Petri网语言的Pumping引理

Petri网语言是Petri网理论的重要组成部分，也是系统行为分析的一种重耍的工具．Petri网语言的Pumping引理反映了Petri网语言的共性，可用来证明某些语言不是Petri网语言，已经证明，当一个Petri网语言可被某个有界Petri网产生时，此语言是正规语言，因此，正规语言的Pumping引理对此语言是有效的，但正规语言的Pumping引理并小适用于所有的Petrl网语言．文中给出了一种Petri网语言的Pumping引理，证明其对任意无空标注的Petri网语言都有效，并且＋正规语言的Pumping引理是此引理的一种特殊形式．利用此Pumping引理可以证明某些语言是不能由Petri网产生的。     

Sur le monoide syntactique de L1 lorsque L est un langage fini

We prove that for every finite monoid M, there exists a finite language L so that M divides the syntactic monoid of L¹. Moreover, one can choose for L a full finite prefix code. The same result for finite group has already been proved by Schützenberger. This result is crucial in the proof of the J.-F. Perrot's theorem that the only variety closed under star operation is the variety of all rational languages.     

上下文无关Petri网语言的Pumping引理

Petri网语言可分为正规Petri网语言、上下文无关Petri网语言和Petri网语言三类,Pumping引理反映了一类语言的共性.对于正规Petri网语言类和Petri网语言类都已给出了其相应的Pumping引理,而对于上下文无关Petri网语言类的Pumping引理却一直未给出.本文通过分析上下文无关Petri网语言的结构性质,给出了上下文无关Petri网语言的Pumping引理,并且正规Petri网语言的Pumping引理是上下文无关Petri网语言的Pumping引理的一种特殊形式,而上下文无关Petri网语言的Pumping引理又是Petri网语言Pumping引理的一种特殊形式,从而完整地解决了三类Petri网语言Pumping引理以及它们之间的关系.     

Permutations are not context-free: An application of the interchange lemma

Let x be a nonempty string and x? another string with the same characters as x, but possibly in a different order. A string of the form xx? is called a permutation. A permutation-containing string is of the form wxx?y. The Interchange Lemma for context-free languages [11] is used to show that the set of permutation-containing strings over a 16 character alphabet is not context-free. The application of the lemma is important since other techniques, such as the pumping lemma and Ogden's lemma cannot show that the set is not context-free. Finally, a collection of open problems is given.     

Approximation of fuzzy context-free grammars

Fuzzy context-free max- grammar (or FCFG, for short), as a straightforward extension of context-free grammar, has been introduced to express uncertainty, imprecision, and vagueness in natural language fragments. Li recently proposed the approximation of fuzzy finite automata, which may effectively deal with the practical problems of fuzziness, impreciseness and vagueness. In this paper, we further develop the approximation of fuzzy context-free grammars. In particular, we show that a fuzzy context-free grammar under max- compositional inference can be approximated by some fuzzy context-free grammar under max-min compositional inference with any given accuracy. In addition, some related properties of fuzzy context-free grammars and fuzzy languages generated by them are studied. Finally, the sensitivity of fuzzy context-free grammars is also discussed.     

可交换上下文无关文法

本文提出了可交换上下文无关文法及其该文法产生的语言——可交换上下文无关语言，证明了正规语言类是可交换上下文无关语言类的一个子集，而可交换上下文无关语言类是上下文无关语言类的一个子集；讨论了可交换上下文无关语言的结构特点，并给出了可交换上下文无关语言的Pumping引理。     

A representation of trees by languages I

A tree can be represented by a language consisting of a suitable coding of its finite branches. We investigate this representation and derive a number of reductions between certain equivalence problems for context-free tree grammars and recursive program schemes and the (open) equivalence problem for DPDA's. This is the first part of this work: it is devoted to technical results on prefix-free languages and strict deterministic grammars. Application to context-free tree grammars will be published in the second part.     

Context-free complexity of finite languages

In this paper the investigation of the theory of grammatical complexity as started by Bucher (1981) and Bucher, Culik II, Maurer and Wotschke (1981) is continued. The basic question we are concerned with is the following: Given some finite language L, what is the smallest number of context-free productions needed to generate L, the so-called (context-free) complexity of L. We strengthen some of the results given by Bucher et al. (1981), the main result being a necessary condition for certain sequences of finite languages to be of sublinear complexity.     

A note on three-dimensional finite automata

This paper investigates the relationships between the accepting powers of three-dimensional six-way finite automata (3-FA's) and three-dimensional five-way Turing machines (5WTM's), where the input tapes of these automata are restricted to cubic ones. A 3-FA (5WTM) can be considered as a natural extension of the two-dimensional four-way finite automaton (two-dimensional three-way Turing machine) to three dimensions. The main results are: (1) n²logn (n³) space is necessary and sufficient for deterministic 5WTM's to simulate deterministic (nondeterministic) 3-FA's; (2) n² (n²) space is necessary and sufficient for nondeterministic 5WTM's to simulate deterministic (nondeterministic) 3-FA's.     

On a special monoid with a single defining relation

We show that no finite union of congruence classes [w], w being an arbitrary element of the free monoid {a, b}¹ with unit 1, is a context-free language if the congruence is defined by the single pair (abbaab, 1). This congruence is neither confluent nor even preperfect. The monoid formed by its congruence classes is a group which has infinitely many isomorphic proper subgroups.     

Inverse star, borders, and palstars

A language L is closed if L=L^?. We consider an operation on closed languages, L−?, that is an inverse to Kleene closure. It is known that if L is closed and regular, then L−? is also regular. We show that the analogous result fails to hold for the context-free languages. Along the way we find a new relationship between the unbordered words and the prime palstars of Knuth, Morris, and Pratt. We use this relationship to enumerate the prime palstars, and we prove that neither the language of all unbordered words nor the language of all prime palstars is context-free.     

Morphismes et bimorphismes d'arbres

For every pair of positive integers n and p, there is a language accepted by a real-time deterministic pushdown automaton with n states and p stack symbols and size O(np), for which every context-free grammar needs at least n²p+1 nonterminals if n>1 (or p non-terminals if n = 1). It follows that there are context-free languages which can be recognized by pushdown automata of size O(np), but which cannot be generated by context-free grammars of size smaller than O(n²p); and that the standard construction for converting a pushdown automaton to a context-free grammar is optimal in the sense that it infinitely often produces grammars with the fewest number of nonterminals possible.     

Undecidability of the first-order arithmetic A[P(x), 2x, x + 1]

It is shown that the first-order arithmetic A[P(x), 2x, x + 1] with two functions 2x, x + 1 and a monadic predicate symbol P(x) is undecidable, by using a kind of two-dimensional finite automata, called finite causal ω²-systems. From this immediately follows R.M. Robinson's result, which says that the monadic second-order theory with two functions 2x, x + 1 is undecidable. This is also considered as an improvement on H. Putnam's result about the undecidability of the first-order arithmetic with addition and a monadic predicate symbol.     

Complexity of certain decision problems about congruential languages

Given an arbitrary finite Church-Rosser Thue system S, it is shown that the question of whether a given congruence class is finite is undecidable, and the question of whether every congruence class is finite is not even semidecidable (in fact, Π₂-complete). It is shown that the question of whether a given (resp. every) congruence class is a context-free language is at least as hard. Also, given a finite rewriting system over a commutative monoid, the question of whether every congruence class is finite is shown to be tractable. This contrasts with the known result that the question of whether a given congruence class is finite requires space at least exponential in the square root of the input length.     

Finite automata with multiplication

A finite automaton with multiplication (FAM) is a finite automaton with a register which is capable of holding any positive rational number. The register can be multiplied by any of a fixed number of rationals and can be tested for value 1. Closure properties and decision problems for various types of FAM's (e.g. two-way, one-way, nondeterministic, deterministic) are investigated. In particular, it is shown that the languages recognized by two-way deterministic FAM's are of tape complexity log n and time complexity n³. Some decision problems related to vector addition systems are also studied.     

Generateurs algebriques et systemes de paires iterantes

The notion of rational transduction is a valuable tool to compare the structures of different languages, in particular context-free languages.The explanation of this is a powerful property of rational transductions with respect to certain iterative pairs [8] and systems of iterative pairs (we define this notion in this paper), in a context-free language. (Intuitively, systems of iterative pairs describe combinations of simultaneous iterative pairs in a context-free language.)This property is the so-called Transfer Theorem, whose terms are:Let A and B be two context-free languages and let T be a rational transduction such that T(B) = A. If A has a strict system of iterative pairs σ, then B has a strict system of iterative pairs σ′, of the same type than σ.(This theorem has been proved in [8] for iterative pairs and we prove it here for systems of iterative pairs.)This theorem means that any combination of certain iterative pairs in the image language by a rational transduction must appear, in a similar way, in the source language.The main result of this paper is obtained by using the previous Transfer Theorem. This result is a characterization of context-free generators i.e. generators of the rational cone or, equivalently [10], of the full-AFL of context-free languages.     

Le cylindre des langages linéaires

Define a cylinder to be a family of languages which is closed under inverse homomorphisms and intersection with regular sets. A number of well-known families of languages are cylinders:

—CFL, the family of context-free languages, is a principal cylinder, i.e. the smallest cylinder containing a languageL _O described in [6].

—the family of deterministic context-free languages is proved to be a nonprincipal cylinder in [7].

—the family of unambiguous context-free languages is a cylinder: to prove that it is not principal seems to be a very hard problem.

In this paper we prove that Lin, the family of linear context-free languages, is a nonprincipal cylinder. This is achieved in the standard way by exhibiting a sequence of languages S_n, n∈N, such that Lin is the union of all the principal cylinders generated by these languages and is not the union of any finite number of these cylinders. This leaves open the problem raised by Sheila Greibach of whether there exists a languageL such that every linear context-free language is the image ofL in some inverse gsm mapping.     

Formes de langages et de grammaires

Summary This paper is devoted to the study of context-free languages over infinite alphabets. This work can be viewed as a new attempt to study families of grammars, replacing the usual grammar forms and giving a new point of view on these questions. A language over an infinite alphabet or I-language appears as being a model for a family of usual languages; an interpretation is an homomorphism from the infinite alphabet to any finite alphabet. Using this notion of interpretation we can associate to each family of I-languages an image, called its shadow, which is a family of usual languages.The closure properties of families, generalizing to infinite alphabets the family of context-free languages, lead to define rational transductions between infinite alphabets or I-transductions, and then, families of I-languages closed under I-transductions, or I-cones. We study here relations between the closure properties of a family of I-languages and these of its shadow. As a result, we obtain that any union closed rational cone of context-free languages, principal or not, is the shadow of a principal I-cone.This work leads to new results about the classical theory of context-free languages. For instance, we prove that any principal rational cone of context-free languages can be generated by a context-free language, whose grammar has only 6 variables. This work also leads to more general considerations about the adequacy of some generating devices to the generated languages. It appears that the context-free grammars are fair, in a sense that we define, for generating context-free languages but that non-expansive context-free grammars are not for generating non-expansive context-free languages. This point of view raises a number of questions.