## Context-Free Languages, Coalgebraically (2011)

### BibTeX

@MISC{Winter11context-freelanguages,,

author = {Joost Winter and Marcello M. Bonsangue and Jan Rutten},

title = {Context-Free Languages, Coalgebraically},

year = {2011}

}

### OpenURL

### Abstract

We give a coalgebraic account of context-free languages using the functor D(X) =2 × X 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. Furthermore, the three characterizations can serve as the basis for the definition of a general coalgebraic notion of context-freeness, which we see as the ultimate long-term goal of the present study.

### Citations

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- Rutten
(Show Context)
Citation Context ... basic definitions of coalgebras, deterministic automata and context-free grammars. A more extensive coalgebraic treatment of languages, automata and regular expressions can be found, for example, in =-=[13,12,7]-=-. A coalgebra for an endofunctor F:Set → Set consists of a carrier set C together with a map c:C → FC. The functor F is usually called the type of the coalgebra. In this paper we will be concerned wit... |

309 |
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(Show Context)
Citation Context ...characterizations, which we see as yet another direction for future research. Related Work. In contrast to regular languages, equality of context-free languages is known to be an undecidable property =-=[6]-=-. This may explain why not so much algebraic or coalgebraic work has been devoted to study the theory of context-free languages. The first, and only, coalgebraic treatment of context-free languages we... |

201 | A completeness theorem for Kleene algebras and the algebra of regular events
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(Show Context)
Citation Context ...ly, the starting point is Kozen’s complete characterization of regular languages in terms of Kleene algebras, idempotent semirings equipped with a star-operation satisfying some fixed point equations =-=[8]-=-. In [9,2], Kleene algebras have been extended with a least fixed point operator to axiomatize fragments ofContext-Free Languages, Coalgebraically 361 the theory of context-free languages. We take a ... |

88 |
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(Show Context)
Citation Context ...ns on context-free grammars and languages, and give only the definitions and results we need in the rest of this paper. For a more comprehensive study of context-free grammars and languages, see e.g. =-=[10]-=-. A context-free grammar, orCFG, on a finite alphabet A is a pair (X, p), where X is a finite set of nonterminals, or variables, and p:X → Pω((A + X) ∗ ) is a function describing the production rules.... |

66 | Automata and coinduction (an exercise in coalgebra
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(Show Context)
Citation Context ...regular expressions, which constitute a D-coalgebra by means of the so-called Brzozowski derivatives. Thus the coalgebraic picture of regular languages and regular expressions is well-understood (cf. =-=[12]-=- for details). Moreover the picture is so elementary that it has recently been possible [15] to generalize it to a large class of other systems, including Mealy machines, labelled transition systems, ... |

55 | Behavioural differential equations: a coinductive calculus of streams, automata, and power series
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(Show Context)
Citation Context ... X. 4 Context-Free Languages via Equations We will now look at a characterization of context-free languages in terms of systems of behavioural differential equations, analogous to those introduced in =-=[14]-=-. The idea is to define context-free languages by means of equations that involve output values and derivatives for each alphabet symbol, using a simple language with only variables, choice and sequen... |

42 | On generalised coinduction and probabilistic specification formats: distributive laws
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Citation Context .... Awell-formed system of equations for X consists of one well-formed equation for each x ∈ X. Equivalently, a well-formed system of equations over X (for a fixed A) can be seen as a mapping f:X → DTX =-=[1]-=- where, writing f(x) = 〈o(x),δ(x)〉, we define o(x) and, for each a ∈ A, xa = δ(x)(a) by the values specified by the system of equations. Before defining what a solution of a system of equations is, we... |

42 |
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Citation Context ... in [17,15] for a large variety of coalgebras, including the one for deterministic automata. The observation that context-free languages can be seen as solutions to systems of equations dates back to =-=[3]-=-. 2 Coalgebras and Deterministic Automata In this section we give the basic definitions of coalgebras, deterministic automata and context-free grammars. A more extensive coalgebraic treatment of langu... |

25 |
A new normal-form theorem for context-free phrase structure grammars
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Citation Context ...phabet symbol, and t ∈ (A + X) ∗ is a (possibly empty) sequence of nonterminal and alphabet symbols. The main difference between weak Greibach normal form and the usual notion of Greibach normal form =-=[4]-=- is that here, t is not a string over X but over (A + X), and hence may contain both nonterminal and alphabet symbols. Clearly, every CFG in Greibach normal form is also in weak Greibach normal form. ... |

25 | Decidability of DPDA equivalence
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Citation Context ...ntextfree languages in terms of pushdown automata [6,10], and the study of coinductive decision procedures for bisimilarity of deterministic pushdown automata, a problem that is known to be decidable =-=[18]-=-. Acknowledgements. We would like to thank Alexandra Silva, for valuable suggestions and discussions. Furthermore, we would like to thank the anonymous referees for their corrections and suggestions f... |

14 | Non-deterministic Kleene coalgebras
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(Show Context)
Citation Context ...olutions in terms of regular expressions with unique fixed points (in Sect. 5). Regular expressions with the Kleene star replaced by a unique right-recursive fixed point operator have been studied in =-=[17,15]-=- for a large variety of coalgebras, including the one for deterministic automata. The observation that context-free languages can be seen as solutions to systems of equations dates back to [3]. 2 Coal... |

13 |
A bialgebraic review of deterministic automata, regular expressions and languages
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(Show Context)
Citation Context ... basic definitions of coalgebras, deterministic automata and context-free grammars. A more extensive coalgebraic treatment of languages, automata and regular expressions can be found, for example, in =-=[13,12,7]-=-. A coalgebra for an endofunctor F:Set → Set consists of a carrier set C together with a map c:C → FC. The functor F is usually called the type of the coalgebra. In this paper we will be concerned wit... |

12 | Context-free languages via coalgebraic trace semantics
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(Show Context)
Citation Context ...uch algebraic or coalgebraic work has been devoted to study the theory of context-free languages. The first, and only, coalgebraic treatment of context-free languages we are aware of, is presented in =-=[5]-=-. In this paper context-free languages are described indirectly, as the result of flattening finite skeletal parsed trees. The authors study context-free grammars as coalgebras for a functor different... |

10 | Generalizing the powerset construction, coalgebraically - Silva, Bonchi, et al. - 2010 |

3 |
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(Show Context)
Citation Context ...starting point is Kozen’s complete characterization of regular languages in terms of Kleene algebras, idempotent semirings equipped with a star-operation satisfying some fixed point equations [8]. In =-=[9,2]-=-, Kleene algebras have been extended with a least fixed point operator to axiomatize fragments ofContext-Free Languages, Coalgebraically 361 the theory of context-free languages. We take a similar ap... |

3 |
Towards Kleene Algebra with Recursion
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(Show Context)
Citation Context ...starting point is Kozen’s complete characterization of regular languages in terms of Kleene algebras, idempotent semirings equipped with a star-operation satisfying some fixed point equations [8]. In =-=[9,2]-=-, Kleene algebras have been extended with a least fixed point operator to axiomatize fragments ofContext-Free Languages, Coalgebraically 361 the theory of context-free languages. We take a similar ap... |

3 |
A sound and complete calculus for finite stream circuits
- Milius
- 2010
(Show Context)
Citation Context ...= μx.τx. We now go on by defining the notions of single syntactic substitutions and chains of syntactic substitutions: these definitions can be seen as a formalization of the corresponding notions in =-=[11]-=-, [17], and [15]. Given an association of expressions σx to variables x, an expression t ′ is a single syntactic substitution of t, ift ′ is obtained by replacing (syntactically) a single occurrence o... |

1 |
Kleene Coalgebra. Ph.D. thesis. Radboud Universiteit Nijmegen (2010
- Silva
(Show Context)
Citation Context ...ivatives. Thus the coalgebraic picture of regular languages and regular expressions is well-understood (cf. [12] for details). Moreover the picture is so elementary that it has recently been possible =-=[15]-=- to generalize it to a large class of other systems, including Mealy machines, labelled transition systems, and various probabilistic automata. In the present paper, we will develop in part a similar ... |