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

### Cached

### Download Links

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}

}

### OpenURL

### 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

### Citations

637 | Linear logic
- Girard
- 1987
(Show Context)
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-... |

579 |
Untersuchungen über das logische Schließen
- Gentzen
- 1935
(Show Context)
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... |

230 | Languages that capture complexity classes
- Immerman
- 1987
(Show Context)
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-... |

181 | A new recursion-theoretic characterization of the polytime functions
- Bellantoni, Cook
- 1992
(Show Context)
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... |

161 | Linear logic: Its syntax and semantics
- Girard
(Show Context)
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... |

94 |
Computability and Complexity from a Programming Perspective
- Jones
- 1997
(Show Context)
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... |

89 | Proof-nets: The parallel syntax for proof-theory. In: Logic and Algebra
- Girard
- 1996
(Show Context)
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... |

72 | Linear types and non-size-increasing polynomial time computation
- Hofmann
- 1999
(Show Context)
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 ... |

59 | Light affine logic
- Asperti
- 1998
(Show Context)
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-... |

54 | From proof nets to interaction nets
- Lafont
- 1995
(Show Context)
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... |

30 |
Proofs and Types, volume 7 of Cambridge Tracts in Theoret Computer Science
- Girard, Lafont, et al.
- 1989
(Show Context)
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
- Boas, P
- 1990
(Show Context)
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... |

23 |
The typed lambda calculus is not elementary recursive
- Statman
- 1979
(Show Context)
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... |

12 |
Definierbare Funktionen im λ-Kalkül mit Typen. Arkhiv für mathematische Logik und Grundlagenforschung
- Schwichtenberg
- 1976
(Show Context)
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 |
Affine Lambda-calculus and polytime strong normalization
- Light
- 2001
(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
- Asperti, Roversi
(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 |
Light affine logic as a programming language: a first contribution
- ROVERSI
(Show Context)
Citation Context |

6 |
Soft linear logic and polynomial time’, Theoretical Computer Science 318(1), 163– 180
- Lafont
- 2004
(Show Context)
Citation Context |

5 | An analysis of the Core-ML language: Expressive power and type reconstruction
- Kanellakis, Hillebrand, et al.
- 1994
(Show Context)
Citation Context ...ventions is that the input determines certain structure in the function—namely, the power of some iterator—which is explicitly coded as a λ-term. In later research, Hillebrand, Kanellakis and Mairson =-=[15]-=- attempted to liberate Statman’s expressiveness results from Schwichtenberg’s bottleneck by merging simply-typed λ-calculus with simple database constructs, allowing λ-terms to compute functions betwe... |

2 | Fault-TolerantParallel Computation - Kanellakis, Shvartsman - 1997 |

1 | 6. Jean-Yves Girard. Linear logic: its syntaxand semantics - Sci - 1987 |

1 | Proof-nets: The parallel syntaxfor proof-theory - Girard - 1996 |

1 | volume 820 of LNCS - Automata, Programming - 1994 |

1 | Helmut Schwichtenberg. Definierbare Funktionen im λ–Kalkül mit Typen - Sci - 2000 |

1 | 21. Kazushige Terui. Light affine lambda calculus and polytime strong normalization - Sci - 1979 |