## Abstract Complexity in Type Theory

### BibTeX

@MISC{Williams_abstractcomplexity,

author = {Ryan Williams},

title = {Abstract Complexity in Type Theory},

year = {}

}

### OpenURL

### Abstract

### Citations

146 |
A machine-independent theory of the complexity of recursive functions
- Blum
- 1967
(Show Context)
Citation Context ...elop an abstract complexity theory based on type-theoretical programming language semantics. We provide two axioms for complexity measures in type theory that have direct analogues with Blum's axioms =-=[2]-=- for classifying the complexity of partial recursive functions. However, our treatment is strictly more general than Blum's in several respects. Rather than being defined in terms of a family of parti... |

93 |
Computability and Complexity From a Programming Perspective
- Jones
- 1997
(Show Context)
Citation Context ...omplexity concepts). We have not encountered in the literature an abstract treatment of complexity in a type theoretic environment, though there are a scant number of relatives. As a reference, Jones =-=[10]-=- gives a nice overview of complexity using tools from programming language semantics, proving several of Blum's theorems in the special case where runtime is the complexity measure. Constable and Boro... |

30 | The Partial Lambda Calculus
- Moggi
- 1988
(Show Context)
Citation Context ...h all canonical terms. So for example, ((*x : N: x) 3) is not of type value, but (*x : N: x) is. 2.1 Partial Types Partial types were introduced by Constable and Smith [5] (with related work by Moggi =-=[12]-=-) to reason about partial recursive functions in a type theoretic setting; that is, functions that do not necessary halt on all inputs. Partial types may be thought of as types that include all non-ha... |

27 |
Computational Complexity and the Existence of Complexity Gaps
- Borodin
- 1972
(Show Context)
Citation Context ...ur definition captures Blum's in several striking ways; we demonstrate proofs of recursive relatedness, arbitrarily complex programs for a function, infinite complexity hierarchies, and a gap theorem =-=[3]-=- within our formulation. Furthermore, following previous work of Constable and Crary [6, 7], our formulation of abstract complexity elucidates the distinction between expressions as functions and expr... |

21 |
Programming languages capturing complexity classes
- Hofmann
- 2000
(Show Context)
Citation Context ...plexity has focused on finding model-independent characterizations of certain types of resource bounds, such as the polynomial time or polynomial space computable functions (for example, [13, 1]; cf. =-=[9]-=- for a survey of programming languages that precisely express various complexity concepts). We have not encountered in the literature an abstract treatment of complexity in a type theoretic environmen... |

20 | Computational foundations of basic recursive function theory
- Constable, Smith
- 1993
(Show Context)
Citation Context ...t is the subset type of term with all canonical terms. So for example, ((*x : N: x) 3) is not of type value, but (*x : N: x) is. 2.1 Partial Types Partial types were introduced by Constable and Smith =-=[5]-=- (with related work by Moggi [12]) to reason about partial recursive functions in a type theoretic setting; that is, functions that do not necessary halt on all inputs. Partial types may be thought of... |

11 |
Indexing of subrecursive classes
- Kozen
- 1978
(Show Context)
Citation Context ...le by measuring the overhead provided by the "diagonalizing function" h = *a : A: q((ref (r(paq)))(a)) from Theorem 6.1. In addition, ideas from Kozen's axiomatic work on the power of diagonalization =-=[11]-=- may be translated over to this setting, which could admit further progress in the development of complexity in our language. This work is still in progress. In our final conference submission, we wil... |

10 | Computational complexity and induction for partial computable functions in type theory
- Constable, Crary
- 1998
(Show Context)
Citation Context ...ve relatedness, arbitrarily complex programs for a function, infinite complexity hierarchies, and a gap theorem [3] within our formulation. Furthermore, following previous work of Constable and Crary =-=[6, 7]-=-, our formulation of abstract complexity elucidates the distinction between expressions as functions and expressions as algorithms, i.e. extensional semantics versus intensional. Previous work bringin... |

6 | Admissibility of Fixpoint Induction over Partial Types
- Crary
- 1998
(Show Context)
Citation Context ...ed point operator is used in the language. For any f : A ! A, f ix(f ) : A, and f (f ix(f )) 7! f ix(f ). To be valid, the fixed point operator inference requires that the type A be admissible. Crary =-=[8]-=- has shown that admissibility is preserved for a very wide range of types, which includes the vast majority of types considered in practice. It is also the case that quoted types are also admissible i... |

3 |
A note on complexity measures for inductive classes in constructive type theory
- Constable
- 1998
(Show Context)
Citation Context ...ve relatedness, arbitrarily complex programs for a function, infinite complexity hierarchies, and a gap theorem [3] within our formulation. Furthermore, following previous work of Constable and Crary =-=[6, 7]-=-, our formulation of abstract complexity elucidates the distinction between expressions as functions and expressions as algorithms, i.e. extensional semantics versus intensional. Previous work bringin... |

2 |
Subrecursive Programming Languages Part I: Efficiency and Program Structure
- Constable, Borodin
- 1972
(Show Context)
Citation Context ...es a nice overview of complexity using tools from programming language semantics, proving several of Blum's theorems in the special case where runtime is the complexity measure. Constable and Borodin =-=[4]-=- formulated many of the original abstract complexity theorems within a simple programming language. 2 Notation/Type Theory The type theory we will use is a partial object type theory introduced by Con... |

1 |
Higher Type Recursion
- Bellantoni, Niggl, et al.
- 2000
(Show Context)
Citation Context ...tions and complexity has focused on finding model-independent characterizations of certain types of resource bounds, such as the polynomial time or polynomial space computable functions (for example, =-=[13, 1]-=-; cf. [9] for a survey of programming languages that precisely express various complexity concepts). We have not encountered in the literature an abstract treatment of complexity in a type theoretic e... |