## Conditional Circular Coinductive Rewriting with Case Analysis (2002)

Citations: | 18 - 1 self |

### BibTeX

@MISC{Goguen02conditionalcircular,

author = {Joseph Goguen and Kai Lin and Grigore Rosu},

title = {Conditional Circular Coinductive Rewriting with Case Analysis},

year = {2002}

}

### Years of Citing Articles

### OpenURL

### Abstract

We argue for an algorithmic approach to behavioral proofs, review the hidden algebra approach, develop circular coinductive rewriting for conditional goals, extend it with case analysis, and give some examples.

### Citations

298 | Universal coalgebra: A theory of systems
- Rutten
(Show Context)
Citation Context ...ations not behavioral and/or with more than one hidden argument; [20, 19] give proofs. Since final algebras need not exist in this setting, Γ -behavioral equivalence cannot use them as coalgebra does =-=[22, 16, 15]-=-. Theorem 1. Given a hidden subsignature Γ of Σ and a hidden Σ-algebra A, then Γ -behavioral equivalence is the largest hidden Γ -congruence on A. Behavioral satisfaction of conditional equations can ... |

228 | A tutorial on (co)algebras and (co)induction
- Jacobs, Rutten
- 1997
(Show Context)
Citation Context ...ations not behavioral and/or with more than one hidden argument; [20, 19] give proofs. Since final algebras need not exist in this setting, Γ -behavioral equivalence cannot use them as coalgebra does =-=[22, 16, 15]-=-. Theorem 1. Given a hidden subsignature Γ of Σ and a hidden Σ-algebra A, then Γ -behavioral equivalence is the largest hidden Γ -congruence on A. Behavioral satisfaction of conditional equations can ... |

111 | A hidden agenda
- Goguen, Malcolm
(Show Context)
Citation Context ...for states, with states behaviorally equivalent iff they are indistinguishable under a given set of experiments; we have formalized this as hidden algebra, originating in [9] and further developed in =-=[11, 20, 18, 19]-=- and other papers. While standard equational proof techniques like induction are suitable for data, coinduction or context induction is generally needed for non-trivial behavioral properties, typicall... |

72 |
Types as theories
- Goguen
- 1991
(Show Context)
Citation Context ...s for data from hidden sorts for states, with states behaviorally equivalent iff they are indistinguishable under a given set of experiments; we have formalized this as hidden algebra, originating in =-=[9]-=- and further developed in [11, 20, 18, 19] and other papers. While standard equational proof techniques like induction are suitable for data, coinduction or context induction is generally needed for n... |

52 | Observational logic
- Hennicker, Bidoit
- 1999
(Show Context)
Citation Context ...s, as At : A → (A (var(t)−{⋆}) → A) with the obvious meaning. 2.2 Behavioral Specification and Satisfaction We generalize the hidden algebra of [9, 11] to include variants such as observational logic =-=[1, 3, 14]-=- and coherent hidden algebra [6, 7]. See [19] for a detailed presentation of variants, history, many other concepts, and proofs for some results mentioned here. Two important variants of behavioral lo... |

46 | Circular Coinductive Rewriting
- Goguen, Lin, et al.
- 2000
(Show Context)
Citation Context ...g. Hence our recent research concerns coinduction algorithms that require no human intervention. The languages we know that support automated behavioral reasoning are Spike [2], CafeOBJ [6], and BOBJ =-=[10]-=-, the first based on context induction, and the other two on forms of coinduction. The powerful coinduction algorithm now in BOBJ has developed through several stages. The first restricts ordinary ter... |

46 |
Context induction: a proof principle for behavioural abstractions and algebraic implementations, Formal Aspects of Computing 3 (4
- Hennicker
- 1991
(Show Context)
Citation Context ...rst name more intuitive because from a coalgebraic point of view, coinduction is all about finding circularities. Another way to look at circular coinduction is through the lense of context induction =-=[13]-=-. To clarify this discussion, we replace the operations [⋆] by C[⋆]. Then our rule says that to show (∀X) t = t ′ if c, one can assume (∀X) C[t] = C[t ′ ] if c and then show (∀X, W ) C[δ(t, W )] = C[δ... |

42 | Hiding more of hidden algebra
- Goguen, Rosu
- 1999
(Show Context)
Citation Context ...r t ′ , and then the substitution also comes for free. Our most powerful rule is Circular Coinduction, but first we recall the important notion of cobasis, originating in [20] and later simplified in =-=[18, 12, 21]-=-. For this paper, a cobasis ∆ is a subset of Γ that generates enough experiments, in the sense that no Γ -experiment can distinguish two states that cannot be distinguished by these experiments. Findi... |

35 | Behavioral coherence in object-oriented algebraic specification
- Diaconescu
- 1998
(Show Context)
Citation Context ...h the obvious meaning. 2.2 Behavioral Specification and Satisfaction We generalize the hidden algebra of [9, 11] to include variants such as observational logic [1, 3, 14] and coherent hidden algebra =-=[6, 7]-=-. See [19] for a detailed presentation of variants, history, many other concepts, and proofs for some results mentioned here. Two important variants of behavioral logic are the fixed data and the loos... |

34 | Observer Complete Definitions are Behaviourally Coherent
- Bidoit, Hennicker
- 1999
(Show Context)
Citation Context ...Γ and all hidden constants are behaviorally congruent for B. Of course, E may be such that other operations are also congruent. An easy criterion for congruence is given in [20] and is generalized in =-=[4, 21]-=-; [20] also shows that congruent operations can be added to or removed from Γ as desired when no equation in E has hidden sorted equalities in its condition (which is a common situation). 3 Behavioral... |

33 |
CafeOBJ Report: The Language, Proof Techniques, and Methodologies for Object-Oriented Algebraic Specification
- Diaconescu, Futatsugi
- 1998
(Show Context)
Citation Context ... model checking. Hence our recent research concerns coinduction algorithms that require no human intervention. The languages we know that support automated behavioral reasoning are Spike [2], CafeOBJ =-=[6]-=-, and BOBJ [10], the first based on context induction, and the other two on forms of coinduction. The powerful coinduction algorithm now in BOBJ has developed through several stages. The first restric... |

30 | Mongruences and cofree coalgebras
- Jacobs
- 1995
(Show Context)
Citation Context ...ations not behavioral and/or with more than one hidden argument; [20, 19] give proofs. Since final algebras need not exist in this setting, Γ -behavioral equivalence cannot use them as coalgebra does =-=[22, 16, 15]-=-. Theorem 1. Given a hidden subsignature Γ of Σ and a hidden Σ-algebra A, then Γ -behavioral equivalence is the largest hidden Γ -congruence on A. Behavioral satisfaction of conditional equations can ... |

27 | Hidden congruent deduction
- Rosu, Goguen
- 2000
(Show Context)
Citation Context ...for states, with states behaviorally equivalent iff they are indistinguishable under a given set of experiments; we have formalized this as hidden algebra, originating in [9] and further developed in =-=[11, 20, 18, 19]-=- and other papers. While standard equational proof techniques like induction are suitable for data, coinduction or context induction is generally needed for non-trivial behavioral properties, typicall... |

25 | Incompleteness of behavioral logics
- Buss, Ro¸su
(Show Context)
Citation Context ... or context induction is generally needed for non-trivial behavioral properties, typically requiring extensive human intervention. This is not surprising, since behavioral satisfaction is Π 0 2 -hard =-=[5]-=-, so that no algorithm can prove [or disprove] all behaviorally true [or false] statements. However, successful technology transfer requires placing less demand on users, as illustrated by the success... |

24 | Observational proofs with critical contexts
- Berregeb, Bouhoula, et al.
- 1998
(Show Context)
Citation Context ...he success of model checking. Hence our recent research concerns coinduction algorithms that require no human intervention. The languages we know that support automated behavioral reasoning are Spike =-=[2]-=-, CafeOBJ [6], and BOBJ [10], the first based on context induction, and the other two on forms of coinduction. The powerful coinduction algorithm now in BOBJ has developed through several stages. The ... |

18 |
Behavioral theories and the proof of behavioral properties
- Bidoit, Hennicker
- 1996
(Show Context)
Citation Context ...s, as At : A → (A (var(t)−{⋆}) → A) with the obvious meaning. 2.2 Behavioral Specification and Satisfaction We generalize the hidden algebra of [9, 11] to include variants such as observational logic =-=[1, 3, 14]-=- and coherent hidden algebra [6, 7]. See [19] for a detailed presentation of variants, history, many other concepts, and proofs for some results mentioned here. Two important variants of behavioral lo... |

18 |
Towards the One-Tiered Design of Data Types and Transition Systems
- Padawitz
- 1998
(Show Context)
Citation Context ...∼ a ′ i for i = 1...n. An operation σ is Γ -behaviorally congruent for A iff it is congruent for ≡Γ Σ . We often write just congruent for behaviorally congruent (a similar notion is given by Padawitz =-=[17]-=-). A hidden Γ -congruence on A is a (V ∪ H)-equivalence on A which is the identity on visible sorts and for which each operation in Γ is congruent. 1 This is called “coherent” in [7], where the concep... |

17 | Observational specifications and the indistinguishability assumption
- Bernot, Bidoit, et al.
- 1995
(Show Context)
Citation Context ...s, as At : A → (A (var(t)−{⋆}) → A) with the obvious meaning. 2.2 Behavioral Specification and Satisfaction We generalize the hidden algebra of [9, 11] to include variants such as observational logic =-=[1, 3, 14]-=- and coherent hidden algebra [6, 7]. See [19] for a detailed presentation of variants, history, many other concepts, and proofs for some results mentioned here. Two important variants of behavioral lo... |

12 | Circular Coinduction - Rosu, Goguen - 2001 |

6 | Behavioral coinductive rewriting
- Rosu
- 1999
(Show Context)
Citation Context ...for states, with states behaviorally equivalent iff they are indistinguishable under a given set of experiments; we have formalized this as hidden algebra, originating in [9] and further developed in =-=[11, 20, 18, 19]-=- and other papers. While standard equational proof techniques like induction are suitable for data, coinduction or context induction is generally needed for non-trivial behavioral properties, typicall... |

5 | Observer complete de are behaviourally coherent - Bidoit, Hennicker - 1999 |

2 |
Rewriting can verify distributed real-time systems { how to verify in CafeOBJ
- Futatsugi, Ogata
- 2001
(Show Context)
Citation Context ...rotocol and the Petersen mutual exclusion algorithm; in addition, we reduced the proof score for liveness of a real-time asynchronous data transmission protocol done in CafeOBJ by Futatsugi and Ogata =-=[8]-=-, by a factor of about ten. Circularities have also been essential for many non-trivial examples, but here is a simple example, proving an identity that is familiar in functional programming, and also... |

2 | Observational speci and the indistinguishability assumption - Bernot, Bidoit, et al. - 1995 |