# No modular iterative way to get joint distribution from covariance matrix

Suppose $A$, and $B$ are events with
$$\Pr(A)=3/12,\quad \Pr(B)=4/12,\quad\Pr(A\cap B)=0$$
Suppose $A’$ and $B’$ are events with
$$\Pr(A’)=3/12,\quad\Pr(B’)=8/12,\quad\Pr(A’\cap B’)=1/12$$
Notice that the covariance matrix $M$ for the Bernoulli random variables $1_A$, $1_B$ is the same as the one for $1_{A’}$, $1_{B’}$.
Now suppose we wanted to take any given joint distribution giving covariance matrix $M$ and extend it to the covariance matrix for $A$, $B$, $C$, where $\Pr(C)=\Pr(C\setminus(A\cup B))=5/12$.

We claim this is impossible if we are given the joint distribution of $A’$ and $B’$. That is, we claim there is no choice of probabilities concerning $C’$ that will give the right covariance matrix.
Note that $\Pr(C’)\in\{5/12, 7/12\}$ is necessary since $\mathrm{Var}(1_C)=\mathrm{Var}(1_{C’})$ is necessary. Moreover
$$0-(3/12)(5/12)=\mathrm{Cov}(A,C)=\mathrm{Cov}(A’,C’)=E(1_{A’}1_{C’})-E(1_{A’})E(1_{C’})=\Pr(A’\cap C’)-(3/12)(5/12\text{ or }7/12)$$
so
$$\Pr(A’\cap C’)=(3/12)(0\text{ or }2/12)=0\text{ or }6/144$$
Similarly for $B’$,
$$0-(4/12)(5/12)=\mathrm{Cov}(B,C)=\mathrm{Cov}(B’,C’)=E(1_{B’}1_{C’})-E(1_{B’})E(1_{C’})=\Pr(B’\cap C’)-(8/12)(5/12\text{ or }7/12)$$
so
$$\Pr(B’\cap C’)=-20/144+(40\text{ or }56)/144= (20\text{ or }36)/144$$
The choice $\Pr(C’)=5/12$. $\Pr(A’\cap C’)=0$, $\Pr(B’\cap C’)=20/144$ gives $\Pr(A’\cup B’\cup C’)=156/144>1$, contradiction.

The other choice $\Pr(C’)=7/12$, $\Pr(A’\cap C’)=6/144$, $\Pr(B’\cap C’)=36/144$ gives $\Pr(C’\setminus(A’\cup B’))\ge 84/144-6/144-36/144=42/144=7/24$, so $\Pr(A’\cup B’\cup C’)\ge 10/12 + 7/24 > 1$, also contradiction.
Q.E.D.

# 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.

# Shift registers fool finite automata @ WoLLIC 2017

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.

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

# 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

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

The paper “Kolmogorov structure functions for automatic complexity in computational statistics” appeared in the Lecture Notes in Computer Science proceedings of the conference COCOA 2014, Maui, Hawaii.
The paper then appeared in the journal Theoretical Computer Science 2015.

The ideas are implemented in the Structure function calculator.

A new paper Few paths, fewer words: model selection with automatic structure functions has been conditionally accepted for publication in Experimental Mathematics.

Some slides

# Covering the computable sets

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.

# A Conflict Between Some Semantic Conditions

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.