## Combining and Representing Logical Systems Using Model-Theoretic Parchments (1997)

Venue: | In Recent Trends in Algebraic Development Techniques, volume 1376 of LNCS |

Citations: | 18 - 5 self |

### BibTeX

@INPROCEEDINGS{Mossakowski97combiningand,

author = {Till Mossakowski and Andrzej Tarlecki},

title = {Combining and Representing Logical Systems Using Model-Theoretic Parchments},

booktitle = {In Recent Trends in Algebraic Development Techniques, volume 1376 of LNCS},

year = {1997},

pages = {349--364},

publisher = {Springer}

}

### OpenURL

### Abstract

. The paper addresses important problems of building complex logical systems and their representations in universal logics in a systematic way. We adopt the model-theoretic view of logic as captured in the notions of institution and of parchment (an algebraic way of presenting institutions). We propose a new, modified notion of parchment together with parchment morphisms and representations. In contrast to the original parchment definition and our earlier work, in model-theoretic parchments introduced here the universal semantic structure is distributed over individual signatures and models. We lift formal properties of the categories of institutions and their representations to this level: the category of model-theoretic parchments is complete, and their representations may be put together using categorical limits as well. However, model-theoretic parchments provide a more adequate framework for systematic combination of logical systems than institutions. We indicate how the necessar...

### Citations

515 | Institutions: Abstract model theory for specification and programming - Goguen, Burstall - 1992 |

266 |
Abstract and Concrete Categories
- Adámek, Herrlich, et al.
(Show Context)
Citation Context ...-tuned to the need of individual signatures and models. 2 Preliminaries Basic knowledge of category theory is assumed throughout the paper; for the standard notions and facts we refer for instance to =-=[1]-=-. We briefly recall some concepts and facts on relational structures and sources in their categories (cf. [1]). Relational algebraic signatures \Sigma = (S; OP;REL), consist of a set of sort symbols s... |

234 | First-Order Modal Logic
- Fitting, Mendelsohn
- 1999
(Show Context)
Citation Context ...by ordinary parchments of [5] or -parchments of [10]. To show the extra flexibility of model-theoretic parchments, we sketch a model-theoretic parchment for a simple ground (no variables) modal logic =-=[3]-=-, which cannot be naturally presented as either a parchment or -parchment. The model theoretic parchment GML = (Sign; Mod;L;Gj ) of ground modal logic is defined as follows: -- The category Sign of si... |

179 |
General Logics
- Meseguer
- 1989
(Show Context)
Citation Context ...ally supported by KBN grant 8 T11C 018 11. arrow: institution morphisms [6] that capture how one institution is built over another and institution representations [15] (or simple maps of institutions =-=[8]-=-) that capture how one institution is encoded in another. In [15] the role of these notions and the interplay between them have been studied. The category of institutions was discussed as a rudimentar... |

105 | A categorical manifesto
- Goguen
- 1991
(Show Context)
Citation Context ...his yields the category Ins of institutions and institution morphisms. Theorem2 [14]. The category Ins of institutions and institution morphisms is complete. The idea of combining things via colimits =-=[4]-=- applies to institutions as objects in Ins --- since we write institution morphisms from a richer institution to poorer one, limits rather than colimits provide a tool for combination of institutions.... |

55 | Moving between logical systems
- Tarlecki
- 1996
(Show Context)
Citation Context ...itations. ?? This work has been partially supported by KBN grant 8 T11C 018 11. arrow: institution morphisms [6] that capture how one institution is built over another and institution representations =-=[15]-=- (or simple maps of institutions [8]) that capture how one institution is encoded in another. In [15] the role of these notions and the interplay between them have been studied. The category of instit... |

50 |
Bits and pieces of the theory of institutions
- Tarlecki
- 1986
(Show Context)
Citation Context ...en them have been studied. The category of institutions was discussed as a rudimentary framework for systematic construction of logical systems via limits (which always exist in this category --- see =-=[14], [16]). M-=-aps between institution representations, consisting of an institution morphism and an extra "fitting" component have been introduced. Representations related by such maps can be combined via... |

48 |
Axioms for abstract model theory
- Barwise
- 1974
(Show Context)
Citation Context ...itutions and institution morphisms Any specification formalism is usually based on some notion of signature, model, sentence and satisfaction. These are the usual ingredients of abstract model theory =-=[2]-=- and are the essence of Goguen and Burstall's notion of institution [6]. An institution I = (Sign; Sen; Mod; j=) consists of -- a category Sign of signatures, -- a functor Sen: Sign \Gamma! Set giving... |

44 |
A study in the foundations of programming methodology: Specifications, institutions, charters and parchments
- Goguen, Burstall
- 1985
(Show Context)
Citation Context ...es are simply united in the combination rather then being properly combined, and the features of the combined logics do not really interact in the result. A possible solution is to move to parchments =-=[5]-=-, certain algebraic presentations of institutions providing an abstract syntax and evaluator-based semantics for sentences and therefore potentially more useful for logic combination [9]. In [10] we h... |

41 |
Some fundamental algebraic tools for the semantics of computation. Part III: Indexed categories
- Tarlecki, Burstall, et al.
- 1991
(Show Context)
Citation Context ...m have been studied. The category of institutions was discussed as a rudimentary framework for systematic construction of logical systems via limits (which always exist in this category --- see [14], =-=[16]). Maps be-=-tween institution representations, consisting of an institution morphism and an extra "fitting" component have been introduced. Representations related by such maps can be combined via categ... |

28 |
Computing in Horn Clause Theories
- Padawitz
- 1988
(Show Context)
Citation Context ...For each signature \Sigma 2 jAlgSigj, we have a category Str(\Sigma) of \Sigma -structures and \Sigma -homomorphisms, where homomorphisms only preserve, but not necessarily reflect the relations, see =-=[12]-=-. A \Sigma -homomorphism h: A \Gamma! B is closed if it also reflects the relations, that is, for all R : w 2 REL, h \Gamma1 w [RB ] = RA . Str(\Sigma) is cocomplete. Each signature morphism oe: \Sigm... |

15 | Using limits of parchments to systematically construct institutions of partial algebras
- Mossakowski
- 1996
(Show Context)
Citation Context ...to parchments [5], certain algebraic presentations of institutions providing an abstract syntax and evaluator-based semantics for sentences and therefore potentially more useful for logic combination =-=[9]-=-. In [10] we have introduced -parchments, a slightly modified notion of parchment, and lifted the main notions of arrow and their properties from institutions to -parchments. One justification for wor... |

13 |
Galleries and institutions
- Mayoh
- 1985
(Show Context)
Citation Context ...L;Gj ) as follows: -- for \Sigma 2 jSignj, let P (\Sigma) = (L; M;G); then Mod(\Sigma) = M, L(\Sigma) = L and Gj \Sigma = G, and 6 The terminology comes from a similar presentation of institutions in =-=[7]-=-, cf. also [16]. 7 Byswe denote the usual multiplication of natural transformations by functors. -- for oe: \Sigma \Gamma! \Sigma 0 2 Sign, let P (oe) = (ff; fi; g): (L(\Sigma); Mod(\Sigma); Gj \Sigma... |

12 | W.: Combining and Representing Logical Systems
- Mossakowski, Tarlecki, et al.
- 1997
(Show Context)
Citation Context ...ments [5], certain algebraic presentations of institutions providing an abstract syntax and evaluator-based semantics for sentences and therefore potentially more useful for logic combination [9]. In =-=[10]-=- we have introduced -parchments, a slightly modified notion of parchment, and lifted the main notions of arrow and their properties from institutions to -parchments. One justification for working with... |

3 |
The first order parchment
- Stefaneas
- 1992
(Show Context)
Citation Context ...y more practical "universal model-theoretic parchment", let us consider as a universal parchment the model-theoretic parchment FOL for first-order logic (with equality), following the presen=-=tation in [13]-=-. In particular (other details are irrelevant here): -- Sign FOL has as objects theories (\Sigma; Ax ) constisting of an AlgSig-signature \Sigma plus some first-order axioms Ax , -- Mod FOL (\Sigma; A... |