Searcharxiv⌕ Search

arXiv subjects

Merlin Carl

Publications and source records attributed to Merlin Carl.

48 records · Page 3Linked to original sources
arXiv2015

Recognizable sets and Woodin cardinals: Computation beyond the constructible universe

We call a subset of an ordinal $λ$ recognizable if it is the unique subset $x$ of $λ$ for which some Turing machine with ordinal time and tape, which halts for all subsets of $λ$ as input, halts with the final state $0$. Equivalently, such a set is the unique subset $x$ which satisfies a given $Σ_1$ formula in $L[x]$. We prove several results about sets of ordinals recognizable from ordinal parameters by ordinal time Turing machines. Notably we show the following results from large cardinals. (1) Computable sets are elements of $L$, while recognizable objects with infinite time computations appear up to the level of Woodin cardinals. (2) A subset of a countable ordinal $λ$ is in the recognizable closure for subsets of $λ$ if and only if it is an element of $M^{\infty}$, where $M^{\infty}$ denotes the inner model obtained by iterating the least measure of $M_1$ through the ordinals, and where the recognizable closure for subsets of $λ$ is defined by closing under relative recognizability for subsets of $λ$.

math.LO↗
arXiv2015

Randomness and Degree Theory for Infinite Time Register Machines

A concept of randomness for infinite time register machines (ITRMs) is defined and studied. In particular, we show that for this notion of randomness, computability from mutually random reals implies computability and that an analogue of van Lambalgen's theorem holds. This is then applied to obtain results on the structure of ITRM-degrees. Finally, we consider autoreducibility for ITRMs and show that randomness implies non-autoreducibility.

math.LO↗
arXiv2015

The Lost Melody Phenomenon

A typical phenomenon for machine models of transfinite computations is the existence of so-called lost melodies, i.e. real numbers $x$ such that the characteristic function of the set $\{x\}$ is computable while $x$ itself is not (a real having the first property is called recognizable). This was first observed by J. D. Hamkins and A. Lewis for infinite time Turing machine, then demonstrated by P. Koepke and the author for $ITRM$s. We prove that, for unresetting infinite time register machines introduced by P. Koepke, recognizability equals computability, i.e. the lost melody phenomenon does not occur. Then, we give an overview on our results on the behaviour of recognizable reals for $ITRM$s. We show that there are no lost melodies for ordinal Turing machines or ordinal register machines without parameters and that this is, under the assumption that $0^{\sharp}$ exists, independent of $ZFC$. Then, we introduce the notions of resetting and unresetting $α$-register machines and give some information on the question for which of these machines there are lost melodies.

math.LO↗
arXiv2015

A Note on Always Decidable Propositional Forms

We ask the following question: If all instantiations of a propositional formula $A(x_1,...,x_n)$ in $n$ propositional variables are decidable in some sufficiently strong recursive theory, does it follow that $A$ is tautological or contradictory? and answer it in the affirmative. We also consider the following related question: Suppose that for some propositional formula $A(x_1,...,x_n)$, there is a Turing program $P$ such that $P([ϕ_{1}],...,[ϕ_{n}])\downarrow=1$ iff $\mathbb{N}\models A(ϕ_{1},...,ϕ_{n})$ and otherwise $P([ϕ_{1}],...,[ϕ_{n}])\downarrow=0$ (where $[ϕ]$ denotes the Gödel number of $ϕ$), does it follow that the truth value of $A(ϕ_{1},...,ϕ_{n})$ is independent of $ϕ_1,...,ϕ_{n}$ and hence that $A$ is tautological or contradictory?

math.LO↗
arXiv2014

A Note on the Decidability of the Necessity of Axioms

A typical kind of question in mathematical logic is that for the necessity of a certain axiom: Given a proof of some statement $ϕ$ in some axiomatic system $T$, one looks for minimal subsystems of $T$ that allow deriving $ϕ$. In particular, one asks whether, given some system $T+ψ$, $T$ alone suffices to prove $ϕ$. We show that this problem is undecidable unless $T+\negψ$ is decidable.

math.LO↗
arXiv2014

Infinite computations with random oracles

We consider the following problem for various infinite time machines. If a real is computable relative to large set of oracles such as a set of full measure or just of positive measure, a comeager set, or a nonmeager Borel set, is it already computable? We show that the answer is independent from ZFC for ordinal time machines (OTMs) with and without ordinal parameters and give a positive answer for most other machines. For instance, we consider, infinite time Turing machines (ITTMs), unresetting and resetting infinite time register machines (wITRMs, ITRMs), and α-Turing machines for countable admissible ordinals α.

math.LO↗
arXiv2014

Algorithmic Randomness for Infinite Time Register Machines

A concept of randomness for infinite time register machines (ITRMs), resembling Martin-Löf-randomness, is defined and studied. In particular, we show that for this notion of randomness, computability from mutually random reals implies computability and that an analogue of van Lambalgen's theorem holds.

math.LO↗
arXiv2013

Models of true arithmetic are integer parts of nice real closed fields

Exploring further the connection between exponentiation on real closed fields and the existence of an integer part modelling strong fragments of arithmetic, we demonstrate that each model of true arithmetic is an integer part of an exponential real closed field that is elementary equivalent to the reals with exponentiation.

math.LO↗
arXiv2013

Towards a Church-Turing-Thesis for Infinitary Computations

We consider the question whether there is an infinitary analogue of the Church-Turing-thesis. To this end, we argue that there is an intuitive notion of transfinite computability and build a canonical model, called Idealized Agent Machines ($IAM$s) of this which will turn out to be equivalent in strength to the Ordinal Turing Machines defined by P. Koepke.

math.LO↗
arXiv2013

Optimal Results on ITRM-recognizability

Exploring further the properties of ITRM-recognizable reals, we provide a detailed analysis of recognizable reals and their distribution in Gödels constructible universe L. In particular, we show that, for unresetting infinite time register machines, the recognizable reals coincide with the computable reals and that, for ITRMs, unrecognizables are generated at every index bigger than the first limit of admissibles. We show that a real r is recognizable iff it is $Σ_{1}$-definable over $L_{ω_ω^{CK,r}}$, that $r\in L_{ω_ω^{CK,r}}$ for every recognizable real $r$ and that either all or no real generated over an index stage $L_γ$ are recognizable.

math.LO↗
arXiv2012

The distribution of ITRM-recognizable reals

Infinite Time Register Machines ($ITRM$'s) are a well-established machine model for infinitary computations. Their computational strength relative to oracles is understood, see e.g. Koepke (2009), Koepke and Welch (2011) and Koepke and Miller (2008). We consider the notion of recognizability, which was first formulated for Infinite Time Turing Machines in Hamkins and Lewis (200) and applied to $ITRM$'s in Carl et al. (2010). A real $x$ is $ITRM$-recognizable iff there is an $ITRM$-program $P$ such that $P^{y}$ stops with output 1 iff $y=x$, and otherwise stops with output 0. In Carl et al. (2010), it is shown that the recognizable reals are not contained in the computable reals. Here, we investigate in detail how the $ITRM$-recognizable reals are distributed along the canonical well-ordering $<_{L}$ of Gödel's constructible hierarchy $L$. In particular, we prove that the recognizable reals have gaps in $<_{L}$, that there is no universal $ITRM$ in terms of recognizability and consider a relativized notion of recognizability.

math.LO↗