On languages of automaton counter machines

A class of formal languages (ACML) acceptable by automaton counter machines is considered. This class is shown to be close with respect to the operations of union, regular intersection, concatenation, infinite iteration, homomorphism, and inverse homomorphism. It follows from here that this class is a full abstract family of languages [7] with all the properties following from this. Furthermore, the ACML is close with respect to intersection and substitution but is not closed with respect to complement and reverse. For the ACML class, the problems of emptiness and recognition of words of a language given by an automaton counter machine are decidable, but the problems of inclusion and equivalence of languages are undecidable. A comparison with other classes of languages (regular, context-free, context-sensitive, and Petri-net languages) is performed.     

An infinite hierarchy of intersections of context-free languages

The class of languages expressible as the intersection ofk context-free languages is shown to be properly contained within the class of languages expressible as the intersection ofk + 1 context-free languages. Hence an infinite hierarchy of classes of languages is exhibited between the class of context-sensitive languages and the class of context-free languages.     

Representation theorems using DOS languages

It is demonstrated that every context-free language is a homomorphic image of the intersection of two DOS languages and that every recursively enumerable language is the homomorphic image of the intersection of three DOS languages. It is also proved that by increasing the number of components in the intersections of DOS languages one gets an infinite hierarchy of classes of languages within the class of context-sensitive languages.     

Context-free languages can be accepted with absolutely no space overhead

We study Turing machines that are allowed absolutely no space overhead. The only work space the machines have, beyond the fixed amount of memory implicit in their finite-state control, is that which they can create by cannibalizing the input bits’ own space. This model more closely reflects the fixed-sized memory of real computers than does the standard complexity-theoretic model of linear space. Though some context-sensitive languages cannot be accepted by such overhead-free machines, we show that all context-free languages can be accepted nondeterministically in polynomial time with absolutely no space overhead, and that all deterministic context-free languages can be accepted deterministically in polynomial time with absolutely no space overhead.     

Extension of tabled 0L-systems and languages

This paper introduces a new family of languages which originated from a study of some mathematical models for the development of biological organisms. Various properties of this family are established and in particular it is proved that it forms a full abstract family of languages. It is compared with some other families of languages which have already been studied and which either originated from the study of models for biological development or belong to the now standard Chomsky hierarchy. A characterization theorem for context-free languages is also established.This research has been supported by NSF Grant GJ 998.     

On the structure of context-sensitive grammars

It is well known that the family of context-sensitive grammars generate languages which are not context-free and that it is undecidable whether a context-sensitive grammar generates a context-free language. However, the mechanism by which the use of context allows a non-context-free language to be generated is not well understood. In this paper we attempt to focus on this problem by surveying some of the results which speak to two questions: (i) What constraints can be placed on the form of the rules of context-sensitive grammars without restricting the weak generative capacity? (ii) What (nontrivial) constraints can be placed on the form of the rules of context-sensitive grammars such that only context-free languages will be generated?     

The Boolean closure of linear context-free languages

Closures of linear context-free languages under Boolean operations are investigated. The intersection closure and the complementation closure are incomparable. By closing these closures under further Boolean operations we obtain several new language families. The hierarchy obtained by such closures of closures is proper up to a certain level, where it collapses to the Boolean closure which, in turn, is incomparable with several closures of the family of context-free languages. The Boolean closure of the linear context-free languages is properly contained in the Boolean closure of the context-free languages. A characterization of a class of non-unary languages that cannot be expressed as a Boolean formula over the linear context-free languages is presented.     

Learning nonregular languages: a comparison of simple recurrent networks and LSTM

In response to Rodriguez's recent article (2001), we compare the performance of simple recurrent nets and long short-term memory recurrent nets on context-free and context-sensitive languages.     

Description of developmental languages using recurrence systems

Recurrence systems have been devised to describe formally certain types of biological developments. A recurrence system specifies a formal language associated with the development of an organism. The family of languages defined by recurrence systems is an extension of some interesting families of languages, including the family of context-free languages. Some normal-form theorems are proved and the equivalence of the family of recurrence languages to a previously studied family of developmental languages (EOL-languages) is shown. Various families of developmental and other formal languages are characterized using recurrence systems. Some closure properties are also discussed.     

Regular patterns, regular languages and context-free languages

In this paper we consider two questions. First we consider whether every pattern language which is regular can be generated by a regular pattern. We show that this is indeed the case for extended (erasing) pattern languages if alphabet size is at least four. In all other cases, we show that there are patterns generating a regular language which cannot be generated by a regular pattern. Next we consider whether there are pattern languages which are context-free but not regular. We show that, for alphabet size 2 and 3, there are both erasing and non-erasing pattern languages which are context-free but not regular. On the other hand, for alphabet size at least 4, every erasing pattern language which is context-free is also regular. It is open at present whether there exist non-erasing pattern languages which are context-free but not regular for alphabet size at least 4.     

Cooperating/distributed grammar systems with hypothesis languages

Abstract

Motivated by the blackboard model of artificial intelligence we introduce the concept of context-free cooperating/distributed grammar systems with hypothesis languages. We prove that these grammar systems have the same generative power as context-sensitive grammars.     

𝒫𝒮-regular languages

In this paper, we introduce a class of generalized regular languages, namely 𝒫𝒮-regular languages, and give some characterizations of such generalized regular languages. As applications of the results, we obtain some characterizations of regular languages. Also, we consider the closure properties of the class of 𝒫𝒮-regular languages, and the relationship among 𝒫𝒮-regular languages, context-free languages and context-sensitive languages.     

Context-free languages over infinite alphabets

In this paper we introduce context-free grammars and pushdown automata over infinite alphabets. It is shown that a language is generated by a context-free grammar over an infinite alphabet if and only if it is accepted by a pushdown automaton over an infinite alphabet. Also the generated (accepted) languages possess many of the properties of the ordinary context-free languages: decidability, closure properties, etc.. This provides a substantial evidence for considering context-free grammars and pushdown automata over infinite alphabets as a natural extension of the classical ones. Received November 27, 1995 / March 4, 1997     

On the languages generated by context-sensitive fuzzy grammars

It is proved that the family of the languages generated by 1-fold fuzzy grammars with context-sensitive rules and grade larger than λ,0≦λ<1, is equal to the family of the context sensitive languages. Also, one shows that the family of the languages generated by fuzzy grammars with context-sensitive rules and grade larger than λ,0≦λ<1, is included in the family of the context-sensitive languages.     

LSTM recurrent networks learn simple context-free andcontext-sensitive languages

Previous work on learning regular languages from exemplary training sequences showed that long short-term memory (LSTM) outperforms traditional recurrent neural networks (RNNs). We demonstrate LSTMs superior performance on context-free language benchmarks for RNNs, and show that it works even better than previous hardwired or highly specialized architectures. To the best of our knowledge, LSTM variants are also the first RNNs to learn a simple context-sensitive language, namely a(n)b(n)c(n).     

Quasi-realtime languages

The quasi-realtime languages are seen to be the languages accepted by nondeterministic multitape Turing machines in real time. The family of quasi-realtime languages forms an abstract family of languages closed under intersection, linear erasing, and reversal. It is identical with the family of languages accepted by nondeterministic multitape Turing machines in linear time. Every quasi-realtime language can be accepted in real time by a nondeterministic one stack, one pushdown store machine, and can be expressed as the length-preserving homomorphic image of the intersection of three context-free languages.The research reported in this paper was announced at the ACM Symposium on the Theory of Computing, Marina del Rey, California, May, 1969. This research has been supported in part by Air Force Cambridge Research Laboratories, Office of Aerospace Research, USAF, under Contract F19628-68-C-0029.     

Two complementary operations inspired by the DNA hairpin formation: Completion and reduction

We consider two complementary operations: Hairpin completion introduced in [D. Cheptea, C. Martin-Vide, V. Mitrana, A new operation on words suggested by DNA biochemistry: Hairpin completion, in: Proc. Transgressive Computing, 2006, pp. 216–228] with motivations coming from DNA biochemistry and hairpin reduction as the inverse operation of the hairpin completion. Both operations are viewed here as formal operations on words and languages. We settle the closure properties of the classes of regular and linear context-free languages under hairpin completion in comparison with hairpin reduction. While the class of linear context-free languages is exactly the weak-code image of the class of the hairpin completion of regular languages, rather surprisingly, the weak-code image of the class of the hairpin completion of linear context-free languages is a class of mildly context-sensitive languages. The closure properties with respect to the hairpin reduction of some time and space complexity classes are also studied. We show that the factors found in the general cases are not necessary for regular and context-free languages. This part of the paper completes the results given in the earlier paper, where a similar investigation was made for hairpin completion. Finally, we briefly discuss the iterated variants of these operations.     

Lower bound technique for length-reducing automata

The class of growing context-sensitive languages (GCSL) was proposed as a naturally defined subclass of context-sensitive languages whose membership problem is solvable in polynomial time. Growing context-sensitive languages and their deterministic counterpart called Church–Rosser languages (CRL) complement the Chomsky hierarchy in a natural way, as the classes filling the gap between context-free languages and context-sensitive languages. They possess characterizations by a natural machine model, length-reducing two-pushdown automata (lrTPDA). We introduce a lower bound technique for lrTPDAs. Using this technique, we prove the conjecture of McNaughton, Narendran and Otto that the set of palindromes is not in CRL. As a consequence we obtain that CFL∩coCFL as well as UCFL∩coUCFL are not included in CRL, where UCFL denotes the class of unambiguous context-free languages; this solves an open problem posed by Beaudry, Holzer, Niemann and Otto. Another corollary is that CRL is a strict subset of GCSL∩coGCSL.