Transfinite machine models

Welch, Philip D.

doi:10.1017/cbo9781107338579.015

Cited by 6 publications

(2 citation statements)

References 88 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Secondly, an infinite time Turing machine (ITTM) [14] is a generalisation of Turing computability involving infinite time or space. Welsh provides an overview in [53] and Dag Normann studies non-montone inductive definitions and the connection to ITTMs in [27].…”

Section: On Higher-order Computationmentioning

confidence: 99%

Between Turing and Kleene

Sanders

2021

Logical Foundations of Computer Science

View full text Add to dashboard Cite

Turing's famous 'machine' model constitutes the first intuitively convincing framework for computing with real numbers. Kleene's computation schemes S1-S9 extend Turing's approach to computing with objects of any finite type. Both frameworks have their pros and cons and it is a natural question if there is an approach that marries the best of both the Turing and Kleene worlds. In answer to this question, we propose a considerable extension of the scope of Turing's approach. Central is a fragment of the Axiom of Choice involving continuous choice functions, going back to Kreisel-Troelstra and intuitionistic analysis. Put another way, we formulate a relation 'is computationally stronger than' involving third-order objects that overcomes (many of) the pitfalls of the Turing and Kleene frameworks.

show abstract

Section: On Higher-order Computationmentioning

confidence: 99%

Between Turing and Kleene

Sanders

2021

Logical Foundations of Computer Science

View full text Add to dashboard Cite

show abstract

“…The ITTM's were introduced by Hamkins and Kidder, but first appeared in published form in [5]. For a recent survey, see [17]. A more ambitious, but vague, conjecture is that there is some total, countably based functional Ψ of type 3 such that Kleene-computations relative to Ψ somehow reflects computations using ITTM's.…”

Section: Speculations On Functionals Of Typementioning

confidence: 99%

Functionals of Type 3 as Realisers of Classical Theorems in Analysis

Normann

2018

Lecture Notes in Computer Science

View full text Add to dashboard Cite

We investigate the relative computational strength of combinations of four higher order functionals, the jump and hyperjump seen as functionals of type 2 and realisers for the compactness of Cantor space and the Lindelöf property of Baire space seen as functionals of type 3. We compare them with the closure operator for non-monotone inductive definitions of sets of integers, also seen as a functional of type 3.

show abstract

The Recognizability Strength of Infinite Time Turing Machines with Ordinal Parameters

Carl

Schlicht

2017

Unveiling Dynamics and Complexity

View full text Add to dashboard Cite

Infinite time Turing machine models with tape length α, denoted Tα, strengthen the machines of Hamkins and Kidder [HL00] with tape length ω. A new phenomenon is that for some countable ordinals α, some cells cannot be halting positions of Tα given trivial input. The main open question in [Rin14] asks about the size of the least such ordinal δ.We answer this by providing various characterizations. For instance, δ is the least ordinal with any of the following properties:• For some ξ < α, there is a T ξ -writable but not Tα-writable subset of ω.• There is a gap in the Tα-writable ordinals.• α is uncountable in L λα . Here λα denotes the supremum of Tα-writable ordinals, i.e. those with a Tαwritable code of length α.We further use the above characterizations, and an analogue to Welch's submodel characterization of the ordinals λ, ζ and Σ, to show that δ is large in the sense that it is a closure point of the function α → Σα, where Σα denotes the supremum of the Tα-accidentally writable ordinals.

show abstract

Transfinite machine models

Cited by 6 publications

References 88 publications

Between Turing and Kleene

Between Turing and Kleene

Functionals of Type 3 as Realisers of Classical Theorems in Analysis

The Recognizability Strength of Infinite Time Turing Machines with Ordinal Parameters

Contact Info

Product

Resources

About