## Calculating invariants as coreflexive bisimulations (2008)

Citations: | 3 - 3 self |

### BibTeX

@MISC{Barbosa08calculatinginvariants,

author = {Luís S. Barbosa and José N. Oliveira and Alexandra M. Silva},

title = {Calculating invariants as coreflexive bisimulations},

year = {2008}

}

### OpenURL

### Abstract

Invariants, bisimulations and assertions are the main ingredients of coalgebra theory applied to computer systems engineering. In this paper we reduce the first to a particular case of the second and show how both together pave the way to a theory of coalgebras which regards invariant predicates as types. An outcome of such a theory is a calculus of invariants ’ proof obligation discharge, a fragment of which is presented in the paper. The approach has two main ingredients: one is that of adopting relations as “first class citizens” in a pointfree reasoning style; the other lies on a synergy found between a relational construct, Reynolds ’ relation on functions involved in the abstraction theorem on parametric polymorphism and the coalgebraic account of bisimulation and invariants. In this process, we provide an elegant proof of the equivalence between two different definitions of bisimulation found in coalgebra literature (due to B. Jacobs and Aczel & Mendler, respectively) and their instantiation to the classical Park-Milner definition popular in process algebra.

### Citations

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Citation Context ...om that of bisimulations 6 . We will address this one first. 4 Calculating bisimulations Let us first show how the classical definition of bisimulation used in process algebra (due to Milner and Park =-=[17]-=-) can be retrieved from (13) simply by instantiating F to the powerset relator PX = {S |S ⊆ X}. We need the universal property of the powertranspose isomorphism Λ f = ΛR ≡ R = ∈ ·f (14) which converts... |

364 |
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(Show Context)
Citation Context ...� Υ ����� ��� ��〈f,g〉 ��� f ��� ���g ��� Φ Clearly, the proof-obligations associated to the two projections π1 · (Ψ × Υ) ⊆ Ψ · π1 , π2 · (Ψ × Υ) ⊆ Υ · π2 are instances of Reynolds abstraction theorem =-=[18, 22, 2]-=-: f f (33) GA � FA is polymorphic ≡ 〈∀ R :: f(GR ← FR)f〉 (34) So there is nothing to prove. To show that 〈f, g〉 is indeed an arrow in Pred we need to recall the universal property of relational splits... |

329 | Theorems for free
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Citation Context ...� Υ ����� ��� ��〈f,g〉 ��� f ��� ���g ��� Φ Clearly, the proof-obligations associated to the two projections π1 · (Ψ × Υ) ⊆ Ψ · π1 , π2 · (Ψ × Υ) ⊆ Υ · π2 are instances of Reynolds abstraction theorem =-=[18, 22, 2]-=-: f f (33) GA � FA is polymorphic ≡ 〈∀ R :: f(GR ← FR)f〉 (34) So there is nothing to prove. To show that 〈f, g〉 is indeed an arrow in Pred we need to recall the universal property of relational splits... |

162 |
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(Show Context)
Citation Context ...U — see diagram on the right. π1 R �� π2 � ������� ���� X ρ Y S p �� R q p ′ q ′ R Next we want want to check (13) against another (also coalgebraic) definition of bisimulation due to Aczel & Mendler =-=[1]-=-: given two coalgebras c : X → F(X) and d : Y → F(Y ) an F-bisimulation is a relation R ⊆ X × Y which can be extended to a coalgebra ρ such that projections π1 and π2 lift to F-coalgebra morphisms. (S... |

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Citation Context ...⫴ is an interleaving operator defined by c ⫴ d def = δ · (c × d) whenever F has a distributive law δ : FΦ × FΨ −→ F(Φ × Ψ) corresponding to the Kleisli composition of F’s left and right strength (see =-=[12]-=- for details). The calculation of (37) follows: FΦ � c c Φ ∧ FΨ Ψ ≡ { (30) twice } c · Φ ⊆ FΦ · c ∧ c · Ψ ⊆ FΨ · c ⇒ { monotonicity of product and composition } δ · (c · Φ × d · Ψ) ⊆ δ · (FΦ · c × FΨ ... |

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Citation Context ...lication proof of this equivalence see Backhouse and Hoogendijk’s work on final dialgebras [6]. A proof of the same result is implicit in Corollary 3.1 of [19] which invokes a result by Carboni et al =-=[8]-=- on extending functors to relators. 12 See [15], where this view of proof obligations is actually extended to arbitrary binary relations. This is suitable for specification languages such as eg. VDM, ... |

27 |
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Citation Context ... respectively. By comparing (13) against (9) we conclude that invariants are special cases of bisimulations: exactly those which are coreflexive relations, cf. diagram (b). 4 The concept of a relator =-=[5]-=- extends that of a functor to relations: F A describes a parametric type while FR is a relation from F A to F B provided R is a relation from A to B. Relators are monotone and commute with composition... |

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Citation Context ...ary step in the evolution of the system (i.e., a state transition). The possible outcomes of such steps are captured by notation FX, where functor F acts as a shape for the system’s interface. Jacobs =-=[10]-=- identifies three cornerstones in the theory of coalgebras: invariants, bisimilarity and assertions. About the first he writes: an important aspect of formally establishing the safety of systems is to... |

18 |
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Citation Context ... universal property in 11 For a longer bi-implication proof of this equivalence see Backhouse and Hoogendijk’s work on final dialgebras [6]. A proof of the same result is implicit in Corollary 3.1 of =-=[19]-=- which invokes a result by Carboni et al [8] on extending functors to relators. 12 See [15], where this view of proof obligations is actually extended to arbitrary binary relations. This is suitable f... |

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(Show Context)
Citation Context ...d(F)(P). (2) Pred(F)(P) stands for the lifting of predicate P via functor F. (We will spell out the meaning of this construct very soon.) Our approach will be to reason about (2) via the PF-transform =-=[16]-=- — a transformation of first order predicate formulæ into pointfree binary relation formulæ [7] which will enable us to blend the concept of invariant with that of bisimulation in a quite handy way. (... |

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(Show Context)
Citation Context ... 〈ν Ψ : : Φ · ○cΨ〉 (43) We end this section by showing how the PF-transform (and in particular the replacement of intersection of coreflexives by composition (42)) together with the fixpoint calculus =-=[3]-=- speed up derivation of laws in such a logic. The law we have chosen to calculate is Lemma 4.2.9(ii) of [10]: ✷Φ ⊆ ✷✷Φ. We drop subscript c of ○c (for economy of notation) and calculate: ✷Φ ⊆ ✷✷Φ ≡ { ... |

10 |
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(Show Context)
Citation Context ...ebras which are R-bisimilar. But it is the calculational power implicit in (18) which really justifies the recasting of (13) in terms of Reynolds’ arrow combinator. This has been studied in detail in =-=[2]-=- (if not earlier), a paper which derives elegant and manageable PF-properties such, for instance id ← id = id (19) (R ← S) ◦ = R ◦ ← S ◦ (20) R ← S ⊆ V ← U ⇐ R ⊆ V ∧ U ⊆ S (21) k(f ← g)h ≡ k · g = f ·... |

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(Show Context)
Citation Context ...ets are included in Set, in order to verify whether a particular universal property in 11 For a longer bi-implication proof of this equivalence see Backhouse and Hoogendijk’s work on final dialgebras =-=[6]-=-. A proof of the same result is implicit in Corollary 3.1 of [19] which invokes a result by Carboni et al [8] on extending functors to relators. 12 See [15], where this view of proof obligations is ac... |

2 | Pointfree foundations for (generic) lossless decomposition
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(Show Context)
Citation Context ...algebra c. This modal operator is easily shown to be the upper adjoint of Galois connection πcΦ ⊆ Ψ ≡ Φ ⊆ ○cΨ (41) whose lower adjoint is the projection operator πcΦ def = c·Φ·c ◦ which is central to =-=[14]-=- in studying the PF-refactoring of data dependency theory (a part of database theory). From this, one immediately infers that ○c is monotonic and distributes over conjunction: ○c(Φ·Ψ) = (○cΦ)·(○cΨ). N... |

2 |
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(Show Context)
Citation Context ...)h ≡ k · g = f · h (22) 7 The pointwise definition of simulation is better preceived once S ·R ⊆ R ·U is re-written into R ⊆ S \(R · U), recall (16) — similarly for the other conjunct. Matteo Vaccari =-=[21]-=- performs a calculation similar to the above starting directly from this pointwise definition. UCalculating invariants 7 From property (21) we learn that the combinator is monotonic on the left hand ... |

1 |
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(Show Context)
Citation Context ...x)〉 Converting predicates P and Q to coreflexives Φ and Ψ, respectively, and making explicit the supremum implict in the existential quantification one gets, ✷Φ = 〈 ⋃ Ψ : Ψ ⊆ ○cΨ : Ψ ⊆ Φ〉 = { trading =-=[4]-=- } 〈 ⋃ Ψ :: Ψ ⊆ ○cΨ ∧ Ψ ⊆ Φ〉 = { ∩-universal } 〈 ⋃ Ψ :: Ψ ⊆ ○cΨ ∩ Φ〉 = { ∩ of coreflexives is composition (42) } 〈 ⋃ Ψ : : Ψ ⊆ Φ · ○cΨ〉 which leads to a greatest (post)fixpoint definition: ✷Φ = 〈ν Ψ :... |

1 |
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(Show Context)
Citation Context ... surely there is much work to be done before this becomes of practical use. On the theory side, the authors are aware of a connection between the “predicates as objects” approach and Frege structures =-=[9]-=- 14 . Quoting this reference: A Frege structure is a lambda structure F on the set A together with a designated subset of A whose elements are called propositions (...) the propositional connectives a... |

1 |
Invariants as coreflexive bisimulations — in a coalgebraic setting
- Oliveira
- 2006
(Show Context)
Citation Context ...ere exists a coalgebra a whose carrier is the ”graph” of bisimulation R and which is such that projections π1 and π2 lift to the corresponding coalgebra morphisms. 9 The proof of (29) can be found in =-=[13]-=-. 10 It is a standard result that every R can be factored in a tabulation R = r · s ◦ [7]. An obvious and easy to check tabulation is r, s := π1, π2 [13], which boils down to pairwise equality of pair... |

1 |
Theory and applications of the PF-transform, Feb. 2008. Tutorial at LerNET’08
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(Show Context)
Citation Context ...se and Hoogendijk’s work on final dialgebras [6]. A proof of the same result is implicit in Corollary 3.1 of [19] which invokes a result by Carboni et al [8] on extending functors to relators. 12 See =-=[15]-=-, where this view of proof obligations is actually extended to arbitrary binary relations. This is suitable for specification languages such as eg. VDM, where the inclusion of a subtyping mechanism wh... |

1 | Coalgebraic foundations of linear systems (an exercise in stream calculus
- Rutten
(Show Context)
Citation Context ...tems able to take into account data persistence and continued interaction.2 L.S. Barbosa, J.N. Oliveira and A.M. Silva Coalgebra theory, widely acknowledged as the mathematics of state-based systems =-=[20]-=-, provides an adequate modeling framework for such systems. The basic insight in coalgebraic modelling is that of representing state-based systems by functions of type p : X −→ FX (1) which, for every... |