## Context-free languages via coalgebraic trace semantics (2005)

Venue: | International Conference on Algebra and Coalgebra in Computer Science (CALCO’05), volume 3629 of Lect. Notes Comp. Sci |

Citations: | 12 - 8 self |

### BibTeX

@INPROCEEDINGS{Hasuo05context-freelanguages,

author = {Ichiro Hasuo and Bart Jacobs},

title = {Context-free languages via coalgebraic trace semantics},

booktitle = {International Conference on Algebra and Coalgebra in Computer Science (CALCO’05), volume 3629 of Lect. Notes Comp. Sci},

year = {2005},

pages = {213--231},

publisher = {Springer}

}

### OpenURL

### Abstract

Abstract. We show that, for functors with suitable mild restrictions, the initial algebra in the category of sets and functions gives rise to the final coalgebra in the (Kleisli) category of sets and relations. The finality principle thus obtained leads to the finite trace semantics of nondeterministic systems, which extends the trace semantics for coalgebras previously introduced by the second author. We demonstrate the use of our technical result by giving the first coalgebraic account on contextfree grammars, where we obtain generated context-free languages via the finite trace semantics. Additionally, the constructions of both finite and possibly infinite parse trees are shown to be monads. Hence our extension of the application domain of coalgebras identifies several new mathematical constructions and structures. 1

### Citations

367 |
Three Models for Description of Language
- Chomsky
(Show Context)
Citation Context ... 2000); D.3.1, F.4.2, F.4.3 (ACM CR 1998) 1 Introduction Context-free grammars and context-free languages are undoubtedly among the most fundamental notions in computer science. Introduced by Chomsky =-=[Cho56]-=-, they have come to serve as a theoretical basis for formal (programming) languages [ASU86]. This paper presents the ¯rst steps in a coalgebraic analysis of those notions. In a sense it extends previo... |

316 | Universal coalgebra: A theory of systems - Rutten |

287 |
Elements of the Theory of Computation
- Lewis, Papadimitriou
- 1981
(Show Context)
Citation Context ...grammars In this section we formulate context-free grammars and context-free languages, from a coalgebraic perspective. For more about traditional treatment of those notions the reader is referred to =-=[LP81]-=-. In fact, what we are interested in here is not a language, i.e. a set of (°at) strings, but a set of parsed strings equipped with a tree structure, called skeletal parse trees [ASU86]. A context-fre... |

240 | A tutorial on (co)algebras and (co)induction
- Jacobs, Rutten
- 1997
(Show Context)
Citation Context ...a3 S2 and the trivial tree is mapped to the empty string ". Proof. The ¯rst part is by induction on the depth. The second part is shown similarly to that §! carries the ¯nal (§£¡)-coalgebra; see e.g. =-=[JR97]-=-. Remark 3.2. We can think of the functor (§ + ¡)¤ as a \signature" in a traditional sense. Let ? be a fresh symbol, and for each s 2 (§ + f?g)¤ let ksk denote the number of ?'s appearing in s. Then w... |

136 |
Terminal coalgebras in well-founded set theory, Theoret
- Barr
- 1993
(Show Context)
Citation Context ...ce t0. Let the relations Rn F n+1X £ F nX be de¯ned inductively by R0 = (1£ g) ¡1(2FX) ; Rn+1 = RelF (Rn) ; and the set U be de¯ned by U = fu 2 Y n<! FnX j 8n < !: hun+1; uni 2 Rng : As presented in =-=[Bar93]-=-, the carrier A of the initial F -algebra is obtained as a colimit (denoted by below) of the initial sequence of F , and the carrier Z of the ¯nal F -coalgebra is as a limit (denoted by ¿) of the te... |

62 | A semantics of shape - Jay - 1995 |

50 | Infinite Trees and Completely Iterative Theories: A Coalgebraic View, Theoret - Aczel, Adámek, et al. |

44 | A 2-categorical approach to change of base and geometric morphisms I, Cahiers Topologie Geom. DiHerentielle Categoriques 32 - Carboni, Kelly, et al. - 1991 |

42 | On Generalized Coinduction and Probabilistic Speci¯cation Formats: Distributive Laws in Coalgebraic Modelling
- Bartels
- 2004
(Show Context)
Citation Context ...ng diagram in SetsP commutes: FPX FP t FPZ X g t Z f¡gFZ ± ; which says that t is a morphism of FP -coalgebras. 6The use of a distributive law in coalgebraic settings is investigated elaborately in =-=[Bar04]-=-. 11 Theorem 4.6 (Main result of [Jac04b]). Let g : X ! PFX a coalgebra. 1. A trace of g always exists, but need not be unique. In other words, the coalgebra f¡gFZ ± : Z ! FPZ is weakly ¯nal in CoAl... |

26 | Introduction to coalgebra: Towards mathematics of states and observations. http://www.cs.ru.nl/B.Jacobs/CLG/JacobsCoalgebraIntro. pdf. Draft book - Jacobs - 2007 |

23 | Toposes, Triples and Theories, volume 278 of Grundleheren der math - Barr, Wells - 1985 |

21 | Trace semantics for coalgebras
- Jacobs
- 2004
(Show Context)
Citation Context ...ts of all the (possibly) in¯nite behavior. We give a solution by presenting the novel technique of ¯nite trace semantics for coalgebras. It builds on (ordinary, possibly in¯nite) trace semantics from =-=[Jac04b]-=- but the domain of the semantics is now the powerset of the initial algebra, not that of the ¯nal coalgebra. It is shown that ¯nite trace semantics is uniquely determined by a corecursive characteriza... |

6 |
On the greatest ¯xed point of a set functor
- Ad¶amek, Koubek
- 1995
(Show Context)
Citation Context ... the above characterization of z. We use Lemma 5.7 to establish that z 2 t(x). Recall the construction of the initial F -algebra as a colimit, as presented after Theorem 4.6. It is standard (see e.g. =-=[AK95]-=-) that the structure map ® and its inverse ®¡1 are characterized as the unique arrows which make the following diagram commute for each m < !. A ®¡1»=Fm0 m Fm¡1 Fmh Fm+10 m+1 Fm FA ® Since the car... |

4 | Relating two approaches to coinductive solution of recurisve equations - Jacobs - 2004 |

4 | Behavioural di®erential equations: a coinductive calculus of streams, automata, and power series - Rutten |

3 | A bialgebraic review of regular expressions, deterministic automata and languages - Jacobs - 2005 |

1 | Available free for downloading at http://www.cwru.edu/artsci/math/wells/pub/ttt.html - Springer-Verlag - 1983 |