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 is to appear in the Lecture Notes in Computer Science volume of the conference Unconventional Computation and Natural Computation (UCNC) 2017.

I participated in the workshop on Algorithmic Randomness in Singapore and the conference on Computability, Complexity and Randomness.

With host Frank Stephan and fellow participant Sebastiaan Terwijn we wrote a paper entitled Covering the recursive sets which appeared in Lecture Notes in Computer Science, Conference on Computability in Europe, 2015, and has now been published in Annals of Pure and Applied Logic.

*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, to appear 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.

*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.*

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.

All talks will be in Keller Hall 314.

## Scientific program

Thursday February 12

- 10:30-11:20am Olga Kharlampovich (CUNY — Hunter College):
*Elementary classification questions for groups and algebras I: Groups.*
- 11:30am-1:20pm: CASUAL LUNCH. Suggestion: Manoa Gardens
- 1:30-2:20pm Free discussion

Friday February 13

- 9:30-10:20am Bakhadyr Khoussainov (University of Auckland):A quest for algorithmically random algebraic structures
- 11:30-12:20pm Alexei Miasnikov (Stevens Institute of Technology):
*Elementary classification questions for groups and algebras II: Associative and Lie algebras.*
- 12:30-1:20pm BRIEF LUNCH. Suggestion: Sustainability Courtyard
- 1:30-2:20pm Paul Kim Long V. Nguyen (University of Hawaii — Leeward Community College):
*$\Sigma^0_3$-completeness of subdirect irreducibility*

## Abstract (Kharlampovich and Miasnikov)

We consider some fundamental model-theoretic questions that can be asked about a given algebraic structure (a group, a ring, etc.), or a class of structures, to understand its principal algebraic and logical properties. These Tarski type questions include: elementary classification and decidability of the first-order theory.

In the case of free groups we proved that two non-abelian free groups of different ranks are elementarily equivalent, classified finitely generated groups elementarily equivalent to a finitely generated free group (also done by Sela) and proved decidability of the first-order theory.

We describe partial solutions to Tarski’s problems in the class of free associative and Lie algebras of finite rank and some open problems. In particular, we will show that unlike free groups, two free associative algebras of finite rank over the same field are elementarily equivalent if and only if they are isomorphic. Two free associative algebras of finite rank over different infinite fields are elementarily equivalent if and only if the fields are equivalent in the weak second order logic, and the ranks are the same. We will also show that for an infinite field the theory of a free associative algebra is undecidable.

## Things to do in Honolulu

Pow!Wow! art festival

Travel: parking, getting to campus from your hotel, and beaches.

## Professor of Mathematics, University of Hawaii at Manoa