## Reasoning about knowledge in linear logic: modalities and complexity (2004)

Venue: | Logic, Epistemology, and the Unity of Science |

Citations: | 5 - 5 self |

### BibTeX

@INPROCEEDINGS{Marion04reasoningabout,

author = {Mathieu Marion and Mehrnouche Sadrzadeh},

title = {Reasoning about knowledge in linear logic: modalities and complexity},

booktitle = {Logic, Epistemology, and the Unity of Science},

year = {2004},

publisher = {Kluwer}

}

### OpenURL

### Abstract

In a recent paper, Jean-Yves Girard commented that ”it has been a long time since philosophy has stopped intereacting with logic”[17]. Actually, it has not

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Citation Context ...at Fagin and Halpern introduced a logic of ’awareness’ [?], where an agent knows explicitely φ if she is aware that φ and knows implicitely that φ.) Secondly, Levesque uses the situation semantics of =-=[3]-=- to deal with the explicit knowledge of agents. In a given situation, some formulas will have a truth-value assigned to them but it is possible some other formulas can have no truth-value. Levesque al... |

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Citation Context ...ed tools and one such tool is the proof assistant Coq, which is a higher-order logic based on the calculus of constructions, whose ancestors are to be found in de Bruijn’s Automath, Girard’s system F =-=[13]-=- and Martin-Löf’s intuitionistic type theory [28]. The calculus of constructions enables us constructively to encode other logics in Coq. These logics are treated as object logics vs the metalogic, wh... |

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Citation Context ...ty cannot be eluded. The problem of logical omniscience in epistemic logic is a perfect case of an idealization in the above sense. It is usually presented in Hintikka’s original Hilbert-style system =-=[19]-=-. We shall present a Gentzen-style version below. Informally the problem is this: if an agent knows that p and knows that ’p implies q’, deductive closure requires that the agent also knows that q. Th... |

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Citation Context ...o the whole argument is seriously to take into account the physical cost of the proof, which is precisely the initial motivation for the development of linear logic, according 3sto Girard (see, e.g., =-=[16]-=-). The optional discharge in the case of the introduction rule for implication has the structural rule of weakening on the left as its counterpart. Relevant logic and linear logic both reject it, as o... |

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Citation Context ...ed. Although the topic was hardly dealt with a mere 25 years ago, computational complexity is by now a well-studied phenomenon, with proof-theoretical measures of complexity, see, e.g., the survey in =-=[33]-=-. One obvious proposal here would be to limit oneself to polynomial time. (This has been suggested, e.g., in [26] for epistemic logic, because of applications to cognitive science.) There are candidat... |

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Citation Context ...ed phenomenon, with proof-theoretical measures of complexity, see, e.g., the survey in [33]. One obvious proposal here would be to limit oneself to polynomial time. (This has been suggested, e.g., in =-=[26]-=- for epistemic logic, because of applications to cognitive science.) There are candidates for this in linear logic, such as the Bounded Linear Logic (BLL) of [18], in which the use of exponentials is ... |

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Citation Context ... Further reasons for its inadequacy in the epistemic context will surface below. The only attempt that we know of at developing an epistemic logic by using a substructural logic is by Hector Levesque =-=[24, 25]-=-, who used relevant logic. 6sLevesque’s approach is original in many respects. First, he distinguishes between ’implicit’ and ’explicit’ knowledge. According to Levesque, the possible worlds semantics... |

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Citation Context ... not settle the matter entirely, as one could attempt to re-phrase the puzzle (without any loss of meaning) so that it could be expressed in MMLL and use the intuitionistic version of . introduced in =-=[4]-=-, but this is mere speculation. Therefore, in order to be able to solve the puzzle, we had to work with the full modal linear logic. So we had to add lists to both sides of our sequents: D1 � ‘A ⊢ D2 ... |

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Citation Context ...ments easily spring to mind), that we chose to deal with 18sthe problem from a computational viewpoint and we chose to have an encoding in Coq. The computational approach to linear logic initiated in =-=[1]-=- links it to functional programming languages and the key here is an extension the Curry-Howard isomorphism [21], which establishes a correspondence between linear logic proofs and computer programs a... |

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Citation Context ...L1A L1B ⊢ L1B −◦ L L1A, L1A −◦ L1B ⊢ L1B 3d congruence L1A, L1(A −◦ B) ⊢ L1B One of the standard puzzles for multi-modal epistemic logic is ’wise men’ or ’King, three wise men and 5 hats’ puzzle (see =-=[11]-=-, p. 12): a king has three wise men and 5 hats: 2 green and 3 red. He asks the wise men to close their eyes and puts a hat on the head of each of them. Then asks them to open their eyes and poses a qu... |

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Citation Context ...encoded in Coq’s higher-order logic; these encodings allows us to state and prove theorems using facilities of this proof assistant. Intuitionistic linear logic has been previously encoded in Coq, in =-=[32]-=-, by associating the constructs of Coq together with linear logic proofs. Modal logic, too, has been previously encoded in Coq in [23]. Our encoding method will be based on that of [12], in which the ... |

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Citation Context ...object logics vs the metalogic, which is the higher-order logic of Coq. The key point here is that, once theorem-proving in these logics becomes automated in Coq, then one has constructive programmes =-=[5]-=-. Hence, the encoding of our two modal linear logics in Coq provides us with automated proofs which lead to constructive programs. Proof automation has not been examined in this paper but, as mentione... |

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Citation Context ... own ideas about ’impossible possible worlds’ [?], based on Rantala’s ’urn’ models [?, ?], to the syntactical solutions of [?, ?], Parikh’s ’knowledge algorithms’ [?, ?], and the logic of ’awareness’ =-=[12]-=-. This is not the place for a critical evaluation of these alternatives. It should be pointed out, however, that number of these solutions can be characterized by the wish to adhere come what may to t... |

3 | degree in signal processing in 1996, both from Lule3 University of Technology, Lule, Sweden. He spent six months as a Researcher with the Division
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Citation Context ...ng system and a general proof strategy. Guidelines for context-handling are mentioned and used succesfully in [32]. The general proof strategy can be found in the linear logic programming approach in =-=[20]-=-. Once proofs are automated, Coq provides a mechanism for construction of programmes out of automated theorem proving. As we just pointed out, automated proofs in Coq are constructed proofs that are c... |

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Citation Context ...didate for further semantical and philosophical elaboration. There are two main strategies for introducing modalities within linear logic. First, one could interpret exponentials as modalities, as in =-=[2]-=-, the system would be a form of linear S4 with indexed modalities. However, the resulting connectives would both control contraction and weakening and serve as modalities. They would appear in both st... |

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Citation Context ... Further reasons for its inadequacy in the epistemic context will surface below. The only attempt that we know of at developing an epistemic logic by using a substructural logic is by Hector Levesque =-=[24, 25]-=-, who used relevant logic. 6sLevesque’s approach is original in many respects. First, he distinguishes between ’implicit’ and ’explicit’ knowledge. According to Levesque, the possible worlds semantics... |

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Citation Context ...ld appear in both structural and modal rules; a very bizarre cocktail. Secondly, one could add modalities to fragments of linear logic. Some such combinations and their semantics have been studied in =-=[27]-=-.In what follows, we shall assess only two of them, namely the multi-modal linear logic or MMLL of [22], which has been inspired by and shown to have applications in reasoning about location-dependent... |

1 |
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Citation Context ...gents reasoning about knowledge without conceiving of them as omniscient. There are no reasons, except philosophical prejudice, not to explore the substructural world. We prefer here to follow Dubucs =-=[9]-=- and try and look for a weaker logic relatively to which agents can be said to be omniscient, without this omniscience being problematic. In that paper, Dubucs only discussed intuitionistic logic as a... |

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Citation Context ...hieu Marion and Mehrnouche Sadrzadeh Université du Québec à Montréal ∗ In a recent paper, Jean-Yves Girard commented that ”it has been a long time since philosophy has stopped intereacting with logic”=-=[17]-=-. Actually, it has not been such a ”long” time since, e.g., Dag Prawitz and Michael Dummett developed philosophical arguments within the paradigm of Gentzen-style systems in favour of the adoption of ... |

1 |
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Citation Context ...e. (This has been suggested, e.g., in [26] for epistemic logic, because of applications to cognitive science.) There are candidates for this in linear logic, such as the Bounded Linear Logic (BLL) of =-=[18]-=-, in which the use of exponentials is bounded in advance, or the more recent Light Linear Logic (LLL) of [17], that has a (locally) polynomial-time cut-elimination. Both BLL and LLL are strong enough ... |

1 |
Epistemic Logic in Higher Order Logic
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Citation Context ...ntuitionistic linear logic has been previously encoded in Coq, in [32], by associating the constructs of Coq together with linear logic proofs. Modal logic, too, has been previously encoded in Coq in =-=[23]-=-. Our encoding method will be based on that of [12], in which the system to be encoded is treated as the object logic and Coq’s Calculus of Constructions (CC) as the metalogic. We are here encoding fo... |