## BOOLEAN PRECONGRUENCES

### BibTeX

@MISC{Levy_booleanprecongruences,

author = {Paul Blain Levy},

title = {BOOLEAN PRECONGRUENCES},

year = {}

}

### OpenURL

### Abstract

Abstract. We investigate the possible ways of ordering terms of ground type in a nondeterministic (or deterministic) language that contains erroneous behaviours such as divergence, crash or deadlock. We see that the ordering at boolean type, called a “boolean precongruence”, is key: it determines the ordering at other ground types, and induces a contextual preorder. We examine the circumstances in which amb is monotone, and in which the ordering at Sierpinski type or even zero type suffices. Each boolean precongruence gives a way of lifting relations, leading to a power-poset construction. We obtain a notion of simulation, and give general conditions for when a

### Citations

586 |
The Theory and Practice of Concurrency
- Roscoe
- 1997
(Show Context)
Citation Context ...s six ordered models, and they can be regarded as contextual preorders for the NDBPs shown in Fig. 8. Although CSP does not have a boolean type in our sense, the full abstraction results presented in =-=[Ros98]-=- for the equivalences can be adapted to the preorders. Curiously, the three rows correspond to LOWER, UPPER and SMASH, which are NDBPs for divergence alone. Each column extends these to deadlock in a ... |

395 |
LCF considered as a programming language
- Plotkin
- 1977
(Show Context)
Citation Context ...⊥, ⊑A) is monotone for any sets A and I. We omit the proof, as we give a more general theorem below—Lemma 4.4(2). Remark 2.24. Deterministic parallel operators such as parallel-or and parallel-exists =-=[Plo77]-=- do not cause the same problems as amb. They are monotone wrt all of the DBPs and NDBPs for divergence (assuming the empty set is excluded).16 PAUL BLAIN LEVY 3. Contexts and Tests 3.1. Contextual Pr... |

291 | The linear time–Branching time spectrum - Glabbeek - 1990 |

214 | A powerdomain construction - Plotkin - 1976 |

120 |
A domain equation for bisimulation
- Abramsky
- 1991
(Show Context)
Citation Context ...equivalently when it is a LOWER simulation and its converse is an UPPER simulation • a CONVEX simulation [Las98, Mor98, Ong93, Pit01] (or pre-bisimulation [HP80, Mil81, Wal90] or partial bisimulation =-=[Abr91]-=-) when it is both a LOWER simulation and an UPPER simulation • a REFINEMENT simulation [How96, Las98, Mor98, Pit01] when its converse is an INCLUSION simulation • a LOWER CONGRUENCE simulation or lowe... |

105 | Proving congruence of bisimulation in functional programming languages - Howe - 1996 |

96 | Equality in lazy computation systems - Howe - 1989 |

96 | The linear time–branching time spectrum I; the semantics of concrete, sequential processes - Glabbeek - 2001 |

78 | Power Domains - Smyth - 1978 |

76 |
On observing nondeterminism and concurrency
- Milner
- 1979
(Show Context)
Citation Context ...sition systems with divergence and deadlock, the paper [AH92] defines simulation corresponding to the NDBP illustrated in Fig. 12.34 PAUL BLAIN LEVY 4.4. Modal Logic For Similarity. It is well known =-=[HM80]-=- that similarity and bisimilarity can be characterized using Hennessy-Milner logic. We want to adapt this theory in the presence of divergence and other errors. For a finite set E, the formulas for E-... |

56 | Modal logics for communicating systems - Stirling - 1987 |

49 | Fully abstract semantics for observably sequential languages - Cartwright, Curien, et al. - 1994 |

47 | Bisimulations and divergence - Walker - 1988 |

40 | Termination, deadlock, and divergence
- Aceto, Hennessy
(Show Context)
Citation Context ...ither lower nor upper bisimilar ACETO-HENNESSY {t,f} {t,f,c} {t} {t,c} {t,f,d}={t,f,d,c} {f} {f,c} {t,d}=t,d,c} {c} {f,d}={f,d,c} {d}={d,c} Figure 12: NDBP for divergence (d) and deadlock (c) used in =-=[AH92]-=- equivalently when R and its converse are INCLUSION simulations, equivalently when R and its converse are REFINEMENT simulations, equivalently when R and its converse are CONVEX simulations. We illust... |

31 | Relational Reasoning about Functions and Nondeterminism
- Lassen
- 1998
(Show Context)
Citation Context ...bras. Beck’s theorem tells us that K is invertible as an unenriched functor. K is locally order-reflecting because U ⊑ is, so its inverse is a Poset-enriched functor. (2) Similar. 4.3. Simulation. In =-=[Las98]-=-, a nondeterministic language with divergence is studied, and various notions of simulation for a language are defined, named “lower”, “upper” and “convex”. We now see how they can be systematically d... |

29 | A logic for coalgebraic simulation - Baltag - 2000 |

25 | Unbounded Nondeterminism in CSP - Roscoe - 1988 |

21 | A term model for CCS
- HENNESSY, PLOTKIN
- 1980
(Show Context)
Citation Context ...mpty, K must be nonempty, but L is empty so we must have {t, f} + ∅ ⊑ {t} + ∅. By the dual argument {t} + ∅ ⊑ {t, f} + ∅, contradicting our consistency assumption by Lemma 2.7. (d) Trivial. Following =-=[HT00]-=-, we can express the general construction R ↦→⊑R in terms of the special case A ↦→⊑A. Proposition 4.3. Let A and B be sets and A R �� B a relation. We write A π ������������ graph(R) π ′ ���� ���� ���... |

19 | Equivalences on observable processes - Ulidowski - 1992 |

15 |
A representable approach to finite nondeterminism
- Anderson, Power
- 1997
(Show Context)
Citation Context ...econgruence of interest on our language L, then Prop. 2.19 tells us that the ordering at each ground type is determined by the NDBP (or DBP) provided by �. Remark 2.20. A similar analysis is given in =-=[AP97]-=-, Thm. 17 (attributed to Gordon Plotkin), in the setting of NDBCs for divergence. Our generalization is to consider preorders rather than equivalence relations.BOOLEAN PRECONGRUENCES 15 INCONSISTENT ... |

14 |
An alternative order for the failures model
- Roscoe
- 1988
(Show Context)
Citation Context ...ted), stable failure set, divergence set and infinite trace set, each saturated with extensions of divergences. For I and U, as well as refinement, a stronger ordering called “definedness” is studied =-=[Ros92]-=-. That gives six ordered models, and they can be regarded as contextual preorders for the NDBPs shown in Fig. 8. Although CSP does not have a boolean type in our sense, the full abstraction results pr... |

8 | Locally boolean domains - Laird |

7 | Bidomains and full abstraction for countable nondeterminism
- Laird
- 2006
(Show Context)
Citation Context ...ntersection with LOWER (viz. CONVEX). We do not list the NDBPs for divergence and crash, as they are numerous. But in [Lai06, Lai07b, Lai09], five NDBPs are studied, displayed in Fig. 6. (Warning: in =-=[Lai06]-=-, both the may and the costable preorders are reversed.) The following is repeatedly useful. Lemma 2.7. Let ⊑ be a NDBP for a finite set E. Then for any D, D ′ ⊆ E, we have Proof. (⇒) is obvious, and ... |

3 | Bistable biorders: A sequential domain theory
- Laird
(Show Context)
Citation Context ...9 tells us that any preorder determined by contexts of ground type must arise in this way. We note some preorders appearing in the denotational literature that turn out to be contextual preorders. In =-=[Lai07a]-=-, a deterministic language with divergence and crash is studied, and a model described using three orderings ≤ E , ≤ B , ↕. These are in fact the contextual preorders induced, respectively, by BIPOINT... |

1 |
Termination, deadlock and divergence in the lambda-calculus with multiplicities
- Boudol, Laneve
- 1995
(Show Context)
Citation Context ... false {inl true} ⊑ {inl false} Both are called INCONSISTENT. Any other boolean precongruence is said to be consistent. Remark 2.8. A deterministic calculus with divergence and deadlock is studied in =-=[BL95]-=-, but it does not include sequencing. For this reason, the “vertical” ordering presented there is not a DBP, although the “standard” and “flat” orderings correspond to DOUBLE POINTED and STABLE respec... |

1 | Full abstraction for recursive types with control and countable nondeterminism - Laird - 2007 |

1 | Nondeterminism and observable sequentiality - Laird - 2009 |

1 |
Labelled transition systems over a game graph
- Lassen, Levy
- 2009
(Show Context)
Citation Context ...+ Errors(x ′) • if {t, f} + Errors(x) ̸⊑ {t} + Errors(x ′ ) then x a y implies that there is y ′ such that x ′ a �� y ′ and y R y ′ 2 More generally, we can consider LTSs over a game graph of actions =-=[LL09]-=-. All of our theory works in that more general situation.� � � � � 32 PAUL BLAIN LEVY • if {t} + Errors(x) ̸⊑ {t, f} + Errors(x ′ ) then x ′ a such that x �� y and y R y ′. (2) The greatest simulatio... |

1 |
Divergence-least semantics of amb is Hoare. 3rd APPSEM
- Lassen, Levy, et al.
(Show Context)
Citation Context ...dmit this generalization. Other variants are where the language is deterministic, or provides McCarthy’s amb. The non-monotonicity of amb wrt certain orderings is notorious, so—continuing the work of =-=[LLP05]-=-—we look at which orders are acceptable. Each boolean precongruence gives rise to a contextual preorder, and in Sect. 3.1 we see some instances of these preorders that have appeared in the literature.... |

1 |
A modal characterization of observable machine-behaviour
- workshop, Germany
- 1981
(Show Context)
Citation Context ... simulations, equivalently when R and its converse are REFINEMENT simulations, equivalently when R and its converse are CONVEX simulations. We illustrate these distinctions with an example taken from =-=[Pit01]-=-. Consider the two root nodes in Fig. 11. They are neither lower bisimilar nor upper bisimilar. But they are mutually convex similar, by the following calculation: a.diverge or a.a.a.diverge = a.diver... |