## Normal Forms for Partitions and Relations (1999)

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Venue: | Recent Trends in Algebraic Development Techniques, volume 1589 of Lect. Notes in Comp. Science |

Citations: | 14 - 11 self |

### BibTeX

@INPROCEEDINGS{Bruni99normalforms,

author = {Roberto Bruni and Fabio Gadducci and Ugo Montanari},

title = {Normal Forms for Partitions and Relations},

booktitle = {Recent Trends in Algebraic Development Techniques, volume 1589 of Lect. Notes in Comp. Science},

year = {1999},

pages = {31--47},

publisher = {Springer Verlag}

}

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### Abstract

Recently there has been a growing interest towards algebraic structures that are able to express formalisms different from the standard, tree-like presentation of terms. Many of these approaches reveal a specific interest towards their application in the "distributed and concurrent systems" field, but an exhaustive comparison between them is difficult because their presentations can be quite dissimilar. This work is a first step towards a unified view, which is able to recast all those formalisms into a more general one, where they can be easily compared. We introduce a general schema for describing a characteristic normal form for many algebraic formalisms, and show that those normal forms can be thought of as arrows of suitable concrete monoidal categories.

### Citations

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Citation Context ...ons of objects (corresponding to the swappings of wires in the wire and box presentation). Then, the models of the resulting symmetric theory of \Sigma are just suitable symmetric monoidal categories =-=[21]-=- (i.e., also equipped with the \Sigma -structure). Definition 3 (Symmetric Monoidal Categories). A monoidal category is a triple hC;\Omega ; ei, where C is the underlying category,\Omega : C \Theta C ... |

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Citation Context ...able processes for Petri nets in [10], and the description of a basic network algebra for data-flows in [3, 34]. 2.1 Enriching the Monoidal Structure The constructive definition of algebraic theories =-=[20]-=- as enriched monoidal categories dates back to the mid-Seventies [17, 27], even if it has received a new stream of attention in these days. In our opinion, it separates very nicely the auxiliary struc... |

150 | Elements of interaction
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Citation Context ...from the flownomial calculus of Stefanescu [6, 34], to the bicategories of processes of Walters [18, 19], to the pre-monoidal categories of Power and Robinson [28], to the action structures of Milner =-=[24]-=-, to the interaction categories of Abramsky [1], to the sharing graphs of Hasegawa [16] and to the gs-monoidal categories of Corradini and Gadducci [7, 8], just to mention a few (see also [9, 11, 15, ... |

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Citation Context ...4], to the bicategories of processes of Walters [18, 19], to the pre-monoidal categories of Power and Robinson [28], to the action structures of Milner [24], to the interaction categories of Abramsky =-=[1]-=-, to the sharing graphs of Hasegawa [16] and to the gs-monoidal categories of Corradini and Gadducci [7, 8], just to mention a few (see also [9, 11, 15, 29]). All these structures can be seen as enric... |

104 | Algebraic Semantics of Imperative Programs - Goguen, Malcolm - 1996 |

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Citation Context ...e-like presentation of terms. They range from the flownomial calculus of Stefanescu [6, 34], to the bicategories of processes of Walters [18, 19], to the pre-monoidal categories of Power and Robinson =-=[28]-=-, to the action structures of Milner [24], to the interaction categories of Abramsky [1], to the sharing graphs of Hasegawa [16] and to the gs-monoidal categories of Corradini and Gadducci [7, 8], jus... |

71 | An introduction to term graph rewriting
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Citation Context ...ms of a wire and box diagram. We propose a schema for describing normal forms for this kind of structures, generalizing the one in [12] (and that bears some similarity to the equational term graph of =-=[2]-=-), thus obtaining a universal framework where each structure finds its unique standard representation. We describe distributed spaces as sets of assignments over sets of variables, distinguishing betw... |

65 | The tile model
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Citation Context ... Milner [24], to the interaction categories of Abramsky [1], to the sharing graphs of Hasegawa [16] and to the gs-monoidal categories of Corradini and Gadducci [7, 8], just to mention a few (see also =-=[9, 11, 15, 29]-=-). All these structures can be seen as enrichments of symmetric monoidal categories, which give the basis for the description of a distributed environment in terms of a wire and box diagram. We propos... |

55 | Contextual nets
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Citation Context ... F op is so. We denote by MShCat the category of match-share functors. Match-share categories have been introduced in [14], and used to embed the algebraic properties of processes for contextual nets =-=[25]-=-. They are the basis for a class of categories where suitable models of partition-based structures can live. Definition 7 (Part-Monoidal Categories). A part-monoidal category is an 8-tuple hC;\Omega ;... |

55 |
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Citation Context ... Milner [24], to the interaction categories of Abramsky [1], to the sharing graphs of Hasegawa [16] and to the gs-monoidal categories of Corradini and Gadducci [7, 8], just to mention a few (see also =-=[9, 11, 15, 29]-=-). All these structures can be seen as enrichments of symmetric monoidal categories, which give the basis for the description of a distributed environment in terms of a wire and box diagram. We propos... |

45 | Recursion from cyclic sharing: traced monoidal categories and models of cyclic lambda-calculi
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Citation Context ... Walters [18, 19], to the pre-monoidal categories of Power and Robinson [28], to the action structures of Milner [24], to the interaction categories of Abramsky [1], to the sharing graphs of Hasegawa =-=[16]-=- and to the gs-monoidal categories of Corradini and Gadducci [7, 8], just to mention a few (see also [9, 11, 15, 29]). All these structures can be seen as enrichments of symmetric monoidal categories,... |

42 | Bicategories of processes
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Citation Context ...osed, for expressing formalisms different from the ordinary tree-like presentation of terms. They range from the flownomial calculus of Stefanescu [6, 34], to the bicategories of processes of Walters =-=[18, 19]-=-, to the pre-monoidal categories of Power and Robinson [28], to the action structures of Milner [24], to the interaction categories of Abramsky [1], to the sharing graphs of Hasegawa [16] and to the g... |

39 |
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Citation Context ...enote by SMCat the category of symmetric functors. Among the uses of symmetric monoidal categories as a semantics framework, we recall the characterization of concatenable processes for Petri nets in =-=[10]-=-, and the description of a basic network algebra for data-flows in [3, 34]. 2.1 Enriching the Monoidal Structure The constructive definition of algebraic theories [20] as enriched monoidal categories ... |

37 | F.: An algebraic presentation of term graphs, via gsmonoidal categories
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(Show Context)
Citation Context ... the axiomatization of algebraic theories: Terms and term graphs differ for two axioms, representing, in a categorical setting, the naturality of transformations for copying and discharging arguments =-=[8]-=-. Many other mathematical structures have been proposed, for expressing formalisms different from the ordinary tree-like presentation of terms. They range from the flownomial calculus of Stefanescu [6... |

36 | Tile Logic for Synchronized Rewriting of Concurrent Systems - Bruni - 1999 |

34 | A 2-categorical presentation of term graph rewriting
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(Show Context)
Citation Context ...binson [28], to the action structures of Milner [24], to the interaction categories of Abramsky [1], to the sharing graphs of Hasegawa [16] and to the gs-monoidal categories of Corradini and Gadducci =-=[7, 8]-=-, just to mention a few (see also [9, 11, 15, 29]). All these structures can be seen as enrichments of symmetric monoidal categories, which give the basis for the description of a distributed environm... |

33 | Process and term tile logic - Bruni, Meseguer, et al. - 1998 |

32 | Mapping tile logic into rewriting logic - Meseguer, Montanari - 1998 |

30 | An inductive view of graph transformation
- Gadducci, Heckel
- 1997
(Show Context)
Citation Context ... possible, where only some of the operators are considered, and their mixed compositions are differently axiomatized, ranging from the match-share categories of [14] to the dgs-monoidal categories of =-=[13, 18]-=-. Here we just sketch a survey of the categorical framework, and briefly comment their role in the literature and the main differences between similar models. We start with what we call a r-monoidal c... |

28 |
Span(Graph): A categorical algebra of transition systems
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- 1997
(Show Context)
Citation Context ...osed, for expressing formalisms different from the ordinary tree-like presentation of terms. They range from the flownomial calculus of Stefanescu [6, 34], to the bicategories of processes of Walters =-=[18, 19]-=-, to the pre-monoidal categories of Power and Robinson [28], to the action structures of Milner [24], to the interaction categories of Abramsky [1], to the sharing graphs of Hasegawa [16] and to the g... |

22 |
Towards a new algebraic foundation of flowchart scheme theory
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(Show Context)
Citation Context ...8]. Many other mathematical structures have been proposed, for expressing formalisms different from the ordinary tree-like presentation of terms. They range from the flownomial calculus of Stefanescu =-=[6, 34]-=-, to the bicategories of processes of Walters [18, 19], to the pre-monoidal categories of Power and Robinson [28], to the action structures of Milner [24], to the interaction categories of Abramsky [1... |

19 | An axiomatization of the algebra of Petri net concatenable processes. Theoret
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(Show Context)
Citation Context ...! 0 and applying the definition of parallel and sequential composition. The initiality result relies on previous characterization results for symmetric monoidal categories as suitable Petri processes =-=[31, 30]-=-, to which our spaces are equivalent. 2 4.2 GS-Monoidal As illustrated in Section 2, gs-monoidal categories are symmetric monoidal categories enriched with suitable transformations for copying and dis... |

18 | A bi-categorical axiomatisation of concurrent graph rewriting - Gadducci, Heckel, et al. |

17 | Graph rewriting, constraint solving and tiles for coordinating distributed systems. applied categorical structures
- Montanari, Rossi
- 1999
(Show Context)
Citation Context ...ccount of term graph rewriting. It is in this setting that we recover the intuitive interpretation of copying and discharging as suitable operations over graphical structures. Also the open graphs of =-=[26]-=- form a free gs-monoidal category: The one generated by the one-sorted signature \Sigma such that \Sigma h;k = ; if k 6= 0, and \Sigma h;0 = L h for h 2 lIN, where L = fL h g h2lIN is the set of label... |

15 |
Network algebra for synchronous and asynchronous data ow
- Stefanescu
- 1994
(Show Context)
Citation Context ...mmetric monoidal categories as a semantics framework, we recall the characterization of concatenable processes for Petri nets in [10], and the description of a basic network algebra for data-flows in =-=[3, 34]-=-. 2.1 Enriching the Monoidal Structure The constructive definition of algebraic theories [20] as enriched monoidal categories dates back to the mid-Seventies [17, 27], even if it has received a new st... |

14 |
Tiles for Concurrent and Located Calculi
- Ferrari, Montanari
- 1997
(Show Context)
Citation Context ...ive the basis for the description of a distributed environment in terms of a wire and box diagram. We propose a schema for describing normal forms for this kind of structures, generalizing the one in =-=[12]-=- (and that bears some similarity to the equational term graph of [2]), thus obtaining a universal framework where each structure finds its unique standard representation. We describe distributed space... |

14 | Axioms for contextual net processes
- Gadducci, Montanari
(Show Context)
Citation Context ... of data. Several combinations are then possible, where only some of the operators are considered, and their mixed compositions are differently axiomatized, ranging from the match-share categories of =-=[14]-=- to the dgs-monoidal categories of [13, 18]. Here we just sketch a survey of the categorical framework, and briefly comment their role in the literature and the main differences between similar models... |

11 |
Classes of finite relations as initial abstract data types
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- 1994
(Show Context)
Citation Context ...heir role in the literature and the main differences between similar models. We start with what we call a r-monoidal category: One of the various extensions, albeit with a different name, proposed in =-=[5, 33, 34]-=-. Definition 5 (R-Monoidal Categories). A r-monoidal category is an 8-tuple hC;\Omega ; e; fl; r; !; \Delta; yi such that hC;\Omega ; e; fl; r; !i and hC op ;\Omega op ; e; fl op ; \Delta op ; y op i ... |

11 |
On flowchart theories: Part II. The nondeterministic case
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Citation Context ...heir role in the literature and the main differences between similar models. We start with what we call a r-monoidal category: One of the various extensions, albeit with a different name, proposed in =-=[5, 33, 34]-=-. Definition 5 (R-Monoidal Categories). A r-monoidal category is an 8-tuple hC;\Omega ; e; fl; r; !; \Delta; yi such that hC;\Omega ; e; fl; r; !i and hC op ;\Omega op ; e; fl op ; \Delta op ; y op i ... |

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11 | Representation theorems for Petri nets - Meseguer, Montanari, et al. - 1997 |

8 |
On the Semantics of Petri Nets: Processes, Unfolding and Infinite Computations
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Citation Context ...! 0 and applying the definition of parallel and sequential composition. The initiality result relies on previous characterization results for symmetric monoidal categories as suitable Petri processes =-=[31, 30]-=-, to which our spaces are equivalent. 2 4.2 GS-Monoidal As illustrated in Section 2, gs-monoidal categories are symmetric monoidal categories enriched with suitable transformations for copying and dis... |

8 |
Algebra of flownomials
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(Show Context)
Citation Context ...8]. Many other mathematical structures have been proposed, for expressing formalisms different from the ordinary tree-like presentation of terms. They range from the flownomial calculus of Stefanescu =-=[6, 34]-=-, to the bicategories of processes of Walters [18, 19], to the pre-monoidal categories of Power and Robinson [28], to the action structures of Milner [24], to the interaction categories of Abramsky [1... |

7 | On partial algebras - Hoenke - 1977 |

6 | Functorial semantics for multi-algebras - Corradini, Gadducci - 1999 |

5 |
On partial recursive definitions and programs
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(Show Context)
Citation Context ... network algebra for data-flows in [3, 34]. 2.1 Enriching the Monoidal Structure The constructive definition of algebraic theories [20] as enriched monoidal categories dates back to the mid-Seventies =-=[17, 27]-=-, even if it has received a new stream of attention in these days. In our opinion, it separates very nicely the auxiliary structure from the \Sigma -structure (better than the ordinary description inv... |

4 |
Universal algebra in s-monoidal categories
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(Show Context)
Citation Context ... network algebra for data-flows in [3, 34]. 2.1 Enriching the Monoidal Structure The constructive definition of algebraic theories [20] as enriched monoidal categories dates back to the mid-Seventies =-=[17, 27]-=-, even if it has received a new stream of attention in these days. In our opinion, it separates very nicely the auxiliary structure from the \Sigma -structure (better than the ordinary description inv... |

2 |
A tile-based coordination view of the asynchronous -calculus
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(Show Context)
Citation Context ... Milner [24], to the interaction categories of Abramsky [1], to the sharing graphs of Hasegawa [16] and to the gs-monoidal categories of Corradini and Gadducci [7, 8], just to mention a few (see also =-=[9, 11, 15, 29]-=-). All these structures can be seen as enrichments of symmetric monoidal categories, which give the basis for the description of a distributed environment in terms of a wire and box diagram. We propos... |

2 | Maude: Specification and programming in rewriting logic. Available at http://maude.csl.sri.com/manual - Clavel, Dur'an, et al. - 1999 |

1 |
Functorial semantics for multi-algebras. Presented at WADT'98
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Citation Context |