regularity (7)--the interest in regular sets stemming from the fact that they have been considered as "finite state languages" (3). Two basic results are proved. The first (Theorem 2.1) permits derivation of (sequential) definability-preserving operations from other (sequential) definability-preserving opera- tions. The second (Theorem 3.1) asserts that a sequential machine always changes a de- finable set into a definable set (but not necessarily a sequentially definable set into a se- quentially definable set). Using these two results, a large number of specific operations-- many occurring in data processing--arc shown to preserve definability and, depending on the operation, sequential definability. In addition to the two main results there occur in appendices A and B necessary conditions for a set to be sequentiMly definable and definable respectively. The former is the first known necessary condition for sequentially definable sets which differs from those for definable sets. It is anticipated that many of the operations considered here will be useful in later studies of problem oriented languages. 1. Preliminaries
In Representation of Events in Nerve Nets and Finite Automata [3], S. C. Kleene obtains a number of interesting results. The most important of these, his analysis and synthesis theorems (theorems 5 and 3), are obscured both by the complexity of his basic concepts and by the nature of the elements used in his nets. In this paper we give a new formulation and new proofs of Kleene’s analysis and synthesis theorems, in which we presuppose no acquaintance with Kleene’s work. We use simpler basic concepts and construct our nets out of more familiar and convenient elements (see section 4). The simplified formulation and proofs should make these important results more widely accessible. Some detailed comments on Kleene’s ideas are made in section 7.