## Coinductive Characterizations of Applicative Structures (1998)

Venue: | MATH. STRUCTURES IN COMP. SCI. 9(4):403–435 |

Citations: | 3 - 0 self |

### BibTeX

@INPROCEEDINGS{Honsell98coinductivecharacterizations,

author = {Furio Honsell and Marina Lenisa},

title = {Coinductive Characterizations of Applicative Structures},

booktitle = {MATH. STRUCTURES IN COMP. SCI. 9(4):403–435},

year = {1998},

pages = {403--435},

publisher = {}

}

### OpenURL

### Abstract

We discuss new ways of characterizing, as maximal fixed points of monotone operators, observational congruences on -terms and, more in general, equivalences on applicative structures. These characterizations naturally induce new forms of coinduction principles, for reasoning on program equivalences, which are not based on Abramsky's applicative bisimulation. We discuss in particular, what we call, the cartesian coinduction principle, which arises when we exploit the elementary observation that functional behaviours can be expressed as cartesian graphs. Using the paradigm of final semantics, the soundness of this principle over an applicative structure can be expressed easily by saying that the applicative structure can be construed as a strongly extensional coalgebra for the functor (P( \Theta )) \Phi (P( \Theta )). In this paper, we present two general methods for showing the soundenss of this principle. The first applies to approximable applicative structures. Many c.p.o. -models in...

### Citations

1118 |
The Lambda Calculus: its Syntax and Semantics
- Barendregt
- 1984
(Show Context)
Citation Context ...dix B we recall basic facts about final coalgebras. We assume the reader familiar with basic concepts and results in -calculus, final semantics, and non-wellfounded Set Theory. The reader may consult =-=[Bar84], [RT93, L-=-en98], and [FH83] respectively, for more details. The authors would like to thank A.Quattrocchi for her help. 2 The "coinductive characterization" problem The basic notions we shall be conce... |

215 | A filter lambda model and the completeness of type assignment - Barendregt, Coppo, et al. - 1984 |

201 |
Lectures on Non-Well-Founded Sets
- Aczel
- 1987
(Show Context)
Citation Context ... ) such that the ordered applicative structure (D; vD ; ffl D ) is a \Phi pEq D -coinductive applicative structure. 3 The final perspective Following the Final Semantics Paradigm, introduced by Aczel =-=[Acz88]-=-, and further developed in [AM89, RT93, Rut96, Tur96, Rut96, Len98], coinduction principles can receive a very neat categorical explanation. It is interesting to point out that many of the monotone op... |

162 | A final coalgebra theorem - Aczel, Mendler - 1989 |

133 | Full Abstraction in the Lazy Lambda Calculus
- Abramsky, Ong
- 1993
(Show Context)
Citation Context ... M 2 g "s0 . The evaluation + l is the least binary relation over 0 \Theta V al l satisfying the rules: x:M + l x:M M + l x:P P [N=x] + l Q MN + l Q Classical fi-reduction is correct w.r.t. ���=-=� l (see [AO93]-=-). This is the reduction strategy of lazy functional languages. A computationally adequate ordered -model forsl is the model D l , studied in [AO93]. The model D l is the quadruple (D l ; v l ; ffl l ... |

105 | Proving congruence of bisimulation in functional programming languages
- Howe
- 1996
(Show Context)
Citation Context ...C ) 0 \Theta ( C ) 0 j M + oe , N + oeg. This method, which we call congruence candidate method, was introduced in [Len98]. It is inspired by the congruence candidate method used in [How89] (see also =-=[How96]), for s-=-howing that �� oe for lazy strategies is \Phi Eq oe -coinductive. The congruence candidate method will be used here for showing that the -theories �� oe , for oe = h; l; v; p, are Eq oe -carte... |

74 | Rutten: Bisimulation for Probabilistic Transition Systems: A Coalgebraic Approach - Vink, M - 1997 |

54 |
Set theory with free construction principles. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 4 e série
- Forti, Honsell
- 1983
(Show Context)
Citation Context ...ely related to the existence of set-theoretical applicative structures in non-wellfounded theories of sets, where the Foundation Axiom is replaced by the Antifoundation Axiom X 1 of Forti and Honsell =-=[FH83]-=-. Set-theoretical applicative structures are applicative structures whose points contain the set-theoretic description of their functional behaviour, so that application can be rendered by the usual s... |

37 | A coinduction principle for recursive data types based on bisimulation - Fiore - 1996 |

36 | From operational semantics to domain theory - Mason, Smith, et al. - 1996 |

27 |
An approximation theorem for topological lambda models and the topological incompleteness of lambda calculus
- Honsell, Rocca
- 1992
(Show Context)
Citation Context ...tice that, if Eq is a (pre)equivalence, then the relation �� Eq is a (pre)congruence. All -congruences are induced by an equivalence relation with just two equivalence classes, i.e.: Proposition 4=-=.1 ([HR92]) Any -c-=-ongruence ��`sC \ThetasC is induced by a suitable set V ` ( C ) 0 in the following sense: M �� N , 8C[ ]: (C[M ]; C[N ] 2 ( C ) 0 =) (C[M ] 2 V () C[N ] 2 V)) : Proof Just take V = fx:xPQ j P ... |

26 |
Type Theories, Normal Forms and D1-Lambda-Models. Information and Computation 72
- COPPO, DEZANI-CIANCAGLINI, et al.
- 1987
(Show Context)
Citation Context ...M 0 1 : : : M 0 n M +n N x:M +n x:N M [N=x]M 1 : : : Mn +n P (x:M)NM 1 : : : Mn +n P fi-reduction is correct w.r.t. ��n . A computationally adequate ordered -model for n is the model D n , studied=-= in [CDZ87]-=-. The model D n is the quadruple (D n ; v n ; ffl D n ; [[ ]] n ), where D n is the inverse limit solution of the equation D ' [D ! D] in the category CPO, obtained starting with the initial domain D ... |

18 |
Ronchi Della Rocca S., “Operational, Denotational and Logical Description: a case study
- Egidi, Honsell
- 1992
(Show Context)
Citation Context ... v =) (M [N=x]) ae * v g: The reduction strategy ! v is the one implemented by the SECD machine of Landin and used in ML. A computationally adequate ordered -model forsv is the model D v , studied in =-=[EHR92]-=-. The model D v is the quadruple (D v ; v v ; ffl v D ; [[ ]] v ), where D v is the inverse limit initial solution of the equation D ' [D !? D]? in the category CPO? , and ffl v D and [[ ]] v are defi... |

15 | Final semantics for a higher order concurrent language - Lenisa - 1996 |

14 | The lazy lambda calculus. Research Topics - Abramsky - 1990 |

11 | On the Foundations of Final Semantics - Rutten - 1993 |

10 |
Final semantics for untyped lambdacalculus
- Honsell, Lenisa
- 1995
(Show Context)
Citation Context ...ar in a few paragraphs, in this paper, we shall call this principle cartesian coinduction, and we shall call cartesian both those applicative structures and those -theories, for which it is sound. In =-=[HL95]-=-, cartesian coinduction was proved sound only for the call-by-value -calculus and many open problems concerning it were raised. In [Len98], it was proved sound also for the lazy -calculus. A first obj... |

9 | Semantic Techniques for Deriving Coinductive Characterizations of Observational Equivalences for -calculi - Lenisa |

9 | A note on logical relations between semantics and syntax - Pitts - 1997 |

4 | A uniform syntactical method for proving coinduction principles in -calculi - Lenisa - 1997 |

3 |
Themes in Final Semantics, Ph.D. Thesis TD-6/98, Dipartimento di Informatica, Universit`a di
- Lenisa
- 1998
(Show Context)
Citation Context ...perators of the same shape, in the line of [AO93, EHR92], and exploit the fact that the observational behaviour of functional programs needs to be tested essentially only in applicative contexts (see =-=[Len98]-=-). In [HL95, Section 3], the authors considered for the first time another kind of monotone operator, thereby introducing a new form of coinduction principle for establishing observational equivalence... |

2 |
Semantical analysis of perpetual strategies
- Honsell, Lenisa
- 1999
(Show Context)
Citation Context ...ith N 2s0 and N + p , is correct w.r.t.sp . The characterization of perpetual-redexes, i.e. �� p " = fi , is given by: fh(x:M)N; M [N=x]i j 8ae 2 [V ar ! V al p ]: N ae * p =) (M [N=x]) ae * =-=p g: See [HL98] for mor-=-e informations. Notice that �� p is not very well behaved, we do not have, for instance, that if M ! p N then M �� p N . Consider for example (x:(xy:x)xx) and x:x. A we did in the case of ! h ... |

2 | call-by-value and the lambda calculus - Call-by-name - 1977 |

1 | Bisimilarity as a Theory of Functional - Gordon - 1995 |

1 | coalgebra: a theory of systems, Report CS-R9652 - Universal - 1996 |

1 | Operational Semantics and its Denotational Dual - Functorial - 1996 |

1 | A Non-wellfounded sets Non-wellfounded sets are elements of a Universe of a Zermelo Fraenkel-like settheory ZF ffi X 1 . ZF ffi X 1 is the theory consisting of the axioms extensionality - Turi - 1998 |