COSC341

Theory of Computing


Alex Gavryushkin


2019 Semester 1

Table of contents

Handouts

Information

  • COSC341 website
  • Assessment
    • Assignment 1: 10% due Thursday 28 March
    • Assignment 2: 10% due Friday 19 April
    • Assignment 3: 10% due Friday 24 May
    • Final exam: 70% on TBA

Lecturer

GitHub: @bioDS

Twitter: @bioDS_lab

Lecture 1

Introduction

Who cares about theory of computing?
I do!


ML = Machine Learning
CoSt = Computational Statistics
AS = Applied Statistics

Who cares about theory of computing?
You do!


Garey and Johnson. Computers and Intractability.

Why biological data science?

Because the skills are highly transferable:

  • Nosy data
  • Visualisation
  • Communication
  • High-performance computing

Why is it suddenly a thing?

Sets

A set is a collection of objects, completely determined by the objects. Two sets are equal if they contain the same objects.

$X = \{1, 2, 5\} = \{2, 5, 1\} = \{5, 2, 1, 5, 1\}$

$X = \{x \mid P(x)\}$, where $P(x)$ is a logical condition on $x$.

$\mathbb N = \{0, 1, \ldots \}$

$X = \{x \mid (\exists y \in \mathbb N) x = y^2\} \stackrel{?}{=} \{x \mid x = y^2\}$

$9 \in X?$

Empty set
$\varnothing$
is a set!

You will forget this. Yes, you.

Operations on sets


ML = Data $\cap$ Algorithms

(Statistics$\setminus$DS) $\cap$ Algorithms $\neq \varnothing$

$\overline{\mbox{Data} \cup \mbox{Algorithms} \cup \mbox{Statistics}} = ?$

Lecture 2

Sets, relations, functions

Subsets, ordered tuples, relations, and functions

$A \subseteq B \iff (\forall x) (x \in A \Rightarrow x \in B)$

$\mathcal P(A) = \{X \mid X \subseteq A\}$

$(a, b) = (c, d) \iff (a = c~\&~b = d)$

$A \times B = \{(a, b) \mid a \in A, b \in B\}$

$P \subseteq A \times B$ is a relation (from $A$ to $B$, or on $A$ if $A = B$)

$f \subseteq A \times B$ is a function if $(f(a) = b ~\&~ f(a) = c) \Rightarrow b = c$

Types of functions



Let $f : A \to B$. The function $f$ is called

injective if $f(a) = f(b) \Rightarrow a = b$

surjective if $(\forall y \in B)(\exists x \in A)f(x) = y$

bijective if $f$ is injective and surjective.

Molecular biology: crash course

Classical genetics

Modern genomics

Reality

Partitions and equivalence relations



$A = B \sqcup C$ is a partition if $A = B \cup C$ and $B \cap C = \varnothing$

A relation $\sim$ on $A$ is called an equivalence relations if $\sim$ is:

  • Reflexive $(\forall x \in A)~x \sim x$
  • Symmetric $(\forall x, y \in A)~(x \sim y \Rightarrow y \sim x)$
  • Transitive $(\forall x, y, z \in A)~(x \sim y ~\&~ y \sim z \Rightarrow x \sim z)$

Theorem 1

A partition of a set $A$ is the same thing as an equivalence relation on $A$.

In other words: Let $A$ be a set. Then
  1. A partition $A_1 \sqcup A_2 \sqcup \ldots$ of the set $A$ is defined by an equivalence relation on $A$.
  2. An equivalence relation $\sim$ on the set $A$ defines a partition on $A$.
Proof: Exercise for those who skipped the lecture.

Lecture 3

Cardinality

Two sets $A$ and $B$ have the same cardinality if there exists a bijection $f : A \to B$, written $|A| = |B|$.

Set $A$ has smaller cardinality than $B$ if there exists a (total) injection $f : A \to B$, written $|A| \leqslant |B|$.

Theorem: $|A| = |B| \iff (|A| \leqslant |B| ~\&~ |B| \leqslant |A|)$.

$|\mathbb N| \stackrel{?}{=} |\mathbb Z| \stackrel{?}{=} |\mathbb Q| \stackrel{?}{=} |\mathbb R|$

Theorem 2

$|\mathbb N| < |\mathbb R|$

Lecture 4

Finite state automata

DFA


A deterministic finite state automaton, $M$, consists of:
  • A finite set, $Q$, of states
  • A finite set $\Sigma$ called the alphabet
  • A total function $\delta: Q \times \Sigma \to Q$ called the transition function
  • A distinguished state $q_0 \in Q$ called the initial state
  • A subset $F \subseteq Q$ called the final or accepting states.

Given a word $w = w_0 w_1 \dots w_{n-1} \in \Sigma^*$ the computation carried out by $M$ on input $w$ is a sequence of states $q_0, q_1, q_2, \dots, q_n$ defined as follows: \[ q_1 = \delta(q_0, w_0), \: q_2 = \delta(q_1, w_1), \: \dots ,\: q_n = \delta(q_{n-1}, w_{n-1}) \]

We say that $M$ accepts or recognises $w$ if $q_n \in F$ and otherwise it rejects $w$.

The language of $M$, $L(M)$ is just the set of strings in $\Sigma^*$ that $M$ accepts.

Lecture 5

Non-deterministic automata

NFA

Relax the definition of the transition function to become a transition relation. \[ \delta \subseteq Q \times \Sigma \cup \{\lambda\} \times Q \]

$\{ab\}$ in English alphabet

Examples

Design an automaton that recognises the following languages in alphabet $\{a,b\}$

$L_1 = \{w | w = xaaybbbz, \mbox{ where } x,y,z \in\{a,b\}^*\}$

$L_2 = \{w | w \mbox{ contains $aa$ or $bbb$}\}$

$L_3 = \{w | w \mbox{ contains $aa$ and $bbb$}\}$

Lecture 6

NFA = DFA, pumping lemma

Theorem 3

The classes of languages recognised by DFA and NFA coincide.
$L = \{w \in\{a, b\}^* \mid w \mbox{ contains $abba$ or $aa$}\}$

Theorem 3

The classes of languages recognised by DFA and NFA coincide.

Proof: Go to the lecture (or read in the notes).

Lecture 7

Pumping lemma

$L = \{a^nb^n \mid n \in \mathbb{N}\}$

Pumping lemma

Let $L$ be an automatic language. Then there exists a positive integer $k$ such that if $z \in L$, $|z| \geq k$ then for some $u$, $v$ and $w$:
$ \begin{eqnarray*} z &=& u \cdot v \cdot w \\ |u| + |v| &\leq& k \\ |v| &>& 0 \\ uv^iw &\in& L \quad \mbox{for all $i \geq 0$}. \end{eqnarray*} $

Proof: Haven't we just proven it?

$L = \{w \in \{a, b\} \mid \mbox{$w$ contains an odd number of $a$'s}$
$\mbox{and an even number of $b$'s}\}$

Lecture 8

Pushdown automata and context free grammars

A pushdown automaton, $M$, consists of:
  • A finite set $Q$ of states
  • A finite set $\Sigma$ called the input alphabet (lower case letters)
  • A finite set $\Gamma$ called the stack alphabet (upper case letters)
  • A relation $\delta \subseteq Q \times \Sigma \cup \{\lambda\} \times \color{blue}{\Gamma \cup \{\lambda\}} \times Q \times \color{blue}{\Gamma \cup \{\lambda\}}$
    called the transition relation
  • A distinguished state $q_0 \in Q$ called the initial state
  • A subset $F \subseteq Q$ called the final or accepting states.

The stack should be empty to accept a string!

A context-free grammar, $G$, consists of:
  • A finite set $V$ of nonterminals (or variables)
    (upper case letters)
  • A finite set $\Sigma$ called the alphabet (or terminals)
    (lower case letters), $\Sigma \cap V = \varnothing$
  • A finite set $P$ of production rules,
    which are function from $V$ to $(V \cup \Sigma)^*$
  • A distinguished $S \in V$ called the start symbol

Examples
$S \to aS, S \to bS, S \to \lambda$

$S \to aS, S \to bT, T \to bT, T \to \lambda$

$\{a^nb^n \mid n \in \mathbb{N}\}$

Danger zone!

We recommend that you don't scroll past this slide.

Genome phasing

Schwarz, Roland F., Anne Trinh, Botond Sipos, James D. Brenton, Nick Goldman, and Florian Markowetz. 2014. “Phylogenetic Quantification of Intra-Tumour Heterogeneity.” PLoS Computational Biology 10 (4): e1003535.
[Corrected]

Formalising biology

  • Let $G$ be the set of binary (quaternary in reality) strings of length $n$.
  • Elements of $G$ are called genotypes.
  • A function $w: G \to \mathbb R^+$ is called a fitness landscape.
  • For $g \in G$, $w(g)$ is called fitness of the genotype $g$.

Scalability: synthetic lethal pairs

How many pairs of genes are there in the human genome?

Gene-gene interactions

Epistasis

is defined as the deviation from the additive expectation of allelic effects: $$u_{11} = w_{00} + w_{11} - (w_{01} + w_{10})$$

Understanding three-way interactions

Marginal epistasis?

$\small u_{\color{blue}{0}11} = w_{\color{blue}{0}00} + w_{\color{blue}{1}00} + w_{\color{blue}{0}11} + w_{\color{blue}{1}11} − (w_{\color{blue}{0}01} + w_{\color{blue}{1}01}) − (w_{\color{blue}{0}10} + w_{\color{blue}{1}10})$

Total three-way interaction?

$\small u_{111} = w_{000} + w_{011} + w_{101} + w_{110} - (w_{001} + w_{010} + w_{100} + w_{111})$

Conditional epistasis?

$\small e = w_{\color{blue}{0}00} − w_{\color{blue}{0}01} − w_{\color{blue}{0}10} + w_{\color{blue}{0}11}$

What if no (credible) fitness measurements are available?


Image: Wikipedia

Mutation fitness graph


Ogbunugafor et al. Malar. J. 2016

Rank orders. The simplest case.

$\small u_{11} = w_{00} + w_{11} - (w_{01} + w_{10})$

Exercise: Dyck word algorithm

$$ \begin{align} \small u_{011} =~ & w_{000} + w_{100} + w_{011} + w_{111} − \\ & w_{001} - w_{101} − w_{010} - w_{110} \end{align} $$

$$ w_{111} > w_{011} > w_{101} > w_{010} > w_{000} > w_{110} > w_{100} > w_{001} $$

$$ w_{111} > w_{011} > w_{100} > w_{000} > w_{001} > w_{101} > w_{010} > w_{110} $$

A way to quantify uncertainties!

Crona, Gavryushkin, Greene, and Beerenwinkel. Inferring genetic interactions from comparative fitness data. eLife, 2017.

Connection between rank orders and mutation graphs

$\small u_{11} = w_{00} + w_{11} - (w_{01} + w_{10})$

Theorem. A partial order (e.g. fitness graph) implies epistasis if and only if all linear extensions compatible with the partial order do.

Mutation graph

Mutation graph

Mutation graph

Mutation graph

Mutation graph


Lienkaemper, Lamberti, Drain, Beerenwinkel, and Gavryushkin. The geometry of partial fitness orders and an efficient method for detecting genetic interactions. Journal of Mathematical Biology, 2018.

4. Online algorithms to improve computational performance


  • The traditional way is to make the algorithm more efficient

  • When the same algorithm has to be re-run routinely, we can economise by making the algorithm slower and doing more!

  • This approach is known as online

Online algorithms

Online algorithms

Online algorithms

Online algorithms

Online algorithms

Online algorithms

Online algorithms

Online algorithms

Online algorithms

Homework:

Find an efficient online algorithm to detect genetic interactions from