## Mathematical models of computational and combinatorial structures. Invited address for Foundations (2005)

Venue: | of Software Science and Computation Structures (FOSSACS 2005 |

Citations: | 12 - 6 self |

### BibTeX

@INPROCEEDINGS{Fiore05mathematicalmodels,

author = {Marcelo P. Fiore},

title = {Mathematical models of computational and combinatorial structures. Invited address for Foundations},

booktitle = {of Software Science and Computation Structures (FOSSACS 2005},

year = {2005},

pages = {25--46},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

Abstract. The general aim of this talk is to advocate a combinatorial perspective, together with its methods, in the investigation and study of models of computation structures. This, of course, should be taken in conjunction with the wellestablished views and methods stemming from algebra, category theory, domain theory, logic, type theory, etc. In support of this proposal I will show how such an approach leads to interesting connections between various areas of computer science and mathematics; concentrating on one such example in some detail. Specifically, I will consider the line of my research involving denotational models of the pi calculus and algebraic theories with variable-binding operators, indicating how the abstract mathematical structure underlying these models fits with that of Joyal’s combinatorial species of structures. This analysis suggests both the unification and generalisation of models, and in the latter vein I will introduce generalised species of structures and their calculus. These generalised species encompass and generalise various of the notions of species used in combinatorics. Furthermore, they have a rich mathematical structure (akin to models of Girard’s linear logic) that can be described purely within Lawvere’s generalised logic. Indeed, I will present and treat the cartesian closed structure, the linear structure, the differential structure, etc. of generalised species axiomatically in this mathematical framework. As an upshot, I will observe that the setting allows for interpretations of computational calculi (like the lambda calculus, both typed and untyped; the recently introduced differential lambda calculus of Ehrhard and Regnier; etc.) that can be directly seen as translations into a more basic elementary calculus of interacting agents that compute by communicating and operating upon structured data.

### Citations

1071 | A calculus of mobile processes
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- 1992
(Show Context)
Citation Context ...ame) generator n; it can be explicitly described as the category of finite cardinals and injections (with tensor product given by addition, initial unit by 1 Readers not familiar with the pi calculus =-=[38]-=- can safely skip Subsect. 1.1, or read it after Subsect. 1.3. Readers only interested in the combinatorial aspects can safely restrict their attention to Subsect. 1.2.sthe empty set, and generator by ... |

1005 | Categories for the Working Mathematician - Lane - 1971 |

313 | Higher-order abstract syntax - Pfenning, Elliott - 1988 |

231 | A new approach to abstract syntax with variable binding
- Gabbay, Pitts
- 2002
(Show Context)
Citation Context ... an interesting place. Indeed, it has been used both for giving denotational models of dynamically generated names [46, 47] and for modelling and reasoning about abstract syntax with variable binding =-=[20]-=-. Further, it is closely related to the category of species Set B [11, 35], which in turn has also been considered as a model of abstract syntax with linear variablebinding [49]. These models are by n... |

201 |
Sheaves in Geometry and Logic. A First Introduction to Topos Theory
- Lane, Moerdijk
- 1992
(Show Context)
Citation Context ...hed upon in Sect. 1 and their applications should not be considered in isolation for they are closely related. In this respect, there is a submodel of Set I , the so-called Schanuel topos (see, e.g., =-=[32, 24]-=-), that occupies an interesting place. Indeed, it has been used both for giving denotational models of dynamically generated names [46, 47] and for modelling and reasoning about abstract syntax with v... |

190 |
Combinatorial Species and Tree-like Structures, Encyclopedia of Mathematics and its
- Bergeron, Labelle, et al.
- 1997
(Show Context)
Citation Context ... An equational proof of the beta isomorphism � 〈������� ◦ ��������� ◦ 〉 � ����� ⊓ � � �� � . � (1) Definition of composition. (2) Definition of evaluation. (3) Density formula. (4) Law of extensions. =-=(5)-=- Law of extensions and definition of pairing. (6) Law of extensions and definition of projection. (7) Density formula and definition of composition. (8) Law of extensions and definitions of projection... |

179 |
Sketches of an elephant: a topos theory compendium
- Johnstone
- 2002
(Show Context)
Citation Context ...hed upon in Sect. 1 and their applications should not be considered in isolation for they are closely related. In this respect, there is a submodel of Set I , the so-called Schanuel topos (see, e.g., =-=[32, 24]-=-), that occupies an interesting place. Indeed, it has been used both for giving denotational models of dynamically generated names [46, 47] and for modelling and reasoning about abstract syntax with v... |

175 | An abstract view of programming languages - Moggi - 1989 |

168 |
The essence of Algol
- Reynolds
- 1981
(Show Context)
Citation Context ...ic generation of names required the traditional denotational models to be indexed (or parameterised) by the set of the known names of a process. Naturally, and in the vein of previous models of store =-=[45, 42]-=- (see also [39, § 4.1.4]), this was formalised by considering models in functor categories; that is, mathematical universes S V of V-variable S-objects (functors V �� S) and V-variable S-maps (natural... |

161 |
Une théorie combinatoire des séries formelles
- Joyal
- 1981
(Show Context)
Citation Context ...in the chronological order in which I got familiar with them during my research. These are: denotational models of the pi calculus [17, 19] (Subsect. 1.1), Joyal’s combinatorial species of structures =-=[25, 26]-=- (Subsect. 1.2), and algebraic models ofsequational theories with variable-binding operators [18, 19, 13] (Subsect. 1.3). 1 My intention here is not to treat any of them in full detail, but rather to ... |

155 |
Abstract syntax and variable binding
- Fiore, Plotkin, et al.
- 1999
(Show Context)
Citation Context ...nal models of the pi calculus [17, 19] (Subsect. 1.1), Joyal’s combinatorial species of structures [25, 26] (Subsect. 1.2), and algebraic models ofsequational theories with variable-binding operators =-=[18, 19, 13]-=- (Subsect. 1.3). 1 My intention here is not to treat any of them in full detail, but rather to give an outline of the most relevant structures present in each model in such a way as to make explicit a... |

117 |
Metric spaces, generalized logic and closed categories
- Lawvere
(Show Context)
Citation Context ...ures. On the other hand, however, I depart from the traditional combinatorial treatment in that the calculus is axiomatically built on top of the mathematical framework of Lawvere’s generalised logic =-=[28]-=- (see Fig. 1 in Subsect. 2.7 for an example). This yields new algebraic proofs, even for the restriction of generalised species to their basic form recalled in Subsect. 1.2. In passing, I will remark ... |

103 |
On closed categories of functors
- Day
- 1970
(Show Context)
Citation Context ...ttention to Subsect. 1.2.sthe empty set, and generator by the singleton). Through the Yoneda embedding, the generator provides an object of names N = y(n) in SetI[n] and, by Day’s tensor construction =-=[8, 23]-=-, the symmetric tensor product provides a (multiplication) symmetric tensor product �⊕ on SetI[n] given by P �⊕Q = �U1,U2∈I[n] � I[n] P(U1) × Q(U2) × I[n] (U1 ⊕ U2, ) P, Q ∈ Set � with y(O) as unit. (... |

84 |
Foncteurs analytiques et espèces de structures
- Joyal
- 1986
(Show Context)
Citation Context ...velopment of the differential calculus. Further, the setting also provides graph-like models of the lambda calculus, fixed-point operators, etc. As it is the case for the basic notion of species (see =-=[26]-=-), generalised species of structures can be equivalently seen as generalised analytic functors (of which generalised species are the coefficients) between categories of variable sets. From this point ... |

65 |
Frege structures and the notions of proposition, truth and set
- Aczel
- 1980
(Show Context)
Citation Context ...rs. These general remarks will become clear after reading the rest of the section, where the approach is exemplified. The original approach of [18] was conceived for the framework of binding algebras =-=[1]-=- (see also [50]), where term judgements are subject to the admissible rules of weakening, contraction, and permutation. Thus, the natural notion of morphism between contexts is that provided by any re... |

56 |
Names and Higher-Order Functions
- Stark
- 1994
(Show Context)
Citation Context ...submodel of Set I , the so-called Schanuel topos (see, e.g., [32, 24]), that occupies an interesting place. Indeed, it has been used both for giving denotational models of dynamically generated names =-=[46, 47]-=- and for modelling and reasoning about abstract syntax with variable binding [20]. Further, it is closely related to the category of species Set B [11, 35], which in turn has also been considered as a... |

54 | From finite sets to Feynman diagrams
- Baez, Dolan
- 2001
(Show Context)
Citation Context ...introduce the operators of creation and annihilation of particles in the quantum systems that these spaces model and establish their commutation laws. In this line of thought and further motivated by =-=[6, 3]-=-, I was considering Feynman diagrams in the context of generalised species when a computational interpretation of my previous calculations became apparent. The outcome of these investigations will be ... |

50 | D.: Semantics of name and value passing - Fiore, Turi - 2001 |

47 |
Sets for mathematics
- Lawvere, Rosebrugh
- 2003
(Show Context)
Citation Context ... (functors V �� S) and V-variable S-maps (natural transformations) between them. In this context, S provides a basic model of denotations; whilst V gives an appropriate model of variation (see, e.g., =-=[30]-=-). In the example at hand, S is a suitable category of domains (or simply the category of sets, if considering the finite pi calculus) and V is the category I of finite sets (or finite subsets of a fi... |

46 | The differential lambda-calculus
- Ehrhard, Regnier
(Show Context)
Citation Context ...ar maps. Interestingly, a certain commutation law between abstraction and linear application (used on differentiation) entails the beta rule of the differential lambda calculus of Ehrhard and Regnier =-=[10]-=- as an isomorphism. 3 Concluding Remarks and Research Perspectives I have drawn a line of investigation concerning models of computational and combinatorial structures. The general common theme of the... |

44 |
Type algebras, functor categories and block structure
- Oles
- 1985
(Show Context)
Citation Context ...ic generation of names required the traditional denotational models to be indexed (or parameterised) by the set of the known names of a process. Naturally, and in the vein of previous models of store =-=[45, 42]-=- (see also [39, § 4.1.4]), this was formalised by considering models in functor categories; that is, mathematical universes S V of V-variable S-objects (functors V �� S) and V-variable S-maps (natural... |

34 |
Isomorphisms of types: from *-calculus to in-formation retrieval and language design. Birkhauser
- Cosmo
- 1995
(Show Context)
Citation Context ... of the type theory (or programs of the programming language). Such a study has already been considered, though for entirely different reasons, under the broad heading of type isomorphism; see, e.g., =-=[9]-=-. Besides applications in computer science, one interest in this subject lies in its connections to various areas of mathematics. Indeed, it is related to Tarski’s high school algebra problem in mathe... |

22 |
Qualitative Distinctions between some Toposes of Generalized Graphs
- Lawvere
- 1989
(Show Context)
Citation Context ...or presheaves). Recall that there is a universal Yoneda embedding y : V � � �� V� given by y(v) = V( , v). For small categories V and W, we have the following important adjoint situations (see, e.g., =-=[29, 12]-=-) � � y V � �� � V� �� Lan F �� ∼= � F ���� # ����������� �� �� ⊣ �� �� �� F∗ �W � obtained by left Kan extension, where and � � y V f �� W � � Lan ∼= y f! �� �V �� ⊣ f ⊣ ∗ �� �� �� �W F # (P) = � v∈V... |

20 |
A universal property of the convolution monoidal structure
- Im, Kelly
- 1986
(Show Context)
Citation Context ...ttention to Subsect. 1.2.sthe empty set, and generator by the singleton). Through the Yoneda embedding, the generator provides an object of names N = y(n) in SetI[n] and, by Day’s tensor construction =-=[8, 23]-=-, the symmetric tensor product provides a (multiplication) symmetric tensor product �⊕ on SetI[n] given by P �⊕Q = �U1,U2∈I[n] � I[n] P(U1) × Q(U2) × I[n] (U1 ⊕ U2, ) P, Q ∈ Set � with y(O) as unit. (... |

19 |
Semantic analysis of normalisation by evaluation for typed lambda calculus
- Fiore
- 2002
(Show Context)
Citation Context ...nal models of the pi calculus [17, 19] (Subsect. 1.1), Joyal’s combinatorial species of structures [25, 26] (Subsect. 1.2), and algebraic models ofsequational theories with variable-binding operators =-=[18, 19, 13]-=- (Subsect. 1.3). 1 My intention here is not to treat any of them in full detail, but rather to give an outline of the most relevant structures present in each model in such a way as to make explicit a... |

15 |
Une combinatoire du pléthysme
- Bergeron
- 1987
(Show Context)
Citation Context ...!T �� Set, T, T ′ ∈ !T) for which p[idT ] = p and p[σ][τ] = p[σ · τ] for all p ∈ P(T) and σ : T �� T ′ , τ : T ′ �� ′′ T in !T. Examples of generalised species in combinatorics abound: permutationals =-=[25, 4]-=- are CP-species for CP the groupoid of finite cyclic permutations, partitionals [40] aresB ∗ -species for B ∗ the groupoid of non-empty finite sets. Further examples are coloured permutationals [34], ... |

14 |
open maps and bisimulation
- Cattani, Winskel, et al.
- 2005
(Show Context)
Citation Context ...tures is not yet in place. The analysis of Sect. 1 suggests both the unification and generalisation of models, and in the latter vein I motivated and introduced generalised species of structures; see =-=[2, 36, 7]-=- for relevant related work. These generalised species extend various of the notions of species used in combinatorics and also their respective calculi. Indeed, they come equipped with an (heterogeneou... |

14 | The role of Michael Batanin’s monoidal globular categories, in Higher Category Theory
- Street
- 1998
(Show Context)
Citation Context ... a structure q of type Q over the set U of n (place-holder) tokens for the structures pi (1 ≤ i ≤ n) in P(ui). Monoids for this composition tensor product are known as (symmetric) operads (see, e.g., =-=[48]-=-). An important part of the theory of species (on which I can only refer the reader to [25] here) is that they can be equivalently seen as analytic endofunctors on Set (of which species are the coeffi... |

12 | Remarks on isomorphisms in typed lambda calculi with empty and sum types
- Fiore, Cosmo, et al.
- 2002
(Show Context)
Citation Context ...ications in computer science, one interest in this subject lies in its connections to various areas of mathematics. Indeed, it is related to Tarski’s high school algebra problem in mathematical logic =-=[15]-=-, to the word problem in quotient polynomial semirings in computational algebra [16, 14], and to Thompson’s groups in group theory [forthcoming joint work with Tom Leinster]. The rest of the paper pro... |

12 |
On clubs and data-type constructors
- Kelly
- 1992
(Show Context)
Citation Context ...T) × P •T (U) (Q, P ∈ Set B[x] ) . This so-called composition (or substitution) operation ◦ on species yields a (highly nonsymmetric) monoidal closed structure on SetB[x] with unit X = y(x) (see also =-=[27]-=-). Translating it to SetB we obtain, whenever P(∅) = ∅, that (Q ◦ P)(U) ∼ = � � U∈Part(U) Q(U) × u∈U P(u) (Q, P ∈ SetB , U ∈ B) where the disjoint sum is taken over the set Part(U) of partitions of U.... |

11 | Isomorphisms of generic recursive polynomial types
- Fiore
- 2004
(Show Context)
Citation Context ...o various areas of mathematics. Indeed, it is related to Tarski’s high school algebra problem in mathematical logic [15], to the word problem in quotient polynomial semirings in computational algebra =-=[16, 14]-=-, and to Thompson’s groups in group theory [forthcoming joint work with Tom Leinster]. The rest of the paper provides another example of the fruitfulness of the approach advocated here. Specifically, ... |

9 |
A fully abstract domain model for the pi-calculus
- Stark
- 1996
(Show Context)
Citation Context ... explicit and apparent the similarities that run through them all. 1.1 Denotational Models of the Pi Calculus The main ingredient leading to the construction of denotational models of the pi calculus =-=[17, 47]-=- was recognising that its feature allowing for the dynamic generation of names required the traditional denotational models to be indexed (or parameterised) by the set of the known names of a process.... |

7 |
A fully-abstract model for the pi-calculus
- Fiore, Moggi, et al.
- 1996
(Show Context)
Citation Context ...in retrospective three mathematical models of computation structures in the chronological order in which I got familiar with them during my research. These are: denotational models of the pi calculus =-=[17, 19]-=- (Subsect. 1.1), Joyal’s combinatorial species of structures [25, 26] (Subsect. 1.2), and algebraic models ofsequational theories with variable-binding operators [18, 19, 13] (Subsect. 1.3). 1 My inte... |

7 |
Abstract syntax and variable binding for linear binders
- Tanaka
- 2000
(Show Context)
Citation Context ... with variable binding [20]. Further, it is closely related to the category of species Set B [11, 35], which in turn has also been considered as a model of abstract syntax with linear variablebinding =-=[49]-=-. These models are by no means the only relevant for applications, and a fully systematic theory providing, for instance, constructions of models of variation that are guaranteed to properly model spe... |

5 |
Colored species, c-monoids and plethysm
- Méndez, Nava
- 1993
(Show Context)
Citation Context ...25, 4] are CP-species for CP the groupoid of finite cyclic permutations, partitionals [40] aresB ∗ -species for B ∗ the groupoid of non-empty finite sets. Further examples are coloured permutationals =-=[34]-=-, and species on graphs and digraphs [33]. A fundamental property of the free symmetric (strict) monoidal completion is that it comes equipped with canonical natural coherent equivalences as shown bel... |

4 |
Species on digraphs
- Méndez
- 1996
(Show Context)
Citation Context ... of finite cyclic permutations, partitionals [40] aresB ∗ -species for B ∗ the groupoid of non-empty finite sets. Further examples are coloured permutationals [34], and species on graphs and digraphs =-=[33]-=-. A fundamental property of the free symmetric (strict) monoidal completion is that it comes equipped with canonical natural coherent equivalences as shown below. 1 O ∼= �� !0 , !C1 × !C2 ⊗ � �� !(C1 ... |

4 | Symmetric monoidal completions and the exponential principle among labeled combinatorial structures. Theory and applications of categories
- Menni
- 2003
(Show Context)
Citation Context ...tures is not yet in place. The analysis of Sect. 1 suggests both the unification and generalisation of models, and in the latter vein I motivated and introduced generalised species of structures; see =-=[2, 36, 7]-=- for relevant related work. These generalised species extend various of the notions of species used in combinatorics and also their respective calculi. Indeed, they come equipped with an (heterogeneou... |

4 |
The adjoints to the derivative functor on species
- Rajan
- 1993
(Show Context)
Citation Context ...Q on a set of tokens U is to decompose U in sets of tokens U1 and U2, and put a structure of type P on U1 and a structure of type Q on U2.sDifferentiation. We have the following situation (see, e.g., =-=[44]-=-) B[x] ◦ ⊕x �� B[x] ◦� � � � y Lan ∼= y ·X �� SetB[x] �� ⊣ d/dx ⊣ �� �� �� B[x] Set which yields a differentiation operator d/dx = ( ⊕ x) ∗ , arising as closed structure � since ( ⊕ x)! ∼ = · X for X ... |

3 |
Notes on combinatorial functors. Draft available electronically
- Fiore
- 2001
(Show Context)
Citation Context ...tional models of dynamically generated names [46, 47] and for modelling and reasoning about abstract syntax with variable binding [20]. Further, it is closely related to the category of species Set B =-=[11, 35]-=-, which in turn has also been considered as a model of abstract syntax with linear variablebinding [49]. These models are by no means the only relevant for applications, and a fully systematic theory ... |

3 |
categories and combinatorics
- Plethysm
- 1985
(Show Context)
Citation Context ...nd σ : T �� T ′ , τ : T ′ �� ′′ T in !T. Examples of generalised species in combinatorics abound: permutationals [25, 4] are CP-species for CP the groupoid of finite cyclic permutations, partitionals =-=[40]-=- aresB ∗ -species for B ∗ the groupoid of non-empty finite sets. Further examples are coloured permutationals [34], and species on graphs and digraphs [33]. A fundamental property of the free symmetri... |

2 | Semantical analysis in higher order abstract syntax - Hofmann - 1999 |

2 |
About I-quantifiers
- Menni
(Show Context)
Citation Context ...tional models of dynamically generated names [46, 47] and for modelling and reasoning about abstract syntax with variable binding [20]. Further, it is closely related to the category of species Set B =-=[11, 35]-=-, which in turn has also been considered as a model of abstract syntax with linear variablebinding [49]. These models are by no means the only relevant for applications, and a fully systematic theory ... |

2 |
What’s in a name
- Milner
- 2003
(Show Context)
Citation Context ...generic names into sets, and the condition on the unit being initial allows for the ability of generating new names. This intuitive view is consistent with that of Needham’s pure names [41] (see also =-=[37]-=-); viz., those that can only be tested for equality with other ones, and indeed one can also formally recast the category I in these terms. The fundamental mathematical structure of S I required for m... |

2 |
A Framework for Binding Operators
- Yong
- 1992
(Show Context)
Citation Context ...ral remarks will become clear after reading the rest of the section, where the approach is exemplified. The original approach of [18] was conceived for the framework of binding algebras [1] (see also =-=[50]-=-), where term judgements are subject to the admissible rules of weakening, contraction, and permutation. Thus, the natural notion of morphism between contexts is that provided by any renaming of varia... |

1 |
Higher-dimensional algebra III: � -categories and the algebra of opetopes
- Baez, Dolan
- 1998
(Show Context)
Citation Context ...tures is not yet in place. The analysis of Sect. 1 suggests both the unification and generalisation of models, and in the latter vein I motivated and introduced generalised species of structures; see =-=[2, 36, 7]-=- for relevant related work. These generalised species extend various of the notions of species used in combinatorics and also their respective calculi. Indeed, they come equipped with an (heterogeneou... |

1 |
Proof nets and Feynman diagrams. Available from the second author
- Blute, Panangaden
- 1998
(Show Context)
Citation Context ...introduce the operators of creation and annihilation of particles in the quantum systems that these spaces model and establish their commutation laws. In this line of thought and further motivated by =-=[6, 3]-=-, I was considering Feynman diagrams in the context of generalised species when a computational interpretation of my previous calculations became apparent. The outcome of these investigations will be ... |

1 |
Rough notes on presheaves. Manuscript available electronically
- Fiore
- 2001
(Show Context)
Citation Context ...or presheaves). Recall that there is a universal Yoneda embedding y : V � � �� V� given by y(v) = V( , v). For small categories V and W, we have the following important adjoint situations (see, e.g., =-=[29, 12]-=-) � � y V � �� � V� �� Lan F �� ∼= � F ���� # ����������� �� �� ⊣ �� �� �� F∗ �W � obtained by left Kan extension, where and � � y V f �� W � � Lan ∼= y f! �� �V �� ⊣ f ⊣ ∗ �� �� �� �W F # (P) = � v∈V... |

1 |
An objective representation of the Gaussian integers
- Fiore, Leinster
(Show Context)
Citation Context ...o various areas of mathematics. Indeed, it is related to Tarski’s high school algebra problem in mathematical logic [15], to the word problem in quotient polynomial semirings in computational algebra =-=[16, 14]-=-, and to Thompson’s groups in group theory [forthcoming joint work with Tom Leinster]. The rest of the paper provides another example of the fruitfulness of the approach advocated here. Specifically, ... |

1 |
Distributed Systems, chapter 12
- Needham
- 1993
(Show Context)
Citation Context ...der batches of generic names into sets, and the condition on the unit being initial allows for the ability of generating new names. This intuitive view is consistent with that of Needham’s pure names =-=[41]-=- (see also [37]); viz., those that can only be tested for equality with other ones, and indeed one can also formally recast the category I in these terms. The fundamental mathematical structure of S I... |