## LAL is square: Representation and expressiveness in light affine logic (2002)

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Venue: | In Proc. Workshop on Implicit Computational Complexity |

Citations: | 3 - 2 self |

### BibTeX

@INPROCEEDINGS{Neergaard02lalis,

author = {Peter Møller Neergaard and Harry G. Mairson},

title = {LAL is square: Representation and expressiveness in light affine logic},

booktitle = {In Proc. Workshop on Implicit Computational Complexity},

year = {2002}

}

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### Abstract

Abstract. We focus on how the choice of input-output representation has a crucial impact on the expressiveness of so-called “logics of polynomial time. ” Our analysis illustrates this dependence in the context of Light Affine Logic (LAL), which is both a restricted version of Linear Logic, and a primitive functional programming language with restricted sharing of arguments. By slightly relaxing representation conventions, we derive doubly-exponential expressiveness bounds for this “logic of polynomial time. ” We emphasize that squaring is the unifying idea that relates upper bounds on cut elimination for LAL with lower bounds on representation. Representation issues arise in the simulation of DTIME[2 2n], where we construct a uniform family of proof-nets encoding a Turing Machine; specifically, the dependence on n only affects the number of enclosing boxes. A related technical improvement is the simulation of DTIME[n k]indepthO(log k) LAL proof-nets. The resulting upper bounds on cut elimination then satisfy the properties of a

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Citation Context ...lynomial slowdown in the simulation of any polynomial computation. 1 Introduction: Representation and Squaring Matters Computer scientists are now defining the next 700 languages of polynomial time 1 =-=[3, 13, 12, 8, 1,18, 16]-=-. These languages recast computation and complexity in settings familiar to programmers, rather than in terms of Turing Machines ⋆ Supported by the Danish Research Agency grants 1999-114-0027 and 642-... |

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Citation Context ...nductively: The basis is a single wire corresponding to the axiom ϕ ⊢ ϕ. It has two proof-net ports with input (and output!) types ϕ and ϕ ⊥ . The cut-rule implements the Cut-rule of Sequent Calculus =-=[4]-=- and joints a proof-net with output ϕ with a proof-net where one of the inputs is ϕ; this gives a proof-net with the remaining inputs as input and the second proof-net’s output as output. The multipli... |

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Citation Context ...lynomial slowdown in the simulation of any polynomial computation. 1 Introduction: Representation and Squaring Matters Computer scientists are now defining the next 700 languages of polynomial time 1 =-=[3, 13, 12, 8, 1,18, 16]-=-. These languages recast computation and complexity in settings familiar to programmers, rather than in terms of Turing Machines ⋆ Supported by the Danish Research Agency grants 1999-114-0027 and 642-... |

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Citation Context ...)) · · ·)) → ∗ x1⊗(c ′ x2(c ′ x3(· · ·(c ′ xn(c ′ 0 nil)) · · ·))) . Given a term L : ID representing a configuration, we now have L[Bool⊗α]C : §(((Bool⊗α) ⊸ (Bool⊗α))⊗State ⊗((Bool⊗α) ⊸ (Bool⊗α))) , =-=(3)-=- where we for each of the two tape halves can get the head and tail by providing a base case (we will use the pair of false and the empty list). Using the above constructions, we can define a transiti... |

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Citation Context ... Specifically, we analyze Light Affine Logic (LAL) and derive doubly-exponential bounds on expressiveness. LAL [1, 2] (and its predecessor Light Linear Logic [8]) is primitive version of Linear Logic =-=[5, 6]-=- where duplication is restricted. Consequently, the paper contains some terse technical results which might only be of interest to LAL aficionados. In particular we make clear the essential and fundam... |

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Citation Context ...ynomial time algorithms can be expressed in the language (completeness) and any program can be evaluated in polynomial time (soundness). In neither case it is not necessarily in the standard way.ssee =-=[14]-=- for an elaboration). Such studies have the potential of aiding compilers in providing target code with resource guarantees. In this context, our paper has a two-fold purpose: on the philosophical lev... |

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Citation Context ...e. Sect. 5 shows LAL’s completeness for doubly exponential time. 2 What You Need to Know About LAL Proof-Nets In this section we will briefly recall the central aspects of Linear Logic [5], proofnets =-=[7, 17]-=-, and LAL [2]. The section is not intended to be self-contained and readers are encouraged to consult the references for more details. Like many other logical systems, Linear Logic can be seen both as... |

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Citation Context ...rmalization. This shows that the LAL as a logic has the power of exponential time rather than polynomial time. We conjecture that other logics of polynomial time, for example Hofmann’s modal calculus =-=[12, 11]-=- and Lafont’s Soft Linear Logic [16], are prone to thessame game as illustrated here: to be able to handle the bounds of an arbitrary polynomial n k the systems must have the flexibility of letting k ... |

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Citation Context ...lynomial slowdown in the simulation of any polynomial computation. 1 Introduction: Representation and Squaring Matters Computer scientists are now defining the next 700 languages of polynomial time 1 =-=[3, 13, 12, 8, 1,18, 16]-=-. These languages recast computation and complexity in settings familiar to programmers, rather than in terms of Turing Machines ⋆ Supported by the Danish Research Agency grants 1999-114-0027 and 642-... |

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Citation Context ...e. Sect. 5 shows LAL’s completeness for doubly exponential time. 2 What You Need to Know About LAL Proof-Nets In this section we will briefly recall the central aspects of Linear Logic [5], proofnets =-=[7, 17]-=-, and LAL [2]. The section is not intended to be self-contained and readers are encouraged to consult the references for more details. Like many other logical systems, Linear Logic can be seen both as... |

31 |
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Citation Context ... → α → α reappear in LAL as Int = ∀α.!(α ⊸ α) ⊸ §(α ⊸ α). The System F type of lists of elements of type T, ∀α.(T → α → α) → α → α is ∀α.!(T ⊸ α ⊸ α) ⊸ §(α ⊸ α). For a compendium of such codings, see =-=[10]-=-. Following the intuitions behind the construction of lists, LAL is powerful enough to simulate the transition function of a TM. We represent the TM tape as two lists: one with the tape to the left of... |

28 |
Machine models and simulations
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Citation Context ...cuits in circuit complexity. In full detail we use the aforementioned lemma to derive the following consequences: – We resolve an open question recently posed by Terui [21] by providing a first-class =-=[22]-=- simulation of Turing Machines in LAL2 . The standard way to represent a DTIME[nk ] results in proof-nets of depth k+7. Computation via normalization is then bounded as O(|X| 2k+7), which is not tight... |

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Citation Context ...mulation is a fixed polynomial of the original Turing Machinestime and constant space resources as the transducer used with the standard conventions, for example in [2]. – LAL has a “Statman theorem” =-=[20]-=-: given two LAL proof-nets, do they have the same normal form? This problem is complete for DTIME[22n]. Even if LAL is the logic of polynomial time, reasoning about proofs in this logic certainly is n... |

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Citation Context ...he only representable functions of type Int → Int → · · · → Int are those functions called the “extended polynomials,” defined by constants, addition, multiplication, and a conditional branch on zero =-=[19]-=-. One would imagine, similarly, that functions of type Into→o → Into→o → · · · → Int would characterize “extended exponentials,” with similar generalizations to higher types. As the order 6 (degree of... |

12 |
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(Show Context)
Citation Context ...scent of the idea of uniform circuits in circuit complexity. In full detail we use the aforementioned lemma to derive the following consequences: – We resolve an open question recently posed by Terui =-=[21]-=- by providing a first-class [22] simulation of Turing Machines in LAL2 . The standard way to represent a DTIME[nk ] results in proof-nets of depth k+7. Computation via normalization is then bounded as... |

9 |
Intuitionistic light affine logic (proof-nets, normalization complexity, expressive power
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(Show Context)
Citation Context ...ight variations in the conventions make the languages complete for super-polynomial time. Specifically, we analyze Light Affine Logic (LAL) and derive doubly-exponential bounds on expressiveness. LAL =-=[1, 2]-=- (and its predecessor Light Linear Logic [8]) is primitive version of Linear Logic [5, 6] where duplication is restricted. Consequently, the paper contains some terse technical results which might onl... |

8 |
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Citation Context |

6 |
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Citation Context |

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