## Equivalences among Various Logical Frameworks of Partial Algebras (1996)

Venue: | Computer Science Logic. 9th Workshop, CSL'95. Paderborn |

Citations: | 17 - 7 self |

### BibTeX

@INPROCEEDINGS{Mossakowski96equivalencesamong,

author = {Till Mossakowski},

title = {Equivalences among Various Logical Frameworks of Partial Algebras},

booktitle = {Computer Science Logic. 9th Workshop, CSL'95. Paderborn},

year = {1996},

pages = {403--433},

publisher = {Springer LNCS}

}

### OpenURL

### Abstract

We examine a variety of liberal logical frameworks of partial algebras. Therefore we use simple, conjunctive and weak embeddings of institutions which preserve model categories and may map sentences to sentences, finite sets of sentences, or theory extensions using unique existential quantifiers, respectively. They faithfully represent theories, model categories, theory morphisms, colimit of theories, reducts etc. Moreover, along simple and conjunctive embeddings, theorem provers can be re-used in a way that soundness and completeness is preserved. Our main result states the equivalence of all the logical frameworks with respect to weak embeddability. This gives us compilers between all frameworks. Thus it is a chance to unify the different branches of specification using liberal partial logics. This is important for reaching the goal of formal interoperability of different specification languages for software development. With formal interoperability, a specification can contain part...

### Citations

486 |
Institutions: Abstract Model Theory for Specification and Programming
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- 1992
(Show Context)
Citation Context ...y, we will start with the logical framework of partial algebras with universal Horn sentences, introduced by Burmeister [4, 5] and others. This seems to be the strongest framework among the "libe=-=ral" [16, 37]-=- logical frameworks admitting initial and free constructions which are important for specification languages with module concepts [11]. Now there are various restrictions of this logical framework, na... |

226 | Fundamentals of Algebraic Specification 1 - Ehrig, Mahr - 1985 |

208 | Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations
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Citation Context ...heories are called left exact sketches (see [19]). ut 3.5 Order-Sorted Algebra with Sort Constraints Finally, COSASC(=!=) is the institution of coherent order sorted signatures, algebras and theories =-=[17] with-=- conditional sort constraints [18]. Order signatures are triples \Sigma = (S; ; OP ), wheresis a partial order on S and OP may contain the same operation with different arities and coarities ("ov... |

202 |
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- 1994
(Show Context)
Citation Context ...all of the above mentioned logical frameworks with respect to weak embeddings of institutions. This extends the following results: The equivalence of Hep-theories and left exact sketches is proved in =-=[1]-=- by showing that both have exactly the locally finitely presentable categories as model categories. Limit-theories are shown to have exactly locally finitely presentable categories as model categories... |

169 |
General logics
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- 1989
(Show Context)
Citation Context ...q q q q q q q q q q q q q q q q q 88 Th 0 // 9! OO Fig. 1. Functors and adjunctions between different categories of logical frameworks 2.6 Re-Use of Theorem Provers The following theorem is proved in =-=[8]-=-: Theorem6. Let : I \Gamma! J be a simple map of institutions with surjective model components (all simple embeddings have this property). Then semantical entailment is preserved: \Gamma j= I \Sigma '... |

143 |
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- 1977
(Show Context)
Citation Context ...o for other frameworks. 1 Introduction Logical frameworks are widely used in computer science, in particular for the specification of correct software systems with the help of specification languages =-=[6, 38]-=-. Each specification language is based on an underlying logical framework, consisting of signatures, models, theories, sentences and so on. Now the large variety of existing specification languages an... |

75 |
A partial functions version of Church’s simple theory of types
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Citation Context ...falsehood of relations are not distinguished (as it would be possible within three-valued logics). An overview over different semantics and a justification of the two-valued semantics can be found in =-=[13]-=-. Signatures \Sigma = (S; OP;POP;REL) consist of (finite sets) of sort symbols s 2 S, total operation symbols op: s 1 ; : : : ; s n \Gamma! s 2 OP , partial operation symbols pop: s 1 ; : : : ; s n \G... |

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(Show Context)
Citation Context ...3, 19], Coste's limit theories [9] and Meseguer's and Goguen's order-sorted theories with sort constraints [18]. It is a folklore theorem, that Hep-theories are "the same" as left exact sket=-=ches, see [15, 32]. And sort constrain-=-ts of order sorted algebra are introduced to capture "the same" partiality as partial algebras with universal Horn sentences. But how to make this "the same" precise and how to stu... |

45 |
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(Show Context)
Citation Context ... for specification languages with module concepts [11]. Now there are various restrictions of this logical framework, namely Reichel's Heptheories [33], called essentially algebraic theories by Freyd =-=[14]-=-, theories formed with ECE-equations [4], Jarzembski's weak varieties of partial algebras [22], and theories formed with strong equations [4, 20, 23, 31]. Further, there are other logical frameworks c... |

42 |
Toposes, triples and theories. Grundlehren der mathematischen Wissenschaften 278
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(Show Context)
Citation Context ...ties of partial algebras [22], and theories formed with strong equations [4, 20, 23, 31]. Further, there are other logical frameworks capturing partiality as well. We only mention left exact sketches =-=[3, 19], Coste's -=-limit theories [9] and Meseguer's and Goguen's order-sorted theories with sort constraints [18]. It is a folklore theorem, that Hep-theories are "the same" as left exact sketches, see [15, 3... |

38 |
B.: Fundamentals of Algebraic Specification 2
- Ehrig, Mahr
- 1990
(Show Context)
Citation Context ... This seems to be the strongest framework among the "liberal" [16, 37] logical frameworks admitting initial and free constructions which are important for specification languages with module=-= concepts [11]-=-. Now there are various restrictions of this logical framework, namely Reichel's Heptheories [33], called essentially algebraic theories by Freyd [14], theories formed with ECE-equations [4], Jarzembs... |

22 |
Relationships between Logical Formalisms
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(Show Context)
Citation Context ...ism, colimit of theories, reduct etc., taken in one institution, can be faithfully simulated in the other. With this, it is easy to show that institution independent specification language constructs =-=[34, 7]-=- are preserved in both directions. We now recall three different notions of map of institution from [25, 30]. 2.3 Simple Maps of Institutions An institution I = (Sign; sen; Mod; j=) induces the catego... |

20 |
Partial algebras – survey of a unifying approach towards a two-valued model theory for partial algebras. Algebra Universalis
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(Show Context)
Citation Context ...o compare logical frameworks. Considering the kind of logical frameworks to study, we will start with the logical framework of partial algebras with universal Horn sentences, introduced by Burmeister =-=[4, 5] and other-=-s. This seems to be the strongest framework among the "liberal" [16, 37] logical frameworks admitting initial and free constructions which are important for specification languages with modu... |

16 |
A model theoretic approach to partial algebras. Akademie
- Burmeister
- 1986
(Show Context)
Citation Context ...ive embeddings and three weak embeddings among them: 1. Relational Partial Conditional Existence-Equational Logic RP (R e =! R e =) [4] 2. Partial Conditional Existence Equational Logic P ( e =! e =) =-=[4, 5]-=- 3. Partial Existentially-Conditioned Existence-Equational Logic P (D ! e =) [4, 21, 22] 4. Partial Logic With Strong Equations P ( s =) [4, 20, 23, 31] 5. Hierarchical Equationally Partial Theories H... |

14 |
Multiparadigm Specification Languages: a First Attempt at Foundations
- Astesiano, Cerioli
- 1994
(Show Context)
Citation Context ...different branches of specification of partial algebras from a semantical point of view. Further, one can combine specifications written in different logical frameworks by multiparadigm specification =-=[2]-=- using the embeddings. The ten simple and two conjunctive embeddings of institutions embed also sentences faithfully, so entailment systems and theorem provers can be re-used along them [8]. Future wo... |

9 |
spectra and sheaf representation
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- 1979
(Show Context)
Citation Context ...nd theories formed with strong equations [4, 20, 23, 31]. Further, there are other logical frameworks capturing partiality as well. We only mention left exact sketches [3, 19], Coste's limit theories =-=[9] and Meseg-=-uer's and Goguen's order-sorted theories with sort constraints [18]. It is a folklore theorem, that Hep-theories are "the same" as left exact sketches, see [15, 32]. And sort constraints of ... |

9 |
On recent trends in algebraic specification
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- 1989
(Show Context)
Citation Context ...Preliminaries How can the notion of logical framework be formalized? In this section, we recall some answers to this problem. 2.1 Specification Frames Specification frames by Ehrig, Pepper and Orejas =-=[12]-=- formalize abstract theories and models of theories, while there are no notions of sentence and satisfaction: A specification frame F = (Th F ; Mod F ) consists of 1. a category Th F of theories 2. a ... |