One-Tape Turing Machine Variants and Language Recognition

We present two restricted versions of one-tape Turing machines. Both characterize the class of context-free languages. In the first version, proposed by Hibbard in 1967 and called limited automata, each tape cell can be rewritten only in the first $d$ visits, for a fixed constant $d\geq 2$. Furthermore, for $d=2$ deterministic limited automata are equivalent to deterministic pushdown automata, namely they characterize deterministic context-free languages. Further restricting the possible operations, we consider strongly limited automata. These models still characterize context-free languages. However, the deterministic version is less powerful than the deterministic version of limited automata. In fact, there exist deterministic context-free languages that are not accepted by any deterministic strongly limited automaton.Comment: 20 pages. This article will appear in the Complexity Theory Column of the September 2015 issue of SIGACT New

Verifying Time Complexity of Deterministic Turing Machines

We show that, for all reasonable functions $T(n)=o(n\log n)$, we can algorithmically verify whether a given one-tape Turing machine runs in time at most $T(n)$. This is a tight bound on the order of growth for the function $T$ because we prove that, for $T(n)\geq(n+1)$ and $T(n)=\Omega(n\log n)$, there exists no algorithm that would verify whether a given one-tape Turing machine runs in time at most $T(n)$. We give results also for the case of multi-tape Turing machines. We show that we can verify whether a given multi-tape Turing machine runs in time at most $T(n)$ iff $T(n_0)< (n_0+1)$ for some $n_0\in\mathbb{N}$. We prove a very general undecidability result stating that, for any class of functions $\mathcal{F}$ that contains arbitrary large constants, we cannot verify whether a given Turing machine runs in time $T(n)$ for some $T\in\mathcal{F}$. In particular, we cannot verify whether a Turing machine runs in constant, polynomial or exponential time.Comment: 18 pages, 1 figur

Descriptive Complexity of Deterministic Polylogarithmic Time and Space

We propose logical characterizations of problems solvable in deterministic polylogarithmic time (PolylogTime) and polylogarithmic space (PolylogSpace). We introduce a novel two-sorted logic that separates the elements of the input domain from the bit positions needed to address these elements. We prove that the inflationary and partial fixed point vartiants of this logic capture PolylogTime and PolylogSpace, respectively. In the course of proving that our logic indeed captures PolylogTime on finite ordered structures, we introduce a variant of random-access Turing machines that can access the relations and functions of a structure directly. We investigate whether an explicit predicate for the ordering of the domain is needed in our PolylogTime logic. Finally, we present the open problem of finding an exact characterization of order-invariant queries in PolylogTime.Comment: Submitted to the Journal of Computer and System Science

Strongly Universal Quantum Turing Machines and Invariance of Kolmogorov Complexity

We show that there exists a universal quantum Turing machine (UQTM) that can simulate every other QTM until the other QTM has halted and then halt itself with probability one. This extends work by Bernstein and Vazirani who have shown that there is a UQTM that can simulate every other QTM for an arbitrary, but preassigned number of time steps. As a corollary to this result, we give a rigorous proof that quantum Kolmogorov complexity as defined by Berthiaume et al. is invariant, i.e. depends on the choice of the UQTM only up to an additive constant. Our proof is based on a new mathematical framework for QTMs, including a thorough analysis of their halting behaviour. We introduce the notion of mutually orthogonal halting spaces and show that the information encoded in an input qubit string can always be effectively decomposed into a classical and a quantum part.Comment: 18 pages, 1 figure. The operation R is now really a quantum operation (it was not before); corrected some typos, III.B more readable, Conjecture 3.15 is now a theore