A Fully Equational Proof of Parikh's Theorem (original) (raw)

On Some Properties of Context-Free

this article. One of the goals of such investigations of sublanguages of Q is to nally encounter a tting tool to solve the problem of its context-freeness. In this line of research also topics like primitive languages generated by small context-free grammars or having the structure (ab , and languages having certain semilinear properties etc. have been investigated. As a small selection of such material we have added the references [2], [13] and [12] to the bibliography. There a wide variety of further references can be found

Coalgebraic Characterizations of Context-Free Languages

Logical Methods in Computer Science, 2013

We provide a coalgebraic account of context-free languages, using functors D := 2 × (−) A for deterministic automata over an alphabet A, in three different but equivalent ways: (i) by viewing context-free grammars as D-coalgebras; (ii) by defining a format for behavioural differential equations (w.r.t. D) for which the unique solutions are precisely the context-free languages; and (iii) as the D-coalgebra of generalized regular expressions in which the Kleene star is replaced by a unique fixed point operator. In all cases, semantics is defined by the unique homomorphism into the final coalgebra of all languages, paving the way for coinductive proofs of context-free language equivalence. This presentation can be seen categorically as an instance of the generalized powerset construction [SBBR10]. Furthermore, we generalize this approach from context-free languages to formal power series in general, by generalizing to functors D := S × (−) A for automata with output values over an arbitrary semiring S.

Parikh-Bounded Languages

1981

A string y is in C(x), the commutative image of a string x, if y is a permutation of the symbols in x. A language L is Parikh-bounded if L contains a bounded language B and all x in L have a corresponding y in B such that x is in C(y). The central result in this paper is that if L is context-free it is also Parikh-bounded. Parikh's theorem follows as a corollary. If L is not bounded but is a Parikh-bounded language closed under intersection with regular sets, then for any positive integer k there is an x in L such that #(C(x) ∩ L) ≥ k. The notion of Parikh-discreteness is introduced.

Comparisons of Parikh's condition to other conditions for context-free languages

Theoretical Computer Science, 1998

In this paper we first compare Parikh's condition to various pumping conditions ~ Bar-Hillel's pumping lemma, Ogden's condition and Bader-Moura's condition; secondly, to interchange condition; and finally, to Sokolowski's and Grant"s conditions. In order to carry out these comparisons we present some properties of Parikh's languages. The main result is the orthogonality of the previously mentioned conditions and Parikh's condition. 0 1998-Elsevier Science B.V. All rights reserved

Free profinite locally idempotent and locally commutative semigroups

Journal of Pure and Applied Algebra, 2001

This paper is concerned with the structure of semigroups of implicit operations on the pseudovariety LSl of finite locally idempotent and locally commutative semigroups. We depart from a general result of Almeida and Weil to give two descriptions of these semigroups: the first in terms of infinite words, and the second in terms of infinite and bi-infinite words. We then derive some applications.

On strongly context-free languages

Discrete Applied Mathematics, 2000

We investigate the context-free languages whose complements are also context-free. We call them strongly context-free languages. The family of strongly linear languages is similarly deÿned. After examining the closure properties of the family of strongly context-free languages, we prove that any slender context-free language is strongly linear. We then show that there are languages of a bounded complexity in terms of the number of non-terminals or productions necessary to generate them, whereas the complexity of their complements is arbitrarily large. ?

Equational theories of tropical semirings

Theoretical Computer Science, 2003

This paper studies the equational theory of various exotic semirings presented in the literature. Exotic semirings are semirings whose underlying carrier set is some subset of the set of real numbers equipped with binary operations of minimum or maximum as sum, and addition as product. Two prime examples of such structures are the (max, +) semiring and the tropical semiring. It is shown that none of the exotic semirings commonly considered in the literature has a finite basis for its equations, and that similar results hold for the commutative idempotent weak semirings that underlie them. For each of these commutative idempotent weak semirings, the paper offers characterizations of the equations that hold in them, decidability results for their equational theories, explicit descriptions of the free algebras in the varieties they generate, and relative axiomatization results.

Free profinite R-trivial, locally idempotent and locally commutative semigroups

Semigroup Forum, 1999

This paper is concerned with the structure of implicit operations on R ∩ LJ 1 , the pseudovariety of all R-trivial, locally idempotent and locally commutative semigroups. We give a unique factorization statement, in terms of component projections and idempotent elements, for the implicit operations on R ∩ LJ 1. As an application we give a combinatorial description of the languages that are both R-trivial and locally testable. A similar study is conducted for the pseudovariety DA ∩ LJ 1 of locally idempotent and locally commutative semigroups in which each regular D-class is a rectangular band.