## Incremental pattern-based coinduction for process algebra and its Isabelle formalization

Citations: | 3 - 1 self |

### BibTeX

@MISC{Popescu_incrementalpattern-based,

author = {Andrei Popescu and Elsa L. Gunter},

title = {Incremental pattern-based coinduction for process algebra and its Isabelle formalization},

year = {}

}

### OpenURL

### Abstract

Abstract. We present a coinductive proof system for bisimilarity in transition systems specifiable in the de Simone SOS format. Our coinduction is incremental, in that it allows building incrementally an a priori unknown bisimulation, and pattern-based, in that it works on equalities of process patterns (i.e., universally quantified equations of process terms containing process variables), thus taking advantage of equational reasoning in a “circular ” manner, inside coinductive proof loops. The proof system has been formalized and proved sound in Isabelle/HOL. 1

### Citations

3313 |
Communication and Concurrency
- Milner
- 1989
(Show Context)
Citation Context ...n [7, 10] for first-order logic and the µ-calculus, respectively.) The second major source of inspiration was the notion of coinduction proofs up to bisimilarity and arbitrary contexts, introduced in =-=[11, 25]-=- and developed in [35, 36]. This idea also appears in a general coalgebraic setting in [5] and is illustrated by extensive examples in, e.g., [34]. The convenience of performing unrestricted equationa... |

463 |
The pi-calculus: a Theory of Mobile Processes
- Sangiorgi, Walker
- 2001
(Show Context)
Citation Context ...logic and the µ-calculus, respectively.) The second major source of inspiration was the notion of coinduction proofs up to bisimilarity and arbitrary contexts, introduced in [11, 25] and developed in =-=[35, 36]-=-. This idea also appears in a general coalgebraic setting in [5] and is illustrated by extensive examples in, e.g., [34]. The convenience of performing unrestricted equational reasoning relies essenti... |

198 |
Bisimulation can’t be traced
- Bloom, Istrail, et al.
- 1995
(Show Context)
Citation Context ... the aforementioned “closed” version. 6Because of the particular format of the rules, bis is a congruence on terms. This is in fact true for rule formats more expressive than the one considered here =-=[6, 15, 33]-=-. However, we shall need to exploit a stronger property specific to the de Simone format, namely: whenever θ is a congruence, it follows that θ ∩ (Retr θ) is also a congruence. Let, for any relation θ... |

192 |
Structured operational semantics and bisimulation as a congruence
- Groote, Vaandrager
- 1992
(Show Context)
Citation Context ...- [ ] :: term × (var ⇒ term) ⇒ term, such that T [σ] is the term obtained from T by substituting all its variables X by σ X. Next we represent the meta-SOS notion of a transition-system specification =-=[15, 27]-=-. Given any type α, the type α ftrans, of formal α-transitions, consists of pairs, written k � l, with k, l :: α, where k is called the source and l the target. We fix a type act, of actions, ranged o... |

125 |
A complete inference system for a class of regular behaviours
- Milner
- 1984
(Show Context)
Citation Context ...ding the goal, without requiring an a priori constructed bisimulation relation. Our results apply to a wide class of process algebras. Related work. Unique fixpoint induction for CCS and its variants =-=[24, 17, 26]-=- is an early notion of proof-theoretic circularity for coinduction applicable to situations where circularity is explicit in the SOS by means of (guarded) fixpoint equations. We conjecture that unique... |

83 | The Bisimulation Proof Method
- Sangiorgi
(Show Context)
Citation Context ...ion θ, congCl θ be its congruence closure. From the above, we infer a powerful “up to” coinduction rule (that is, up to bisimilarity and up to arbitrary contexts), due to de Simone [11] and Sangiorgi =-=[35]-=-, improving on traditional coinduction: Theorem 1. For all θ :: rel, if θ ⊆ Retr(congCl(θ ∪ bis)), then θ ⊆ bis. 3 The raw coinductive proof system We now present the core of our original theoretical ... |

39 | Structured Proofs in Isar/HOL - Nipkow - 2003 |

37 | Locales - a sectioning concept for Isabelle - Kammüller, Wenzel, et al. - 1999 |

30 |
The Maude system
- Clavel, Durán, et al.
- 1999
(Show Context)
Citation Context ...ntroduced in [14] in the behavioral specification language BOBJ [1], and then also implemented axiomatically in Isabelle under the “supervision” of the CoCASL specification language [16] and in Maude =-=[9]-=- as the circular coiductive prover CIRC [21, 20, 32]. A comparison of our proof system with CC is somewhat difficult to sketch, as it has to deal with different technical settings and to balance the a... |

29 | Proof systems for message-passing process algebras
- Hennessy, Lin
- 1996
(Show Context)
Citation Context ...ding the goal, without requiring an a priori constructed bisimulation relation. Our results apply to a wide class of process algebras. Related work. Unique fixpoint induction for CCS and its variants =-=[24, 17, 26]-=- is an early notion of proof-theoretic circularity for coinduction applicable to situations where circularity is explicit in the SOS by means of (guarded) fixpoint equations. We conjecture that unique... |

21 |
Higher-level synchronizing devices in meije-sccs
- Simone
- 1985
(Show Context)
Citation Context ...le continue, obtaining arbitrarily large proof trees. We show that deductions such as the above are sound for a wide class of process algebras – those specifiable by SOS rules in the de Simone format =-=[11]-=-. Our results have been given a formalization in Isabelle/HOL [3], which was desirable for two reasons: first, the very technical constructions (especially in Sec. 4) and arguments (in both Secs. 3 an... |

21 | Bisimilarity of open terms
- Rensink
- 2000
(Show Context)
Citation Context ...onvenience of performing unrestricted equational reasoning relies essentially on the “up to” coinduction principle, Theorem 1. Other related work includes frameworks for bisimilarity of open terms in =-=[30, 8, 4]-=- (also building on the seminal work from [11]), where open terms are considered universally quantified, as we do in this paper for universal bisimilarity. Our soundness result for ⊢ w.r.t. universal b... |

21 |
Processes as terms: Non-well-founded models for bisimulation
- Rutten
- 1992
(Show Context)
Citation Context ... the aforementioned “closed” version. 6Because of the particular format of the rules, bis is a congruence on terms. This is in fact true for rule formats more expressive than the one considered here =-=[6, 15, 33]-=-. However, we shall need to exploit a stronger property specific to the de Simone format, namely: whenever θ is a congruence, it follows that θ ∩ (Retr θ) is also a congruence. Let, for any relation θ... |

20 |
Elements of stream calculus (an extensive exercise in coinduction
- Rutten
(Show Context)
Citation Context ...isimilarity and arbitrary contexts, introduced in [11, 25] and developed in [35, 36]. This idea also appears in a general coalgebraic setting in [5] and is illustrated by extensive examples in, e.g., =-=[34]-=-. The convenience of performing unrestricted equational reasoning relies essentially on the “up to” coinduction principle, Theorem 1. Other related work includes frameworks for bisimilarity of open te... |

17 | Generalised coinduction
- Bartels
(Show Context)
Citation Context ...nspiration was the notion of coinduction proofs up to bisimilarity and arbitrary contexts, introduced in [11, 25] and developed in [35, 36]. This idea also appears in a general coalgebraic setting in =-=[5]-=- and is illustrated by extensive examples in, e.g., [34]. The convenience of performing unrestricted equational reasoning relies essentially on the “up to” coinduction principle, Theorem 1. Other rela... |

16 |
An application of co-inductive types in Coq: Verification of the alternating bit protocol
- Giménez
- 1996
(Show Context)
Citation Context ...future research, to our knowledge not pursued so far). Finally, in Coq [2], the interaction between its general-purpose support for building proofs and its coinductive types (as illustrated, e.g., in =-=[13]-=-) also leads to a form of incremental coinduction whose relationship with our approach is yet to be understood. Future work. The de Simone SOS format is already fairly general, covering a wide range o... |

15 | Cyclic proofs for first-order logic with inductive definitions
- Brotherston
- 2005
(Show Context)
Citation Context ...nted the main difficulty in our soundness proof. 14system has the advantage of “purity”. 3 (Here we should also remark some less related work: circular systems in logical form were also developed in =-=[7, 10]-=- for first-order logic and the µ-calculus, respectively.) The second major source of inspiration was the notion of coinduction proofs up to bisimilarity and arbitrary contexts, introduced in [11, 25] ... |

13 | Bisimilarity congruences for open terms and term graphs via tile logic
- Bruni, Frutos-Escrig, et al.
- 2000
(Show Context)
Citation Context ...onvenience of performing unrestricted equational reasoning relies essentially on the “up to” coinduction principle, Theorem 1. Other related work includes frameworks for bisimilarity of open terms in =-=[30, 8, 4]-=- (also building on the seminal work from [11]), where open terms are considered universally quantified, as we do in this paper for universal bisimilarity. Our soundness result for ⊢ w.r.t. universal b... |

12 |
CIRC: A circular coinductive prover
- Lucanu, Rosu
- 2007
(Show Context)
Citation Context ...cation language BOBJ [1], and then also implemented axiomatically in Isabelle under the “supervision” of the CoCASL specification language [16] and in Maude [9] as the circular coiductive prover CIRC =-=[21, 20, 32]-=-. A comparison of our proof system with CC is somewhat difficult to sketch, as it has to deal with different technical settings and to balance the advantages of both generality and specialization. To ... |

11 | Sos formats and meta-theory: 20 years after
- Mousavi, Reniers, et al.
(Show Context)
Citation Context ...- [ ] :: term × (var ⇒ term) ⇒ term, such that T [σ] is the term obtained from T by substituting all its variables X by σ X. Next we represent the meta-SOS notion of a transition-system specification =-=[15, 27]-=-. Given any type α, the type α ftrans, of formal α-transitions, consists of pairs, written k � l, with k, l :: α, where k is called the source and l the target. We fix a type act, of actions, ranged o... |

11 |
Circular coinduction: A proof theoretical foundation
- Rosu, Lucanu
- 2009
(Show Context)
Citation Context ...cation language BOBJ [1], and then also implemented axiomatically in Isabelle under the “supervision” of the CoCASL specification language [16] and in Maude [9] as the circular coiductive prover CIRC =-=[21, 20, 32]-=-. A comparison of our proof system with CC is somewhat difficult to sketch, as it has to deal with different technical settings and to balance the advantages of both generality and specialization. To ... |

8 |
A bisimulation-based method for proving the validity of equations in gsos languages
- Aceto, Cimini, et al.
(Show Context)
Citation Context ...al, due to the need to avoid name overlapping and compose side-conditions. Here, we count on its understanding by examples and by its abstract properties, but App. A gives the general definition. (In =-=[6, 4]-=-, where what we call “matched derived rules” are called “ruloids”, mdr is not even defined, but rather the existence of such an operator satisfying suitable properties (essentially our below soundness... |

8 |
Automatic Verification Of Distributed System: The Process Algebra Approach.” Formal Methods
- Inverardi, Priami
- 1996
(Show Context)
Citation Context ..., where the desired bisimulation relation needs to be fed by the user. Descriptions of more or less automatic software tools for proving bisimilarity in process algebra abound in the literature – see =-=[18, 22]-=- for overviews. While most of these tools are dedicated to (and optimized for) particular process algebras (and many to finite-state systems), ECRINS [12] is based precisely on generic process algebra... |

8 | A mechanized theory of the pi-calculus in HOL
- Melham
- 1994
(Show Context)
Citation Context ...ction for generalization is the allowance of bindings in the syntax of terms, including π-calculus-like bindings featuring scope extrusion (thus generalizing HOL-based settings for π-calculus such as =-=[23, 31]-=-). In our proof system for universal bisimilarity, ⊢, one has to come up with 3 In a sense, what these freezing operators do is to guard against coinduction up-to, not sound in general. So again, our ... |

8 |
A fully adequate shallow embedding of the π-calculus in Isabelle/HOL with mechanized syntax analysis
- Röckl, Hirschkoff
- 2003
(Show Context)
Citation Context ...ction for generalization is the allowance of bindings in the syntax of terms, including π-calculus-like bindings featuring scope extrusion (thus generalizing HOL-based settings for π-calculus such as =-=[23, 31]-=-). In our proof system for universal bisimilarity, ⊢, one has to come up with 3 In a sense, what these freezing operators do is to guard against coinduction up-to, not sound in general. So again, our ... |

5 | Iterative Circular Coinduction for CoCasl
- Hausmann, Mossakowski, et al.
- 2005
(Show Context)
Citation Context ...ications. It was introduced in [14] in the behavioral specification language BOBJ [1], and then also implemented axiomatically in Isabelle under the “supervision” of the CoCASL specification language =-=[16]-=- and in Maude [9] as the circular coiductive prover CIRC [21, 20, 32]. A comparison of our proof system with CC is somewhat difficult to sketch, as it has to deal with different technical settings and... |

5 |
CIRC: A behavioral verification tool based on circular coinduction
- Lucanu, Goriac, et al.
- 2009
(Show Context)
Citation Context ...cation language BOBJ [1], and then also implemented axiomatically in Isabelle under the “supervision” of the CoCASL specification language [16] and in Maude [9] as the circular coiductive prover CIRC =-=[21, 20, 32]-=-. A comparison of our proof system with CC is somewhat difficult to sketch, as it has to deal with different technical settings and to balance the advantages of both generality and specialization. To ... |

4 | µ-calculus with explicit points and approximations
- Dam, Gurov
(Show Context)
Citation Context ...nted the main difficulty in our soundness proof. 14system has the advantage of “purity”. 3 (Here we should also remark some less related work: circular systems in logical form were also developed in =-=[7, 10]-=- for first-order logic and the µ-calculus, respectively.) The second major source of inspiration was the notion of coinduction proofs up to bisimilarity and arbitrary contexts, introduced in [11, 25] ... |

3 |
Proving process calculi translations in ECRINS: The pureLOTOS → MEIJE example
- Doumenc, Madelaine, et al.
- 1990
(Show Context)
Citation Context ...ss algebra abound in the literature – see [18, 22] for overviews. While most of these tools are dedicated to (and optimized for) particular process algebras (and many to finite-state systems), ECRINS =-=[12]-=- is based precisely on generic process algebra in de Simone format, meaning that the results of this paper on incremental coinduction apply directly to that setting (and, interestingly, a form of coin... |

2 |
Verification tools from the CONCUR project. http://wwwsop.inria.fr/meije/papers/concur-tools
- Madelaine
(Show Context)
Citation Context ..., where the desired bisimulation relation needs to be fed by the user. Descriptions of more or less automatic software tools for proving bisimilarity in process algebra abound in the literature – see =-=[18, 22]-=- for overviews. While most of these tools are dedicated to (and optimized for) particular process algebras (and many to finite-state systems), ECRINS [12] is based precisely on generic process algebra... |

2 |
On process equivalence = equation solving in ccs
- Monroy, Bundy, et al.
(Show Context)
Citation Context ...ding the goal, without requiring an a priori constructed bisimulation relation. Our results apply to a wide class of process algebras. Related work. Unique fixpoint induction for CCS and its variants =-=[24, 17, 26]-=- is an early notion of proof-theoretic circularity for coinduction applicable to situations where circularity is explicit in the SOS by means of (guarded) fixpoint equations. We conjecture that unique... |

2 |
Weak bisimilarity coalgebraically
- Popescu
- 2009
(Show Context)
Citation Context ... with guarded recursion, and even de Simone systems under weak bisimilarity (since for them weak bisimilarity can be regarded as strong bisimilarity for trace-based de Simone systems, as suggested in =-=[29]-=-). An extension of our incremental coinductive technique to more general formats such as GSOS [6] or tyft/tyxt [15] is of course desirable. Another direction for generalization is the allowance of bin... |

2 |
Automating soundness proofs
- Weerdenburg
(Show Context)
Citation Context ...simulation under formal hypotheses from [11, 30]; on the other, w.r.t. to the relation from [4] (which is essentially universal bisimilarity in any conservative extension of the SOS system). Finally, =-=[37]-=- discusses bisimilarity proofs in a mildly specialized Gentzen system for FOL. All works cited in this paragraph discuss non-incremental proof systems, where the desired bisimulation relation needs to... |