## A Judgmental Analysis of Linear Logic (2003)

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@MISC{Chang03ajudgmental,

author = {Bor-Yuh Evan Chang and Kaustuv Chaudhuri and Frank Pfenning},

title = {A Judgmental Analysis of Linear Logic},

year = {2003}

}

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### Abstract

We reexamine the foundations of linear logic, developing a system of natural deduction following Martin-L of's separation of judgments from propositions. Our construction yields a clean and elegant formulation that accounts for a rich set of multiplicative, additive, and exponential connectives, extending dual intuitionistic linear logic but differing from both classical linear logic and Hyland and de Paiva's full intuitionistic linear logic. We also provide a corresponding sequent calculus that admits a simple proof of the admissibility of cut by a single structural induction. Finally, we show how to interpret classical linear logic (with or without the MIX rule) in our system, employing a form of double-negation translation.

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Citation Context ... (contr! and contr?), respectively. Sequents of the form !∆, A CLL −−−→ A, ?Ω that follow from weak∗ and init we justify as init∗ . 4.1. The intuitionistic fragment of classical linear logic. Gentzen =-=[14]-=- observed that one way to characterize intuitionistic logic is by restricting classical sequents to have an empty or unitary right-hand side, but without changing any of the rules. This was generalize... |

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Citation Context ...ar from ordinary logic, and some linear hypotheses as described in the preceding section. We write this as a judgment with two zones containing assumptions, following the style introduced by Andreoli =-=[3]-=-. A1 valid, . . . , Aj valid � �� � Γ ; B1 true, . . . , Bk true ⊢ C true � �� � ∆ The use of the semi-colon to separate the two kinds of hypotheses is now standard practice. Our definition of validit... |

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Citation Context ...mit more than one true proposition would violate either linearity or the intuitionistic interpretation. Yet, multiple conclusions do not necessarily conflict with natural deduction (see, for example, =-=[28]-=-), even for intuitionistic [34] and linear logics [22]. Indeed, we can readily incorporate such an approach in our judgmental framework by introducing a new judgment form, C1 poss | · · · | Ck poss, o... |

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Citation Context ...program and the goal. Intuitionistic versions of linear logic have been used to explore interesting phenomena in functional computation (see, for example, [24, 1, 6, 39, 20, 2, 5]), logic programming =-=[19]-=-, and logical frameworks [9]. In this paper, we analyze linear logic in an inherently asymmetric natural deduction formulation following Martin-Löf’s methodology of separating judgments from propositi... |

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Citation Context ...ogramming maintains an asymmetry between the program and the goal. Intuitionistic versions of linear logic have been used to explore interesting phenomena in functional computation (see, for example, =-=[24, 1, 6, 39, 20, 2, 5]-=-), logic programming [19], and logical frameworks [9]. In this paper, we analyze linear logic in an inherently asymmetric natural deduction formulation following Martin-Löf’s methodology of separating... |

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Citation Context ...onistic versions of linear logic have been used to explore interesting phenomena in functional computation (see, for example, [24, 1, 6, 39, 20, 2, 5]), logic programming [19], and logical frameworks =-=[9]-=-. In this paper, we analyze linear logic in an inherently asymmetric natural deduction formulation following Martin-Löf’s methodology of separating judgments from propositions [25]. We require minimal... |

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Citation Context ...ce fills an important gap in the judgmental description of the MIX rules. Much related work exists, so we only briefly touch upon it here. We view the judgmental reconstruction of modal and lax logic =-=[31]-=- as the motivation for this approach. We also owe much to Polakow’s development of ordered logic [32], which employs linear and ordered hypothetical judgments, but does not introduce possibility and i... |

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Citation Context ...d logical frameworks [9]. In this paper, we analyze linear logic in an inherently asymmetric natural deduction formulation following Martin-Löf’s methodology of separating judgments from propositions =-=[25]-=-. We require minimal judgmental notions – linear hypothetical judgments, categorical judgments, and ordinary hypothetical judgments suffice to explain a full range of intuitionistic linear proposition... |

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Citation Context ..., which employs linear and ordered hypothetical judgments, but does not introduce possibility and its associated connectives. JILL contains, as fragments, both dual intuitionistic linear logic (DILL) =-=[4]-=- and hereditary Harrop logic underlying linear logic programming [19]. The contribution of JILL with respect to these systems is the judgmental account, which gives rise to the new ?, ⊥ and ¬ connecti... |

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Citation Context ...he negation of the translation of A. It is more economical and allows some further applications if we instead parameterize the translation by a propositional parameter p. This idea is due to Friedman =-=[13]-=- in the case of ordinary classical and intuitionistic logic, and was also employed by Lamarche [23] in the setting of game semantics. It is convenient to introduce a parametric negation ∼pA, defined t... |

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Citation Context ... −−−→ Ω CLL −−−→ A, Ω1 ∆2, B CLL −−−→ Ω2 ∆1, ∆2, A ⊸ B CLL −−−→ Ω1, Ω2 ¬R ∆ CLL −−−→ A, Ω ∆, ¬A CLL −−−→ Ω ⊸L Figure 3. Sequent calculus for classical linear logic CLL CLL enjoys cut-elimination (see =-=[35, 38, 29]-=-), although this property is not needed here. We write weak∗ and contr∗ for repeated applications of weakening (weak! and weak?) and contraction rules (contr! and contr?), respectively. Sequents of th... |

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Citation Context ...ogramming maintains an asymmetry between the program and the goal. Intuitionistic versions of linear logic have been used to explore interesting phenomena in functional computation (see, for example, =-=[24, 1, 6, 39, 20, 2, 5]-=-), logic programming [19], and logical frameworks [9]. In this paper, we analyze linear logic in an inherently asymmetric natural deduction formulation following Martin-Löf’s methodology of separating... |

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Citation Context ...ogramming maintains an asymmetry between the program and the goal. Intuitionistic versions of linear logic have been used to explore interesting phenomena in functional computation (see, for example, =-=[24, 1, 6, 39, 20, 2, 5]-=-), logic programming [19], and logical frameworks [9]. In this paper, we analyze linear logic in an inherently asymmetric natural deduction formulation following Martin-Löf’s methodology of separating... |

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Citation Context ...rucural cut-admissibility proof for a sequent calculus formulation of JILL. Related structural cut-elimination proofs have appeared for intuitionistic and classical logic [30], classical linear logic =-=[29]-=-, and ordered logic [33], but these did not incorporate possibility and related connectives (?, ⊥, ¬, and ⊞). To our knowledge, the double-negation translation from classical into intuitionistic linea... |

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Citation Context ... we show how to interpret classical linear logic (with or without the MIX rule) in our system, employing a form of double-negation translation. §1. Introduction. Central to the design of linear logic =-=[15]-=- are the beautiful symmetries exhibited by the classical sequent calculus. This has led to applications in the areas of concurrent computation and games among others, where the symmetry captures a rel... |

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Citation Context ...nding cut-admissibility proof—become quite large; Howe has a detailed proof for such a single-zoned sequent system for intuitionistic linear logic (without possibility, negation or modal disjunction) =-=[21]-=-. Our proof of cut admissibility is more direct, using lexicographic induction on the relevant derivations. Theorem 3. The above principles of cut are admissible rules in JILL.s14 BOR-YUH EVAN CHANG, ... |

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Citation Context ...in this paper. Lamarche has previously given a more complex double-negation translation from classical linear logic into intuitionistic linear logic using a one-sided sequent calculus with polarities =-=[23]-=-, which is essentially a one-sided version of Girard’s LU [16]. This paper is organized as follows. In Sec. 2, we describe natural deduction for JILL in terms of the required judgmental notions. In pa... |

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Citation Context ...y proof for a sequent calculus formulation of JILL. Related structural cut-elimination proofs have appeared for intuitionistic and classical logic [30], classical linear logic [29], and ordered logic =-=[33]-=-, but these did not incorporate possibility and related connectives (?, ⊥, ¬, and ⊞). To our knowledge, the double-negation translation from classical into intuitionistic linear logic that can optiona... |

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Citation Context ...of this paper; we present below the cases for the various propositional constants and modal connectives. There are exactly seven distinguishable modal operators – none, !, ?, !?, ?!, !?! and ?!? (see =-=[36]-=- for a more complete discussion); for the propositional constants, there are some further equivalences: (1) ?⊤ ≡ ⊤ (2) 1 ≡ !⊤ ≡ !?⊤ ≡ !?!⊤ (3) ?1 ≡ ?!⊤ ≡ ?!?⊤ (4) 0 ≡ !0 (5) ⊥ = ?0 ≡ ?!0 ≡ ?!?0 (6) !⊥... |

22 | Proof theory for full intuitionistic linear logic, bilinear logic, and mix categories. Theory and Applications of Categories
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Citation Context ...tives in isolation. On the other hand, multiplicative disjunction � in FILL seems closer to its classical counterpart than our modal disjunction ⊞; furthermore, FILL has a clear categorical semantics =-=[10]-=- that we have not yet explored for JILL. We present a strucural cut-admissibility proof for a sequent calculus formulation of JILL. Related structural cut-elimination proofs have appeared for intuitio... |

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Citation Context ...ion would violate either linearity or the intuitionistic interpretation. Yet, multiple conclusions do not necessarily conflict with natural deduction (see, for example, [28]), even for intuitionistic =-=[34]-=- and linear logics [22]. Indeed, we can readily incorporate such an approach in our judgmental framework by introducing a new judgment form, C1 poss | · · · | Ck poss, on the right-hand side of a hypo... |

16 |
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Citation Context ...ll therefore satisfy this condition. Thes12 BOR-YUH EVAN CHANG, KAUSTUV CHAUDHURI, AND FRANK PFENNING notion of a verification can thus be formalized directly on natural deductions (see, for example, =-=[37]-=-). We take a different approach in this paper and formalize verifications as cut-free proofs in the sequent calculus. Not only is it immediately evident that the sequent calculus satisfies our conditi... |

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Citation Context ...n of sequent derivations in classical linear logic with a unitary right-hand side. This restriction is sometimes taken as a definition of “intuitionistic linear logic”; for example, see Girard et al. =-=[17, 18]-=-, or Troelstra [38] who additionally permits an empty right-hand side, allowing a definition of ⊥ and ¬. A similar observation was made by Girard for his Logic of Unity (LU), where he characterized in... |

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Citation Context ...he contribution of JILL with respect to these systems is the judgmental account, which gives rise to the new ?, ⊥ and ¬ connectives and a modal disjunction, ⊞. Full intuitionistic linear logic (FILL) =-=[22]-=- does not have additives 1 and does not proceed via a judgmental account. It requires either proof terms [7] or occurrence labels [8] in order to formulate the rules for linear implication, which make... |

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Citation Context ... rules or induction measures are required. Similar structural proofs for cut-admissibility have been demonstrated for classical linear logic [29], classical and intuitionistic uniform sequent calculi =-=[26, 27]-=- and ordered logic [33]. Sequents have valid interpretations as natural deduction judgments, which we state as a soundness theorem for the sequent calculus. Theorem 4 (Soundness of =⇒ wrt ⊢). 1. If Γ ... |

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Citation Context ...d logical frameworks [9]. In this paper, we analyze linear logic in an inherently asymmetric natural deduction formulation following Martin-Lof's methodology of separating judgments from propositions =-=[25]-=-. We require minimal judgmental notions -- linear hypothetical judgments, categorical judgments, and ordinary hypothetical judgments su#ce to explain a full range of intuitionistic linear propositiona... |

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Citation Context ...y not as a primitive judgment, but as a categorical judgment derived from truth in the absence of linear hypotheses. Similar notions of categorical judgment have been used in a wide variety of logics =-=[12, 31]-=-. If · � A true, then A valid. For the resource interpretation of judgments, A valid means that one can achieve A true without consuming any resources. Dually, an assumption A valid lets one generate ... |

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Citation Context ...hout any need for the possibility judgment; a similar approach was taken by Troelstra for the system ILZ, which is a sequent-style presentation of an intuitionistic fragment of classical linear logic =-=[38]-=-. Instead of taking this path, we relate this new hypothetical judgment form to the possibility judgment in JILL. Thus we obtain a right-weakening rule: Γ ; ∆ ⊢ · Γ ; ∆ ⊢ C poss poss′ The pair of rule... |

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Citation Context ...n of sequent derivations in classical linear logic with a unitary right-hand side. This restriction is sometimes taken as a definition of “intuitionistic linear logic”; for example, see Girard et al. =-=[17, 18]-=-, or Troelstra [38] who additionally permits an empty right-hand side, allowing a definition of ⊥ and ¬. A similar observation was made by Girard for his Logic of Unity (LU), where he characterized in... |

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of linear calculi
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Citation Context ... rules or induction measures are required. Similar structural proofs for cut-admissibility have been demonstrated for classical linear logic [29], classical and intuitionistic uniform sequent calculi =-=[26, 27]-=- and ordered logic [33]. Sequents have valid interpretations as natural deduction judgments, which we state as a soundness theorem for the sequent calculus. Theorem 4 (Soundness of =⇒ wrt ⊢). 1. If Γ ... |

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Citation Context ...red for JILL. We present a strucural cut-admissibility proof for a sequent calculus formulation of JILL. Related structural cut-elimination proofs have appeared for intuitionistic and classical logic =-=[30]-=-, classical linear logic [29], and ordered logic [33], but these did not incorporate possibility and related connectives (?, ⊥, ¬, and ⊞). To our knowledge, the double-negation translation from classi... |

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Citation Context ...y not as a primitive judgment, but as a categorical judgmentsderived from truth in the absence of linear hypotheses. Similar notions of categorical judgment have been used in a wide variety of logics =-=[12, 31]-=-. If # A true, then A valid. For the resource interpretation of judgments, A valid means that one can achieve A true without consuming any resources. Dually, an assumption A valid lets one generate as... |