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MA201: Mathematical Logic and Theory of Computation

Unit 2: Modeling Computation   The advance of first-order logic as a way to formalize mathematical truth inspired several efforts in the 1930s to give a technical definition for what an effective process – an algorithm, or a method for computing – ought to look like.  Several of these definitions turned out to be equivalent.  In the next two decades, these definitions became the dominant mathematical model and the dominant metaphor behind the invention of the digital computer.
 
Our first attempt to model computation uses a somewhat weaker definition, one that takes seriously the limitations on memory and storage space that a computer has, and that lends itself to making comprehensive pictures of the model.  This model of computation is called the
finite-state automaton, which proves to be very useful but very restrictive.
 
We then move on to the equivalent definitions given by several people in the 1930s, which abstract the time and memory restrictions in real computers – and give the most convincing negative results.  After all, if you couldn’t solve the problem even with unbounded time and memory, you couldn’t solve it in a more realistic model, either.  We will explore in some detail how to encode these computations as natural numbers, and how to use these encodings to prove impossibility results, including Gödel Incompleteness (the theorem that a certain kind of first-order structure cannot be described by a computable set of first-order axioms), and to construct examples of counterintuitive behavior in computation.
 
Central questions to keep in mind throughout this unit:  What is a computer?  What features are included or abstracted in each model?  Is there a psychological or philosophical difference between the mathematically equivalent models?  What is the importance of the counterexamples (e.g., the Gödel sentence and the incomplete set)?

Unit 2 Time Advisory
This unit should take approximately 55 hours to complete.

☐    Subunit 2.1: 15 hours

☐    Sub-subunit 2.1.1: 2 hours

☐    Sub-subunit 2.1.2: 2 hours

☐    Sub-subunit 2.1.3: 2 hours

☐    Sub-subunit 2.1.4: 2 hours

☐    Sub-subunit 2.1.5: 2 hours

☐    Sub-subunit 2.1.6: 3 hours

☐    Sub-subunit 2.1.7: 2 hours

☐    Subunit 2.2: 16 hours

☐    Sub-subunit 2.2.1: 3 hours

☐    Sub-subunit 2.2.2: 2 hours

☐    Sub-subunit 2.2.3: 2 hours

☐    Sub-subunit 2.2.4: 3 hours

☐    Sub-subunit 2.2.5: 1 hour

☐    Sub-subunit 2.2.6: 3 hours

☐    Sub-subunit 2.2.7: 2 hours

☐    Subunit 2.3: 7 hours

☐    Sub-subunit 2.3.1: 2 hours

☐    Sub-subunit 2.3.2: 2 hours

☐    Sub-subunit 2.3.3: 3 hours

☐    Subunit 2.4: 11 hours

☐    Sub-subunit 2.4.1: 1 hour

☐    Sub-subunit 2.4.2: 2 hours

☐    Sub-subunit 2.4.3: 3 hours

☐    Sub-subunit 2.4.4: 3 hours

☐    Sub-subunit 2.4.5: 2 hours

☐    Subunit 2.5: 6 hours

☐    Sub-subunit 2.5.1: 2 hours

☐    Sub-subunit 2.5.2: 4 hours

Unit2 Learning Outcomes
Upon successful completion of this unit, the student will be able to: - Give examples of languages recognized by finite-state automata. - Prove the equivalence of deterministic and non-deterministic finite automata. - Apply the pumping lemma to show that some computations are impossible for finite automata. - Construct a Turing machine to compute a function. - Use Church’s Thesis to argue that a function is computable. - Distinguish primitive recursive functions from recursive functions. - Define functions in the lambda calculus. - Give an example of a computably enumerable set that is not computable. - Give an example of a non-computable set. - Explain the meaning of the Gödel Incompleteness Theorems. - Construct a solution to Post’s Problem.

2.1 Finite Automata and Regular Languages   2.1.1 Deterministic Finite Automata   - Reading: Carleton University: Anil Maheshwari and Michiel Smid’s Introduction to the Theory of Computation: “Chapter 2, Sections 2.1 and 2.2” Link: Carleton University: Anil Maheshwari and Michiel Smid’s Introduction to the Theory of Computation: “Chapter 2, Sections 2.1 and 2.2” (PDF)

 Instructions: Read from the beginning of Chapter 2 to the end of
section 2.2.  Observe the usual issues about definitions.  Make more
examples, and some near-misses.  Verify that each example satisfies
the definition.  Remember that it was said (by no less an authority
than statistician George Box), “Essentially, all models are wrong,
but some are useful.”  What aspects of a computer does the
definition of a finite automaton capture well?  What aspects does it
capture poorly?  

 This reading should take approximately 1 hour.  

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  • Assignment: Carleton University: Anil Maheshwari and Michiel Smid’s Introduction to the Theory of Computation: “Exercise 2.1” Link: Carleton University: Anil Maheshwari and Michiel Smid’s Introduction to the Theory of Computation: “Exercise 2.1” (PDF)

    Instructions: Go to the Exercises starting on page 78 and solve any three items from Exercise 2.1.

    This assignment should take approximately 1 hour.

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2.1.2 Regular Operations   - Reading: Carleton University: Anil Maheshwari and Michiel Smid’s Introduction to the Theory of Computation: “Section 2.3” Link: Carleton University: Anil Maheshwari and Michiel Smid’s Introduction to the Theory of Computation: “Section 2.3” (PDF)

 Instructions: Read all of section 2.3.  You could think of the
relationship between regular operations and finite automata as an
image of the relationship between syntax and semantics in model
theory.  The hope, in the end, is that the two will coincide – that
is, that the set of regular languages (those accepted by finite
automata) will be generated somehow by regular operations.  This
works out, in the end, but it takes some time to get to this proof.
 Again, make sure you’re reading actively through all these
definitions and bookkeeping proofs.  

 This reading should take approximately 2 hours.  

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2.1.3 Non-Deterministic Finite Automata   - Reading: Carleton University: Anil Maheshwari and Michiel Smid’s Introduction to the Theory of Computation: “Section 2.4” Link: Carleton University: Anil Maheshwari and Michiel Smid’s Introduction to the Theory of Computation: “Section 2.4” (PDF)

 Instructions: Read all of section 2.4.  Non-deterministic automata
may – even should – feel a little contrived.  They originated as a
guess about what kind of “magic” we could add to an automaton in
order to allow it to do more.  The “magic” involved here is that
there are two (or more) possibilities for the “right” next step, and
the automaton is given the ability to “magically” know which one is
right.  In the end, it doesn’t matter for automata, but with other
models of computation, this turns out to be a big issue – perhaps
the biggest open problem in theoretical computing today.  

 This reading should take approximately 1 hour.  

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  • Assignment: Carleton University: Anil Maheshwari and Michiel Smid’s Introduction to the Theory of Computation: “Exercise 2.2” Link: Carleton University: Anil Maheshwari and Michiel Smid’s Introduction to the Theory of Computation: “Exercise 2.2” (PDF)

    Instructions: Go to page 79 and solve Exercise 2.2.

    This assignment should take approximately 1 hour.

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2.1.4 Equivalence of DFAs and NFAs   - Reading: Carleton University: Anil Maheshwari and Michiel Smid’s Introduction to the Theory of Computation: “Section 2.5” Link: Carleton University: Anil Maheshwari and Michiel Smid’s Introduction to the Theory of Computation: “Section 2.5” (PDF)

 Instructions: Read all of section 2.5.  If you have properly
understood the definitions of NFAs and DFAs, you should intuitively
believe that the main theorem of this section is impossible.  The
result should be surprising.  Suppose you had a magical ability to
always guess right.  It should help you, shouldn’t it?  And yet this
result says that (at least in this context) it really doesn’t.  You
should perhaps read the proof of equivalence with a specific NFA in
mind, perhaps one that you think would be hard for a DFA to emulate.
 In any case, read skeptically.  

 This reading should take approximately 2 hours.  

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2.1.5 Closure Under the Regular Operations   - Reading: Carleton University: Anil Maheshwari and Michiel Smid’s Introduction to the Theory of Computation: “Section 2.6” Link: Carleton University: Anil Maheshwari and Michiel Smid’s Introduction to the Theory of Computation: “Section 2.6” (PDF)

 Instructions: Read section 2.6.  The goal here is to develop an
“algebra” to represent what automata are doing.  The roots of this
sort of analysis go back to Noam Chomsky’s attempt to explain how
such a complex thing as language could be biologically hard-wired,
and could be independent of the many wildly different natural
languages in use.  Try, in each proof, to draw the state diagram of
the “proof” (i.e., the DFA witnessing closure under that operation)
before either looking at the figure or reading the proof.  Then look
at the figure provided, if there is one, and then read the proof.  

 This reading should take approximately 2 hours.  

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2.1.6 Regular Expressions and Regular Languages   - Reading: Carleton University: Anil Maheshwari and Michiel Smid’s Introduction to the Theory of Computation: “Sections 2.7 and 2.8” Link: Carleton University: Anil Maheshwari and Michiel Smid’s Introduction to the Theory of Computation: “Sections 2.7 and 2.8” (PDF)

 Instructions: Read sections 2.7 and 2.8.  This is a continuation of
the work in the previous reading to develop an algebra to describe
what automata do.  The regular expressions are quintessential forms
of “grammatically correct sentences” in the language.  In English,
for instance, we have a common form SVO, where S is a subject, V is
a transitive verb, and O is an object.  Whether you would want to
say it or not, any such construction (with a few extra details) is a
legitimate English sentence.  Of course, the “languages” we care
about in computation are a bit more abstract, but it’s helpful to
think of them as languages anyway.  

 This reading should take approximately 2 hours.  

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  • Assignment: Carleton University: Anil Maheshwari and Michiel Smid’s Introduction to the Theory of Computation: “Exercise 2.8” Link: Carleton University: Anil Maheshwari and Michiel Smid’s Introduction to the Theory of Computation: “Exercise 2.8” (PDF)

    Instructions: Go to page 81 and solve Exercise 2.8.

    This assignment should take approximately 1 hour.

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2.1.7 The Pumping Lemma and Non-Regular Languages   - Reading: Carleton University: Anil Maheshwari and Michiel Smid’s Introduction to the Theory of Computation: “Section 2.9” Link: Carleton University: Anil Maheshwari and Michiel Smid’s Introduction to the Theory of Computation: “Section 2.9” (PDF)

 Instructions: Read all of section 2.9.  While the pumping lemma is
a theorem usually stated in the form, “For all regular languages A,”
the real importance is in the contrapositive: “If A does not have a
certain property, then A is not regular.”  It’s really a way of
proving that DFAs cannot do certain things; it’s a result about the
limits of computers, insofar as finite automata are good models for
them.  

 This reading should take approximately 1 hour.  

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  • Assignment: Carleton University: Anil Maheshwari and Michiel Smid’s Introduction to the Theory of Computation: “Exercise 2.22” Link: Carleton University: Anil Maheshwari and Michiel Smid’s Introduction to the Theory of Computation: “Exercise 2.22” (PDF)

    Instructions: Go to page 84 and solve Exercise 2.22.

    This assignment should take approximately 1 hour.

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2.2 Turing Machines and Related Models   2.2.1 Turing Machines   - Reading: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Chapter 2 Introduction and Sections 2.1 and 2.2” Link: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Chapter 2 Introduction and Sections 2.1 and 2.2” (PDF)

 Instructions: Near the bottom of the page, find the bullet point,
“Computability and Incompleteness,” and click on the PDF link for
“Lecture notes.”  Read from the beginning of Chapter 2 through the
end of section 2.2.  This is a much stronger model than finite
automata (i.e., one of these can do things that a DFA can’t).
 Again, think of the big modeling questions: Where does the Turing
machine definition go right as a model of computation (e.g., where
is it better than a DFA)?  Where does it go wrong?  

 This reading should take approximately 2 hours.  

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  • Assignment: University of Manchester: Andrea Schalk’s Turing Machines: “Section 3” Link: University of Manchester: Andrea Schalk’s Turing Machines: “Section 3” (PDF or PS)

    Instructions: In the “Course notes” section, click on the link for the format you want of “Section 3: Turing machines.”  Go to page 54 and solve both parts of Exercise 23.

    This assignment should take approximately 1 hour.

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2.2.2 Primitive Recursion   - Reading: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Sections 2.3 and 2.4” Link: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Sections 2.3 and 2.4” (PDF)

 Instructions: Near the bottom of the page, find the bullet point,
“Computability and Incompleteness,” and click on the PDF link for
“Lecture notes.”  Read sections 2.3 and 2.4.  The great miracle of
the general-purpose computer in the 1930s was the diversity of
approaches that produced the same thing.  Primitive recursion was
the approach of Gödel, whose motivation was more philosophical than
technical.  As usual, make up examples as you read.  Check the
examples given.  Make near-miss non-examples.  

 This reading should take approximately 2 hours.  

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2.2.3 Recursive Functions   - Reading: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Section 2.5” Link: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Section 2.5” (PDF)

 Instructions: Near the bottom of the page, find the bullet point,
“Computability and Incompleteness,” and click on the PDF link for
“Lecture notes.”  Read section 2.5.  While the definitions are of
great importance and should be given the usual treatment, the proof
that begins the section will become the real star of the show.  It
is known as a “diagonal argument” (can you see why?) and is the
central technique in all computability theory, although it appears
in many guises.  

 This reading should take approximately 2 hours.  

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2.2.4 Recursive is Equivalent to Turing Computable   - Reading: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Section 2.6” Link: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Section 2.6” (PDF)

 Instructions: Near the bottom of the page, find the bullet point,
“Computability and Incompleteness,” and click on the PDF link for
“Lecture notes.”  Read section 2.6.  Read the proof critically.  It
is important that, in the long run, you believe that partial
recursive functions and partial Turing computable functions are the
same.  Pick some examples where the theorem seems hard to believe.
 Follow through the proof with these examples, to see how it catches
them.  

 This reading should take approximately 3 hours.  

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2.2.5 Church’s Thesis   - Reading: Stanford Encyclopedia of Philosophy: B. Jack Copeland’s “The Church-Turing Thesis” Link: Stanford Encyclopedia of Philosophy: B. Jack Copeland’s “The Church-Turing Thesis” (HTML)

 Instructions: Read the article.  The real importance for your
learning of the proof in the previous section is that it helps you
believe the extremely important Church-Turing Thesis (sometimes
called Church’s Thesis).  The point here is that if we believe the
Church-Turing Thesis, then we believe that the concept of
“computation” is mathematically well-defined – and that, for
instance, we can prove impossibility results for computation without
fear that some new idea of what computation means will come and
supersede our result.  

 People often attempt things that are known to be impossible,
thinking that there is an analogy of mathematical impossibilities
(things nobody will ever be able to do) with technical
impossibilities (things we can’t imagine how anybody would ever do).
 People who try to trisect angles, for instance, might say that
“people used to think airplanes were impossible, too.”  The
Church-Turing Thesis shows that any impossibility results that we
prove about computation are like the impossibility of angle
trisection (a genuine proof of permanent impossibility), not like
the “impossibility” of flight (a matter of technological
distance).  

 This reading should take approximately 1 hour.  

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2.2.6 Theorems on Computability   - Reading: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Section 2.7” Link: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Section 2.7” (PDF)

 Instructions: Near the bottom of the page, find the bullet point,
“Computability and Incompleteness,” and click on the PDF link for
“Lecture notes.”  Read section 2.7.  These theorems all have
important uses, but the real importance at the moment is that you
get used to the kinds of things you can do with a formally defined
model of computation.  Work through each proof.  See if you can
anticipate the next step in each proof.  Then reflect on the
importance of Kleene’s Normal Form Theorem: What does it mean for
computer hardware?  

 This reading should take approximately 3 hours.  

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2.2.7 The Lambda Calculus   - Reading: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Section 2.8” Link: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Section 2.8” (PDF)

 Instructions: Near the bottom of the page, find the bullet point,
“Computability and Incompleteness,” and click on the PDF link for
“Lecture notes.”  Read section 2.8.  This added model of
computability (equivalent to Turing computability and partial
recursion) is of particular importance for computing: it points the
way (in Unit 3 of this course) to a link between formal proofs and
computation.  If you want to be impressed by Unit 3 – or even to
understand its importance – you would do well to make sure you’re
convinced of the proofs in this section.  Also, lambda calculus is
the model of computation that has been turned most explicitly into a
programming language.  Most of the so-called “functional” languages
(e.g., FORTRAN) are derived from lambda calculus.  

 This reading should take approximately 2 hours.  

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2.3 Computability Theory   2.3.1 Computably Enumerable Sets   - Reading: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Chapter 3 Introduction and Sections 3.1 and 3.2” Link: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Chapter 3 Introduction and Sections 3.1 and 3.2” (PDF)

 Instructions: Near the bottom of the page, find the bullet point,
“Computability and Incompleteness,” and click on the PDF link for
“Lecture notes.”  Read from the beginning of Chapter 3 through the
end of section 3.2.  The non-computability of K<sub>0</sub> is our
second big encounter with the diagonal argument.  It is the
archetypal proof of which all other non-computability proofs are
either imitations or applications.  It also, in itself, places a
kind of limit on automated software verification: If you can’t check
whether a program halts, then you can’t check much.  

 This reading should take approximately 1 hour.  

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  • Assignment: University of Ottawa: Pieter Hofstra’s Recursion Theory: “Exercise 3.23” Link: University of Ottawa: Pieter Hofstra’s Recursion Theory: “Exercise 3.23” (PS)

    Instructions: Under the heading of “Lecture Notes,” click on the link to download the file.  Solve Exercise 3.23 on page 77.

    This assignment should take approximately 1 hour.

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2.3.2 Reducibility and Rice’s Theorem   - Reading: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Section 3.3” Link: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Section 3.3” (PDF)

 Instructions: Near the bottom of the page, find the bullet point,
“Computability and Incompleteness,” and click on the PDF link for
“Lecture notes.”  Read section 3.3.  This is the way one applies the
non-computability of K<sub>0</sub> to prove non-computability of
other things.  The results of this section justify the definition of
reducibility.  

 This reading should take approximately 2 hours.  

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2.3.3 The Fixed-Point Theorem   - Reading: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Sections 3.4 and 3.5” Link: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Sections 3.4 and 3.5” (PDF)

 Instructions: Near the bottom of the page, find the bullet point,
“Computability and Incompleteness,” and click on the PDF link for
“Lecture notes.”  Read sections 3.4 and 3.5.  To a mathematician, a
fixed-point theorem is a very natural thing – if you’re dealing with
continuous functions in some sort of geometric context.  Here, the
point is very technical – and it would not be worth studying if it
were not useful for so many other results, some of which we’ll see
in later sections.  

 This reading should take approximately 2 hours.  

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  • Assignment: University of Ottawa: Pieter Hofstra’s Recursion Theory: “Exercise 3.11” Link: University of Ottawa: Pieter Hofstra’s Recursion Theory: “Exercise 3.11” (PS)

    Instructions: Under the heading of “Lecture Notes,” click on the link to download the file.  Solve Exercise 3.11 on page 76.

    This assignment should take approximately 1 hour.

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2.4 Incompleteness   2.4.1 Historical Background   - Reading: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Section 4.1” Link: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Section 4.1” (PDF)

 Instructions: Near the bottom of the page, find the bullet point,
“Computability and Incompleteness,” and click on the PDF link for
“Lecture notes.”  Read section 4.1.  Many people consider the
Incompleteness Theorems to be the beginning of modern logic, and
certainly the beginning of its involvement with computation.  The
formulation of the theorems, though, is more than a little recondite
outside its historical context.  If we aren’t careful, the
Incompleteness Theorems can sound like either shallow philosophy or
uninteresting technical details in a part of math that not many
people care about any more.  It is neither, and the point of this
section is to help you navigate this.  

 This reading should take approximately 1 hour.  

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2.4.2 Background in Logic   - Reading: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Section 4.2” Link: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Section 4.2” (PDF)

 Instructions: Near the bottom of the page, find the bullet point,
“Computability and Incompleteness,” and click on the PDF link for
“Lecture notes.”  Read section 4.2.  Many points will be familiar
from our study of model theory, but it is good to read them to fill
in details to which we didn’t attend then.  

 This reading should take approximately 2 hours.  

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2.4.3 Representability in Q   - Reading: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Section 4.3” Link: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Section 4.3” (PDF)

 Instructions: Near the bottom of the page, find the bullet point,
“Computability and Incompleteness,” and click on the PDF link for
“Lecture notes.”  Read section 4.3.  The point here is to make a
model-theoretic definition of computability, equivalent to the
others we have just seen.  The ultimate goal is to use this as a way
to carry out a diagonal argument.  Many of the points here are
technical and difficult, but they are fundamental for understanding
the theorems, which are, collectively, one of the great intellectual
accomplishments of human history.  

 This reading should take approximately 3 hours.  

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2.4.4 The First Incompleteness Theorem   - Reading: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Section 4.4” Link: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Section 4.4” (PDF)

 Instructions: Near the bottom of the page, find the bullet point,
“Computability and Incompleteness,” and click on the PDF link for
“Lecture notes.”  Read section 4.4.  Much has been made of this
theorem in the popular literature.  What do you make of it?  What is
its practical meaning?  

 This reading should take approximately 3 hours.  

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2.4.5 The Second Incompleteness Theorem   - Reading: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Section 4.7” Link: Carnegie Mellon University: Jeremy Avigad’s Computability and Incompleteness: “Section 4.7” (PDF)

 Instructions: Near the bottom of the page, find the bullet point,
“Computability and Incompleteness,” and click on the PDF link for
“Lecture notes.”  Read section 4.7.  What does the Second
Incompleteness Theorem say that we already knew from the First?
 What does it tell us that is new?  

 This reading should take approximately 2 hours.  

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2.5 Turing Reduction and Post’s Problem   2.5.1 Post’s Problem and Reducibilities   - Reading: National University of Singapore: Frank Stephan’s Recursion Theory: “Chapter 4” Link: National University of Singapore: Frank Stephan’s Recursion Theory: “Chapter 4” (PDF or PS).

 Instructions: In the section labeled “Lecture Notes,” click on the
link to the file format of your choice.  Read Chapter 4, and solve
Exercises 4.4, 4.7, and 4.13.  One can still start a very
interesting (and sometimes heated) conversation among experts by
asking for a “natural” solution to Post’s problem.  Many people
believe that the constructions of solutions that we will see (and
many others more recent) have an *ad hoc* feel to them – as if Post
might say (though he didn’t), “Yes, but that’s not really what I was
looking for.”  

 This reading should take approximately 2 hours.  

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2.5.2 A Solution to Post’s Problem for Enumerable Sets   - Reading: National University of Singapore: Frank Stephan’s Recursion Theory: “Chapter 7” Link: National University of Singapore: Frank Stephan’s Recursion Theory: “Chapter 7” (PDF or PS)

 Instructions: In the section labeled “Lecture Notes,” click on the
link to the file format of your choice.  Read Chapter 7, and solve
Exercises 7.2, 7.3, and 7.5.  Think on the issue of naturality, too.
 Does the set constructed in this section seem intuitively
reasonable to you?  Or is it one of those theorems that you know
must be true because you’ve seen the proof, but that in your heart
you still don’t feel quite right about?  There are experts – experts
at the highest level – on either side of the question.  

 This reading should take approximately 4 hours.  

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