## From X to π; representing the classical sequent calculus

### Cached

### Download Links

Citations: | 12 - 12 self |

### BibTeX

@MISC{Bakel_fromx,

author = {Steffen Van Bakel and Luca Cardelli and Maria Grazia Vigliotti},

title = {From X to π; representing the classical sequent calculus},

year = {}

}

### OpenURL

### Abstract

Abstract. We study the π-calculus, enriched with pairing and non-blocking input, and define a notion of type assignment that uses the type constructor →. We encode the circuits of the calculus X into this variant of π, and show that all reduction (cut-elimination) and assignable types are preserved. Since X enjoys the Curry-Howard isomorphism for Gentzen’s calculus LK, this implies that all proofs in LK have a representation in π.

### Citations

1118 | The lambda calculus: its syntax and semantics - Barendregt - 1984 |

784 | A.D.: A calculus for cryptographic protocols: The spi calculus
- Abadi, Gordon
- 1999
(Show Context)
Citation Context ...nformation) that encompasses implication; – the encoding preserves assignable types, effectively showing that all proofs in LK have a representation in π – to represent LK, π is enriched with pairing =-=[2]-=-. Classical sequents, X , and π The sequent calculus LK, introduced by Gentzen in [15], is a logical system in which the rules only introduce connectives (but on either side of a sequent), in contrast... |

587 |
Communicating and mobile systems: The π-calculus
- Milner
- 1999
(Show Context)
Citation Context ...me of the current continuation, i.e. the name of the hole in the context in which M occurs. Combining the interpretation of λ into X and X into π, we get yet another encoding of the λ-calculus into π =-=[27, 26]-=-, one that preserves assignable simple types; as usual, the interpretation is parametric over a name. Definition 24 (Interpretation of the λ-calculus in π via X ). The mapping ⌈· ⌋ · π : Λ→π is define... |

567 |
Untersuchungen über das logische Schließen
- Gentzen
- 1935
(Show Context)
Citation Context ...-Howard isomorphism for Gentzen’s calculus LK, this implies that all proofs in LK have a representation in π. Introduction In this paper we present an encoding of proofs of Gentzen’s (implicative) LK =-=[15]-=- into the π-calculus [26] that respects cut-elimination, and define a new notion of type assignment for π so that processes will become witnesses for the provable formulae. The encoding of classical l... |

566 | Term rewriting systems - Klop - 1992 |

474 |
The calculus of constructions
- Coquand, Huet
- 1988
(Show Context)
Citation Context ..., and has resulted in a vast and well-investigated area of research, resulting in, amongst others, functional programming languages and much further to system F [17] and the Calculus of Constructions =-=[11]-=-. Abramsky [3, 5] has studied correspondence between multiplicative linear logic and processes, and later moved to the context of game semantics [4]. In fact, all the calculi are applicative in that a... |

390 | Explicit substitutions
- Abadi, Cardelli, et al.
- 1991
(Show Context)
Citation Context ...is, without a cut. This procedure is defined via local reductions of the proof-tree, which has –with some discrepancies– the flavour of term rewriting [25] or the evaluation of explicit substitutions =-=[14, 1]-=-. The calculus X achieves a Curry-Howard isomorphism, first discovered for Combinatory Logic [13], for the proofs in LK by constructing witnesses (called nets) for derivable sequents, without any noti... |

361 | An object calculus for asynchronous communication
- Honda, Tokoro
- 1991
(Show Context)
Citation Context ...en Q · ·· Γ ⊢X ∆. 3 The asynchronous π-calculus with pairing and nesting The notion of asynchronous π-calculus that we consider in this paper is different from other systems studied in the literature =-=[22]-=-. One reason for this change lies directly in the calculus that is going to be interpreted, X : since we are going to model sending and receiving pairs of names as interfaces for functions, we add pai... |

320 |
λµ–calculus: an algorithmic interpretation of classical natural deduction
- Parigot
- 1992
(Show Context)
Citation Context ... Logic [13], for the proofs in LK by constructing witnesses (called nets) for derivable sequents, without any notion of application. In establishing the isomorphism for X , similar to calculi like λµ =-=[28]-=- and λµ ˜µ [12], Roman names are attached to formulae in the left context, and Greek names for those on the right, and syntactic structure is associated to the rules. These correspond to variables and... |

315 |
Functions as Processes
- Milner
- 1992
(Show Context)
Citation Context ...entzen’s calculus LK, this implies that all proofs in LK have a representation in π. Introduction In this paper we present an encoding of proofs of Gentzen’s (implicative) LK [15] into the π-calculus =-=[26]-=- that respects cut-elimination, and define a new notion of type assignment for π so that processes will become witnesses for the provable formulae. The encoding of classical logic into π-calculus is a... |

281 | Computational interpretation of linear logic
- Abramsky
- 1990
(Show Context)
Citation Context ...ted in a vast and well-investigated area of research, resulting in, amongst others, functional programming languages and much further to system F [17] and the Calculus of Constructions [11]. Abramsky =-=[3, 5]-=- has studied correspondence between multiplicative linear logic and processes, and later moved to the context of game semantics [4]. In fact, all the calculi are applicative in that abstraction and ap... |

235 | A formulae-as-types notion of control
- Griffin
- 1990
(Show Context)
Citation Context ...corresponding to arrow introduction and elimination) are the main constructors in the syntax. The link between Classical Logic and continuations and control was first established for the λ C-Calculus =-=[19]-=- (where C stands for Felleisen’s C operator). The introduction-elimination approach is easy to understand and convenient to use, but is also rather restrictive: for example, the handling of negation i... |

209 | and full completeness for multiplicative linear logic
- Abramsky, Jagadeesan
- 1994
(Show Context)
Citation Context ...to system F [17] and the Calculus of Constructions [11]. Abramsky [3, 5] has studied correspondence between multiplicative linear logic and processes, and later moved to the context of game semantics =-=[4]-=-. In fact, all the calculi are applicative in that abstraction and application (corresponding to arrow introduction and elimination) are the main constructors in the syntax. The link between Classical... |

161 | The duality of computation
- Curien, Herbelin
- 2000
(Show Context)
Citation Context ...r the proofs in LK by constructing witnesses (called nets) for derivable sequents, without any notion of application. In establishing the isomorphism for X , similar to calculi like λµ [28] and λµ ˜µ =-=[12]-=-, Roman names are attached to formulae in the left context, and Greek names for those on the right, and syntactic structure is associated to the rules. These correspond to variables and co-variables, ... |

144 |
The System F of Variable Types, fifteen years later
- Girard
- 1986
(Show Context)
Citation Context ...h types) is well studied and understood, and has resulted in a vast and well-investigated area of research, resulting in, amongst others, functional programming languages and much further to system F =-=[17]-=- and the Calculus of Constructions [11]. Abramsky [3, 5] has studied correspondence between multiplicative linear logic and processes, and later moved to the context of game semantics [4]. In fact, al... |

139 | On reduction-based process semantics
- Honda, Yoshida
- 1995
(Show Context)
Citation Context ... but this is strongly needed in our completeness result (Theorem 20); without it, we can at most show a partial result. Moreover, notice that a〈 b, c 〉 | a( x, y ). Q → ∗ π Q[b/x, c/y]Definition 13 (=-=[23]-=-). Barbed contextual simulation is the largest relation ≼π such that P ≼π Q implies: – for each name n, if P ↓ n then Q ⇓ n; – for any context C, if C[P] →π P ′ , then for some Q ′ , C[Q] → ∗ π Q ′ an... |

71 | Categorical Structure of Continuation Passing Style
- Thielecke
- 1997
(Show Context)
Citation Context ...Without this last addition, we cannot model full cut-elimination; this was, for example, also the case with the interpretations defined by Milner [26], Sangiorgi [29], Honda et al [24], and Thielecke =-=[31]-=-, where reduction in the original calculus had to be restricted in order to get a completeness result. Notice that this last extension of π only relates to cut-elimination: that all proofs in LK are r... |

59 | Classical Logic and Computation
- Urban
- 2000
(Show Context)
Citation Context ...lae. The encoding of classical logic into π-calculus is attained by using the intuition of the calculus X , which gives a computational meaning to LK (a first version of this calculus was proposed in =-=[32, 34, 33]-=-; the implicative fragment of X was studied in [8]). X enjoys the Curry-Howard property for LK; it achieves the isomorphism by constructing witnesses, called nets, for derivable sequents. Nets in X ha... |

47 |
Call-by-value is dual to call-by-name
- Wadler
- 2003
(Show Context)
Citation Context ... attached to formulae in the left context, and Greek names for those on the right, and syntactic structure is associated to the rules. These correspond to variables and co-variables, respectively, in =-=[35]-=-, or, alternatively, to Parigot’s λ- and µ-variables [28] (see also [12]). Gentzen’s proof reductions by cut-elimination become the fundamental principle of computation in X . Cuts in proofs are witne... |

41 |
Proofs as processes
- Abramsky
- 1994
(Show Context)
Citation Context ...ted in a vast and well-investigated area of research, resulting in, amongst others, functional programming languages and much further to system F [17] and the Calculus of Constructions [11]. Abramsky =-=[3, 5]-=- has studied correspondence between multiplicative linear logic and processes, and later moved to the context of game semantics [4]. In fact, all the calculi are applicative in that abstraction and ap... |

41 |
The Pi-Calculus
- Sangiorgi, Walker
- 2003
(Show Context)
Citation Context ...there it was shown that it is straightforward to map λµ ˜µ-terms into X whilst preserving reduction, but that it is not possible to do the converse. The π-calculus is equipped with a rich type theory =-=[29]-=-: from the basic type system for counting the arity of channels to sophisticated linear types in [24], which studies a relation between Call-by-Value λµ and a linear π-calculus. Linearisation is used ... |

36 | Strong normalisation of cut-elimination in classical logic
- Urban, Bierman
(Show Context)
Citation Context ...lae. The encoding of classical logic into π-calculus is attained by using the intuition of the calculus X , which gives a computational meaning to LK (a first version of this calculus was proposed in =-=[32, 34, 33]-=-; the implicative fragment of X was studied in [8]). X enjoys the Curry-Howard property for LK; it achieves the isomorphism by constructing witnesses, called nets, for derivable sequents. Nets in X ha... |

33 |
Séquents qu’on calcule : de l’interprétation du calcul des séquents comme calcul de λ-termes et comme calcul de stratégies gagnantes. Thèse d’université, Université Paris 7
- Herbelin
- 1995
(Show Context)
Citation Context ... free, and substitution only takes place on channel names, similar to the renaming feature of X , so cut-elimination is similar to synchronisation. Related work In the past, say before Herbelin’s PhD =-=[20]-=- and Urban’s PhD [32], the study of the relation between computation, programming languages and logic has concentrated mainly on natural deduction systems (of course, exceptions exist [16, 18]). In fa... |

30 | Minimal classical logic and control operators
- Ariola, Herbelin
- 2003
(Show Context)
Citation Context ...at is being manipulated and possibly several alternative ones. Adding ⊥ as pseudo-type (only negation, or A→⊥, is expressed; ⊥→A is not a type), the λµ-calculus corresponds to minimal classical logic =-=[6]-=-. Herbelin has studied the calculus λµ ˜µ as a non-applicative extension of λµ, which gives a fine-grained account of manipulation of sequents [20, 12, 21]. The relation between call-by-name and call-... |

29 | The language X : circuits, computations and Classical Logic
- Bakel, Lengrand, et al.
- 2005
(Show Context)
Citation Context ...λµ and λµ ˜µ, that calculus considers a logic with active formulae, so these calculi do not achieve a direct CurryHoward isomorphism with LK. The relation between X and λµ ˜µ has been investigated in =-=[7, 8]-=-; there it was shown that it is straightforward to map λµ ˜µ-terms into X whilst preserving reduction, but that it is not possible to do the converse. The π-calculus is equipped with a rich type theor... |

22 |
C’est maintenant qu’on calcule: au cœur de la dualité. Mémoire de habilitation, Université Paris 11, Décembre 2005
- Herbelin
(Show Context)
Citation Context ...he λµ-calculus corresponds to minimal classical logic [6]. Herbelin has studied the calculus λµ ˜µ as a non-applicative extension of λµ, which gives a fine-grained account of manipulation of sequents =-=[20, 12, 21]-=-. The relation between call-by-name and call-by-value in the fragment of LK with negation and conjunction is studied in the Dual Calculus [35]; as in calculi like λµ and λµ ˜µ, that calculus considers... |

20 |
de Bruijn. A namefree lambda calculus with facilities for internal definition of expressions and segments. TH-Report 78-WSK-03
- G
- 1978
(Show Context)
Citation Context ...is, without a cut. This procedure is defined via local reductions of the proof-tree, which has –with some discrepancies– the flavour of term rewriting [25] or the evaluation of explicit substitutions =-=[14, 1]-=-. The calculus X achieves a Curry-Howard isomorphism, first discovered for Combinatory Logic [13], for the proofs in LK by constructing witnesses (called nets) for derivable sequents, without any noti... |

16 | Computation with Classical Sequents
- Bakel, Lescanne
(Show Context)
Citation Context ...ained by using the intuition of the calculus X , which gives a computational meaning to LK (a first version of this calculus was proposed in [32, 34, 33]; the implicative fragment of X was studied in =-=[8]-=-). X enjoys the Curry-Howard property for LK; it achieves the isomorphism by constructing witnesses, called nets, for derivable sequents. Nets in X have multiple named inputs and multiple named output... |

15 | On the pi-Calculus and Linear Logic
- Bellin, Scott
- 1994
(Show Context)
Citation Context ... LK. The relation between process calculi and classical logic is an interesting and very promising area of research (similar attempts we made in the context of natural deduction [24] and linear logic =-=[10]-=-). Our aim is to widen further the path to practical application of classical logic in computation by providing an interpretation of classical logic into process algebra, that fully exploits the non-d... |

6 | Strong Normalisation for a Gentzen-like Cut-Elimination Procedure
- Urban
- 2001
(Show Context)
Citation Context ...lae. The encoding of classical logic into π-calculus is attained by using the intuition of the calculus X , which gives a computational meaning to LK (a first version of this calculus was proposed in =-=[32, 34, 33]-=-; the implicative fragment of X was studied in [8]). X enjoys the Curry-Howard property for LK; it achieves the isomorphism by constructing witnesses, called nets, for derivable sequents. Nets in X ha... |

4 |
A new constrcutive logic: classical logic
- Girard
- 1991
(Show Context)
Citation Context ...belin’s PhD [20] and Urban’s PhD [32], the study of the relation between computation, programming languages and logic has concentrated mainly on natural deduction systems (of course, exceptions exist =-=[16, 18]-=-). In fact, these carry the predicate ‘natural’ deservedly; in comparison with, for example, sequent style systems, natural deduction systems are easy to understand and reason about. This holds most s... |

1 |
Extending lambda-mu with first class continuations
- Summers
- 2007
(Show Context)
Citation Context ...t is also rather restrictive: for example, the handling of negation is not as nicely balanced, as is the treatment of contradiction (normally represented by the type ⊥; for a detailed discussion, see =-=[30]-=-). This imbalance can be observed in Parigot’s λµ-calculus [28], an approach for representing classical proofs via a natural deduction system in which there is one main conclusion that is being manipu... |

1 |
Computing with Sequents and Diagrams
- ˇZunić
- 2007
(Show Context)
Citation Context ...tics is studied. Both linear logic and game semantics are outside the scope of this paper, yet we leave for future work the study of the relation of linear X (with explicit weakening and contraction) =-=[36]-=-, and relate that with both game semantics and π without replication. One of the main goals we aimed for with our interpretation was: if α does not occur free in P, and x does not occur free in Q, the... |