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Bjørn Kjos-Hanssen

Publications and source records attributed to Bjørn Kjos-Hanssen.

At least 37 records · Page 2Linked to original sources
arXiv2020

Automatic complexity of Fibonacci and Tribonacci words

For a complexity function $C$, the lower and upper $C$-complexity rates of an infinite word $\mathbf{x}$ are \[ \underline{C}(\mathbf x)=\liminf_{n\to\infty} \frac{C(\mathbf{x}\upharpoonright n)}n,\quad \overline{C}(\mathbf x)=\limsup_{n\to\infty} \frac{C(\mathbf{x}\upharpoonright n)}n \] respectively. Here $\mathbf{x}\upharpoonright n$ is the prefix of $x$ of length $n$. We consider the case $C=\mathrm{A_N}$, the nondeterministic automatic complexity. If these rates are strictly between 0 and $1/2$, we call them intermediate. Our main result is that words having intermediate $\mathrm{A_N}$-rates exist, viz. the infinite Fibonacci and Tribonacci words.

cs.DM↗
arXiv2020

A tractable case of the Turing automorphism problem: bi-uniformly $E_0$-invariant Cantor homeomorphisms

A function $F:2^ω\to 2^ω$ is an $E_0$-isomorphism if for all $x,y\in 2^ω$, we have $xE_0y\iff f(x)E_0 f(y)$, where $xE_0y\iff(\exists a)(\forall n\ge b) x(n)=y(n)$. If such witnesses $a$ for $xE_0 y$ and for $f(x)E_0 f(y)$ depend on each other but not on $x$, $y$, then $F$ is called bi-uniform. It is shown that a homeomorphism of Cantor space which is a bi-uniform $E_0$-isomorphism can induce only the trivial automorphism of the Turing degrees.

math.LO↗
arXiv2020

From eventually different functions to pandemic numberings

A function is strongly non-recursive (SNR) if it is eventually different from each recursive function. We obtain hierarchy results for the mass problems associated with computing such functions with varying growth bounds. In particular, there is no least and no greatest Muchnik degree among those of the form SNR$_f$ consisting of SNR functions bounded by varying recursive bounds $f$. We show that the connection between SNR functions and canonically immune sets is, in a sense, as strong as that between DNR (diagonally non-recursive) functions and effectively immune sets. Finally, we introduce pandemic numberings, a set-theoretic dual to immunity.

math.LO↗
arXiv2020

Nondeterministic automatic complexity of overlap-free and almost square-free words

Shallit and Wang studied deterministic automatic complexity of words. They showed that the automatic Hausdorff dimension $I(\mathbf t)$ of the infinite Thue word satisfies $1/3\le I(\mathbf t)\le 2/3$. We improve that result by showing that $I(\mathbf t)\ge 1/2$. For nondeterministic automatic complexity we show $I(\mathbf t)=1/2$. We prove that such complexity $A_N$ of a word $x$ of length $n$ satisfies $A_N(x)\le b(n):=\lfloor n/2\rfloor + 1$. This enables us to define the complexity deficiency $D(x)=b(n)-A_N(x)$. If $x$ is square-free then $D(x)=0$. If $x$ almost square-free in the sense of Fraenkel and Simpson, or if $x$ is a strongly cube-free binary word such as the infinite Thue word, then $D(x)\le 1$. On the other hand, there is no constant upper bound on $D$ for strongly cube-free words in a ternary alphabet, nor for cube-free words in a binary alphabet. The decision problem whether $D(x)\ge d$ for given $x$, $d$ belongs to $NP\cap E$.

cs.FL↗
arXiv2020

On the complexity of automatic complexity

Generalizing the notion of automatic complexity of individual strings due to Shallit and Wang, we define the automatic complexity $A(E)$ of an equivalence relation $E$ on a finite set $S$ of strings. We prove that the problem of determining whether $A(E)$ equals the number $|E|$ of equivalence classes of $E$ is $\mathsf{NP}$-complete. The problem of determining whether $A(E) = |E| + k$ for a fixed $k\ge 1$ is complete for the second level of the Boolean hierarchy for $\mathsf{NP}$, i.e., $\mathsf{BH}_2$-complete. Let $L$ be the language consisting of all strings of maximal nondeterministic automatic complexity. We characterize the complexity of infinite subsets of $L$ by showing that they can be co-context-free but not context-free, i.e., $L$ is $\mathsf{CFL}$-immune, but not $\mathsf{coCFL}$-immune. We show that for each $ε>0$, $L_ε\not\in\mathsf{coCFL}$, where $L_ε$ is the set of all strings whose deterministic automatic complexity $A(x)$ satisfies $A(x)\ge |x|^{1/2-ε}$.

cs.FL↗
arXiv2020

Automatic complexity of shift register sequences

Let $x$ be an $m$-sequence, a maximal length sequence produced by a linear feedback shift register. We show that $x$ has maximal subword complexity function in the sense of Allouche and Shallit. We show that this implies that the nondeterministic automatic complexity $A_N(x)$ is close to maximal: $n/2-A_N(x)=O(\log^2n)$, where $n$ is the length of $x$. In contrast, Hyde has shown $A_N(y)\le n/2+1$ for all sequences $y$ of length $n$.

cs.FL↗
arXiv2019

Constructing a weak subset of a random set

The tree forcing method given by (Liu 2015) enables the cone avoiding of strong enumeration of a given tree, within a subset or co-subset of an arbitrary given set, provided the given tree does not admit computable strong enumeration. Using this result, we settled and reproduced a series of problems in reverse mathematics. In this paper, we demonstrate cone avoiding results within an infinite subset of a given 1-random set. We show that for any given 1-random set $X$, there exists an infinite subset $Y$ of $X$ such that $Y$ does not compute any real with positive effective Hausdorff dimension, thus answering negatively a question posed by Kjos-Hanssen that whether there exists a 1-random set of which any infinite subset computes some 1-random real. The result is surprising in that the tree forcing technique used on the subset or co-subset seems to heavily rely on subset co-subset combinatorics, whereas this result does not.

math.LO↗
arXiv2019

Planar digraphs for automatic complexity

We show that the digraph of a nondeterministic finite automaton witnessing the automatic complexity of a word can always be taken to be planar. In the case of total transition functions studied by Shallit and Wang, planarity can fail. Let $s_q(n)$ be the number of binary words $x$ of length $n$ having nondeterministic automatic complexity $A_N(x)=q$. We show that $s_q$ is eventually constant for each $q$ and that the eventual constant value of $s_q$ is computable.

cs.FL↗
arXiv2019

The Strength of the Grätzer-Schmidt Theorem

The Grätzer-Schmidt theorem of lattice theory states that each algebraic lattice is isomorphic to the congruence lattice of an algebra. We study the reverse mathematics of this theorem. We also show that the set of indices of computable lattices that are complete is $Π^1_1$-complete; the set of indices of computable lattices that are algebraic is $Π^1_1$-complete; the set of compact elements of a computable lattice is $Π^{1}_{1}$ and can be $Π^1_1$-complete; and the set of compact elements of a distributive computable lattice is $Π^{0}_{3}$, and there is an algebraic distributive computable lattice such that the set of its compact elements is $Π^0_3$-complete.

math.LO↗
arXiv2017

Covering the recursive sets

We give solutions to two of the questions in a paper by Brendle, Brooke-Taylor, Ng and Nies. Our examples derive from a 2014 construction by Khan and Miller as well as new direct constructions using martingales. At the same time, we introduce the concept of i.o. subuniformity and relate this concept to recursive measure theory. We prove that there are classes closed downwards under Turing reducibility that have recursive measure zero and that are not i.o. subuniform. This shows that there are examples of classes that cannot be covered with methods other than probabilistic ones. It is easily seen that every set of hyperimmune degree can cover the recursive sets. We prove that there are both examples of hyperimmune-free degree that can and that cannot compute such a cover.

math.LO↗
arXiv2017

Effective bi-immunity and randomness

We study the relationship between randomness and effective bi-immunity. Greenberg and Miller have shown that for any oracle X, there are arbitrarily slow-growing DNR functions relative to X that compute no ML random set. We show that the same holds when ML randomness is replaced with effective bi-immunity. It follows that there are sequences of effective Hausdorff dimension 1 that compute no effectively bi-immune set. We also establish an important difference between the two properties. The class Low(MLR, EBI) of oracles relative to which every ML random is effectively bi-immune contains the jump-traceable sets, and is therefore of cardinality continuum.

math.LO↗
arXiv2017

Superposition as memory: unlocking quantum automatic complexity

Imagine a lock with two states, "locked" and "unlocked", which may be manipulated using two operations, called 0 and 1. Moreover, the only way to (with certainty) unlock using four operations is to do them in the sequence 0011, i.e., $0^n1^n$ where $n=2$. In this scenario one might think that the lock needs to be in certain further states after each operation, so that there is some memory of what has been done so far. Here we show that this memory can be entirely encoded in superpositions of the two basic states "locked" and "unlocked", where, as dictated by quantum mechanics, the operations are given by unitary matrices. Moreover, we show using the Jordan--Schur lemma that a similar lock is not possible for $n=60$. We define the semi-classical quantum automatic complexity $Q_{s}(x)$ of a word $x$ as the infimum in lexicographic order of those pairs of nonnegative integers $(n,q)$ such that there is a subgroup $G$ of the projective unitary group PU$(n)$ with $|G|\le q$ and with $U_0,U_1\in G$ such that, in terms of a standard basis $\{e_k\}$ and with $U_z=\prod_k U_{z(k)}$, we have $U_x e_1=e_2$ and $U_y e_1 \ne e_2$ for all $y\ne x$ with $|y|=|x|$. We show that $Q_s$ is unbounded and not constant for strings of a given length. In particular, \[ Q_{s}(0^21^2)\le (2,12) < (3,1) \le Q_{s}(0^{60}1^{60}) \] and $Q_s(0^{120})\le (2,121)$.

cs.CR↗
arXiv2016

Few paths, fewer words: model selection with automatic structure functions

We consider the problem of finding an optimal statistical model for a given binary string. Following Kolmogorov, we use structure functions. In order to get concrete results, we replace Turing machines by finite automata and Kolmogorov complexity by Shallit and Wang's automatic complexity. The $p$-value of a model for given data $x$ is the probability that there exists a model with as few states, accepting as few words, fitting uniformly randomly selected data $y$. Deterministic and nondeterministic automata can give different optimal models. For $x=011\, 110\, 110\, 11$, the best deterministic model has $p$-value $0.3$, whereas the best nondeterministic model has $p$-value $0.04$. In the nondeterministic case, counting paths and counting words can give different optimal models. For $x=01100\, 01000$, the best path-counting model has $p$-value $0.79$, whereas the best word-counting model has $p$-value $0.60$.

cs.FL↗
arXiv2016

Models of the Chisholm set

We give a counter-example showing that Carmo and Jones' condition 5(e) may conflict with other conditions on the models in their paper \emph{A new approach to contrary-to-duty obligations}.

cs.LO↗
arXiv2016

Local initial segments of the Turing degrees

Recent results on initial segments of the Turing degrees are presented, and some conjectures about initial segments that have implications for the existence of non-trivial automorphisms of the Turing degrees are indicated.

math.LO↗
arXiv2016

Pricing complexity options

We consider options that pay the complexity deficiency of a sequence of up and down ticks of a stock upon exercise. We study the price of European and American versions of this option numerically for automatic complexity, and theoretically for Kolmogorov complexity. We also consider run complexity, which is a restricted form of automatic complexity.

q-fin.PR↗