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Solutions to Problem Set 1
Winter, 2014
Due: Friday, January 24, beginning of tutorial
1. Prove the duality theorem, as expressed in Exercise 1 on page 4 of the course notes.
Use structural induction on A.
Solution:
We are to prove that if A
CSC 438F/2404F
Notes (S. Cook)
Fall, 2008
Predicate Calculus
(First-Order Logic)
Syntax
A first-order vocabulary (or just vocabulary or language) L is specified by the following:
1) For each n N a set of n-ary function symbols (possibly empty). We use f,
CSC438F/2404F
Solutions to Problem Set 1
Fall, 2016
Due: Friday, September 30, beginning of tutorial
1. Do Exercise 7, page 12 of the Notes: Show that the contraction rules can be derived
from the cut rule (with weakenings and exchanges).
Solution:
To get
CSC438F/2404F
Problem Set 1
Fall, 2016
Due: Friday, September 30, beginning of tutorial
NOTE: Each problem set counts 10% of your mark, and it is important to do your own
work. You may consult with others concerning the general approach for solving proble
CSC 438F/2404F
Notes (S. Cook)
Fall, 2008
Gdels Incompleteness Theorems
o
In the early 1900s there was a drive to nd adequate axiomatic foundations for mathematics.
Russells paradox (If S is the set of all sets that do not contain themselves, does S conta
CSC 438F/2404F
Notes (S. Cook)
Fall, 2008
Completeness of System LK for Predicate Calculus
In general in this section of the notes we assume that every formula A satises the restriction
described on page 27: All free variables of A are from the free varia
CSC 438F/2404F
Notes (S. Cook)
Fall, 2008
Incompleteness and Undecidability
First part: Representing relations by formulas
Our goal now is to prove the Gdel Incompleteness Theorems, and associated undecidability
o
results. Recall that TA (True Arithmetic)
CSC 438F/2404F
Notes (S. Cook)
Fall, 2008
Recursive and Recursively Enumerable Sets
Recursive Sets
For this section, a set means a subset of Nn , where usually n = 1. Thus formally a set is the
same thing as a relation, which is the same as a total 0-1 va
CSC 438F/2404F
Notes (S. Cook)
Fall, 2008
Peano Arithmetic
Goals Now
1) We will introduce a standard set of axioms for the language LA . The theory generated
by these axioms is denoted PA and called Peano Arithmetic. Since PA is a sound,
axiomatizable the
CSC 438F/2404F
Notes (S. Cook)
Fall, 2008
Computability Theory
This section is partly inspired by the material in A Course in Mathematical Logic by Bell
and Machover, Chap 6, sections 1-10.
Other references: Introduction to the theory of computation by Mi
CSC 438F/2404F
Notes (S. Cook)
Fall, 2008
Herbrand Theorem, Equality, and Compactness
The Herbrand Theorem
We now consider a complete method for proving the unsatisability of sets of rst-order
sentences which is an alternative to LK. This forms the basis
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Solutions to Midterm Test
Last Name
February 28, 2014
First Name & Initial
Student No.
NO AIDS ALLOWED. Answer ALL questions on test paper. Use backs of sheets for scratch work.
Total Marks: 40
[5]
1. Give a specic formula A such that
xA |=
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Solutions to Problem Set 3
Winter, 2014
Due: Friday, March 14, beginning of tutorial
1. Show that if f (x) = y[g(x, y) = 0] and g is a computable function, then f is a
computable function. Do this by giving an RM (Register Machine) program f
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Solutions to Problem Set 4
Winter, 2014
Due: Friday, April 4, beginning of tutorial
1. Let A = cfw_x | 5 ran(cfw_x1 )
Let B = cfw_x | dom(cfw_x1 ) is innite
Which of A, Ac , B, B c is recursive? Which is r.e.? Justify your answers. You may u
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Solutions to Problem Set 2
Winter, 2014
Due: Friday, February 14, beginning of tutorial
1. Give an LK proof of the sequent A B, where
A =syn xy x = f y
B =syn xy x = f f y
Start by giving the specic instances of the LK equality axioms EL1,.,
CSC 438F/2404F
Notes (S. Cook)
Fall, 2008
Predicate Calculus
(First-Order Logic)
Syntax
A rst-order vocabulary (or just vocabulary or language) L is specied by the following:
1) For each n N a set of n-ary function symbols (possibly empty). We use f, g, h
CSC 438F/2404F
Notes (S. Cook)
Fall, 2008
Computability Theory
This section is partly inspired by the material in A Course in Mathematical Logic by Bell
and Machover, Chap 6, sections 1-10.
Other references: Introduction to the theory of computation by Mi