## An encoding of partial algebras as total algebras

Venue: | Information Processing Letters |

Citations: | 3 - 2 self |

### BibTeX

@ARTICLE{Diaconescu_anencoding,

author = {Răzvan Diaconescu},

title = {An encoding of partial algebras as total algebras},

journal = {Information Processing Letters},

year = {},

pages = {10--1016}

}

### OpenURL

### Abstract

We introduce a semantic encoding of partial algebras as total algebras through a Horn axiomatization of the existence equality relation interpreted as an algebraic operation. We show that this novel encoding enjoys several important properties that make it a good tool for the execution of partial algebraic specifications through means specific to ordinary algebraic reasoning, such as term rewriting.

### Citations

476 |
Institutions: Abstract model theory for specication and programming
- Goguen, Burstall
- 1992
(Show Context)
Citation Context ... embeddings of classical into intuitionistic logic [11], this has received a new life with the arrival of the soEmail address: Razvan.Diaconescu@imar.ro (Răzvan Diaconescu). called institution theory =-=[7]-=- with its array of model theory oriented category theoretic concepts supporting the systematic studies of logic translations [13] at a higher mathematical level. A characteristic aspect of the institu... |

175 |
On the structure of abstract algebras
- Birkhoff
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(Show Context)
Citation Context ...within the proof of Cor. 4.2, namely that each quasi-existence equation gets mapped by α to a sentence equivalent to a conditional equation, allows the application of the Birkhoff completeness result =-=[2,8,4,5]-=- giving α(ρ) if and only if α(E) ∪ Γ (S,TF,PF) ⊢e (S∪{b},TF ⊕PF) α(ρ). ✷ that α(E) ∪ Γ (S,TF,PF) |= ALG (S∪{b},TF ⊕PF) Example 5.2 Let us illustrate how Cor. 5.1 may be used to perfom quasi-existence ... |

58 | Institution morphisms
- Goguen, Ros¸u
- 2002
(Show Context)
Citation Context ...d, in the sense that they involve the deepest aspects of the logical systems. The translations of partial algebra into logics with total functions appear usually as ‘simple theoroidal comorphisms’ of =-=[9]-=-. One of the most prominent such translation [12] encodes partial operations as total operations but employs also relations for expressing the domains of definition of the partial operations. Another ... |

49 |
A Model Theoretic Oriented Approach to Partial Algebras (Introduction to Theory and Application of Partial Algebras
- Burmeister
- 1986
(Show Context)
Citation Context ... Partial algebra. (abbreviated PA) Here we refer to the partial algebra as used in CASL [12] which represents a slight refinement of the concept of partial algebra as defined in the standard textbook =-=[3]-=-. Another important partial algebra work within computing science is [15]. A partial algebraic signature is a tuple (S,TF,PF), where both (S,TF) and (S,PF) are algebraic signatures such that TFw→s and... |

41 |
Institution-independent Model Theory
- Diaconescu
- 2008
(Show Context)
Citation Context ...pressing the domains of definition of the partial operations. Another important one [12,14] encodes the partial operations as relation symbols. Both of them can be found also in the recent monography =-=[5]-=-. Here we propose a novel translation that, unlike the previous ones, has a pure algebraic nature since it does employ only operation symbols, no relation symbols. Thus any partial algebraic signature... |

34 | Relating Casl with other specification languages: the institution level
- Mossakowski
(Show Context)
Citation Context ...pects of the logical systems. The translations of partial algebra into logics with total functions appear usually as ‘simple theoroidal comorphisms’ of [9]. One of the most prominent such translation =-=[12]-=- encodes partial operations as total operations but employs also relations for expressing the domains of definition of the partial operations. Another important one [12,14] encodes the partial operati... |

15 |
Andrzej Tarlecki. Casl: The Common Algebraic Specification Language
- Astesiano, Bidoit, et al.
(Show Context)
Citation Context ...tions that are partially defined makes partial algebra as one of the important formalisms employed by modern formal methods, a prominent example being the recent algebraic specification language CASL =-=[1]-=-. However, the current mathematical culture, including school mathematics that is responsible for the basic patterns of our mathematical thinking, is strongly biased towards total functions. Reasons a... |

14 |
On the principle of excluded middle
- Kolmogorov
- 1967
(Show Context)
Citation Context ... to the original partial algebra framework. Although such way of ‘doing logic by translation’ has a long tradition starting perhaps with the study of embeddings of classical into intuitionistic logic =-=[11]-=-, this has received a new life with the arrival of the soEmail address: Razvan.Diaconescu@imar.ro (Răzvan Diaconescu). called institution theory [7] with its array of model theory oriented category th... |

12 |
and José Meseguer. Completeness of many-sorted equational logic
- Goguen
- 1982
(Show Context)
Citation Context ...within the proof of Cor. 4.2, namely that each quasi-existence equation gets mapped by α to a sentence equivalent to a conditional equation, allows the application of the Birkhoff completeness result =-=[2,8,4,5]-=- giving α(ρ) if and only if α(E) ∪ Γ (S,TF,PF) ⊢e (S∪{b},TF ⊕PF) α(ρ). ✷ that α(E) ∪ Γ (S,TF,PF) |= ALG (S∪{b},TF ⊕PF) Example 5.2 Let us illustrate how Cor. 5.1 may be used to perfom quasi-existence ... |

10 |
Răzvan Diaconescu, and Andrzej Tarlecki, What is a logic translation
- Mossakowski
(Show Context)
Citation Context ...azvan.Diaconescu@imar.ro (Răzvan Diaconescu). called institution theory [7] with its array of model theory oriented category theoretic concepts supporting the systematic studies of logic translations =-=[13]-=- at a higher mathematical level. A characteristic aspect of the institution theoretic translations is that due to their semantic nature they are rather fine grained, in the sense that they involve the... |

7 |
Petria and Răzvan Diaconescu. Abstract Beth definability in institutions
- Marius
- 2006
(Show Context)
Citation Context ...ost prominent such translation [12] encodes partial operations as total operations but employs also relations for expressing the domains of definition of the partial operations. Another important one =-=[12,14]-=- encodes the partial operations as relation symbols. Both of them can be found also in the recent monography [5]. Here we propose a novel translation that, unlike the previous ones, has a pure algebra... |

4 |
Birkhoff completeness in institutions
- Codescu, Găină
(Show Context)
Citation Context ...within the proof of Cor. 4.2, namely that each quasi-existence equation gets mapped by α to a sentence equivalent to a conditional equation, allows the application of the Birkhoff completeness result =-=[2,8,4,5]-=- giving α(ρ) if and only if α(E) ∪ Γ (S,TF,PF) ⊢e (S∪{b},TF ⊕PF) α(ρ). ✷ that α(E) ∪ Γ (S,TF,PF) |= ALG (S∪{b},TF ⊕PF) Example 5.2 Let us illustrate how Cor. 5.1 may be used to perfom quasi-existence ... |