## On the computational complexity of cut-elimination in linear logic (2003)

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Venue: | In Proceedings of ICTCS 2003, volume 2841 of LNCS |

Citations: | 11 - 0 self |

### BibTeX

@INPROCEEDINGS{Mairson03onthe,

author = {Harry G. Mairson and Kazushige Terui},

title = {On the computational complexity of cut-elimination in linear logic},

booktitle = {In Proceedings of ICTCS 2003, volume 2841 of LNCS},

year = {2003},

pages = {23--36},

publisher = {Springer}

}

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### Abstract

Abstract. Given two proofs in a logical system with a confluent cutelimination procedure, the cut-elimination problem (CEP) is to decide whether these proofs reduce to the same normal form. This decision problem has been shown to be ptime-complete for Multiplicative Linear Logic (Mairson 2003). The latter result depends upon a restricted simulation of weakening and contraction for boolean values in MLL; in this paper, we analyze how and when this technique can be generalized to other MLL formulas, and then consider CEP for other subsystems of Linear Logic. We also show that while additives play the role of nondeterminism in cut-elimination, they are not needed to express deterministic ptime computation. As a consequence, affine features are irrelevant to expressing ptime computation. In particular, Multiplicative Light Linear Logic (MLLL) and Multiplicative Soft Linear Logic (MSLL) capture ptime even without additives nor unrestricted weakening. We establish hierarchical results on the cut-elimination problem for MLL(ptime-complete), MALL(coNP-complete), MSLL(EXPTIME-complete), and for MLLL (2EXPTIME-complete). 1

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Citation Context ...plete Problem Now we show that the following conp-complete problem is logspace reducible to CEP for IMALL: Logical Equivalence Problem: Given two boolean formulas, are they logically equivalent? (cf. =-=[GJ78]-=-) By Theorem 2, every boolean formula C with n variables can be translated into a term tC of type B (n) −◦ B in O(log |C|) space. For each 1 ≤ k ≤ n, let tak ≡ λf.λx1 · · · xk−1.〈f true x1 · · · xk−1,... |

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Citation Context ...able (say, by a Turing machine) in the class, and show how each can be coded as a fixed proof (program) in the logic. For example, Light Linear Logic has been shown to so represent ptime computations =-=[Gir98]-=-, and the use of additives in that proof was replaced by unrestricted weakening in Light Affine Logic [Asp98,AR02]. We improve these results to show that such weakening is also unnecessary: Multiplica... |

65 | Intuitionistic light affine logic
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(Show Context)
Citation Context ...ltiplicative fragment IMLLL of Light Linear Logic is already expressive enough to represent all polynomial time functions; it needs neither additives (as in [Gir98]) nor unrestricted weakening (as in =-=[Asp98]-=-). Since our concern is not normalization but representation, we do not need to introduce a proper term calculus with the polynomial time normalizationproperty (see [Asp98] and [Ter01] for such term ... |

53 | Soft linear logic and polynomial time
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(Show Context)
Citation Context ...tion 4), where we also prove that deciding CEP is complete for doubly-exponential time. Finally, in Section 5 we show similar characterizations of exponential time in Multiplicative Soft Linear Logic =-=[Laf01]-=-. 2 Expressivity of Multiplicatives 2.1 Weakening in MLL We restrict our attention to the intuitionistic (−◦, ∀) fragment IMLL of MLL, although all the results in this section carry over to the full c... |

23 |
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Citation Context .... The canonical ptime-complete decision problem is the following: Circuit Value Problem: Given a boolean circuit C with n inputs and 1 output, and truth values x = x1, . . . , xn, is x accepted by C? =-=[Lad75]-=- Using the above coding of boolean operations, the problem is logspace reducible to CEP for IMLL: Theorem 2 (ptime-completeness of IMLL, [Mai03]). There is a logspace algorithm which transforms a bool... |

22 | Deciding provability of linear logic formulas
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Citation Context ...led “light” linear logics). Such an investigation suggests another way to characterize the complexity of linear logics: not only by the complexity of theorem proving (proof search)— see, for example, =-=[Lin95]-=-—but also by the complexity of theorem simplification (proof normalization). Even in intuitionistic multiplicative linear logic (IMLL), which has no exponentials, it is possible to simulate weakening ... |

19 |
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Citation Context ... proofs with a suitably powerful interpreter that can “run” representations of programs in that complexity class. The cut-elimination problem is known to be non-elementary for simply typed λ-calculus =-=[Sta79]-=-, and hence for linear logic. Low order fragments of simply typed λ-calculus are investigated in [Sch01]. In this paper, we consider the decision problem for various weak subsystems of linear logic th... |

13 |
Affine Lambda-calculus and polytime strong normalization
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Citation Context ...akening (as in [Asp98]). Since our concern is not normalization but representation, we do not need to introduce a proper term calculus with the polynomial time normalizationproperty (see [Asp98] and =-=[Ter01]-=- for such term calculi). We rather use the standard λ-calculus and think of IMLLL as a typing system for it. The type assignment rules of IMLLL are those of IMLL with the following: x:B ⊢ t:A x :!B ⊢ ... |

9 | Intuitionistic light affine logic (proof-nets, normalization complexity, expressive power, programming notation - Asperti, Roversi |

3 |
de Falco. The additive multiboxes
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(Show Context)
Citation Context ... to be careful when considering the classical system, which is not confluent as it stands. It could be overcome by adopting Tortora’s proofnet syntax with generalized & boxes, which enjoys confluence =-=[dF03]-=-; see also [MR02].and the reduction rules are extended with πi〈t1, t2〉 −→ ti, for i = 1, 2. Note that some reductions cause duplication (e.g. (λx.〈x, x〉)t −→ 〈t, t〉), thus cut-elimination in IMALL is... |

3 |
BCK-combinators and linear λ -terms have types
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Citation Context ...B) −◦ B, are all included in the class eΠ1. Theorem 1 (eΠ1-Weakening). For any closed eΠ1 type A, there is a term wA of type A −◦ 1. 3 Note that any term of linear λ-calculus has a propositional type =-=[Hin89]-=-; the role of second order quantifers is not to increase the number of typable terms, but to assign a uniform type to structurally related terms.Proof. Without loss of generality, we may assume that ... |

3 | Proofnets and context semantics for the additives
- Mairson, Rival
(Show Context)
Citation Context ...en considering the classical system, which is not confluent as it stands. It could be overcome by adopting Tortora’s proofnet syntax with generalized & boxes, which enjoys confluence [dF03]; see also =-=[MR02]-=-.and the reduction rules are extended with πi〈t1, t2〉 −→ ti, for i = 1, 2. Note that some reductions cause duplication (e.g. (λx.〈x, x〉)t −→ 〈t, t〉), thus cut-elimination in IMALL is no more in linea... |

3 | LAL is square: representation and expressiveness in light affine logic
- Neergaard, Mairson
- 2002
(Show Context)
Citation Context ...e consider a fixed program t, so all the terms tw to be evaluated have a fixed depth. On the other hand, CEP allows the depth to vary, hence we get a characterization of doubly-exponential time as in =-=[NM02]-=-. Theorem 6 (2exptime-completeness of IMLLL). The cut-elimination problem for IMLLL is complete for 2exptime = ⋃ k dtime[22nk ]. 5 Multiplicative Soft Linear Logic and EXPTIME In this section, we show... |

3 | The Complexity of β -Reduction in Low Orders
- Schubert
(Show Context)
Citation Context ...ty class. The cut-elimination problem is known to be non-elementary for simply typed λ-calculus [Sta79], and hence for linear logic. Low order fragments of simply typed λ-calculus are investigated in =-=[Sch01]-=-. In this paper, we consider the decision problem for various weak subsystems of linear logic that have no exponentials, or have very weak forms of them (i.e., the so-called “light” linear logics). Su... |

2 |
Linear lambda calculus and polynomial time
- Mairson
(Show Context)
Citation Context ...nd truth values x = x1, . . . , xn, is x accepted by C? [Lad75] Using the above coding of boolean operations, the problem is logspace reducible to CEP for IMLL: Theorem 2 (ptime-completeness of IMLL, =-=[Mai03]-=-). There is a logspace algorithm which transforms a boolean circuit C with n inputs and m outputs into a term tC of type B n −◦ B m , where the size of tC is O(|C|). As a consequence, the cut-eliminat... |