L9 - 11

Question 1

What does promiscuous interfaces mean when it is regarding the human mind?

Answer 1

Another way of saying different modules can share notes

Question 2

Why are paradoxes the route to defining the limits of these powerful system?

Answer 2

Self-reference - going on forever (infinity) generate paradoxes. They are hard to avoid in formal systems (even those designed to avoid them)

Question 3

Does the set of all sets that do not contain themselves, contain itself?

Answer 3

Russell’s paradox - The sets of all sets that are not members of themselves is a member of itself if it is not a member of itself.

Imagine a barber that shaves everyone that does not shave themselves.
Does he shave himself also? But if he does shave himself – then he is shaving someone that does shave themselves. But if he doesn’t shave himself – then he is not shaving everyone that does not shave themselves.

Question 4

Difference between induction and deduction?

Answer 4

Deduction is from general to specific - using theory/general statement to generate observations
e.g. Dr.Seuss hates cats, provide supporting examples, you can deduce he hates cats

Induction is using observations/questions to generate theories.
e.g. How does Dr.Seuss feel about cats? Cat looks weird and does dumb things. He doesn’t like cats.

Question 5

Does the fundamental limits to formal systems of logic also include arithmetic? What is this limit?

Answer 5

Yes – this limit is that not all true statements can be derived by the system HENCE system is incomplete.

Question 6

T/F – Formal systems are limited because they are inconsistent

Answer 6

F

Question 7

Define algorithm

Answer 7

A process or set of rules to be followed in calculations or other problem-solving operations e.g. Addition

Question 8

Define recursion

Answer 8

Recursion occurs when a thing is defined in terms of itself or of its type. It is essentially defining an infinite set of objects by a finite statement

Question 9

Principia Mathematica – what is it?

Answer 9

Provides the logic foundation of maths and in a way that avoided the paradoxes that arise from self-reference. It could never be completed.

Question 10

Truth is bigger than proof - you can prove things by informal systems like principia Mathematica but there will always be things that are true that you cannot prove

Answer 10

Truth is bigger than proof - you can prove things by informal systems like principia Mathematica but there will always be things that are true that you cannot prove

Question 11

What is Godel’s first incompleteness theorem? What about his second one?

Answer 11

First - no logical system can capture all the truths of maths
Second – logical maths system will contain inconsistency (no logical system for maths could, by its own devices, be shown to be free from inconsistency)

Question 12

Did Godel show that maths is inconsistent?

Answer 12

No but he did show that if you want to be sure your maths is going to be free of contradiction, then it will, necessary, be incomplete

Question 13

Godel: One cannot claim - with any certainty, of any formal system that all conceptual considerations are representable in it as there are things that are true that you just cannot get to in that formal system

Answer 13

Godel: One cannot claim - with any certainty, of any formal system that all conceptual considerations are representable in it as there are things that are true that you just cannot get to in that formal system

Question 14

Define proposition

Answer 14

A sentence which is either true or false

Question 15

Define axiom

Answer 15

A statement or proposition which is regarded as being established, accepted, or self-evidently true

Question 16

Define composite numbers

Answer 16

A whole number that can be divided evenly by numbers other than 1 or itself

Question 17

Define prime number

Answer 17

A whole number that can only be divided evenly by 1 or itself

Question 18

Godel numbering

Answer 18

Converted the logic of arithmetic/axiomatic math into a system of math that talked about itself

Question 19

Turing’s formal definition of computation/algorithm

Answer 19

Any idea, process or method that can be implemented on a TM

Question 20

What is a Turing machine?

Answer 20

Basically a computer. It is a math model of a hypothetical computing machine which can use a predefined set of rules to determine a result form a set of input variables

Question 21

How is the minimum specification of the TM also the maximum specification?

Answer 21

Any algorithmic process that can be performed, can be performed on a TM.

Question 22

A universal TM

Answer 22

A computer. A TM that can take as input the instructions for how any particular TM works, thus it can emulate any TM. It’s like having a tape (full description of what one TM does with a certain input) and then playing that tape on another TM e.g .Apple computer simulating Windows PC

Question 23

What is the halting problem?

Answer 23

A limitation of UTMs, where it is impossible for any program to solve the halting problem – how could we tell the difference between a computation that is taking a long time vs one that will never finish?

Question 24

How is the halting problem a similar result to Gödel’s theorem?

Answer 24

It shows that there are computational problems (ie questions that can be encoded as computer programs) for which we are unable to know if they are computable or not.

Question 25

Are neurons – which are essentially computational units, an example of a TM?

Answer 25

Yes (If you want to argue that they are not TMs, you will need to explain why it is that they can be completely implemented on a computer ie a universal Turning machine!)

Question 26

What is a binary number system in computer?

Answer 26

Computers use binary - the digits 0 and 1 - to store data. A binary digit, or bit, is the smallest unit of data in computing. It is represented by a 0 or a 1. Binary numbers are made up of binary digits (bits), eg the binary number 1001.

Question 27

Cellular automata (CA)

Answer 27

Cells in a grid and their interactions. It uses simple rules to generate complex patterns. It is analogous to formal systems, whose axioms and rules of inference are very simple, yet vast bodies of mathematical theory can be derived from the repeated (recursive) activity of the system.

Question 28

What is Wolfram’s classification of 1-Dimensional CA behaviour?

Answer 28

Spatially stable, sequence of stable or periodic structures, chaotic aperiodic behaviour, complicated localized structures

Question 29

T/F - Simple cellular automata (like Wolfram’s rule 30) can generate something as complex as a random pattern

Answer 29

T

Question 30

Is rule 110 random/non-random, periodic/non-periodic

Answer 30

Non-random and non-periodic

Question 31

Artificial Neural Networks (ANNs) – how do back-propagation algorithms work?

Answer 31

Basically improving gradually every single time.

By making tiny adjustments in the direction that would lower the error between the input and the
desired output. They can propagate meaningfully through lots of layers of hidden units. Al the mathematics (vast numbers of differential equations) that must be solved for all nodes includes lots of re-usable results, and the computation scales with the size of the network (not exponentially with the size of the network, which would significantly limit their utility).

Question 32

Deep ANNs is able to outperform humans at identifying different images and make less mistakes, it can also mimic visual networks (extracting details like colours and edges from images) and recognise patters (such as an individual person – not just visually but also by a way of behaving) and predict who should be given bail (hence by used by governments)– why are they so good?

Answer 32

They are trained on a massive data set – endless amounts of images and data on the internet

Question 33

In what ways are artificial neural networks like biological neural networks?

Answer 33

Have properties that behave like a neuron – emulating the physical way a neuron works rather than a computer simulating a neuron

Question 34

What is one thing that human reasoning entails which is not readily explicable as a known computational process?

Answer 34

Insight - “jumping out of the system” where we stop searching and take a global view

Question 35

Real numbers have different infinity to natural numbers – proven by the diagonal exceptions

Answer 35

T