# Automatic Complexity 2014-2019

The Simons Foundation under the program Collaboration Grants for Mathematicians (#315188 to Bjørn Kjos-Hanssen, grant title “Automatic Complexity”) supported my travel during 2014–2019.

Year Trips
2014 – 2015 COCOA 2014 (Maui); U. Washington; Varieties of Algorithmic Information; CCR, Heidelberg; Haidar to Arizona Winter School
2015 – 2016 ASL Annual Meeting, UConn (myself and Beros); SIAM Meeting on Discrete Math, Atlanta
2016 – 2017 WoLLIC, London; UCNC, Arkansas
2017 – 2018 Workshop on Computability Theory, Waterloo
2018 – 2019 Computability Theory and Foundations of Mathematics, Tokyo; Joint Math Meetings, Baltimore

#### Nine subprojects

Here are papers produced about automatic complexity, many with Master’s students.
Conferences in parentheses are those with no published proceedings.
Students or consultants in parentheses discussed the topic but were not coauthors.

Student/consultant Conference Journal
Hyde (MA, 2013) COCOON 2014 Elec. J. Combinatorics (2015)
COCOA 2014 Theoretical Computer Science (2015)
Alikhani (MA, 2014); Pakravan, Saadat (MSc Fin.Eng., 2013) Algorithmic Finance (2015)
(written at CCR 2015) Theory of Computing Systems (2017)
(Castiglione 2015) WoLLIC 2017 Discrete Mathematics (2018)
(Kobayashi 2016) (VAI 2015) Experimental Mathematics (2019)
(Huggins 2016) UCNC 2017
Liu (MA, 2017) (ALH 2018)
Yogi (MA, 2018)

For instance, I gave a talk in the Seattle Probability Seminar organized by Soumik Pal and Chris Burdzy at the University of Washington Department of Mathematics.

Title:
Kolmogorov structure functions for automatic complexity

Abstract:

We study an analogue of Kolmogorov’s notion of structure function, introduced in 1973, with Kolmogorov complexity replaced by Shallit and Wang’s (2001) automatic complexity. We discuss the prospects for using it for model selection in statistics. We prove an upper bound which is piecewise smooth, related to the binary entropy function, and appears to be fairly sharp based on numerical evidence.

The paper is loosely coupled with the following software:
- Complexity Guessing Game
- Complexity Option Game
- Structure Function Calculator
These are gathered in some slides.com slides.

# Long words with given complexity

Kaui Yogi completed his Master project in Spring 2018. With Achilles Beros we have a new paper on The number of long words with given automatic complexity based in part on Yogi’s project work.

# Deontic Logic and proof assistants

Damir Dzhafarov, Stefan Kaufmann, Bjørn Kjos-Hanssen, Dave Ripley, et al., at the 2016 ASL Annual Meeting at UConn.

Slides

José Carmo and Andrew J.I. Jones have studied contrary-to-duties obligations in a series of papers.

They develop a logical framework for scenarios such as the following:

1. There ought to be no dog.
2. If there is a dog, there ought to be a fence.

One conjecture from Carmo and Jones 1997 was refuted in a rather technical way in my 1996 term paper at University of Oslo.
The conjecture stated that one could simply add the condition
$\DeclareMathOperator{\pii}{ob}$
$$(Z \in \pii(X)) \land (Y \subseteq X) \land (Y \cap Z \ne \emptyset ) \rightarrow (Z \in \pii(Y )) \tag{5e}$$
for the conditional obligation operator ob.
In a follow-up paper (2001) they argued that (5e) could be added by weakening some other conditions.
In a new paper in Studia Logica, and presented at the Association for Symbolic Logic Annual Meeting 2016 at UConn, I argue that (5d) and (5e) are in conflict with each other. The argument is a generalization and strengthening of the 1996 argument.

2018: Benzmüller et al. have implemented Carmo and Jones’ logic in the proof assistant Isabelle.

# Automorphism(s) of the Turing degrees

Two papers on restrictions on automorphisms of Turing and truth-table degrees appeared in the Downey Festschrift for the Computability and Complexity Symposium 2017 in honour of Rod Downey’s 60th birthday.

One, “Permutations of the integers induce only the trivial automorphism of the Turing degrees”, is to appear in Bulletin of Symbolic Logic (2018), and will be presented at Workshop on Computability Theory in Waterloo, Ontario.

Slides

Image Credit: By Giligone – My own work using a Sony a 200, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=4801373

# Shift registers fool finite automata @ WoLLIC and in Discrete Mathematics

LFSRs (linear feedback shift registers) are popular pseudorandomness generators.
In a new project we show that they generate output (often called $m$-sequences) of maximal (nondeterministic path-based) automatic complexity. At this point we have an experimental result, one which would have probability $2^{-93}$ to occur “by chance”, as well as a theoretical but sub-optimal result.

Moreover, an $m$-sequence of length 31 provides an example of a word $x$ such that
$$A^-(x)=A_N(x)+2$$
where $A_N$ is nondeterministic path-based automatic complexity, and $A^-$ is non-total deterministic automatic complexity.
Such an example (where $A^-(x)-A_N(x)>1$) was not known before the consideration of LFSRs in this area. That consideration was an idea of Jason Castiglione.

The paper has been accepted at WoLLIC 2017 and in journal form for Discrete Mathematics (2018).

This project was presented at the poster session of the SIAM Conference on Discrete Mathematics 2016 in Atlanta, Georgia. The session was otherwise dominated by interesting work on RNA pseudoknots and chord diagrams (3 out of 6 posters) which in the case of the work of the Biocomplexity Institute researchers Ricky Chen and Thomas Li involves modeling with multiply context-free grammars.

WoLLIC 2017 slides

# On the complexity of automatic complexity

The paper “On the complexity of automatic complexity” is to appear in Theory of Computing Systems, in a post-conference journal issue for Computability, Complexity and Randomness 2015 held in Heidelberg, Germany.

While it is not known whether the set of strings of maximal nondeterministic automatic complexity is NP-complete (hence the paper is called “On the complexity…” rather than just “The complexity…”), the paper shows that the more general problem for automatic complexity of equivalence relations is NP-complete. It is also shown that the set of highly complex strings is not context-free.

# Superposition as memory @ UCNC17

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 $n=60$.

Details in the paper: Superposition as memory: unlocking quantum automatic complexity which appeared in the Lecture Notes in Computer Science volume of the conference Unconventional Computation and Natural Computation (UCNC) 2017.

Slides