## Deciding Provability of Linear Logic Formulas (1994)

Venue: | Advances in Linear Logic |

Citations: | 21 - 0 self |

### BibTeX

@INPROCEEDINGS{Lincoln94decidingprovability,

author = {Patrick Lincoln},

title = {Deciding Provability of Linear Logic Formulas},

booktitle = {Advances in Linear Logic},

year = {1994},

pages = {109--122},

publisher = {Cambridge University Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

Introduction There are many interesting fragments of linear logic worthy of study in their own right, most described by the connectives which they employ. Full linear logic includes all the logical connectives, which come in three dual pairs: the exponentials ! and ?, the additives & and \Phi, and the multiplicatives\Omega and . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........ . SRI International Computer Science Laboratory, Menlo Park CA 94025 USA. Work supported under NSF Grant CCR-9224858. lincoln@csl.sri.com http://www.csl.sri.com/lincoln/lincoln.html Patrick Lincoln For the most part we will consider fragments of linear logic built up using these connectives in any combination. For example, full linear logic formulas may employ any connective, while multiplic

### Citations

3831 |
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Citation Context ...ted exchange (commutativity), but with the additional property that exponentials can commute (and thus exponentials enjoy all the structural rules, while other formulas exhibit none) has been studied =-=[47, 32]-=-. It has been found to be undecidable by encoding Turing machine tapes directly in the sequent [32]. Since the sequent comma is not commutative, the entire state of the tape including the current stat... |

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Citation Context ...ut there is a natural counterpart of Fork: the additive conjunction &. The remaining ACM instructions may be encoded using techniques very similar to the well-studied Petri net reachability encodings =-=[8, 18, 38, 9, 16]-=-. The full proof of undecidability is presented in [32]. 2.2 Propositional Multiplicative-Additive Linear Logic The multiplicative-additive fragment of linear logic (MALL) excludes the reusable modals... |

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Citation Context ...eveloping a direct encoding of nondeterministic exponential time Turing machines [33]. This encoding is reminiscent of the standard proof of the pspace-hardness of quantified boolean formula validity =-=[43, 21]-=-, and is related to the logic programming simulation of Turing machines given in [42]. This encoding in first order MALL formulas is somewhat unique in that the computation is read `across the top' of... |

31 |
Normal process representatives
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Citation Context ...ut there is a natural counterpart of Fork: the additive conjunction &. The remaining ACM instructions may be encoded using techniques very similar to the well-studied Petri net reachability encodings =-=[8, 18, 38, 9, 16]-=-. The full proof of undecidability is presented in [32]. 2.2 Propositional Multiplicative-Additive Linear Logic The multiplicative-additive fragment of linear logic (MALL) excludes the reusable modals... |

30 | Constant-Only Multiplicative Linear Logic is NP-Complete
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Citation Context ... necessary and sufficient for provability. It turns out that there is no efficiently computable measure function on this class of formulas, as shown by an encoding of 3-Partition in constant-only MLL =-=[36]-=-. This work points out that the multiplicative constants 1 and ? have very `propositional' behavior. The bottom line is that even for constantonly expressions of MLL deciding provability is np-complet... |

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Citation Context ...ic with unrestricted weakening (sometimes called Affine Logic) have also been studied [32, 7]. Here again the logics are somewhat speculative, although there is a close relationship with direct logic =-=[27, 11]-=-. Some fragments of linear logic with weakening have the same complexity as the same fragment without weakening. For example, just as for linear logic, full first-order affine logic is undecidable, as... |

27 | Proof search in firstorder linear logic and other cut-free sequent calculi
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Citation Context ...t bottom up, the intermediate states appearing directly in the sequents all along the way. 4.2 First Order Multiplicative-Additive Linear Logic Without the exponentials, first order MALL is decidable =-=[35]-=-. Intuitively, this stems from the lack of the ability to copy the instructions for reuse. However, there is no readily apparent decision procedure for this fragment since the quantifier rules allow s... |

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Citation Context ...GammaffiB (which may be read right-to-left as the definition of the connective \Gammaffi). Using the connective \Gammaffi one can define other more specific fragments such as the Horn fragment of MLL =-=[25]-=-, but these results will be largely omitted here. In order to gain an intuition about provability, we will usually be speaking informally of a computational process searching for a proof of a formula ... |

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Citation Context ...em is np-hard by reduction from 3-Partition, a problem which requires a perfect partitioning of groups of objects in much the same way that linear logic requires a complete accounting of propositions =-=[23, 24, 25]-=-. The proof of correctness of the encoding makes heavy use of the `balanced' property of MLL, which states that if a formula is provable in MLL, then the number of positive and negative occurrences of... |

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Citation Context ...y, variants of linear logic with unrestricted contraction are very similar to relevance logic [3]. Urquhart has shown that some propositional variants are undecidable, and has studied other fragments =-=[45, 46]-=-. However, the results regarding relevance logic are very different in character than those described above, since they rely in an essential way on a distributivity that appears in relevance logic but... |

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Citation Context ...er intuitionistic logic (LJ2) sequent can be translated directly into IMLL2, and by adding Patrick Lincoln enough copies of C and W one can preserve provability. By the undecidability of LJ2 shown in =-=[37, 15]-=-, IMLL2 is undecidable. This result can be extended to show the undecidability of pure second order IMALL, but has not yet been extended to pure second order MLL. The decision problem for second order... |

18 |
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Citation Context ...is encoding is reminiscent of the standard proof of the pspace-hardness of quantified boolean formula validity [43, 21], and is related to the logic programming simulation of Turing machines given in =-=[42]-=-. This encoding in first order MALL formulas is somewhat unique in that the computation is read `across the top' of a completed cut-free proof, rather than `bottom up', which is utilized in most of th... |

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Citation Context ... is also sound and complete as a translation into affine logic. Some fragments of affine logic are easier to decide than their linear counterpart. For example, propositional affine logic is decidable =-=[28]-=-, where propositional linear logic is not [32]. Finally, some fragments of affine logic are harder to decide than their linear counterpart. For example, the extended Horn fragment +HL is (pspace-compl... |

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Citation Context ..., 10, 41]. The results here also lead into the study of semantics of linear logic, pointing to deep connections between various fragments of linear logic and familiar structures from computer science =-=[12, 29, 30]-=-. In particular, work has progressed in attempting to find viewpoints where the proof theory of linear logic can be viewed as a machine. For example, Kanovich's results derive from his view of fragmen... |

10 |
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8 |
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Citation Context ...priate function symbols is undecidable [1, 2]. In recent work, pure second order intuitionistic MLL (IMLL2) has been shown to be undecidable, through the encoding of second order intuitionistic logic =-=[34]-=-. The key point is that it is possible to encode contraction and weakening using second order formulas. C \Delta = 8X:X \Gammaffi(X\Omega X) W \Delta = 8X:X \Gammaffi1 [\Sigma ` LJ2 \Delta] \Delta =C;... |

7 |
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Citation Context ...ic with unrestricted weakening (sometimes called Affine Logic) have also been studied [32, 7]. Here again the logics are somewhat speculative, although there is a close relationship with direct logic =-=[27, 11]-=-. Some fragments of linear logic with weakening have the same complexity as the same fragment without weakening. For example, just as for linear logic, full first-order affine logic is undecidable, as... |

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3 |
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Citation Context ...stantiate quantifiers with specific (bounded) terms. Decidability depends only on the last, which is fortunate since neither of the other techniques apply to linear logic in their full generality. In =-=[35, 13]-=- proof systems are developed where the quantifier rules are converted into a form without unbounded guessing. The resulting system generates cut-free proofs of at most exponential size for first-order... |

2 |
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2 |
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Citation Context ...had an impact on the study of proof nets. Later results have generalized this result by providing general translations from arbitrary balanced MLL propositional formulas to constant-only MLL formulas =-=[22, 26]-=-. Together with the np-completeness of propositional MLL, these translations provide an alternate proof of the np-completeness of constant-only MLL. 3.2 Constant-Only Full Linear Logic Amazingly, cons... |

2 |
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Citation Context ...er intuitionistic logic (LJ2) sequent can be translated directly into IMLL2, and by adding Patrick Lincoln enough copies of C and W one can preserve provability. By the undecidability of LJ2 shown in =-=[37, 15]-=-, IMLL2 is undecidable. This result can be extended to show the undecidability of pure second order IMALL, but has not yet been extended to pure second order MLL. The decision problem for second order... |

1 |
Decision Problems for Second Order Linear Logic Without Exponentials
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Citation Context ...y other remaining open problems in this area. 5.2 Higher-Order Linear Logic Amiot has shown that MLL (and MALL) with first and second order quantifiers and appropriate function symbols is undecidable =-=[1, 2]-=-. In recent work, pure second order intuitionistic MLL (IMLL2) has been shown to be undecidable, through the encoding of second order intuitionistic logic [34]. The key point is that it is possible to... |

1 |
Volume 1
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- 1975
(Show Context)
Citation Context ...ded Horn fragment +HL is (pspace-complete) in affine logic, but (np-complete) in linear logic [23]. Finally, variants of linear logic with unrestricted contraction are very similar to relevance logic =-=[3]-=-. Urquhart has shown that some propositional variants are undecidable, and has studied other fragments [45, 46]. However, the results regarding relevance logic are very different in character than tho... |

1 |
Full propositional affine logic is decidable. Email Message 260 to Linear Mailing List
- Artemov, Kopylov
- 1994
(Show Context)
Citation Context ...e some seemingly arbitrary choices to be made, so these fragments are somewhat speculative. Variants of linear logic with unrestricted weakening (sometimes called Affine Logic) have also been studied =-=[32, 7]-=-. Here again the logics are somewhat speculative, although there is a close relationship with direct logic [27, 11]. Some fragments of linear logic with weakening have the same complexity as the same ... |

1 |
Eliminating variables in balanced Horn sequents
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(Show Context)
Citation Context ...had an impact on the study of proof nets. Later results have generalized this result by providing general translations from arbitrary balanced MLL propositional formulas to constant-only MLL formulas =-=[22, 26]-=-. Together with the np-completeness of propositional MLL, these translations provide an alternate proof of the np-completeness of constant-only MLL. 3.2 Constant-Only Full Linear Logic Amazingly, cons... |