## Third-order computation and bounded arithmetic (2006)

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Venue: | University of Wales Swansea |

Citations: | 1 - 1 self |

### BibTeX

@INPROCEEDINGS{Skelley06third-ordercomputation,

author = {Alan Skelley},

title = {Third-order computation and bounded arithmetic},

booktitle = {University of Wales Swansea},

year = {2006}

}

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### Abstract

Abstract. We describe a natural generalization of ordinary computation to a third-order setting and give a function calculus with nice properties and recursion-theoretic characterizations of several large complexity classes. We then present a number of third-order theories of bounded arithmetic whose definable functions are the classes of the EXP-time hierarchy in the third-order setting.

### Citations

151 |
The Intrinsic Computational Difficulty of Functions
- Cobham
- 1965
(Show Context)
Citation Context ...g recursion restricted to E1 ∪ E2. 4. FEXP + is the closure of I under composition, limited 3-comprehension and limited doubling recursion. Proof (sketch). The first point is essentially as in Cobham =-=[4]-=-. FPSPACE + is contained in the closure of FP + ∪P ◦ by limited recursion on E3, composition and limited 3-comprehension: First, a superstring-valued FP + ∪P ◦ function can compute from the input of a... |

42 |
Some classes of recursive functions
- Grzegorczyk
- 1953
(Show Context)
Citation Context ...ursion in the context of number functions is of exponential length, as is limited doubling recursion in our setting. This in turn is reminiscent of E 2 , the second level of the Grzegorczyk hierarchy =-=[9]-=-, which is defined similarly except with an initial function of linear growth rate as opposed to x# ′ y; this was shown by Ritchie [11] to equal the linear space functions. Finally, with limited doubl... |

38 |
Classes of predictably computable functions
- Ritchie
- 1963
(Show Context)
Citation Context ...eminiscent of E 2 , the second level of the Grzegorczyk hierarchy [9], which is defined similarly except with an initial function of linear growth rate as opposed to x# ′ y; this was shown by Ritchie =-=[11]-=- to equal the linear space functions. Finally, with limited doubling recursion on E3, the step function of an exptime Turing machine can be iterated exponentially many times. See [3] for a previous re... |

30 | Theories for complexity classes and their propositional translations
- Cook
- 2005
(Show Context)
Citation Context ... must have distinguished string and superstring free variables Y and X . φ rec then has X free as well as a new variable S. This scheme is analogous to that from [2] and follows the presentation from =-=[7]-=-. The scheme Σ B 0 -superstring-halfrecursion is ∃X φ hrc (S, X ), where φ(Y, X ) ∈ ΣB 0 , and φhrc (S, X ) ≡ ∀Y ≤ |S|(L2(Y, S) ⊃ (X (Y ) ↔ φ(Y, X <Y/2 ))), where X <Y/2 is a chop function returning t... |

28 | Axiomatizations and conservation results for fragments of bounded arithmetic; in Logic and Computation, proceedings of a Workshop held Carnegie-Mellon
- Buss
- 1987
(Show Context)
Citation Context ...riables than the displayed ones, but φ must have distinguished string and superstring free variables Y and X . φ rec then has X free as well as a new variable S. This scheme is analogous to that from =-=[2]-=- and follows the presentation from [7]. The scheme Σ B 0 -superstring-halfrecursion is ∃X φ hrc (S, X ), where φ(Y, X ) ∈ ΣB 0 , and φhrc (S, X ) ≡ ∀Y ≤ |S|(L2(Y, S) ⊃ (X (Y ) ↔ φ(Y, X <Y/2 ))), where... |

19 | Provably total functions in bounded arithmetic theories Ri, U;' and
- Buss, Krajicek, et al.
- 1993
(Show Context)
Citation Context ...ierarchy functions in the third-order setting. Remarks. 1. Uniqueness of the function value is important, otherwise the definable multi-functions are (FEXP (Σexp i−1 )⋄ [wit,poly]) + ; analogously to =-=[1]-=- for U 1 2 . 2. Straightforward. 3. Functions are defined using the superstring-recursion scheme. 4. Straightforward. 5. Equality of theories is by “shortening of cuts”; the theories are conservative ... |

19 |
Propositional representation of arithmetic proofs
- Dowd
- 1979
(Show Context)
Citation Context ...osed under this restriction of limited doubling recursion as the recursion can be unwound with only a polynomial amount of additional space. This characterization is analogous to the one used in Dowd =-=[8]-=-: initial functions closed under limited recursion. Limited recursion in the context of number functions is of exponential length, as is limited doubling recursion in our setting. This in turn is remi... |

10 | The Strength of Replacement in Weak Arithmetic
- Cook, Thapen
- 2004
(Show Context)
Citation Context ...W 0 1 is therefore in some sense a minimal theory for FPSPACE + . 4 Further Research Some particular problems: First, does � W 1 1 prove the general induction of W 1 1 ? One approach is the method of =-=[6]-=-, namely by KPT witnessing, with HW 0 1 as the 9s10 starting point. Second, what about propositional translations of these theories? Partial progress is made in [13]. Third, are any of the theories HW... |

7 | A Safe Recursion Scheme for Exponential time
- Clote
- 1996
(Show Context)
Citation Context ...shown by Ritchie [11] to equal the linear space functions. Finally, with limited doubling recursion on E3, the step function of an exptime Turing machine can be iterated exponentially many times. See =-=[3]-=- for a previous recursion-theoretic characterization of the exponential-time number functions. ⊓⊔ 3 Third-Order Bounded Arithmetic Theories Our main theories are W i 1 and T W i 1, intended to corresp... |

7 | A third-order bounded arithmetic theory for PSPACE
- Skelley
- 2004
(Show Context)
Citation Context ...roach simplifies presentation of the theories and their propositional translations, and furthermore is applicable to complexity classes that previously had no corresponding theories. In previous work =-=[12]-=-, we adapted the second-order viewpoint to PSPACE with the third-order theory W 1 1 . In what follows, we generalize this result in several directions: First, by expanding the notion of computation to... |

2 |
The collapsing hierarchies
- Hartmanis
- 1987
(Show Context)
Citation Context ...urned to the machine by way of a special read-only reply tape (with random access in the case of a superstring-valued oracle). The usual exponential-time hierarchy has definition Σ exp i = NEXPΣp i−1 =-=[10]-=-. This is equal to Σi-TIME(exp), which are the languages computed by exponential time alternating Turing machines with i alternations (starting with existential). Paralleling this definition, we can d... |

2 | Theories and Proof Systems for PSPACE and the EXP-Time Hierarchy
- Skelley
- 2005
(Show Context)
Citation Context ... on a write-only tape, or “by query”, as a predicate with a distinguished string input as the characteristic function of the output bits of the superstring-valued function. Precise definitions are in =-=[13]-=-. We are interested primarily in polynomially bounded functions. In the context of third-order computation, we mean that the polynomial bound applies to the value of a number output or the length of a... |