## The Tile Model (1996)

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Venue: | PROOF, LANGUAGE AND INTERACTION: ESSAYS IN HONOUR OF ROBIN MILNER |

Citations: | 66 - 24 self |

### BibTeX

@INPROCEEDINGS{Gadducci96thetile,

author = {Fabio Gadducci and Ugo Montanari},

title = {The Tile Model},

booktitle = {PROOF, LANGUAGE AND INTERACTION: ESSAYS IN HONOUR OF ROBIN MILNER},

year = {1996},

pages = {133--166},

publisher = {MIT Press}

}

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### Abstract

In this paper we introduce a model for a wide class of computational systems, whose behaviour can be described by certain rewriting rules. We gathered our inspiration both from the world of term rewriting, in particular from the rewriting logic framework [Mes92], and of concurrency theory: among the others, the structured operational semantics [Plo81], the context systems [LX90] and the structured transition systems [CM92] approaches. Our model recollects many properties of these sources: first, it provides a compositional way to describe both the states and the sequences of transitions performed by a given system, stressing their distributed nature. Second, a suitable notion of typed proof allows to take into account also those formalisms relying on the notions of synchronization and side-effects to determine the actual behaviour of a system. Finally, an equivalence relation over sequences of transitions is defined, equipping the system under analysis with a concurrent semantics, ...

### Citations

3643 | Communicating Sequential Processes
- Hoare
- 1985
(Show Context)
Citation Context ...concurrency theory it is quite common to deal with formalisms relying on the notions of side-effect and synchronisation in determining the actual behaviour of a system. Process (Description) Algebras =-=[4, 19, 35]-=- offer a constructive way to describe concurrent systems, considered as structured entities (the agents) interacting by means of some synchronisation mechanism. A system is then a term of an algebra o... |

3430 |
Communication and Concurrency
- Milner
- 1989
(Show Context)
Citation Context ...A Case Study on Process Algebras In this section we show how to recast process algebras and their well-known operational semantics by suitable rewriting systems. In particular, we first introduce ccs =-=[35]-=-, maybe the best known example of these formalisms; and then we define an algebraic rewriting system R ccs which faithfully corresponds to the ccs transition system. Furthermore, we show that, when ap... |

1373 | A structural approach to operational semantics
- Plotkin
- 1981
(Show Context)
Citation Context ... of synchronisation and side-effects in determining the actual behaviour of a given system. We consider as a major breakthrough the introduction of the structural operational semantics (sos) approach =-=[40]-=-: States are terms of a free algebra, whose operators express basic features of a system, and the transition relation is defined by means of inference rules, guided by the structure of the states. Som... |

986 |
Categories for the Working Mathematician
- Lane
- 1971
(Show Context)
Citation Context ...aph theory. Thanks to the functoriality axiom, it is easy to show that each term can be decomposed in a tree-like fashion: A monoidal theory is just an example of a so-called strict monoidal category =-=[32]-=-, for which many representation results are well-known. Now we introduce the more expressive kind of theories we deal with in our paper, gs-monoidal theories. Definition 4 (gs-monoidal theories). Give... |

700 |
Concurrency and automata on infinite sequences
- Park
- 1981
(Show Context)
Citation Context ...g those five-tuples with the same border. Restricting our attention to a transition system allows us to define a suitable notion of behavioural equivalence by the well-known technique of bisimulation =-=[38]-=-. Definition 10 (tile bisimulations). Let R = h\Sigma oe ; \Sigmas; N;Ri be an ars. An equivalence relation j b ` A(\Sigma oe )\ThetaA(\Sigma oe ) is a tile bisimulation for R if, whenever s j b t for... |

604 |
Petri Nets: an Introduction
- REISIG
- 1985
(Show Context)
Citation Context ... we consider formalisms with a compositional structure over states, and the transition relation needs to be inductively defined according to that structure. This is the case for example of Petri nets =-=[42]-=-, where a state is a multi-set of atomic components, and disjoint subsets may evolve simultaneously (i.e., in parallel); of term rewriting systems [26], where states are terms of an algebra, and rewri... |

578 | Term Rewriting Systems
- Klop
- 1992
(Show Context)
Citation Context ...ure. This is the case of formalisms like Petri Nets [Rei85], where a state is a multiset of basic components, and each of them may evolve simultaneously (i.e., in parallel); or term rewriting systems =-=[Klo92]-=-, where states are terms of a free algebra, and rewrites are freely obtained from a basic set of deduction rules. Furthermore, we may need to consider formalisms relying on the use of synchronization ... |

510 |
Conditional Rewriting Logic as a Unified Model of Concurrency
- Meseguer
- 1992
(Show Context)
Citation Context ...of) transitions that identify "computationally equivalent" behaviours. To equip transitions (actually, rewriting steps) with an algebraic structure is also a key point of the rewriting logic=-= approach [33]-=-: A system is considered ? Research partially supported by MURST Project Tecniche Formali per Sistemi Software, by CNR Integrated Project Metodi per Sistemi Connessi mediante Reti, and by ESPRIT Worki... |

419 | Category Theory for Computer Science - Barr, Wells - 1990 |

393 | Basic concepts of enriched category theory - Kelly - 2005 |

384 | Process algebra for synchronous communication - Bergstra, Klop - 1984 |

194 | Initial algebra semantics and continuous algebras - Goguen, Thatcher, et al. - 1977 |

185 |
Functorial Semantics of Algebraic Theories
- Lawvere
- 1963
(Show Context)
Citation Context ...s (related papers range from [20, 39] to the more recent [22, 28]). Then, our notion of algebraic theory can be proved equivalent to the classical definition, dating back to the early work of Lawvere =-=[27, 31]-=-: Hence, a classical result states the equivalence of these theories with the usual term algebras. Proposition 1 (algebraic theories and term algebras). Let \Sigma be a signature. Then for all n; m 2 ... |

172 |
Formal Verification of Parallel Programs
- Keller
- 1976
(Show Context)
Citation Context ...as been an unprecedented flow of proposals, aiming at methodologies to describe the semantics of rule-based computational systems. Widely spread in the field of concurrency theory, transition systems =-=[Kel76]-=- offered a useful tool for recovering suitable descriptions. They are roughly defined as a set of states, representing e.g. the possible memory contents, and a transition relation over states, where e... |

171 |
Petri nets are monoids
- Meseguer, Montanari
- 1990
(Show Context)
Citation Context ...r expressing basic properties of computing devices has been shown by a large amount of work in recent years. In this vein, we could consider the categorical description of Petri nets [42] proposed in =-=[34]-=- as one of the starting point for our work, and gs-monoidal categories [7] as a further enrichment, in order to deal with specific problems involving duplication and erasing of data. Definition 19 (gs... |

153 | Elements of interaction
- Milner
- 1993
(Show Context)
Citation Context ...lationship between so-called sharing graphs structures (as introduced in [18]) and gs-monoidal theories (see Section 2), a framework used to accommodate the behaviour of e.g. Milner's action calculus =-=[36]-=-. As far as categorical models are concerned, 2-categories [25] are the best-known cat-enriched structure in computer science: They are categories such that each hom-set (the set of arrows with same s... |

136 |
Review on the elements of 2-categories
- Kelly, Street
- 1974
(Show Context)
Citation Context ...oduced in [18]) and gs-monoidal theories (see Section 2), a framework used to accommodate the behaviour of e.g. Milner's action calculus [36]. As far as categorical models are concerned, 2-categories =-=[25]-=- are the best-known cat-enriched structure in computer science: They are categories such that each hom-set (the set of arrows with same source and target) is the object-set of another category, whose ... |

91 | Turning SOS rules into equations
- Aceto, Bloom, et al.
- 1994
(Show Context)
Citation Context ..., f; g 2 \Sigma P and a ij ; a 2 ActP . Moreover, all the process variables P i 's, Q jk 's are different, and R j 2 fP 1 : : : Pn ; Q 11 : : : Qnmn g. In the so-called positive gsos format (see e.g. =-=[ABV94]-=-) rules have the form P i a ij \Gamma! Q ij for i = 1 : : : n; j = 1 : : : m i f(P 1 ; : : : ; Pn ) a \Gamma! D[R 1 : : : R k ] where D is a process context in which each process variable can appear a... |

85 |
Computations in orthogonal rewriting systems
- Huet, Levy
- 1991
(Show Context)
Citation Context ... system: Each equivalence class of sequents should intuitively describe the same set of causally unrelated computations. This is not so different in spirit from the well-known permutation equivalence =-=[5, 21]-=-, and there exists in fact a tight correspondence between the two notions [29]. For a few initial considerations about the actual degree of concurrency expressed by the axioms, we refer to [8]. Exampl... |

74 |
Compositionality through an operational semantics of contexts
- Larsen, Xinxin
- 1990
(Show Context)
Citation Context ...nd the transition relation is defined by means of inference rules, guided by the structure of the states. Some of the extensions of this approach that proved fruitful for our view are context systemss=-=[30]-=-, where the transition relation is defined on open terms (that is, terms that may contain free variables), describing partially unspecified component of a system; and structured transition systems [9,... |

69 |
Semantics of weakening and contraction
- Jacobs
- 1992
(Show Context)
Citation Context ...sed into a symmetric monoidal category, together with a family of suitable natural transformations, usually denoted as diagonals and projections (related papers range from [20, 39] to the more recent =-=[22, 28]-=-). Then, our notion of algebraic theory can be proved equivalent to the classical definition, dating back to the early work of Lawvere [27, 31]: Hence, a classical result states the equivalence of the... |

48 | Computational semantics of term rewriting systems - Boudol - 1985 |

43 | Bicategories of processes
- Katis, Sabadini, et al.
- 1997
(Show Context)
Citation Context ...relevance of the computational interpretation of (the different presentations of) gs-monoidal theories, has not received enough attention in the literature, despite a few notable exceptions (see e.g. =-=[10, 23, 28]-=-). In fact, the main point for our discussion is that, although their definition is more involved than the classical, set-theoretical ones, algebraic (and a fortiori gs-monoidal) theories allow a desc... |

40 | Enhanced operational semantics - Degano, Priami - 1996 |

36 | A 2-categorical presentation of term graph rewriting
- Corradini, Gadducci
- 1997
(Show Context)
Citation Context ...ctory chapter of [44]): Each arrow n ! m identifies a term graph t over \Sigma with a specified m-tuple of roots and a specified n-tuple of variables nodes, and arrow composition is graph replacement =-=[7]-=-. This correspondence motivates the acronym, where gs stands for graph substitution. Example 1 (terms and theories). Let us consider the signature \Sigma ffl = S 2 i=0 \Sigma i , where \Sigma 0 = fa; ... |

36 |
First-order categorical logic
- Makkai, Reyes
- 1977
(Show Context)
Citation Context ...s (related papers range from [20, 39] to the more recent [22, 28]). Then, our notion of algebraic theory can be proved equivalent to the classical definition, dating back to the early work of Lawvere =-=[27, 31]-=-: Hence, a classical result states the equivalence of these theories with the usual term algebras. Proposition 1 (algebraic theories and term algebras). Let \Sigma be a signature. Then for all n; m 2 ... |

35 |
An algebraic semantics for structured transition systems and its application to logic programs
- Corradini, Montanari
- 1992
(Show Context)
Citation Context ...from the rewriting logic framework [Mes92], and of concurrency theory: among the others, the structured operational semantics [Plo81], the context systems [LX90] and the structured transition systemss=-=[CM92]-=- approaches. Our model recollects many properties of these sources: first, it provides a compositional way to describe both the states and the sequences of transitions performed by a given system, str... |

34 |
Categorical structures, in
- Street
- 1996
(Show Context)
Citation Context ...nctors. We want now to introduce a finitary structure, which is able to generate a double category with a free construction, starting from a given set of cells. We then present the notion of computad =-=[45, 46]-=-. Definition 23 (computads). A computad is a triple hC; D;Si, where C is a category, D is a graph (whose set of objects coincides with the set of objects of C) and S is a set of cells, each of which h... |

33 |
Flow models of distributed computations: Three equivalent semantics for CCS
- BOUDOL, CASTELLANI
- 1994
(Show Context)
Citation Context ...ts [Win82, GM84]; ii) or on the structure of the proofs of transitions 22 [Old87, DDM88, GM94]; iii) defining the concurrent computations by assigning a residue structure to the CCS transition system =-=[BC91]-=-; and iv) defining the concurrent computations by direct axiomatization [FM90]. However, all the proposed approaches are essentially equivalent (see e.g. [GM94]) in the sense that they yield the same ... |

24 | On the Algebraic Approach to Concurrent Term Rewriting
- Gadducci
- 1996
(Show Context)
Citation Context ... m; and the monoidality axiom id n\Omega m = id n\Omega id m for all n; m 2 IlN. Finally, we present the more expressive kind of theories we deal with in our paper, s-monoidal theories. Introduced in =-=[Gad96]-=-, their use as a suitable specification tool represents an original contribution of the paper. Definition 5 (S-Monoidal Theories). Given a hyper-signature \Sigma S , the associatedss-monoidal theory S... |

23 | Sesqui-pushout rewriting - Corradini, Heindel, et al. - 2006 |

23 |
Towards a new algebraic foundation of flowchart scheme theory
- Stefanescu
- 1990
(Show Context)
Citation Context ...relevance of the computational interpretation of (the different presentations of) gs-monoidal theories, has not received enough attention in the literature, despite a few notable exceptions (see e.g. =-=[10, 23, 28]-=-). In fact, the main point for our discussion is that, although their definition is more involved than the classical, set-theoretical ones, algebraic (and a fortiori gs-monoidal) theories allow a desc... |

22 |
Higher level synchronizing devices in MEIJE{SCCS
- Simone
- 1985
(Show Context)
Citation Context ...` f1 : : : ng. Moreover, all the P i 's, Q j 's are different, except for P k = Q k with k 62 I . Actually, algebraic rewriting systems allow for dealing also with process algebras in deSimone format =-=[12]-=-, i.e, such that all their rules have the form P i a i \Gamma! Q i for i 2 I f(P 1 ; : : : ; Pn ) a \Gamma! D[Q 1 : : : Qn ] for P r(a 1 ; : : : ; an ; a) where f 2 \Sigma P , I ` f1 : : : ng and all ... |

21 | Equational reasoning with 2-dimensional diagrams, Term rewriting (Font Romeux
- Lafont
- 1993
(Show Context)
Citation Context ...sed into a symmetric monoidal category, together with a family of suitable natural transformations, usually denoted as diagonals and projections (related papers range from [20, 39] to the more recent =-=[22, 28]-=-). Then, our notion of algebraic theory can be proved equivalent to the classical definition, dating back to the early work of Lawvere [27, 31]: Hence, a classical result states the equivalence of the... |

20 |
Foundations of equational deduction: A categorical treatment of equational proofs and unification algorithms
- Rydeheard, Stell
- 1987
(Show Context)
Citation Context ...e structure can be informally described as the superposition of a vertical and a horizontal category of cells. It is well-known that 2-categories represent a faithful model for term rewriting systems =-=[8, 41, 43]-=-: The arrows of the underlying category denote terms, and cells denote (equivalence classes of) rewrites. In the paper we show that a similar adequacy result holds for our tile model, if we consider d... |

19 |
Term graph rewriting: theory and practice
- Sleep, Plasmeijer
- 1993
(Show Context)
Citation Context ...the first author [Gad96] it is shown that for ordinary signatures a fundamental property of correspondence holds between s-monoidal theories and term graphs (as defined in the introductory chapter of =-=[EPS93]-=-): each arrow t \Sigma : n ! m identifies a term graph t over \Sigma with a specified m-tuple of roots and a specified n-tuple of variables nodes, and arrow composition is graph replacement. Definitio... |

19 | F.: Graph rewriting and constraint solving for modelling distributed systems with synchronization (extended abstract
- Montanari, Rossi
- 1996
(Show Context)
Citation Context ...lso higher-order formalisms, despite the simple term structure we chose for configurations. In fact, tiles have been used for coordination formalisms equipped with flexible synchronisation primitives =-=[6, 37]-=- and for calculi for mobile processes, like the asynchronous -calculus [16]. Tiles are also convenient for handling concurrent process calculi [17]. These results are then strengthened by the tight re... |

18 | Relating two categorical models of term rewriting
- Corradini, Gadducci, et al.
- 1995
(Show Context)
Citation Context ...e structure can be informally described as the superposition of a vertical and a horizontal category of cells. It is well-known that 2-categories represent a faithful model for term rewriting systems =-=[8, 41, 43]-=-: The arrows of the underlying category denote terms, and cells denote (equivalence classes of) rewrites. In the paper we show that a similar adequacy result holds for our tile model, if we consider d... |

18 |
General associativity and general composition for double categories
- Dawson, PareĢ
- 1993
(Show Context)
Citation Context ...ategory objects in Cat, that is, categories whose classes of objects and arrows actually form categories, and the source, target and identity functions are actually functors (see [3] for a survey, or =-=[11]-=- for some recent results on pasting). They represent an intuitive generalisation of the better known 2-categories, and admit the following naive presentation, adapted from [25]. Definition 20 (double ... |

16 |
An abstract formulation for rewrite systems
- Power
- 1989
(Show Context)
Citation Context ...e structure can be informally described as the superposition of a vertical and a horizontal category of cells. It is well-known that 2-categories represent a faithful model for term rewriting systems =-=[8, 41, 43]-=-: The arrows of the underlying category denote terms, and cells denote (equivalence classes of) rewrites. In the paper we show that a similar adequacy result holds for our tile model, if we consider d... |

15 |
Multiple functors I, Limits relative to double categories
- Bastiani, Ehresmann
- 1974
(Show Context)
Citation Context ...f another category, whose arrows (called cells) are closed under certain composition operators, and are subject to suitable coherence axioms. A more refined enrichment is present in double-categories =-=[BE74]-=-, whose structure can be informally described as the superposition of a vertical and a horizontal category of cells. It is wellknown that 2-categories represent a faithful model for term rewriting sys... |

14 |
Tiles for Concurrent and Located Calculi
- Ferrari, Montanari
- 1997
(Show Context)
Citation Context ...equipped with flexible synchronisation primitives [6, 37] and for calculi for mobile processes, like the asynchronous -calculus [16]. Tiles are also convenient for handling concurrent process calculi =-=[17]-=-. These results are then strengthened by the tight relationship between so-called sharing graphs structures (as introduced in [18]) and gs-monoidal theories (see Section 2), a framework used to accomm... |

13 |
U.: Towards the unification of models for concurrency
- Ferrari, Montanari
- 1990
(Show Context)
Citation Context ...Tr ccs respectively, obtained dropping the proof term: via the embedding j (and identifyingss withs), Tr p is contained in T ccs . Now we define a quaternary relation, called the concurrency relation =-=[FM90], between -=-transitions: whenever the pairs ht 1 ; t 2 i and ht 3 ; t 4 i are related, they actually represents the same two "causally independent" events. There are many different ways to provide a tru... |

13 |
Models of Sharing Graphs
- Hasegawa
- 1997
(Show Context)
Citation Context ...6]. Tiles are also convenient for handling concurrent process calculi [17]. These results are then strengthened by the tight relationship between so-called sharing graphs structures (as introduced in =-=[18]-=-) and gs-monoidal theories (see Section 2), a framework used to accommodate the behaviour of e.g. Milner's action calculus [36]. As far as categorical models are concerned, 2-categories [25] are the b... |

13 |
Higher categories, strings, cubes and simplex equations
- Street
- 1995
(Show Context)
Citation Context ...nctors. We want now to introduce a finitary structure, which is able to generate a double category with a free construction, starting from a given set of cells. We then present the notion of computad =-=[45, 46]-=-. Definition 23 (computads). A computad is a triple hC; D;Si, where C is a category, D is a graph (whose set of objects coincides with the set of objects of C) and S is a set of cells, each of which h... |

12 |
Axiomatizing permutation equivalence in the - calculus
- Laneve, Montanari
- 1992
(Show Context)
Citation Context ...set of causally unrelated computations. This is not so different in spirit from the well-known permutation equivalence [5, 21], and there exists in fact a tight correspondence between the two notions =-=[29]-=-. For a few initial considerations about the actual degree of concurrency expressed by the axioms, we refer to [8]. Example 4 (equating sequents). Let us consider again the ars V . As shown in Example... |

11 | Higher dimensional word problem - Burroni - 1991 |

11 | Enriched Categories as Models of Computation - Gadducci, Montanari - 1995 |

10 | Cpo models for infinite term rewriting - Corradini, Gadducci - 1995 |

8 | Semantics of weakening and contraction. Annals of pure and applied logic - Jacobs - 1994 |