## Bialgebraic methods in structural operational semantics (2007)

Venue: | ENTCS |

Citations: | 5 - 1 self |

### BibTeX

@ARTICLE{Klin07bialgebraicmethods,

author = {Bartek Klin},

title = {Bialgebraic methods in structural operational semantics},

journal = {ENTCS},

year = {2007}

}

### OpenURL

### Abstract

Bialgebraic semantics, invented a decade ago by Turi and Plotkin, is an approach to formal reasoning about well-behaved structural operational specifications. An extension of algebraic and coalgebraic methods, it abstracts from concrete notions of syntax and system behaviour, thus treating various kinds of operational descriptions in a uniform fashion. In this talk, the current state of the art in the area of bialgebraic semantics is presented, and its prospects for the future are sketched. In particular, a combination of basic bialgebraic techniques with a categorical approach to modal logic is described, as an abstract approach to proving compositionality by decomposing modal logics over structural operational specifications. Keywords:

### Citations

1562 | The Definition of Standard ML
- Miller, Tofte, et al.
- 1990
(Show Context)
Citation Context ... for a clear and concise presentation of innumerable ideas and formalisms (see [5] for many examples), it has also been proved a viable option for the description of fully grown programming languages =-=[31]-=-. In the structural operational approach, the semantics of programs (or processes) is described by means of transition systems, induced by inference rules following their syntactic structure. The intu... |

1286 | A structural approach to operational semantics
- Plotkin
- 1981
(Show Context)
Citation Context ...ecifications. Keywords: Structural operational semantics, category theory, algebra, coalgebra, bialgebra 1 Introduction Since its invention in the early 1980’s, Structural Operational Semantics (SOS) =-=[36,35,1]-=- has been one of the most popular and successful frameworks for the formal description of programming languages and process calculi. Not only it has become the formalism of choice for a clear and conc... |

921 |
Categories for the working mathematician
- Lane
- 1998
(Show Context)
Citation Context ...eference). Bialgebras are defined in the language of category theory, so familiarity with basic notions such as category, functor and natural transformation is also recommended (the first chapters of =-=[2,30]-=- are good references). 2.1 Processes as coalgebras In the standard framework of SOS as used e.g. in process algebra [1], structural operational descriptions specify labelled transition systems (LTSs),... |

298 | Universal coalgebra: a theory of systems
- Rutten
(Show Context)
Citation Context ... world of computing. The basics of such a theory were proposed by Turi and Plotkin in [40]. Its beginnings lie in the coalgebraic account of transition systems (see [17] for a gentle introduction and =-=[38]-=- is a good reference), where the notion used to classify various 2sKlin kinds of processes is that of behaviour, modelled as a functor on a category and formally representing the vague concept of comp... |

228 | A tutorial on (co)algebras and (co)induction
- Jacobs, Rutten
- 1997
(Show Context)
Citation Context ... prepared for future changes in the world of computing. The basics of such a theory were proposed by Turi and Plotkin in [40]. Its beginnings lie in the coalgebraic account of transition systems (see =-=[17]-=- for a gentle introduction and [38] is a good reference), where the notion used to classify various 2sKlin kinds of processes is that of behaviour, modelled as a functor on a category and formally rep... |

206 | A new approach to abstract syntax with variable binding
- Gabbay, Pitts
- 2002
(Show Context)
Citation Context ...be found in [38]. Coalgebras for functors on other categories have also been used; for example, in [8], coalgebras for a certain functor on the category NSet of nominal sets and equivariant functions =-=[13]-=- are shown to correspond to a kind of labelled transition systems with name binding. The coalgebraic abstraction allows one to treat many different kinds of systems in a uniform manner. At the same ti... |

194 |
Bisimulation can’t be traced
- Bloom, Istrail, et al.
- 1995
(Show Context)
Citation Context ...ve for the simple laws. It turns out that for B = P(A×−), and for Σ and T corresponding to an algebraic signature, laws of this kind correspond exactly to specifications in the well-known format GSOS =-=[6]-=-; hence the name abstract GSOS. In particular, the three problematic rules (3) can be modelled as distributive laws. In [40], distributive laws dual to (4) were also considered, i.e., natural transfor... |

134 | G.: Towards a mathematical operational semantics
- Turi, Plotkin
- 1997
(Show Context)
Citation Context ...riptions on an abstract level. With such a theory, SOS will hopefully be better prepared for future changes in the world of computing. The basics of such a theory were proposed by Turi and Plotkin in =-=[40]-=-. Its beginnings lie in the coalgebraic account of transition systems (see [17] for a gentle introduction and [38] is a good reference), where the notion used to classify various 2sKlin kinds of proce... |

121 | Structural operational semantics
- Aceto, Fokkink, et al.
- 1999
(Show Context)
Citation Context ...ecifications. Keywords: Structural operational semantics, category theory, algebra, coalgebra, bialgebra 1 Introduction Since its invention in the early 1980’s, Structural Operational Semantics (SOS) =-=[36,35,1]-=- has been one of the most popular and successful frameworks for the formal description of programming languages and process calculi. Not only it has become the formalism of choice for a clear and conc... |

116 | R.: Deriving bisimulation congruences for reactive systems
- Leifer, Milner
(Show Context)
Citation Context ...nd the most important challenges to be taken. 3.1 Relations to reactive systems In modern process algebra, much attention is paid to a semantic framework alternative to SOS, i.e., to reactive systems =-=[27]-=-. Many languages, such as the π-calculus or ambient calculus, are naturally described in the language of reactive systems, where dynamic behaviour is described by an unlabelled reaction relation toget... |

64 | Structural Operational Semantics
- Plotkin
- 1981
(Show Context)
Citation Context ...ecifications. Keywords: Structural operational semantics, category theory, algebra, coalgebra, bialgebra 1 Introduction Since its invention in the early 1980’s, Structural Operational Semantics (SOS) =-=[36,35,1]-=- has been one of the most popular and successful frameworks for the formal description of programming languages and process calculi. Not only it has become the formalism of choice for a clear and conc... |

55 | Modular structural operational semantics
- Mosses
(Show Context)
Citation Context ...vative extensions has also been done. Least progress has been made in the area of language translation and modularity, with the notable exception of Modular Structural Operational Semantics by Mosses =-=[32]-=-, which enjoys better modularity than its classical version when applied to standard cases. However, the issue becomes more complex with the continuing appearance of new syntactic and computational pa... |

54 | Ntyft/ntyxt rules reduce to ntree rules
- Fokkink, Glabbeek
- 1996
(Show Context)
Citation Context ...is the set of finite or infinite trees edge-labelled by A and node-labelled by X, quotiented by strong bisimilarity, and distributive laws (5) correspond to sets of SOS rules in the safe ntree format =-=[11,40]-=-. Both GSOS and safe ntree formats guarantee bisimilarity to be a congruence on the induced LTS. An important contribution of [40] was to show that these congruence properties can be proved on the abs... |

46 | Semantics of name and value passing
- Fiore, Turi
- 2001
(Show Context)
Citation Context ...ow to combine standard GSOS distributive laws with recursive equations to obtain other well-behaved distributive laws. Another bialgebraic approach to recursive equations is [16]. 2.7 Name binding In =-=[9,10]-=-, syntax with variable binding was modelled algebraically in a presheaf category, and the standard SOS description of the π-calculus was shown to fit in the abstract GSOS format there, although no act... |

38 | On generalised coinduction and probabilistic specification formats
- Bartels
- 2004
(Show Context)
Citation Context ...dered as special cases of the more general notion of a distributive law of a monad over a comonad. In [28,29,37], various types of distributive laws are studied on the abstract, categorical level. In =-=[4]-=-, different kinds of distributive laws are studied and related on the concrete example of LTSs; also a complete proof of one-to-one correspondence between abstract GSOS and concrete GSOS specification... |

24 | Distributivity for endofunctors, pointed and copointed endofunctors, monads and comonads
- Lenisa, Power, et al.
(Show Context)
Citation Context ...on. 2.4 Categorical foundations In [40], natural transformations of the type (4) and (5) are considered as special cases of the more general notion of a distributive law of a monad over a comonad. In =-=[28,29,37]-=-, various types of distributive laws are studied on the abstract, categorical level. In [4], different kinds of distributive laws are studied and related on the concrete example of LTSs; also a comple... |

22 | Modular construction of modal logics - Cîrstea, Pattinson - 2004 |

21 | Towards a duality result in the modal logic of coalgebras
- Jacobs
(Show Context)
Citation Context ...me such a theory for SOS, and [27,39] can be seen as attempts to develop such a theory for reactive systems. 3.2 Modal logic decomposition In a coalgebraic approach to modal logics and system testing =-=[15,26,34]-=-, one considers a contravariant adjunction between a category C of processes, where coalgebras for a functor B are systems, and a category D of tests, with a functor L representing the syntax of a log... |

20 | A congruence rule format for name-passing process calculi
- Fiore, Staton
- 2009
(Show Context)
Citation Context ...state and termination. Many other examples of systems modelled as coalgebras for functors on Set can be found in [38]. Coalgebras for functors on other categories have also been used; for example, in =-=[8]-=-, coalgebras for a certain functor on the category NSet of nominal sets and equivariant functions [13] are shown to correspond to a kind of labelled transition systems with name binding. The coalgebra... |

19 | An Abstract Coalgebraic Approach to Process Equivalence for Well-Behaved Operational Semantics
- Klin
- 2004
(Show Context)
Citation Context ...y, in nominal settings the syntax and behaviour 7sKlin functors reside in different categories. The basic bialgebraic setting is suitably generalised to accommodate this. 2.8 Van Glabbeek spectrum In =-=[25,22,24]-=-, abstract GSOS is interpreted in certain fibered categories. This allows one to derive congruence formats for process equivalences other than the canonical coalgebraic notion of bisimilarity. In part... |

14 |
Coalgebras and their logics
- Kurz
- 2006
(Show Context)
Citation Context ...me such a theory for SOS, and [27,39] can be seen as attempts to develop such a theory for reactive systems. 3.2 Modal logic decomposition In a coalgebraic approach to modal logics and system testing =-=[15,26,34]-=-, one considers a contravariant adjunction between a category C of processes, where coalgebras for a functor B are systems, and a category D of tests, with a functor L representing the syntax of a log... |

13 | GSOS for probabilistic transition systems
- Bartels
- 2002
(Show Context)
Citation Context ... example of LTSs; also a complete proof of one-to-one correspondence between abstract GSOS and concrete GSOS specifications is included there. 2.5 Abstract GSOS for probabilistic and timed systems In =-=[3,4]-=-, the abstract GSOS framework is applied to reactive probabilistic systems and probabilistic automata, represented as coalgebras for suitable functors. A congruence format for probabilistic bisimilari... |

13 | J.B.: Testing Semantics: Connecting Processes and Process Logics
- Pavlovic, Mislove, et al.
- 2006
(Show Context)
Citation Context ...me such a theory for SOS, and [27,39] can be seen as attempts to develop such a theory for reactive systems. 3.2 Modal logic decomposition In a coalgebraic approach to modal logics and system testing =-=[15,26,34]-=-, one considers a contravariant adjunction between a category C of processes, where coalgebras for a functor B are systems, and a category D of tests, with a functor L representing the syntax of a log... |

12 | Distributive laws for the coinductive solution of recursive equations
- Jacobs
- 2006
(Show Context)
Citation Context ...ions. There, it is shown how to combine standard GSOS distributive laws with recursive equations to obtain other well-behaved distributive laws. Another bialgebraic approach to recursive equations is =-=[16]-=-. 2.7 Name binding In [9,10], syntax with variable binding was modelled algebraically in a presheaf category, and the standard SOS description of the π-calculus was shown to fit in the abstract GSOS f... |

10 |
Bialgebraic modelling of timed processes
- Kick
- 2002
(Show Context)
Citation Context ...S framework is applied to reactive probabilistic systems and probabilistic automata, represented as coalgebras for suitable functors. A congruence format for probabilistic bisimilarity is derived. In =-=[18,19]-=-, the same framework is applied to processes with timed transitions. Congruence results regarding time bisimilarity are proved, and a congruence format for the case of discrete time is derived. In [20... |

10 |
Adding recursive constructs to bialgebraic semantics
- Klin
- 2004
(Show Context)
Citation Context ...ime is derived. In [20,21], the combination of timing with action is studied more carefully, with insights on combining different behaviours to obtain a modular account of semantics. 2.6 Recursion In =-=[23]-=-, abstract GSOS is studied in a CPO-enriched setting, where recursion is possible to express via straightforward fixpoint constructions. There, it is shown how to combine standard GSOS distributive la... |

10 | From bialgebraic semantics to congruence formats
- Klin
- 2005
(Show Context)
Citation Context ...y, in nominal settings the syntax and behaviour 7sKlin functors reside in different categories. The basic bialgebraic setting is suitably generalised to accommodate this. 2.8 Van Glabbeek spectrum In =-=[25,22,24]-=-, abstract GSOS is interpreted in certain fibered categories. This allows one to derive congruence formats for process equivalences other than the canonical coalgebraic notion of bisimilarity. In part... |

9 |
Category theory for operational semantics
- Lenisa, Power, et al.
(Show Context)
Citation Context ...on. 2.4 Categorical foundations In [40], natural transformations of the type (4) and (5) are considered as special cases of the more general notion of a distributive law of a monad over a comonad. In =-=[28,29,37]-=-, various types of distributive laws are studied on the abstract, categorical level. In [4], different kinds of distributive laws are studied and related on the concrete example of LTSs; also a comple... |

8 | Compositionality of Hennessy-Milner logic through structural operational semantics
- Fokkink, Glabbeek, et al.
- 2003
(Show Context)
Citation Context ... general framework for proving congruence results, of which [22,24] is a special case. On the concrete level, the resulting congruence principle is related to work on decomposing modal logics such as =-=[12]-=-. 3.3 Equations and weak equivalences An important tool in practical operational semantics are equations between process terms, allowing language designers to prepare shorter and more intuitive descri... |

8 |
Rule formats for timed processes
- Kick
- 2002
(Show Context)
Citation Context ...S framework is applied to reactive probabilistic systems and probabilistic automata, represented as coalgebras for suitable functors. A congruence format for probabilistic bisimilarity is derived. In =-=[18,19]-=-, the same framework is applied to processes with timed transitions. Congruence results regarding time bisimilarity are proved, and a congruence format for the case of discrete time is derived. In [20... |

8 | Coalgebraic semantics for timed processes
- Kick, Power, et al.
(Show Context)
Citation Context ...19], the same framework is applied to processes with timed transitions. Congruence results regarding time bisimilarity are proved, and a congruence format for the case of discrete time is derived. In =-=[20,21]-=-, the combination of timing with action is studied more carefully, with insights on combining different behaviours to obtain a modular account of semantics. 2.6 Recursion In [23], abstract GSOS is stu... |

8 | Syntactic formats for free: An abstract approach to process equivalence
- Klin, Sobocinski
- 2003
(Show Context)
Citation Context ...y, in nominal settings the syntax and behaviour 7sKlin functors reside in different categories. The basic bialgebraic setting is suitably generalised to accommodate this. 2.8 Van Glabbeek spectrum In =-=[25,22,24]-=-, abstract GSOS is interpreted in certain fibered categories. This allows one to derive congruence formats for process equivalences other than the canonical coalgebraic notion of bisimilarity. In part... |

6 | Reniers, A hierarchy of SOS rule formats
- Groote, Mousavi, et al.
- 2005
(Show Context)
Citation Context ... that case, a reasonably general theory based on the notion of Transition System Specification (TSS) has been developed, and much progress towards overcoming the above difficulties has been made (see =-=[1,14]-=- for a survey). For most classical process equivalences congruence formats have been provided, i.e., syntactic restrictions on inference rules that guarantee the equivalences compositional. Much work ... |

6 | Congruence for structural congruences, in
- Mousavi, Reniers
- 2005
(Show Context)
Citation Context ...syntactic monads other than the freely generated ones, the development of concrete formats based on this observation is a difficult challenge. One would like to match the more concrete development of =-=[33]-=-. Dually, it is also important to treat weak equivalences such as weak bisimilarity in the bialgebraic setting, providing congruence results for them. This would require a general coalgebraic approach... |

6 | Distributivity for a monad and a comonad
- Power, Watanabe
- 1999
(Show Context)
Citation Context ...on. 2.4 Categorical foundations In [40], natural transformations of the type (4) and (5) are considered as special cases of the more general notion of a distributive law of a monad over a comonad. In =-=[28,29,37]-=-, various types of distributive laws are studied on the abstract, categorical level. In [4], different kinds of distributive laws are studied and related on the concrete example of LTSs; also a comple... |

5 |
Modularity of behaviours for mathematical operational semantics
- Kick, Power
- 2004
(Show Context)
Citation Context ...19], the same framework is applied to processes with timed transitions. Congruence results regarding time bisimilarity are proved, and a congruence format for the case of discrete time is derived. In =-=[20,21]-=-, the combination of timing with action is studied more carefully, with insights on combining different behaviours to obtain a modular account of semantics. 2.6 Recursion In [23], abstract GSOS is stu... |

4 | Well-behaved translations between structural operational semantics
- Watanabe
(Show Context)
Citation Context ...lved before such a framework appears. A theory of well-behaved translations between operational descriptions needs to be developed, and notions of morphisms between distributive laws such as those in =-=[41]-=- seem good steps to this end. Modular construction of SOS descriptions will also require techniques for combining rules based on different types of behaviour. Initial attempts to that end were made in... |

3 |
Abstract syntax with variable binding
- Fiore, Plotkin, et al.
- 1999
(Show Context)
Citation Context ...ow to combine standard GSOS distributive laws with recursive equations to obtain other well-behaved distributive laws. Another bialgebraic approach to recursive equations is [16]. 2.7 Name binding In =-=[9,10]-=-, syntax with variable binding was modelled algebraically in a presheaf category, and the standard SOS description of the π-calculus was shown to fit in the abstract GSOS format there, although no act... |

2 |
Abstract and Concrete Categories,” 2004, available from http://katmat.math.uni-bremen.de/acc
- Adámek, Herrlich, et al.
(Show Context)
Citation Context ...eference). Bialgebras are defined in the language of category theory, so familiarity with basic notions such as category, functor and natural transformation is also recommended (the first chapters of =-=[2,30]-=- are good references). 2.1 Processes as coalgebras In the standard framework of SOS as used e.g. in process algebra [1], structural operational descriptions specify labelled transition systems (LTSs),... |

1 |
Locating reaction with 2-categories, Theoretical Computer Science 333
- Sassone, Sobocinski
- 2003
(Show Context)
Citation Context ...ons between the two formalisms. It seems that sufficiently abstract theories of both approaches is indispensable to that end. The bialgebraic approach will hopefully become such a theory for SOS, and =-=[27,39]-=- can be seen as attempts to develop such a theory for reactive systems. 3.2 Modal logic decomposition In a coalgebraic approach to modal logics and system testing [15,26,34], one considers a contravar... |

1 |
Semantics of name and value passing, in:Proc
- Fiore, Turi
(Show Context)
Citation Context ... how to combine standard GSOS distributive laws with recursive equations to obtain other well-behaved distributive laws. Another bialgebraic approach to recursive equationsis[16]. 2.7 Name binding In =-=[9,10]-=-, syntax with variable binding was modelled algebraically in a presheaf category, and the standard SOS description of the π-calculus was shown to fit in the abstract GSOS format there, although no act... |

1 |
Reniers, A hierarchy of SOS rule formats, in:Proc. SOS’05
- Groote, Mousavi, et al.
- 2005
(Show Context)
Citation Context ... that case, a reasonably general theory based on the notion of Transition System Specification (TSS) has been developed, and much progress towards overcoming the above difficulties has been made (see =-=[1,14]-=- for a survey). For most classical process equivalences congruence formats have been provided, i.e., syntactic restrictions on inference rules that guarantee the equivalences compositional. Much work ... |

1 |
Distributivity for a monad and a comonad, in:Procs
- Power, Watanabe
- 1999
(Show Context)
Citation Context ...ion. 2.4 Categorical foundations In [40], natural transformations of the type (4) and(5) are considered as special cases of the more general notion of a distributive law of a monad over a comonad. In =-=[28,29,37]-=-, various types of distributive laws are studied on the abstract, categorical level. In [4], different kinds of distributive laws are studied and related on the concrete example of LTSs; also a comple... |

1 |
Towards a mathematical operational semantics, in:Proc. LICS’97
- Turi, Plotkin
- 1997
(Show Context)
Citation Context ...n Theoretical Computer Science 175 (2007) 33–43 35 SOS will hopefully be better prepared for future changes in the world of computing. The basics of such a theory were proposed by Turi and Plotkin in =-=[40]-=-. Its beginnings lie in the coalgebraic account of transition systems (see [17] foragentle introduction and [38] is a good reference), where the notion used to classify various kinds of processes is t... |