Optimal Results on Recognizability for Infinite Time Register Machines
نویسنده
چکیده
Exploring further the properties of ITRM -recognizable reals started in [Ca], we provide a detailed analysis of recognizable reals and their distribution in Gödels constructible universe L. In particular, we show that new unrecognizables are generated at every index γ ≥ ω ω . We give a machine-independent characterization of recognizability by proving that a real r is recognizable iff it is Σ1-definable over LωCK,r ω and that r ∈ LωCK,r ω for every recognizable real r and show that either every or no r with r ∈ LωCK,r ω generated over an index stage Lγ is recognizable. Finally, the techniques developed along the way allow us to prove that the halting number for ITRMs is recognizable and that the set of ITRM -computable reals is not ITRM -decidable.
منابع مشابه
Infinite time recognizability from generic oracles and the recognizable jump operator
By a theorem of Sacks, if a real x is recursive relative to all elements of a set of positive Lebesgue measure, x is recursive. This statement, and the analogous statement for non-meagerness instead of positive Lebesgue measure, have been shown to carry over to many models of transfinite computations in [11]. Here, we start exploring another analogue concerning recognizability rather than compu...
متن کاملAn Enhanced Theory of Infinite Time Register Machines
Infinite time register machines (ITRMs) are register machines which act on natural numbers and which are allowed to run for arbitrarily many ordinal steps. Successor steps are determined by standard register machine commands. At limit times a register content is defined as a lim inf of previous register contents, if that limit is finite; otherwise the register is reset to 0. (A previous weaker ...
متن کاملThe Recognizability Strength of Infinite Time Turing Machines with Ordinal Parameters
We study infinite time Turing machines that attain a special state at a given class of ordinals during the computation. We prove results about sets that can be recognized by these machines. For instance, the recognizable sets of natural numbers with respect to the cardinal-detecting infinite time Turing machines introduced in [Hab13] are contained in a countable level of the constructible hiera...
متن کاملPost's Problem for Ordinal Register Machines
We study Post’s Problem for the ordinal register machines defined in [6], showing that its general solution is positive, but that any set of ordinals solving it must be unbounded in the writable ordinals. This mirrors the results in [3] for infinite-time Turing machines, and also provides insight into the different methods required for register machines and Turing machines in infinite time.
متن کاملInfinite Time Register Machines
Infinite time register machines (ITRMs) are register machines which act on natural numbers and which may run for arbitrarily many ordinal steps. Successor steps are determined by standard register machine commands, at limits the register contents are defined as lim inf’s of the previous register contents. We prove that a real number is computable by an ITRM iff it is hyperarithmetic.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- J. Symb. Log.
دوره 80 شماره
صفحات -
تاریخ انتشار 2015