## Bialgebraic Methods and Modal Logic in Structural Operational Semantics (2007)

Venue: | Electronic Notes in Theoretical Computer Science |

Citations: | 10 - 3 self |

### BibTeX

@MISC{Klin07bialgebraicmethods,

author = {Bartek Klin},

title = {Bialgebraic Methods and Modal Logic in Structural Operational Semantics},

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 semantics (SOS). 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 paper, bialgebraic semantics is combined with a coalgebraic approach to modal logic in a novel, general approach to proving the compositionality of process equivalences for languages defined by structural operational semantics. To prove compositionality, one provides a notion of behaviour for logical formulas, and defines an SOS-like specification of modal operators which reflects the original SOS specification of the language. This approach can be used to define SOS congruence formats as well as to prove compositionality for specific languages and equivalences. Key words: structural operational semantics, coalgebra, bialgebra, modal logic, congruence format 1

### Citations

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Citation Context ...cessors altogether. One writes x � a −→ to say that x has no 4sa-successors, and x �−→ means that x has no successors at all. Various equivalences are defined on processes in an LTS, and bisimilarity =-=[45]-=- is the most widely studied. In an LTS (X, A, −→), a relation R ⊆ X × X is a bisimulation if xRy implies: • ∀x a −→ x ′ . ∃y a −→ y ′ . x ′ Ry ′ , and • ∀y a −→ y ′ . ∃x a −→ x ′ . x ′ Ry ′ , and proc... |

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Citation Context ...nality for specific languages and equivalences. Key words: structural operational semantics, coalgebra, bialgebra, modal logic, congruence format 1 Introduction Structural Operational Semantics (SOS) =-=[51,1]-=- is one of the most successful frameworks for the formal description of programming languages and process calculi. There, the behaviour of programs or processes is described by means of transition rel... |

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Citation Context ...ic transition systems they are interpreted as probabilities [28]; in timed transition systems, as action durations [5]; in systems with name passing, they carry information about free and bound names =-=[46]-=-. These different kinds of systems are induced by various kinds of transition system specifications, similar to these described above. However, SOS congruence formats for equivalences on LTSs cannot b... |

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Citation Context ...uivalence. Typically, process equivalences are characterized by modal logics. For example, two processes in an LTS are bisimilar if and only if they satisfy the same formulas in Hennessy-Milner logic =-=[24]-=-, and fragments of that logic characterize other interesting equivalences on LTSs. Several attempts have been made to generalize such logics to coalgebras of arbitrary type [48,40,49,54]. Recently [38... |

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Citation Context ...not hold [40] suggest behavioural equivalence as the more basic notion of canonical process equivalence on coalgebras. For more information on the coalgebraic approach to the theory of processes, see =-=[53,27,25]-=-. 3.2 Terms as algebras In SOS, processes are closed terms over some algebraic signature. It is standard to consider sets of such terms as algebras for certain functors on Set. For example, a signatur... |

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Citation Context ...nd in [53]. Coalgebras for functors on other categories have also been considered; for example, in [16], coalgebras for a certain functor on the category Nom of nominal sets and equivariant functions =-=[21]-=- 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... |

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Citation Context ...n proposed. Such a format is a syntactic restriction on SOS specifications that guarantees a specific equivalence to be compositional on the induced transition system. The most popular format is GSOS =-=[10]-=-, which guarantees the compositionality of bisimilarity, but formats for other equivalences and/or kinds of transition systems have also been studied (see [1,23]). The task of finding a reasonably per... |

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Citation Context ...ines of §4. Constructors nil, a and ⊗ used here will be called spatial modalities, as opposed to behavioural modalities ⊤ and 〈a〉 used in the definition of L. This is motivated by “spatial logics” of =-=[14]-=-, where similar logical operators based on process syntax are present. Intuitively, formulas built from these spatial modalities can check the structure of Σ-terms. 25sOne might now attempt to define ... |

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Citation Context ... the process equivalence and by the kind of transition system. It is the purpose of this paper to provide such a framework. Our approach is based on the categorical framework of bialgebraic semantics =-=[57]-=-, where process syntax is modeled via algebras, and transition systems are viewed as coalgebras. For example, LTSs are coalgebras for the functor (P−) A on the category Set of sets and functions, wher... |

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Citation Context ...nality for specific languages and equivalences. Key words: structural operational semantics, coalgebra, bialgebra, modal logic, congruence format 1 Introduction Structural Operational Semantics (SOS) =-=[51,1]-=- is one of the most successful frameworks for the formal description of programming languages and process calculi. There, the behaviour of programs or processes is described by means of transition rel... |

85 |
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Citation Context ...btained by merging the two approaches, and it is illustrated in §6 on some simple examples. Finally, §7 sketches some related and future work. Some familiarity with basic category theory is expected; =-=[3,44]-=- are good references. The present paper is a full version of extended abstracts [36] and [37], with more detailed explanations provided and with more examples, including the substantial example of de ... |

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Citation Context ..., see [1,23]. In some applications, it is useful to impose additional structure on labels in transition systems. For example, in probabilistic transition systems they are interpreted as probabilities =-=[28]-=-; in timed transition systems, as action durations [5]; in systems with name passing, they carry information about free and bound names [46]. These different kinds of systems are induced by various ki... |

56 |
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Citation Context ... Hennessy-Milner logic [24], and fragments of that logic characterize other interesting equivalences on LTSs. Several attempts have been made to generalize such logics to coalgebras of arbitrary type =-=[48,40,49,54]-=-. Recently [38], based on earlier insights of [11,12,41,50], we have proposed a categorical generalization of modal logics for coalgebras in arbitrary categories. There, the syntax of a logic is model... |

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Citation Context ...ariables that occur in t, with no variable occurring more than once in t. This format guarantees also trace equivalence to be a congruence. On the other hand, an extension of GSOS is the ntree format =-=[19]-=-, where rules of the following form are allowed: {zi ai −→ yi}i∈I {wj � bj f(x1, . . . , xn) −→}j∈J c −→ t where xi and yi are all distinct and are the only variables occurring in the rule (i.e., each... |

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Citation Context ...re, 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 [26]. In =-=[17,18]-=-, syntax with variable binding was modeled 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 actu... |

50 |
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Citation Context ... other functors, called behaviour functors in this context. Coalgebras 2 (1)salso provide a general and abstract notion of bisimilarity (for more information on the coalgebraic theory of systems, see =-=[53]-=-). As it turns out, SOS specifications in the GSOS format are essentially distributive laws of syntax functors over (P−) A . Moreover, the process of inducing an LTS with a syntactic structure on proc... |

42 | On generalised coinduction and probabilistic specification formats: distributive laws
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Citation Context ...tributive laws. This makes bialgebraic semantics a general framework for deriving congruence formats for bisimilarities, parametrized by the kind of transition systems; it was used to this purpose in =-=[7,16,30]-=- for probabilistic, timed and name-passing systems. In this paper, the framework is further parametrized by the notion of process equivalence. Typically, process equivalences are characterized by moda... |

41 | Expressivity of coalgebraic modal logic: The limits and beyond
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Citation Context ... Hennessy-Milner logic [24], and fragments of that logic characterize other interesting equivalences on LTSs. Several attempts have been made to generalize such logics to coalgebras of arbitrary type =-=[48,40,49,54]-=-. Recently [38], based on earlier insights of [11,12,41,50], we have proposed a categorical generalization of modal logics for coalgebras in arbitrary categories. There, the syntax of a logic is model... |

33 |
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Citation Context ...aracterize other interesting equivalences on LTSs. Several attempts have been made to generalize such logics to coalgebras of arbitrary type [48,40,49,54]. Recently [38], based on earlier insights of =-=[11,12,41,50]-=-, we have proposed a categorical generalization of modal logics for coalgebras in arbitrary categories. There, the syntax of a logic is modeled via algebras for an endofunctor, and its semantics via a... |

30 |
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Citation Context ...aracterize other interesting equivalences on LTSs. Several attempts have been made to generalize such logics to coalgebras of arbitrary type [48,40,49,54]. Recently [38], based on earlier insights of =-=[11,12,41,50]-=-, we have proposed a categorical generalization of modal logics for coalgebras in arbitrary categories. There, the syntax of a logic is modeled via algebras for an endofunctor, and its semantics via a... |

29 | Process algebra with timing: Real time and discrete time
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Citation Context ...ose additional structure on labels in transition systems. For example, in probabilistic transition systems they are interpreted as probabilities [28]; in timed transition systems, as action durations =-=[5]-=-; in systems with name passing, they carry information about free and bound names [46]. These different kinds of systems are induced by various kinds of transition system specifications, similar to th... |

28 |
Coalgebraic logic, Annals of Pure and Applied Logic 96
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Citation Context ... Hennessy-Milner logic [24], and fragments of that logic characterize other interesting equivalences on LTSs. Several attempts have been made to generalize such logics to coalgebras of arbitrary type =-=[48,40,49,54]-=-. Recently [38], based on earlier insights of [11,12,41,50], we have proposed a categorical generalization of modal logics for coalgebras in arbitrary categories. There, the syntax of a logic is model... |

26 | Introduction to coalgebra: Towards mathematics of states and observations. http://www.cs.ru.nl/B.Jacobs/CLG/JacobsCoalgebraIntro. pdf. Draft book
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Citation Context ...not hold [40] suggest behavioural equivalence as the more basic notion of canonical process equivalence on coalgebras. For more information on the coalgebraic approach to the theory of processes, see =-=[53,27,25]-=-. 3.2 Terms as algebras In SOS, processes are closed terms over some algebraic signature. It is standard to consider sets of such terms as algebras for certain functors on Set. For example, a signatur... |

26 | Distributivity for endofunctors, pointed and copointed endofunctors, monads and comonads
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Citation Context ...raic approach to modal logic. In [57], natural transformations of the type (18) and (19) are considered as special cases of the more general notion of a distributive law of a monad over a comonad. In =-=[42,43,52]-=-, various types of distributive laws are studied on the abstract, categorical level. In [7], different kinds of distributive laws are studied and related on the concrete example of LTSs; also a comple... |

21 | Compositional SOS and beyond: a coalgebraic view of open systems
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Citation Context ...elated to the present paper, and indeed the following sections can be seen as a refinement of the approach described in [39,35,33]. One should also mention the approach of structured coalgebras (e.g. =-=[15,13]-=-), used for purposes similar to this work and related to the framework of bialgebraic semantics. 4 Coalgebraic modal logic To study HML and other modal logics at the level of generality of distributiv... |

20 | A congruence rule format for name-passing process calculi
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Citation Context ...tributive laws. This makes bialgebraic semantics a general framework for deriving congruence formats for bisimilarities, parametrized by the kind of transition systems; it was used to this purpose in =-=[7,16,30]-=- for probabilistic, timed and name-passing systems. In this paper, the framework is further parametrized by the notion of process equivalence. Typically, process equivalences are characterized by moda... |

20 | A coalgebraic approach to process equivalence and a coinduction principle for traces
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Citation Context ...ed trace and failures equivalences on LTSs were obtained. That work is closely related to the present paper, and indeed the following sections can be seen as a refinement of the approach described in =-=[39,35,33]-=-. One should also mention the approach of structured coalgebras (e.g. [15,13]), used for purposes similar to this work and related to the framework of bialgebraic semantics. 4 Coalgebraic modal logic ... |

18 | R.: Precongruence formats for decorated trace semantics
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Citation Context ...lowed. x a −→ y y b −→ z c f(x) −→ g(z) Many other congruence formats have been studied in the literature. For example, interesting formats for various “decorated trace” equivalences were proposed in =-=[9]-=-. A considerably more complex format for completed trace equivalence was defined in [39,35]. For a detailed study of various congruence formats and their properties, see [1,23]. In some applications, ... |

15 | A Compositional Proof System for the Modal µ-Calculus
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Citation Context ...nal proof systems as in [56] and to techniques for modal logic decomposition as in [20]. Also, the notion of spatial modality used here seems to be related to spatial logics for process calculi as in =-=[4,14]-=-, with a coalgebraic treatment suggested in [47]. The precise nature of these connections needs to be investigated. Several other problems are left open. Importantly, some guidelines for finding the r... |

15 |
Coalgebras and their logics
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Citation Context ...aracterize other interesting equivalences on LTSs. Several attempts have been made to generalize such logics to coalgebras of arbitrary type [48,40,49,54]. Recently [38], based on earlier insights of =-=[11,12,41,50]-=-, we have proposed a categorical generalization of modal logics for coalgebras in arbitrary categories. There, the syntax of a logic is modeled via algebras for an endofunctor, and its semantics via a... |

13 | GSOS for probabilistic transition systems
- Bartels
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(Show Context)
Citation Context ...stributive laws are studied and related on the concrete example of LTSs; also a complete proof of one-toone correspondence between abstract GSOS and concrete GSOS specifications is included there. In =-=[6,7]-=-, 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 |
The linear time - branching time spectrum I
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Citation Context ...mal description of process algebras (see [8] for many examples). For reasoning about processes a suitable notion of process equivalence is needed. Various equivalences on LTSs have been proposed (see =-=[22]-=- for a survey). Bisimilarity is the most widely studied, but other equivalences such as trace equivalence or testing equivalence have also been considered. Several equivalences have also been proposed... |

13 | Distributive laws for the coinductive solution of recursive equations
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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 =-=[26]-=-. In [17,18], syntax with variable binding was modeled algebraically in a presheaf category, and the standard SOS description of the π-calculus was shown to fit in the abstract GSOS format there, alth... |

13 | Testing semantics: Connecting processes and process logics
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(Show Context)
Citation Context |

12 |
A first order coalgebraic model of picalculus early observational equivalence
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(Show Context)
Citation Context ...elated to the present paper, and indeed the following sections can be seen as a refinement of the approach described in [39,35,33]. One should also mention the approach of structured coalgebras (e.g. =-=[15,13]-=-), used for purposes similar to this work and related to the framework of bialgebraic semantics. 4 Coalgebraic modal logic To study HML and other modal logics at the level of generality of distributiv... |

12 | Coalgebraic modal logic beyond sets
- Klin
(Show Context)
Citation Context ...24], and fragments of that logic characterize other interesting equivalences on LTSs. Several attempts have been made to generalize such logics to coalgebras of arbitrary type [48,40,49,54]. Recently =-=[38]-=-, based on earlier insights of [11,12,41,50], we have proposed a categorical generalization of modal logics for coalgebras in arbitrary categories. There, the syntax of a logic is modeled via algebras... |

11 |
Bialgebraic modelling of timed processes
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(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 =-=[29,30]-=-, the same framework is applied to processes with timed transitions. Congruence results regarding timed bisimilarity are proved, and a congruence format for the case of discrete time is derived. In [3... |

11 | From bialgebraic semantics to congruence formats
- Klin
- 2005
(Show Context)
Citation Context ... in the literature. For example, interesting formats for various “decorated trace” equivalences were proposed in [9]. A considerably more complex format for completed trace equivalence was defined in =-=[39,35]-=-. For a detailed study of various congruence formats and their properties, see [1,23]. In some applications, it is useful to impose additional structure on labels in transition systems. For example, i... |

11 | Sequent Calculi for Process Verification: Hennessy-Milner Logic for an Arbitrary GSOS
- Simpson
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(Show Context)
Citation Context ... extending the functors Γ with finite conjunctions as in in §6.4. More interestingly, the SOS-like presentation of logical distributive laws suggests a connection to compositional proof systems as in =-=[56]-=- and to techniques for modal logic decomposition as in [20]. Also, the notion of spatial modality used here seems to be related to spatial logics for process calculi as in [4,14], with a coalgebraic t... |

10 |
Adding recursive constructs to bialgebraic semantics
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- 2004
(Show Context)
Citation Context ... of discrete time is derived. In [31,32], the combination of timing with action is studied more carefully, with insights on combining different behaviours to obtain a modular account of semantics. In =-=[34]-=-, 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... |

8 |
Rule formats for timed processes
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(Show Context)
Citation Context ...tributive laws. This makes bialgebraic semantics a general framework for deriving congruence formats for bisimilarities, parametrized by the kind of transition systems; it was used to this purpose in =-=[7,16,30]-=- for probabilistic, timed and name-passing systems. In this paper, the framework is further parametrized by the notion of process equivalence. Typically, process equivalences are characterized by moda... |

8 | Coalgebraic semantics for timed processes
- Kick, Power, et al.
(Show Context)
Citation Context ...0], the same framework is applied to processes with timed transitions. Congruence results regarding timed bisimilarity are proved, and a congruence format for the case of discrete time is derived. In =-=[31,32]-=-, the combination of timing with action is studied more carefully, with insights on combining different behaviours to obtain a modular account of semantics. In [34], abstract GSOS is studied in a CPO-... |

8 | Syntactic formats for free: An abstract approach to process equivalence
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- 2003
(Show Context)
Citation Context ... in the literature. For example, interesting formats for various “decorated trace” equivalences were proposed in [9]. A considerably more complex format for completed trace equivalence was defined in =-=[39,35]-=-. For a detailed study of various congruence formats and their properties, see [1,23]. In some applications, it is useful to impose additional structure on labels in transition systems. For example, i... |

6 | Reniers, A hierarchy of SOS rule formats
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(Show Context)
Citation Context ... system. The most popular format is GSOS [10], which guarantees the compositionality of bisimilarity, but formats for other equivalences and/or kinds of transition systems have also been studied (see =-=[1,23]-=-). The task of finding a reasonably permissive congruence format for a given equivalence is usually quite demanding, therefore it would be desirable to have a general framework for the derivation of f... |

6 |
Expressivity Results in the Modal Logic of Coalgebras
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Citation Context |

5 |
Modularity of behaviours for mathematical operational semantics
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(Show Context)
Citation Context ...0], the same framework is applied to processes with timed transitions. Congruence results regarding timed bisimilarity are proved, and a congruence format for the case of discrete time is derived. In =-=[31,32]-=-, the combination of timing with action is studied more carefully, with insights on combining different behaviours to obtain a modular account of semantics. In [34], abstract GSOS is studied in a CPO-... |

4 | A noninterleaving model of concurrency based on transition systems with spatial structure
- Monteiro
(Show Context)
Citation Context ... modal logic decomposition as in [20]. Also, the notion of spatial modality used here seems to be related to spatial logics for process calculi as in [4,14], with a coalgebraic treatment suggested in =-=[47]-=-. The precise nature of these connections needs to be investigated. Several other problems are left open. Importantly, some guidelines for finding the right logical behaviour Γ, and for checking wheth... |

3 |
Abstract syntax with variable binding
- Fiore, Plotkin, et al.
- 1999
(Show Context)
Citation Context ...re, 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 [26]. In =-=[17,18]-=-, syntax with variable binding was modeled 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 actu... |

3 |
Bialgebraic semantics and modal logic
- Klin
- 2007
(Show Context)
Citation Context ...ally, §7 sketches some related and future work. Some familiarity with basic category theory is expected; [3,44] are good references. The present paper is a full version of extended abstracts [36] and =-=[37]-=-, with more detailed explanations provided and with more examples, including the substantial example of de Simone format in §6.3. Acknowledgments. The author is grateful to Alexander Kurz and Gordon P... |