## Psi-Calculi in Isabelle

Venue: | In Proc of the 22nd Conference on Theorem Proving in Higher Order Logics (TPHOLs), volume 5674 of LNCS |

Citations: | 11 - 4 self |

### BibTeX

@INPROCEEDINGS{Bengtson_psi-calculiin,

author = {Jesper Bengtson and Joachim Parrow},

title = {Psi-Calculi in Isabelle},

booktitle = {In Proc of the 22nd Conference on Theorem Proving in Higher Order Logics (TPHOLs), volume 5674 of LNCS},

year = {},

pages = {99--114}

}

### OpenURL

### Abstract

Abstract. Psi-calculi are extensions of the pi-calculus, accommodating arbitrary nominal datatypes to represent not only data but also communication channels, assertions and conditions, giving it an expressive power beyond the applied pi-calculus and the concurrent constraint picalculus. We have formalised psi-calculi in the interactive theorem prover Isabelle using its nominal datatype package. One distinctive feature is that the framework needs to treat binding sequences, as opposed to single binders, in an efficient way. While different methods for formalising single binder calculi have been proposed over the last decades, representations for such binding sequences are not very well explored. The main effort in the formalisation is to keep the machine checked proofs as close to their pen-and-paper counterparts as possible. We discuss two approaches to reasoning about binding sequences along with their strengths and weaknesses. We also cover custom induction rules to remove the bulk of manual alpha-conversions. 1

### Citations

3400 | Communicating Sequential Processes - Hoare - 1985 |

3204 | Communication and Concurrency - Milner - 1989 |

1329 | A calculus of communicating systems - Milner - 1980 |

1115 | The Lambda Calculus: Its Syntax and Semantics - Barendregt - 1981 |

995 | A calculus of mobile processes - Milner, Parrow, et al. - 1992 |

785 | A calculus for cryptographic protocols: The spi calculus - Abadi, Gordon - 1997 |

718 |
Isabelle/HOL — A Proof Assistant for HigherOrder Logic
- Nipkow, Paulson, et al.
- 2002
(Show Context)
Citation Context ...er than ingenuity and complicated new constructions they should be amenable to proof mechanisation. Our contribution in this paper is to implement a family of application oriented calculi in Isabelle =-=[12]-=-. The calculi we consider are the so called psi-calculi [5], obtained by extending the basic untyped pi-calculus with the following parameters: (1) a set of data terms, which can function as both2 Je... |

655 | Concurrency and automata on infinite sequences - Park - 1981 |

360 | Process algebra for synchronous communication - Bergstra, Klop - 1984 |

303 | N.: Lambda Calculus Notation with Nameless Dummies, a Tool for Automatic Formula Manipulation.; Indag - Bruijn - 1972 |

286 |
The Lambda-Calculus, its Syntax and Semantics, volume 103
- Barendregt
- 1984
(Show Context)
Citation Context ...channels. The main difficulty in representing calculi such as the lambda-, pi- or psicalculi is to find an efficient treatment of binders. Informal proofs often use the Barendregt variable convention =-=[4]-=-, that everything bound is unique. This convention provides a tractable abstraction when doing proofs involving binders, but it has recently been proven to be unsound in the general case [16]. Theorem... |

273 | Mobile values, new names, and secure communication
- Abadi, Fournet
- 2001
(Show Context)
Citation Context ...ry many cases to check and the temptation to resort to formulations such as “by analogy with . . . ” or “is easily seen. . . ” can be overwhelming. For examples in point, both the applied pi-calculus =-=[1]-=- and the concurrent constraint pi-calculus [8] have recently been discovered to have flaws or incompletenesses in the sense that the claimed compositionality results do not hold [5]. Since such proofs... |

263 | An unsolvable problem of elementary number theory - Church |

239 | Communicating and Mobile Systems: the Pi-Calculus - Milner - 1999 |

206 | A new approach to abstract syntax with variable binding - Gabbay, Pitts - 2002 |

162 | Nominal Logic, A first order theory of names and binding
- Pitts
(Show Context)
Citation Context ...s, but it has recently been proven to be unsound in the general case [16]. Theorem provers have commonly used approaches based on de Bruijn indices [9], higher order abstract syntax, or nominal logic =-=[13]-=-. We use the nominal datatype package in Isabelle [15], and its strategy for dealing with single binders. Recent work by Aydemir et. al. introduce the locally nameless framework [2] which might be an ... |

154 |
de Bruijn. Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the church-rosser theorem
- G
- 1972
(Show Context)
Citation Context ...table abstraction when doing proofs involving binders, but it has recently been proven to be unsound in the general case [16]. Theorem provers have commonly used approaches based on de Bruijn indices =-=[9]-=-, higher order abstract syntax, or nominal logic [13]. We use the nominal datatype package in Isabelle [15], and its strategy for dealing with single binders. Recent work by Aydemir et. al. introduce ... |

137 | Mechanized metatheory for the masses: The PoplMark challenge - Aydemir, Bohannon, et al. - 2005 |

86 | A polyadic pi-calculus: a tutorial - Milner - 1991 |

83 | Engineering formal metatheory
- Aydemir, Charguéraud, et al.
- 2008
(Show Context)
Citation Context ... or nominal logic [13]. We use the nominal datatype package in Isabelle [15], and its strategy for dealing with single binders. Recent work by Aydemir et. al. introduce the locally nameless framework =-=[2]-=- which might be an improvement since the infrastructure is small and elegant. One of our main contributions in the present paper is to extend the strategy to finite sequences of binders. Though it is ... |

82 | Nominal techniques in Isabelle/HOL
- Urban
(Show Context)
Citation Context ...e general case [16]. Theorem provers have commonly used approaches based on de Bruijn indices [9], higher order abstract syntax, or nominal logic [13]. We use the nominal datatype package in Isabelle =-=[15]-=-, and its strategy for dealing with single binders. Recent work by Aydemir et. al. introduce the locally nameless framework [2] which might be an improvement since the infrastructure is small and eleg... |

50 | Explicit fusions - Wischik, Gardner |

47 | Cc-pi: A constraint-based language for specifying service level agreements - Buscemi, Montanari - 2007 |

38 | Locales and locale expressions in Isabelle/Isar
- Ballarin
(Show Context)
Citation Context ...lculus. We also require an entailment relation, a notion of channel equality and composition of assertions. Isabelle has good support for reasoning about parametric systems through the use of locales =-=[3]-=-. 4.1 Substitution properties We require a substitution function on agents. Since terms, assertions and conditions of psi-calculi are parameters, a locale is created to ensure that a set of substituti... |

30 | A brief overview of HOL4 - Slind, Norrish - 2008 |

29 | On the expressive power of polyadic synchronisation in π-calculus - Carbone, Maffeis |

20 | Barendregt’s variable convention in rule inductions
- Urban, Berghofer, et al.
- 2007
(Show Context)
Citation Context ...onvention [4], that everything bound is unique. This convention provides a tractable abstraction when doing proofs involving binders, but it has recently been proven to be unsound in the general case =-=[16]-=-. Theorem provers have commonly used approaches based on de Bruijn indices [9], higher order abstract syntax, or nominal logic [13]. We use the nominal datatype package in Isabelle [15], and its strat... |

17 |
π-calculus in (co)inductive type theories
- Honsell, Miculan, et al.
- 2001
(Show Context)
Citation Context ...ny other work on implementing calculi of this calibre in a proof assistant such as Isabelle. The closest related work are implementations of the basic pi-calculus, by ourselves [6] and also by others =-=[10, 11, 14]-=-. Neither are we aware of any other general technique for multiple binders, other than the yet unpublished work by Berghofer and Urban which we describe in Section 3. The rest of the paper is structur... |

17 | The Isabelle framework - Wenzel, Paulson, et al. |

16 | Psi-calculi: Mobile processes, nominal data, and logic
- BENGTSON, JOHANSSON, et al.
- 2009
(Show Context)
Citation Context ...pplied pi-calculus [1] and the concurrent constraint pi-calculus [8] have recently been discovered to have flaws or incompletenesses in the sense that the claimed compositionality results do not hold =-=[5]-=-. Since such proofs often require stamina and attention to detail rather than ingenuity and complicated new constructions they should be amenable to proof mechanisation. Our contribution in this paper... |

14 |
Formalising the pi-calculus using nominal logic
- Bengtson, Parrow
- 2008
(Show Context)
Citation Context .... We are not aware of any other work on implementing calculi of this calibre in a proof assistant such as Isabelle. The closest related work are implementations of the basic pi-calculus, by ourselves =-=[6]-=- and also by others [10, 11, 14]. Neither are we aware of any other general technique for multiple binders, other than the yet unpublished work by Berghofer and Urban which we describe in Section 3. T... |

11 | Open Bisimulation for the concurrent Constraint Pi-Calculus
- Buscemi, Montanari
- 2008
(Show Context)
Citation Context ...esort to formulations such as “by analogy with . . . ” or “is easily seen. . . ” can be overwhelming. For examples in point, both the applied pi-calculus [1] and the concurrent constraint pi-calculus =-=[8]-=- have recently been discovered to have flaws or incompletenesses in the sense that the claimed compositionality results do not hold [5]. Since such proofs often require stamina and attention to detail... |

10 | B.: Psi-calculi: a framework for mobile processes with nominal data and logic - Bengtson, Johansson, et al. |

8 | C.: Nominal inversion principles - Berghofer, Urban - 2008 |

8 | A full formalisation of pi-calculus theory in the calculus of constructions
- Hirschkoff
- 1997
(Show Context)
Citation Context ...ny other work on implementing calculi of this calibre in a proof assistant such as Isabelle. The closest related work are implementations of the basic pi-calculus, by ourselves [6] and also by others =-=[10, 11, 14]-=-. Neither are we aware of any other general technique for multiple binders, other than the yet unpublished work by Berghofer and Urban which we describe in Section 3. The rest of the paper is structur... |

8 |
A fully adequate shallow embedding of the π-calculus in Isabelle/HOL with mechanized syntax analysis
- Röckl, Hirschkoff
- 2003
(Show Context)
Citation Context ...ny other work on implementing calculi of this calibre in a proof assistant such as Isabelle. The closest related work are implementations of the basic pi-calculus, by ourselves [6] and also by others =-=[10, 11, 14]-=-. Neither are we aware of any other general technique for multiple binders, other than the yet unpublished work by Berghofer and Urban which we describe in Section 3. The rest of the paper is structur... |

8 | Formalizing process calculi - Bengtson - 2010 |

8 | A mechanized theory of the pi-calculus in HOL - Melham - 1994 |

6 | The locally nameless representation - Charguéraud - 2012 |

4 | Broadcast psicalculi with an application to wireless protocols - Borgström, Huang, et al. - 2011 |

4 | I.: pi-calculus in (co)inductive-type theory - Honsell, Miculan, et al. - 2001 |

4 | C.: General bindings and alpha-equivalence - Urban, Kaliszyk - 2012 |

3 | C.: A new foundation for Nominal Isabelle - Huffman, Urban - 2010 |

2 | Simply-typed lambda-calculus with let and tuple patterns. http://isabelle.in.tum.de/repos/ isabelle/file/81e8fdfeb849/src/HOL/Nominal/Examples/Pattern.thy - Berghofer - 2010 |

2 | A short presentation of Coq - Bertot - 2008 |

2 | M.: Implementing spi calculus using nominal techniques - Kahsai, Miculan - 2008 |

1 | Psi-calculi in Isabelle. Archive of Formal Proofs (2012). http://afp.sf.net/entries/Psi Calculi.shtml, Formal proof development - Bengtson |

1 | A.: Sorted psi-calculi with generalised pattern matching - Borgström, Gutkovas, et al. |

1 | A formalisation of the spi calculus in Coq (2007). Email to the Coq-club mailing list sent - Briais |

1 | Retrieved from http://isabelle.in.tum.de - Isabelle |