scholarly journals P ≠ NP ∩ co-NP for Infinite Time Turing Machines

2005 ◽  
Vol 15 (5) ◽  
pp. 577-592 ◽  
Author(s):  
Vinay Deolalikar ◽  
Joel David Hamkins ◽  
Ralf Schindler
Keyword(s):  
Turing Machines ◽  
Infinite Time

Eventually infinite time Turing machine degrees: infinite time decidable reals

2000 ◽  
Vol 65 (3) ◽  
pp. 1193-1203 ◽  
Author(s):  
P.D. Welch
Keyword(s):  
Turing Machine ◽  
Initial Segment ◽  
Upper Bounds ◽  
Turing Degrees ◽  
Turing Machines ◽  
Jump Operator ◽  
Infinite Time

AbstractWe characterise explicitly the decidable predicates on integers of Infinite Time Turing machines, in terms of admissibility theory and the constructible hierarchy. We do this by pinning down ζ, the least ordinal not the length of any eventual output of an Infinite Time Turing machine (halting or otherwise); using this the Infinite Time Turing Degrees are considered, and it is shown how the jump operator coincides with the production of mastercodes for the constructible hierarchy; further that the natural ordinals associated with the jump operator satisfy a Spector criterion, and correspond to the Lζ-stables. It also implies that the machines devised are “Σ2 Complete” amongst all such other possible machines. It is shown that least upper bounds of an “eventual jump” hierarchy exist on an initial segment.

scholarly journals Space and time complexity for infinite time Turing machines

2020 ◽  
Vol 30 (6) ◽  
pp. 1239-1255
Author(s):  
Merlin Carl
Keyword(s):  
Time Complexity ◽  
Space Complexity ◽  
Complexity Classes ◽  
Turing Machines ◽  
Infinite Time ◽  
Space And Time ◽  
Open Questions ◽  
Separation Results ◽  
Ordinal Computability

Abstract We consider notions of space by Winter [21, 22]. We answer several open questions about these notions, among them whether low space complexity implies low time complexity (it does not) and whether one of the equalities P=PSPACE, P$_{+}=$PSPACE$_{+}$ and P$_{++}=$PSPACE$_{++}$ holds for ITTMs (all three are false). We also show various separation results between space complexity classes for ITTMs. This considerably expands our earlier observations on the topic in Section 7.2.2 of Carl (2019, Ordinal Computability: An Introduction to Infinitary Machines), which appear here as Lemma $6$ up to Corollary $9$.

scholarly journals SAFE RECURSIVE SET FUNCTIONS

2015 ◽  
Vol 80 (3) ◽  
pp. 730-762 ◽  
Author(s):  
ARNOLD BECKMANN ◽  
SAMUEL R. BUSS ◽  
SY-DAVID FRIEDMAN
Keyword(s):  
Growth Rate ◽  
Polynomial Growth ◽  
Turing Machines ◽  
Finite Sets ◽  
Infinite Time ◽  
Machine Model ◽  
Set Functions ◽  
Rate Functions ◽  
Primitive Recursive ◽  
Recursive Set

AbstractWe introduce the safe recursive set functions based on a Bellantoni–Cook style subclass of the primitive recursive set functions. We show that the functions computed by safe recursive set functions under a list encoding of finite strings by hereditarily finite sets are exactly the polynomial growth rate functions computed by alternating exponential time Turing machines with polynomially many alternations. We also show that the functions computed by safe recursive set functions under a more efficient binary tree encoding of finite strings by hereditarily finite sets are exactly the quasipolynomial growth rate functions computed by alternating quasipolynomial time Turing machines with polylogarithmic many alternations.We characterize the safe recursive set functions on arbitrary sets in definability-theoretic terms. In its strongest form, we show that a function on arbitrary sets is safe recursive if and only if it is uniformly definable in some polynomial level of a refinement of Jensen's J-hierarchy, relativized to the transitive closure of the function's arguments.We observe that safe recursive set functions on infinite binary strings are equivalent to functions computed by infinite-time Turing machines in time less than ωω. We also give a machine model for safe recursive set functions which is based on set-indexed parallel processors and the natural bound on running times.

An injection from the Baire space to natural numbers

2014 ◽  
Vol 25 (7) ◽  
pp. 1484-1489 ◽  
Author(s):  
ANDREJ BAUER
Keyword(s):  
Baire Space ◽  
Turing Machines ◽  
Natural Numbers ◽  
Infinite Time ◽  
Model Based ◽  
Infinite Sequences

We provide a realizability model based on infinite time Turing machines in which there is an injection from the internal Baire space, the object of infinite sequences of numbers, to the object of natural numbers.

Physical Computational Complexity and First-order Logic

2021 ◽  
Vol 181 (2-3) ◽  
pp. 129-161
Author(s):  
Richard Whyman
Keyword(s):  
Order Theory ◽  
Quantum Computers ◽  
Arbitrary System ◽  
Turing Machines ◽  
Logical System ◽  
Computational Power ◽  
Infinite Time ◽  
First Order ◽  
First Order Theory

We present the concept of a theory machine, which is an atemporal computational formalism that is deployable within an arbitrary logical system. Theory machines are intended to capture computation on an arbitrary system, both physical and unphysical, including quantum computers, Blum-Shub-Smale machines, and infinite time Turing machines. We demonstrate that for finite problems, the computational power of any device characterisable by a finite first-order theory machine is equivalent to that of a Turing machine. Whereas for infinite problems, their computational power is equivalent to that of a type-2 machine. We then develop a concept of complexity for theory machines, and prove that the class of problems decidable by a finite first order theory machine with polynomial resources is equal to 𝒩𝒫 ∩ co-𝒩𝒫.

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