## On focusing and polarities in linear logic and intuitionistic logic (2006)

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Citations: | 6 - 2 self |

### BibTeX

@MISC{Liang06onfocusing,

author = {Chuck Liang and Dale Miller},

title = {On focusing and polarities in linear logic and intuitionistic logic},

year = {2006}

}

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### Abstract

There are a number of cut-free sequent calculus proof systems known that are complete for first-order intuitionistic logic. Proofs in these different systems can vary a great deal from one another. We are interested in providing a flexible and unifying framework that can collect together important aspects of many of these proof systems. First, we suggest that one way to unify these proof systems is to first translate intuitionistic logic formulas into linear logic formulas, then assign a bias (positive or negative) to atomic formulas, and then examine the nature of focused proofs in the resulting linear logic setting. Second, we provide a single focusing proof system for intuitionistic logic and show that these other intuitionistic proof systems can be accounted for by assigning bias to atomic formulas and by inserting certain markers that halt focusing on formulas. Using either approach, we are able to account for proof search mechanisms that allow for forward-chaining (program-directed search), backward-chaining (goaldirected search), and combinations of these two. The keys to developing this kind of proof system for intuitionistic logic involves using Andreoli’s completeness result for focusing proofs and Girard’s notion of polarity used in his LC and LU proof systems. 1

### Citations

668 | Linear Logic
- Girard
- 1987
(Show Context)
Citation Context ...ir91]. Ultimately, the success of our search for an intuitionistic focusing system will result from a better understanding of the notion of polarity: specifically how polarity as defined in LC and LU =-=[Gir93]-=- relate to the operational behavior of focused proofs. The 3sdesign of our systems will be motivated entirely proof theoretically. That is, the reasons for our choices in design will be operational: t... |

385 | Uniform proofs as a foundation for logic programming
- Miller, Nadathur, et al.
- 1991
(Show Context)
Citation Context ...mic or) asynchronous on the right and synchronous on the left. Thus the right-hand side will be decomposed down to an atom before a left rule can be applied, yielding what are known as uniform proofs =-=[MNPS91]-=-. 39sP , Q positive, N, M neutral, other symbols arbitrary unless noted. Structural Rules (Decision and Reaction): [N, Γ] N −→ [R] [N, Γ] −→ [R] Lf [Γ]−P→ [Γ] −→ [P ] Rf [C, Γ], Θ −→ R [Γ], Θ, C −→ R ... |

344 | Logic programming with focusing proofs in linear logic
- Andreoli
- 1992
(Show Context)
Citation Context ... to reason about computation. Thus, our interest here in developing normal forms for cut-free proofs. A principle tool for analyzing proof search is Andreoli’s completeness theorem for focused proofs =-=[And92]-=-. Since that result holds for linear logic, we shall develop a number of translations of intuitionistic logic into linear logic. Girard’s original translation [Gir87] of intuitionistic logic into line... |

306 | Logic programming in a fragment of intuitionistic linear logic
- Hodas, Miller
- 1994
(Show Context)
Citation Context ...PS91, Section 5].) In such settings, intuitionistic formulas can be mapped in two different ways depending on whether or not they occur on the left or right of sequents. For example, Hodas and Miller =-=[HM94]-=- used two translations such that [B ⊃ C] + = ![B] − −◦ [C] + and [B ⊃ C] − = [B] + −◦ [C] − . Notice that such a mapping translates the initial sequent B ⊃ C −→ B ⊃ C to [B] + −◦[C] − −→ ![B] − −◦[C] ... |

273 | Investigations into logical deductions - Gentzen - 1969 |

215 |
Call-by-name, call-by-value and the λ-calculus
- Plotkin
- 1975
(Show Context)
Citation Context ...is approach to proof systems in intuitionistic logic has a parallel with the way that continuation-passing style translations are used to make explicit either call-by-name or call-by-value evaluation =-=[Plo76]-=-: the strategy is encoded in the translation. In the proof search setting, we have a choice of doing goal-directed or program-directed proof search or a combination of both. Which way this is done is ... |

171 |
A new constructive logic: classical logic
- Girard
- 1991
(Show Context)
Citation Context ...both forward and backward chaining. There might be various ways to motivate and validate our design: for example, coherent space semantics is used by some to motivate related proof systems such as LC =-=[Gir91]-=-. Ultimately, the success of our search for an intuitionistic focusing system will result from a better understanding of the notion of polarity: specifically how polarity as defined in LC and LU [Gir9... |

109 | A new deconstructive logic: Linear logic
- Danos, Joinet, et al.
- 1997
(Show Context)
Citation Context ... used throughout our analysis. Where our work diverges from theirs, however, is in the adoptation of Andreoli’s focusing calculus as our main instrument of construction (deconstruction?). The work of =-=[DJS97]-=- was based on an in-depth analysis of cut elimination in classical logic. Their LK η p system essentially reformulates focusing not as a sequent calculus but as a protocol for constructing proofs. Its... |

87 | FORUM: A Multiple-Conclusion Specification Logic
- Miller
- 1996
(Show Context)
Citation Context ...the resulting proofs are sometimes referred to as “top-down” or “goal-directed”: here, one tries to prove the goal immediately and to produce new sub-goals that are attempted next. It was observed in =-=[Mil96]-=- that all of linear logic could be seen as goal-directed search if all atomic formulas were assigned a negative polarity. Changes to the polarities assigned to atoms does not affect provability of a l... |

65 | Cut-elimination for a logic with definitions and induction - McDowell, Miller |

59 |
Mathematical Intuitionism: Introduction to Proof Theory, volume 67 of Translations of Mathematical Monographs
- Dragalin
- 1987
(Show Context)
Citation Context ... calculus, it is reasonable to consider also ⊤ as an alternative to 1 for representing true. ⊤ is indeed the translation of ¬F in LC and LKF. Furthermore, multiple conclusion versions of LJ are known =-=[Dra88]-=- and have also been studied as a variant of LJQ’ [DL06]. It remains to determine if these variations are still essentially consistent with the polarity analysis of LC and LU. 10.3 Focused LU In the ma... |

59 | Rules of definitional reflection - Schroeder-Heister - 1993 |

46 |
Zur intuitionistischen Arithmetik und Zahlentheorie”, Ergebnisse eines mathematischen Kolloquiums
- Gödel
- 1933
(Show Context)
Citation Context ...sed. While it is possible to derive such a system again using linear logic, classical logic can also be embedded within intuitionistic logic using the well-known double-negation translations of Gödel =-=[Göd33]-=-, Gentzen and Kolmogorov, who had the earliest [Kol25]. These translations do not, however, yield significant focusing features: they are not sufficiently sensitive to polarities. The use of double-ne... |

43 | A fixpoint theorem in linear logic. An email posting to the mailing list linear@cs.stanford.edu - Girard - 1992 |

43 | A theory of modules for logic programming
- Miller
- 1986
(Show Context)
Citation Context ...sults in attempts to prove A, it might be valuable to expect that A is present in the context instead of allowing proof search to reprove A (see the discussion on memo-ization in logic programming in =-=[Mil86]-=-). In this note, we follow the atom-based approach, which subsumes the predicate-based approach. 2.3 The meaning of bias One could say that the “meaning” of bias is to simply reduce non-determinism in... |

41 | Cut-elimination and a permutation-free sequent calculus for intuitionistic logic - Dyckho, Pinto - 1998 |

40 | LKQ and LKT: Sequent calculi for second order logic based upon dual linear decompositions of classical implication
- Danos, Joinet, et al.
- 1995
(Show Context)
Citation Context ...essary. However, our interest lies also in preserving the structure of proofs, not just in the traditional presentation of linear logic but also in the focusing calculus LLF. Using the terminology of =-=[DJS95]-=- we seek inductive decoration strategies: i.e., we wish to define one-to-one mappings between focused proofs and unrestricted LJ proofs. We shall soon consider other translations of intuitionistic log... |

39 | Focusing the inverse method for linear logic
- Chaudhuri, Pfenning
- 2005
(Show Context)
Citation Context ...ed how to construct a general strategy for choosing between ∧ + and ∧ − . One possible strategy is derived from an interesting translation from intuitionistic to linear logic explored by Chaudhuri in =-=[Cha06]-=-. Aspects of this translation bare similarities to ours, but it is based on a dimension we have not yet considered: conjunction is translated differently depending on whether or not it is under focus.... |

39 | Proof Search Issues in Some Non-Classical Logics
- Howe
- 1998
(Show Context)
Citation Context ...n this respect: in particular, we show how LJQ’ can be captured inside LJF (Section 8). 4.2 LJT’ The complement to LJQ is LJT, also called MJ [Her95, DP98] (a version with full connectives appears in =-=[How98]-=-). This system supports left-side focus and is noted for its ability to distinguish a stoup inducing a head-variable in lambda-term reduction. The translation for this calculus is interesting in that ... |

35 |
Séquents qu’on calcule: de l’interprétation du calcul des séquents comme calcul de lambda-termes et comme calcul de stratégies gagnantes
- Herbelin
- 1995
(Show Context)
Citation Context ...anslations of intuitionistic logic into linear logic. 4.1 LJQ’ We first consider the proof system LJQ’ for intuitionistic logic that is given in [DL06], which generalizes the LJQ calculus of Herbelin =-=[Her95]-=-, itself a derivative of LKQ [DJS95]. This calculus was designed to include strong characteristics of focusing. There are two style of sequents, Γ → A and Γ ⇒ A, which represent focused and unfocused ... |

33 |
Troelstra and Helmut Schwichtenberg, Basic Proof Theory, 2nd edition
- Anne
- 2000
(Show Context)
Citation Context ...ound in Figure 4. The context Γ is a set in this calculus. Contraction and weakening will not need to be made explicit. This variant of LJ bares some close resemblances to the “G3i” calculus found in =-=[TS96]-=-. However, the additive version of ∧L is used instead of A, B, Γ ⊢I R A ∧ B, Γ ⊢I R ∧L Such variations are common in intuitionistic calculi. One of the aims of this paper is to further clarify these v... |

18 | LJQ: a strongly focused calculus for intuitionistic logic
- Dyckhoff, Lengrand
- 2006
(Show Context)
Citation Context ...ss of these systems within the framework of defining translations of intuitionistic logic into linear logic. 4.1 LJQ’ We first consider the proof system LJQ’ for intuitionistic logic that is given in =-=[DL06]-=-, which generalizes the LJQ calculus of Herbelin [Her95], itself a derivative of LKQ [DJS95]. This calculus was designed to include strong characteristics of focusing. There are two style of sequents,... |

17 |
de Falco. Polarized and focalized linear and classical proofs
- Laurent, Quatrini, et al.
- 2005
(Show Context)
Citation Context ...mulates focusing not as a sequent calculus but as a protocol for constructing proofs. Its connections to polarization and focusing were further explored and extended by Laurent, Quatrini and de Falco =-=[LQdF05]-=- using polarized proof nets. It is likely that these alternative formalisms can be adjusted to intuitionistic logic and replace Andreoli’s sequent calculus as the means to accomplish our goals. We fin... |

13 | On the specification of sequent systems
- Pimentel, Miller
- 2005
(Show Context)
Citation Context ...from focusing. Using the following asymmetrical translation for cut-free proofs, we can provide a link between the two classifications of atoms (cut-elimination for LU was proved in [Vau93]; see also =-=[PM05]-=-): 1. P −1 =!P, P +1 = P , for left-permeable atom P . 2. N −1 = N, N +1 =?N, for right-permeable atom N. 4 Girard also uses the terms “positive” or “+1” for left-permeables and “negative” or “−1” for... |

6 |
Gopalan Nadathur, and Vijay Saraswat. Testing concurrent systems: An interpretation of intuitionistic logic
- Jagadeesan
- 2005
(Show Context)
Citation Context ...ate symbols: atoms then inherent their polarity from the predicate that is their head. Such assignment is stable under first-order substitution. As we shall see in Section 5, the λRCC proof system of =-=[JNS05]-=- can be seen as using this style of assignment. 3. The proof-occurrence-based approach assigns polarities to atoms based on their location within proofs. Two different occurrences of the same atom in ... |

3 |
On the principle of the excluded middle, Matematicheskii sbornik 32
- Kolmogorov
- 1925
(Show Context)
Citation Context ...n using linear logic, classical logic can also be embedded within intuitionistic logic using the well-known double-negation translations of Gödel [Göd33], Gentzen and Kolmogorov, who had the earliest =-=[Kol25]-=-. These translations do not, however, yield significant focusing features: they are not sufficiently sensitive to polarities. The use of double-negation plays the role of a throttle, similar to the ro... |

2 |
Cut elimination for the unified logic
- Vauzeilles
- 1993
(Show Context)
Citation Context ...t arise naturally from focusing. Using the following asymmetrical translation for cut-free proofs, we can provide a link between the two classifications of atoms (cut-elimination for LU was proved in =-=[Vau93]-=-; see also [PM05]): 1. P −1 =!P, P +1 = P , for left-permeable atom P . 2. N −1 = N, N +1 =?N, for right-permeable atom N. 4 Girard also uses the terms “positive” or “+1” for left-permeables and “nega... |

1 | Myriam Quatrini, and Lorenzo Tortora de Falco. Polarized and focalized linear and classical proofs - Laurent - 2005 |