## Algebraic-coalgebraic specification in CoCasl (2006)

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Venue: | J. LOGIC ALGEBRAIC PROGRAMMING |

Citations: | 19 - 8 self |

### BibTeX

@ARTICLE{Mossakowski06algebraic-coalgebraicspecification,

author = {Till Mossakowski and Horst Reichel and Markus Roggenbach and Lutz Schröder},

title = {Algebraic-coalgebraic specification in CoCasl},

journal = {J. LOGIC ALGEBRAIC PROGRAMMING},

year = {2006},

volume = {67},

pages = {2006}

}

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### Abstract

We introduce CoCasl as a simple coalgebraic extension of the algebraic specification language Casl. CoCasl allows the nested combination of algebraic datatypes and coalgebraic process types. We show that the well-known coalgebraic modal logic can be expressed in CoCasl. We present sufficient criteria for the existence of cofree models, also for several variants of nested cofree and free specifications. Moreover, we describe an extension of the existing proof support for Casl (in the shape of an encoding into higher-order logic) to CoCasl.

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Citation Context ...servation congruence, and trace equivalence. Typical system building operations include sequential composition, parallel composition, and nondeterministic choice. For details, refer to [25] (CCS) and =-=[17,48]-=- (CSP). The description of the specifications is organized as follows. In Section 11.1, we define the syntax of both process algebras using (algebraic) datatypes. In Section 11.2, we specify the opera... |

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Citation Context ... basic specifications, so that the language is easily adapted to the extension of the logic envisaged here. We now briefly sketch the many-sorted Casl logic, which can be formalized as an institution =-=[14]-=-. Full details can be found in [31,34]; examples of actual Casl specifications will appear in later sections. A many-sorted Casl signature Σ = (S, TF , PF , P ) consists of a set S of sorts, two S ∗ ×... |

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Citation Context ...t has been obtained in [24]. In summary, the following is known. 36s(i) Cofree coalgebras exist for bounded functors Σ on Set, more generally for accessible functors on locally presentable categories =-=[4,40]-=-. Here, a functor is called (κ-)accessible if it preserves κ-filtered (equivalently: κdirected) colimits for some regular cardinal κ. The category Set n is locally presentable. (ii) Let Σ be a Set-val... |

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Citation Context ...weakly guarded expressions (i.e. expressions in which each variable is directly or indirectly prefixed by an action). It is well-known that for this restricted case, fixed points exist and are unique =-=[26]-=- and even finitely branching. The latter ensures that the fixed points can be expressed in our framework based on finite sets. spec CCS Coalgebraic Semantics = FinAct and Zero and ActionPrefixing and ... |

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Citation Context ..., thus also providing a clear separation of concerns. The Coalgebraic Class Specification Language CCSL [25], developed in close cooperation with the LOOP project [29], is based on the observation of =-=[22]-=- that coalgebras can give a semantics to classes of object–oriented languages. CCSL provides a notation for parameterized class specifications based on final coalgebras. Its semantic is based on a hig... |

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Citation Context ...her operations) from further, derived operations defined on top of this in a conservative way. Note that this is not a purely theoretical question: programming languages such Charity [12] and Haskell =-=[19]-=- support infinite datastructures that correspond to the infinite trees in the behaviour algebras, and one should be able to specify that as many infinite trees as needed for all programs over some dat... |

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Citation Context ...in motivation for introducing the modal logic; essentially, our model existence theorem is further support for the claim that modal formulas play the same role in coalgebra as equations do in algebra =-=[21,22]-=-. Finally, we illustrate the use of CoCasl in a typical reactive setting by means of specifying the syntax and semantics of two prominent process algebras, namely CCS and CSP. These two examples serve... |

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Citation Context ...category of CoAlg(Σ) that is closed under E-sinks, and Σ has a final coalgebra, then B has a final coalgebra that is fully abstract, i.e. an M-subobject of the final Σ-coalgebra. PROOF. (i): Cf. e.g. =-=[21]-=-. (ii): The closure condition implies that B is M-coreflective in CoAlg(Σ) [1]. The coreflection of the final Σ-coalgebra is final in B. ✷ For functors Σ on Set n , equipped with the componentwise fac... |

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Citation Context ...otypes are easily expressible (and partly already available in Isabelle/HOL), structured cofree specifications will be a challenge. Moreover, we expect that recent research about circular coinduction =-=[15]-=- and terminal sequence induction [37] will provide useful tactics for the encoding of CoCasl into Isabelle/HOL. All the specifications shown in this paper are available under 61shttp://www.cofi.info/L... |

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Citation Context ...for Casl (in the shape of an encoding into higher-order logic) to CoCasl. In recent years, coalgebra has emerged as a convenient and suitably general way of modeling the reactive behaviour of systems =-=[26]-=-. While algebraic specification deals with inductive datatypes generated by constructors, coalgebraic specification deals with coinductive process types that are observable by selectors. An important ... |

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Citation Context ...sugar now becomes the formal statement that the modal CoCasl institution can be encoded in the institution of structured specifications over the plain CoCasl institution via an institution comorphism =-=[16]-=-. We recall that a comorphism I → J between institutions I and J consists of a translation Φ of I-signatures into J-signatures, a translation of Σ-sentences into ΦΣ-sentences, and a reduction of ΦΣ-mo... |

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Citation Context ...as suffice for practical purposes. 7 Existence of cofree models The theory of algebraic specification and institutions provides us with a very general characterization of the existence of free models =-=[27]-=-: free models exist for specifications with universally quantified Horn axioms. (Part of) a dual result has been obtained in [15]. In summary, results from [26, 14, 15] guarantee that cofree coalgebra... |

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Citation Context ...t congruence [12]. In terms of tool support, this means the following: – Equality of elements in the cofree datatype is obtained as before by coinductive reasoning (or via terminal sequence induction =-=[21]-=-), the difference with the absolutely free case being that the formulas in the free specification (e.g. associativity, commutativity, and idempotence in the case of finite sets) are now available for ... |

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Citation Context ...pecifications with several non-observable sorts, i.e. for coalgebras over Set n . Even more generally, we have Proposition 32 Let C be a category equipped with a factorization system (E, M) for sinks =-=[1]-=-, and let Σ : C → C be a functor that preserves M, i.e. Σ[M] ⊂ M. Then (i) (E, M) lifts to a factorization structure (U −1 [E], U −1 [M]), also denoted (E, M), on CoAlg(Σ), where U is the forgetful fu... |

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Citation Context ...coalgebra combinations in the literature: [11] uses a simpler logic, CCSL [51] has fewer datatypes available, while hidden algebra such as in BOBJ [46] and reachable-observable algebra such as in COL =-=[7]-=- do not support cofree types. If, for example, streams are not specified as the final (=cofree) model, then there are stream models which do not contain all corecursively definable functions (like the... |

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Citation Context ...idea being that observer operations give rise to modalities that describe the evolution of the system upon application of the observer. Related work, to be discussed at the end of Section 8, includes =-=[18,19,23,51]-=-. The underlying intuition is that the non-observable sorts of a process type form a multi-sorted state space, and that observers either directly produce observable values or effect an evolution of th... |

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Citation Context ...hy there is no cofree types construct). More concretely, this means that, in addition to cogeneratedness, there is also a principle stating that there are enough behaviours, namely all infinite trees =-=[3]-=- (with branching as specified by the observers). In contrast to its dual (no confusion among constructors), the latter principle cannot be expressed in first-order logic; however, a secondorder specif... |

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Citation Context ...rphisms. That is, they are total as functions, and they preserve (but not necessarily reflect) the definedness of partial functions and the satisfaction of predicates. A homomorphism is called closed =-=[9]-=-, if it not only preserves, but also reflects definedness and satisfaction of predicates. A congruence R on a model is an equivalence relation that is compatible with the total and partial functions (... |

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Citation Context ...e systems, process algebra plays a prominent rôle. It has proven to be suitable at the level of requirement specification, at the level of design specifications, and also for formal refinement proofs =-=[6]-=-. Almost all of the underlying concepts of process algebra can be found in the languages CCS and CSP: a type system on the communications; synchronous as well as asynchronous communication; operationa... |

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Citation Context ...tures include first-order logic, partial functions, subsorts, sort generation constraints, and structured and architectural specifications. For the language definition and a full formal semantics cf. =-=[8,34]-=-. An important point here is that the semantics of structured and architectural specifications is independent of the logic employed for basic specifications, so that the language is easily adapted to ... |

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Citation Context ...ox specification into a glass-box one, thus also providing a clear separation of concerns. The Coalgebraic Class Specification Language CCSL [25], developed in close cooperation with the LOOP project =-=[29]-=-, is based on the observation of [22] that coalgebras can give a semantics to classes of object–oriented languages. CCSL provides a notation for parameterized class specifications based on final coalg... |

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Citation Context ...ongruences. 11.3.2 Weak Bisimulation In the specification of weak bisimulation in our setting, we make use of the following characterization in terms of strong bisimulation, reformulating a result of =-=[10]-=- (see also [50] for a general coalgebraic treatment of weak bisimulation in a similar spirit): Theorem 38 (Weak vs. Strong Bisimulation) Let Ti = (Si, si, Act, →i) be transition systems over Act with ... |

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Citation Context ...tion symbols introduced along the way are hidden. We conclude the section with the announced discussion of related work on modal logic for coalgebra (omitting the logic developed in the seminal paper =-=[27]-=-, which is not immediately suitable for use in a specification language due to the presence of infinitary conjunction and the complex nature of its 30smodal operator). The syntax chosen here is largel... |

8 |
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Citation Context ... observers and other operations) from further, derived operations defined on top of this in a conservative way. Note that this is not a purely theoretical question: programming languages such Charity =-=[12]-=- and Haskell [39] support infinite data structures that correspond to the infinite trees in the behaviour algebras, and one should be able to specify that as many infinite trees as needed for all prog... |

8 |
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Citation Context ... necessary to retain the modal formulae explicitly, e.g. in order to pass them on to a modal theorem prover or in order to incorporate CoCasl into a heterogeneous framework such as heterogeneous Casl =-=[28,32]-=-, but most notably in order to integrate modal CoCasl into the institution-independent framework of Casl with regard to structured and architectural specifications. We recall that defining an institut... |

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Citation Context ...nto higher-order logic can be extended to CoCasl. CoCasl is more expressive than other algebra-coalgebra combinations in the literature: [7] uses a simpler logic, while hidden algebra such as in BOBJ =-=[24]-=- and reachable-observable algebra such as in COL [4, 5] do not support cofree types (at least not at the level of basic specifications), which in particular means that corecursive definitions are not ... |

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Citation Context ...ating that CoCasl is an expressive language which is able to deal with reactive systems at an appropriate level. CoCasl is more expressive than other algebra-coalgebra combinations in the literature: =-=[11]-=- uses a simpler logic, CCSL [51] has fewer datatypes available, while hidden algebra such as in BOBJ [46] and reachable-observable algebra such as in COL [7] do not support cofree types. If, for examp... |

4 | CoCasl at work — modelling process algebra
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Citation Context ...voted to the existence theorem for final models. The specifications of CCS and CSP are described in Section 11. This work is an 2sextended version of [45]; the process algebra example has appeared in =-=[33]-=-. 1 Casl The specification language Casl (Common Algebraic Specification Language) has been designed by CoFI, the international Common Framework Initiative for Algebraic Specification and Development.... |

3 |
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Citation Context ...final coalgebras, which are complete sets of possibly infinite behaviours, such as streams or even the real numbers [38]. For algebraic specification, the Common Algebraic Specification Language Casl =-=[13]-=- has been designed as a unifying standard, while for the much younger field of coalgebraic specification there is still a divergence of notions Preprint submitted to Elsevier Science 9 July 2004sand n... |

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admitting final semantics
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Citation Context ...in motivation for introducing the modal logic; essentially, our model existence theorem is further support for the claim that modal formulae play the same role in coalgebra as equations do in algebra =-=[23,24]-=-. Finally, we illustrate the use of CoCasl in a typical reactive setting by means of specifying the syntax and semantics of two prominent process algebras, namely CCS and CSP. These two examples serve... |

3 |
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Citation Context ...] could be expressed in CoCasl as a refinement of a black-box specification into a glass-box one, thus also providing a clear separation of concerns. The Coalgebraic Class Specification Language CCSL =-=[25]-=-, developed in close cooperation with the LOOP project [29], is based on the observation of [22] that coalgebras can give a semantics to classes of object–oriented languages. CCSL provides a notation ... |

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Citation Context ...idea being that observer operations give rise to modalities that describe the evolution of the system upon application of the observer. Related work, to be discussed at the end of Section 8, includes =-=[18,19,23,51]-=-. The underlying intuition is that the non-observable sorts of a process type form a multi-sorted state space, and that observers either directly produce observable values or effect an evolution of th... |

2 |
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Citation Context ... a 1. Here, the ‘always’ stems from the fact that the modal formula is, on the outside, implicitly quantified over all states, i.e. over all elements of type BitStream. Remark 12 The modal µ-calculus =-=[20]-=-, which provides a syntax for least and greatest fixed points of recursive modal predicate definitions, is expressible using free and cofree specifications: µ is expressible by free recursively define... |

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Citation Context ... modal formulas explicitly, e.g. in order to pass them on to a modal theorem prover or in order to incorporate CoCasl into a heterogeneous framework such as the language of the heterogeneous tool set =-=[25]-=-. Since the syntax of modal logic relies to some extent on syntactical mechanisms that are not normally explicitly retained in the semantics of CoCasl specifications, such as in particular cotypes and... |

1 |
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Citation Context ...n (stating equivalence of definedness of its two sides and equality in case of definedness), an existence equation (stating definedness and equality of its two sides), or a predicate application; see =-=[13,5]-=- for details. There is an additional type of sentence that goes beyond first-order logic: a sort generation constraint states that a given set of sorts is generated by a given set of functions, i.e. t... |

1 |
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Citation Context ... effect is that axioms formulated in modal logic indeed describe only the observable behaviour of a system, formally: satisfaction of modal formulae is bisimulation invariant (see Section 10 and e.g. =-=[22,35]-=-). Methodologically, this means that the state space is appropriately encapsulated; a technical advantage is that restriction by modal formulae preserves existence of final models (cf. Section 10). In... |

1 |
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Citation Context ...t [29,28] already provides a parser and static analysis for Casl and CoCasl basic and structured specifications. Concerning proof support, it is planned to extend the coding of Casl into Isabelle/HOL =-=[30]-=- to CoCasl. While cogenerated and cofree cotypes are easily expressible (and partly already available in Isabelle/HOL), structured cofree specifications will be a challenge. Moreover, we expect that r... |

1 |
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Citation Context ...CCSL does not support data type specifications with partial constructors, axioms or equations, i.e. it only supports free types in the sense of Casl. Recently CCSL has been extended by binary methods =-=[28]-=-, which are supported in CoCasl via cogenerated constraints. At the level of proof principles, recent research about circular coinduction [10] and terminal sequence induction [21] is expected to provi... |

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