CSC438S/2404S
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
Computability and Logic
Fall, 2016
Prerequisites for 438: See Calendar
Exclusions for 438: MAT309H, PHL344H
Lectures: MW 4 in SS 2106
Tutorials: F 12 in SS 2106
Tutor: Lalla Mouatadid
Instructor: Stephen Cook, SF 2303C, 416 978-5183, sacook
CSC 438S/2404S
Last Name
Midterm Test
Friday, Oct 30, 2015
First Name & Initial
Student No.
NO AIDS ALLOWED. Answer ALL questions on test paper. Use backs of sheets for scratch work.
Total Marks: 40
[10]
1. Use the definition of |= to prove xf (g(x) = x 6
Marker's Comments for Midterm Test CSC 438h/2404h 2016
The average mark was 27/40. 8 students received 36 (90%) or better.
(28 students took the test.)
Question 1: Most students realized that compactness was a necessary
part of the proof. The maximum mark
CSC438F/2404F
Problem Set 1
Fall, 2015
Due: Friday, October 2, beginning of tutorial
NOTE: Each problem set counts 15% of your mark, and it is important to do your own
work. You may consult with others concerning the general approach for solving problems
CSC438F/2404F
Solutions to Problem Set 2
Fall, 2016
Due: Friday, October 21, beginning of tutorial
1. Let A1 be the formula xy x = f y and let A2 be the formula xy x = f f y.
Give an LK proof of the sequent A1 A2 .
Begin by specifying the specific instanc
CSC438F/2404F
Problem Set 3
Fall, 2016
Due: Friday, November 18, 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 problem
CSC438F/2404F
Solutions to Problem Set 4
Fall, 2016
Due: Monday, Dec 5, beginning of lecture.
1. Let A = cfw_x | dom(cfw_x1 ) PRIMES where PRIMES is the set of prime numbers.
Is A r.e.? Is Ac r.e.? Justify your answers. To show something is not r.e., use
CSC438F/2404F
Problem Set 4
Fall, 2016
Due: Monday, Dec 5, beginning of lecture.
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 problems on
a
CSC438F/2404F
Problem Set 2
Fall, 2015
Due: Friday, October 23, beginning of tutorial
NOTE: Each problem set counts 15% of your mark, and it is important to do your own
work. You may consult with others concerning the general approach for solving problems
CSC 438F/2404F: Computability and Logic
Fall, 2016
Announcements
I have now submitted final marks for both undergraduate and graduate students.
Graduate students can pick up their final exam from my office.
=
Chapter's I and II in Logical Foundations of P
CSC 438F/2404F
Computability and Logic
Fall, 2015
Prerequisites for 438: See Calendar
Exclusions for 438: MAT309H, PHL344H
Lectures: MW 4 in SS 1085
Tutorials: F 12 in SS 1085
Tutor: Robert Robere
Instructor: Stephen Cook, SF 2303C, 416 978-5183, [email protected]
CSC 438F/2404F: Computability and Logic
Fall, 2015
Final marks for undergraduates have been submitted.
They are posted on the secure CDF web site.
(Grad students may send me an email to find out their marks.)
There was a large variation on the final exam
CSC438F/2404F
Problem Set 4
Fall, 2015
Due: Monday, Dec 7, beginning of lecture
NOTE: Each problem set counts 15% of your mark, and it is important to do your own
work. You may consult with others concerning the general approach for solving problems on
as
CSC438F/2404F
Problem Set 2
Fall, 2016
Due: Friday, October 21, 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 problems
CSC438F/2404F
Problem Set 3
Fall, 2015
Due: Friday, November 20, beginning of tutorial
NOTE: Each problem set counts 15% of your mark, and it is important to do your own
work. You may consult with others concerning the general approach for solving problem
CSC 438F/2404F
Solutions to Midterm Test
Friday, Oct 28, 2016
NO AIDS ALLOWED. Answer ALL questions on test paper. Use backs of sheets for scratch work.
Total Marks: 40
[10]
1. Let = cfw_A1 , A2 , . . . be an infinite set of sentences over a language L s
CSC 438F/2404F
Last Name
Midterm Test
Friday, Oct 28, 2016
First Name & Initial
Student No.
NO AIDS ALLOWED. Answer ALL questions on test paper. Use backs of sheets for scratch work.
Total Marks: 40
[10]
1. Let = cfw_A1 , A2 , . . . be an infinite set of
Marking comments on PS1 (CSC 438F 2016)
Quest 1: 5 Quest 2: 10 Quest 3: 15 Quest 4: 15
Total: 45
Average: 35.5/45
In general, everyone did well. The main part where
students lost marks were in questions 1 and 3.
Q1: They derived contraction from other rul
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
CSC438S/2404S
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,.,
CSC438S/2404S
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
CSC438S/2404S
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
CSC 438S/2404S
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 |=
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
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
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
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
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
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
CSC438F/2404F
Solutions to Problem Set 3
Fall, 2016
Due: Friday, November 18, beginning of tutorial
1. Do Exercise 8, page 64 of the NOTES: (Show that #R(x) is primitive recursive if R(x)
is primitive recursive, and show that (x) is primitive recursive.)