## Bialgebraic Operational Semantics and Modal Logic (extended abstract)

### BibTeX

@MISC{Klin_bialgebraicoperational,

author = {Bartek Klin},

title = {Bialgebraic Operational Semantics and Modal Logic (extended abstract)},

year = {}

}

### OpenURL

### Abstract

A novel, general approach is proposed to proving the compositionality of process equivalences on languages defined by Structural Operational Semantics (SOS). The approach, based on modal logic, is inspired by the simple observation that if the set of formulas satisfied by a process can be derived from the corresponding sets for its subprocesses, then the logical equivalence is a congruence. Striving for generality, SOS rules are modeled categorically as bialgebraic distributive laws for some notions of process syntax and behaviour, and modal logics are modeled via coalgebraic polyadic modal logic. Compositionality is proved by providing a suitable notion of behaviour for the logic together with a dual distributive law, reflecting the one modeling the SOS specification. Concretely, the dual laws may appear as SOS-like rules where logical formulas play the role of processes, and their behaviour models logical decomposition over process syntax. The approach can be used either to proving compositionality for specific languages or for defining SOS congruence formats.

### Citations

3463 |
Communication and Concurrency
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- 1989
(Show Context)
Citation Context ...φ 2where a ∈ A, and with semantics defined on a given LTS by: x |= 〈a〉φ ⇐⇒ ∃y ∈ X. x a −→ y, y |= φ and by the standard interpretation of propositional connectives, characterizes strong bisimilarity =-=[23]-=-. Fragments of HML have also been considered; see [12] for a survey. For example, the fragment without conjunction or negation characterizes trace equivalence on arbitrary LTSs, and the same fragment ... |

1385 | A structural approach to operational semantics
- Plotkin
- 2004
(Show Context)
Citation Context ...ion over process syntax. The approach can be used either to proving compositionality for specific languages or for defining SOS congruence formats. Introduction Structural Operational Semantics (SOS) =-=[25, 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... |

1005 |
Categories for the Working Mathematician
- Lane
- 1971
(Show Context)
Citation Context ...btained by merging the two approaches, and it is illustrated in §5 on some simple examples. Finally, §6 sketches some related and future work. Some familiarity with basic category theory is expected; =-=[2, 22]-=- are good references. Acknowledgments. The author is grateful to Alexander Kurz and Gordon Plotkin for fruitful discussions, and to an anonymous referee for insightful comments on the content and pres... |

520 | Algebraic Laws for Nondeterminism and Concurrency
- Henessy, Milner
- 1985
(Show Context)
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 =-=[15]-=-, and fragments of that logic characterize other interesting equivalences on LTSs. Recently [19] we have proposed a categorical generalization of modal logics for coalgebras in arbitrary categories. T... |

330 | Universal coalgebra: A theory of systems
- Rutten
(Show Context)
Citation Context ...as for other functors, called behaviour functors in this context. Coalgebras also provide a general and abstract notion of bisimilarity (for more information on the coalgebraic theory of systems, see =-=[26]-=-). 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... |

201 |
Bisimulation can’t be traced
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- 1995
(Show Context)
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 =-=[7]-=-, which guarantees the compositionality of bisimilarity, but formats for other equivalences and/or kinds of transition systems have also been studied (see [1, 14]). The task of finding a reasonably pe... |

142 | L.: A spatial logic for concurrency (part I
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(Show Context)
Citation Context ...nal proof systems as in [28] and to techniques for modal logic decomposition as in [11]. Also, the notion of spatial modality used here seems to be related to spatial logics for process calculi as in =-=[3, 8]-=-. The precise nature of these connections needs to be studied. Several other problems are left open. Importantly, some guidelines for finding the right logical behaviour Γ are much needed instead of i... |

142 | Towards a Mathematical Operational Semantics
- Turi, Plotkin
- 1997
(Show Context)
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 =-=[30]-=-, where process syntax is modeled via algebras, and transition systems are viewed as coal-gebras. For example, LTSs are coalgebras for the functor (P−) A on the category Set of sets and functions, wh... |

131 | Structural operational semantics
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(Show Context)
Citation Context ...ion over process syntax. The approach can be used either to proving compositionality for specific languages or for defining SOS congruence formats. Introduction Structural Operational Semantics (SOS) =-=[25, 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... |

50 | D.: Semantics of name and value passing
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- 2001
(Show Context)
Citation Context ... is the finite powerset functor. Several other kinds of transition systems are Bcoalgebras for other endofunctors B, called behaviour functors in this context, on Set or on other categories (see e.g. =-=[26, 10]-=-). A coalgebra morphism from h : X → BX to g : Y → BY is a function f : X → Y such that g ◦ f = Bf ◦ h. Dually, syntax is traditionally modeled with algebras [13]. An algebraic signature Σ corresponds... |

42 | On generalised coinduction and probabilistic specification formats: distributive laws
- Bartels
- 2004
(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 =-=[4, 9, 16]-=- 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
- Schröder
- 2008
(Show Context)
Citation Context ... specifications. 3 Coalgebraic modal logic To study HML and other modal logics at the level of generality of distributive laws, we will use the recent approach of [19], inspired by earlier results of =-=[21, 24, 27]-=-. Assume a category C of structures of processes, and a category D of structures of logical formulas, connected by an adjunction F ⊣ Gop : C → Dop . This means that a bijection C(X, GΦ) ∼ = D(Φ, F X) ... |

20 | A congruence rule format for name-passing process calculi
- Fiore, Staton
- 2009
(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 =-=[4, 9, 16]-=- 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... |

18 | R.: Precongruence formats for decorated trace semantics
- Bloom, Fokkink, et al.
- 2004
(Show Context)
Citation Context ... framework easier to apply. 9Some existing work on specific SOS formats and their properties can be rephrased in terms of the present framework. For example, the technique of frozen/liquid positions =-=[6]-=- used to derive formats for decorated trace equivalences corresponds exactly to extending the functors Γ with finite conjunctions as in in §5.3. More interestingly, the SOS-like presentation of logica... |

17 |
Abstract and Concrete Categories. Wiley-Interscience
- Adámek, Herrlich, et al.
- 1990
(Show Context)
Citation Context ...btained by merging the two approaches, and it is illustrated in §5 on some simple examples. Finally, §6 sketches some related and future work. Some familiarity with basic category theory is expected; =-=[2, 22]-=- are good references. Acknowledgments. The author is grateful to Alexander Kurz and Gordon Plotkin for fruitful discussions, and to an anonymous referee for insightful comments on the content and pres... |

15 | A Compositional Proof System for the Modal µ-Calculus
- Andersen, Stirling, et al.
- 1994
(Show Context)
Citation Context ...nal proof systems as in [28] and to techniques for modal logic decomposition as in [11]. Also, the notion of spatial modality used here seems to be related to spatial logics for process calculi as in =-=[3, 8]-=-. The precise nature of these connections needs to be studied. Several other problems are left open. Importantly, some guidelines for finding the right logical behaviour Γ are much needed instead of i... |

15 |
Coalgebras and their logics
- Kurz
(Show Context)
Citation Context ... specifications. 3 Coalgebraic modal logic To study HML and other modal logics at the level of generality of distributive laws, we will use the recent approach of [19], inspired by earlier results of =-=[21, 24, 27]-=-. Assume a category C of structures of processes, and a category D of structures of logical formulas, connected by an adjunction F ⊣ Gop : C → Dop . This means that a bijection C(X, GΦ) ∼ = D(Φ, F X) ... |

13 |
The linear time - branching time spectrum I
- Glabbeek
- 2001
(Show Context)
Citation Context ...mal description of process algebras (see [5] for many examples). For reasoning about processes a suitable notion of process equivalence is needed. Various equivalences on LTSs have been proposed (see =-=[12]-=- 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 | Testing semantics: Connecting processes and process logics
- Pavlović, Mislove, et al.
- 2006
(Show Context)
Citation Context ... specifications. 3 Coalgebraic modal logic To study HML and other modal logics at the level of generality of distributive laws, we will use the recent approach of [19], inspired by earlier results of =-=[21, 24, 27]-=-. Assume a category C of structures of processes, and a category D of structures of logical formulas, connected by an adjunction F ⊣ Gop : C → Dop . This means that a bijection C(X, GΦ) ∼ = D(Φ, F X) ... |

12 | Coalgebraic modal logic beyond sets
- Klin
(Show Context)
Citation Context ...rocesses in an LTS are bisimilar if and only if they satisfy the same formulas in Hennessy-Milner logic [15], and fragments of that logic characterize other interesting equivalences on LTSs. Recently =-=[19]-=- 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 ... |

11 | From bialgebraic semantics to congruence formats
- Klin
- 2005
(Show Context)
Citation Context ...bisimilarity is guaranteed to be a congruence. Another well-known format is de Simone format for trace equivalence. A considerably more complex format for completed trace equivalence was suggested in =-=[20, 17]-=-. For a detailed study of various congruence formats and their properties, see [1, 14]. 2 Bialgebraic operational semantics LTSs can be seen as functions h : X → (PX) A along a the correspondence y ∈ ... |

11 | Sequent Calculi for Process Verification: Hennessy-Milner Logic for an Arbitrary GSOS
- Simpson
- 2004
(Show Context)
Citation Context ... extending the functors Γ with finite conjunctions as in in §5.3. More interestingly, the SOS-like presentation of logical distributive laws suggests a connection to compositional proof systems as in =-=[28]-=- and to techniques for modal logic decomposition as in [11]. Also, the notion of spatial modality used here seems to be related to spatial logics for process calculi as in [3, 8]. The precise nature o... |

8 | Compositionality of Hennessy-Milner logic through structural operational semantics
- Fokkink, Glabbeek, et al.
- 2003
(Show Context)
Citation Context ... §5.3. More interestingly, the SOS-like presentation of logical distributive laws suggests a connection to compositional proof systems as in [28] and to techniques for modal logic decomposition as in =-=[11]-=-. Also, the notion of spatial modality used here seems to be related to spatial logics for process calculi as in [3, 8]. The precise nature of these connections needs to be studied. Several other prob... |

8 |
Rule formats for timed processes
- Kick
- 2002
(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 =-=[4, 9, 16]-=- 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 | Syntactic formats for free: An abstract approach to process equivalence
- Klin, Sobocinski
- 2003
(Show Context)
Citation Context ...bisimilarity is guaranteed to be a congruence. Another well-known format is de Simone format for trace equivalence. A considerably more complex format for completed trace equivalence was suggested in =-=[20, 17]-=-. For a detailed study of various congruence formats and their properties, see [1, 14]. 2 Bialgebraic operational semantics LTSs can be seen as functions h : X → (PX) A along a the correspondence y ∈ ... |

8 | Categorical modeling of structural operational rules: case studies
- Turi
- 1997
(Show Context)
Citation Context ...w axioms. Laws of type (3), (5) or (6) uniquely induce laws of type (7) by appropriate “recursion theorems” [30]. As was observed in [30], proved in detail in [4], and explained on simple examples in =-=[29, 17, 18]-=-, for B = (Pω−) A and for Σ on Set corresponding to an algebraic signature, laws of type (5) correspond to image finite GSOS 3specifications, and hλ is the LTS induced by the corresponding specificat... |

6 | Reniers, A hierarchy of SOS rule formats
- Groote, Mousavi, et al.
- 2005
(Show Context)
Citation Context ...n system. The most popular format is GSOS [7], which guarantees the compositionality of bisimilarity, but formats for other equivalences and/or kinds of transition systems have also been studied (see =-=[1, 14]-=-). 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... |

5 | Bialgebraic methods in structural operational semantics
- Klin
- 2007
(Show Context)
Citation Context ...w axioms. Laws of type (3), (5) or (6) uniquely induce laws of type (7) by appropriate “recursion theorems” [30]. As was observed in [30], proved in detail in [4], and explained on simple examples in =-=[29, 17, 18]-=-, for B = (Pω−) A and for Σ on Set corresponding to an algebraic signature, laws of type (5) correspond to image finite GSOS 3specifications, and hλ is the LTS induced by the corresponding specificat... |

2 |
et al. Initial algebra semantics and continuous algebras
- Goguen, Thatcher
- 1977
(Show Context)
Citation Context ... or on other categories (see e.g. [26, 10]). A coalgebra morphism from h : X → BX to g : Y → BY is a function f : X → Y such that g ◦ f = Bf ◦ h. Dually, syntax is traditionally modeled with algebras =-=[13]-=-. An algebraic signature Σ corresponds to a functor ΣX = ∐ f∈Σ Xar(f) on Set, in the sense that algebras for the signature are exactly Σ-algebras for the functor, i.e, functions g : ΣX → X. An algebra... |