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Theories with Equality and Algebra Exam III | CSE 607, Exams of Engineering

Material Type: Exam; Class: Mathematical Basis for Computing; Subject: Computer Engineering; University: Syracuse University; Term: Spring 2002;

Typology: Exams

Pre 2010

Uploaded on 08/09/2009

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CSE607: Exam-3 S-K Chin 2002 Theories with Equality Exam:
1
CSE607
Theories with Equality and Algebra Exam
Exam#3 April 1, 2002
This is an open-book and open-notes examination. Do all problems in 80 minutes. Make sure you justify your
steps using the formal inference rules to earn full credit. (This includes rewriting rules–this will help you to
avoid silly mistakes). Left justify assumptions and indent goals – it makes evaluating your proofs easier.
1. (15 points) Relations and Orders
(10 points for setting up the proof; 5 points for the proof)
Let R(x, y) be a relation that is
Transitive
Irreflexive
Let S(x, y) be a relation that is
Transitive
Symmetric
Reflexive
Let relation Q be defined as
.. S(x,y)R(x, y)[Q(x, y)yx
Prove there exists a case where Q(x, y) and Q(y, x)are both true.
2. (10 points) Prove the following
)](yz)xz[(y z y.x. G1.
z)(xyzx)(y A3.
x A2.
xex A1.
x
x
A0.
1
x
ex
==
=
=
=
=
3. (15 points) Prove the following
A1. Whoever cooperates with Deena will hire Cindy.
A2. Jenny hires only for friends of Laura.
A3. No one is friend of Kelly and has Cindy as a friend.
G1. If Laura is a friend of Kelly, Jenny will not cooperate with Deena.
A. (10 points of 15) Use the following notation to translate the above statements into symbolic
form:
C(x, y): x cooperates with y
H(x, y): x hires y
F(x, y): x is a friend of y
d: Deena, c: Cindy, j: Jenny, l: Laura, k: Kelly
B. (5 points of 15) Proven the goal G1 given assumptions A1, A2, and A3. Make sure you justify
each step in the tableau.
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CSE607: Exam-3 S-K Chin 2002 Theories with Equality Exam: 1

CSE

Theories with Equality and Algebra Exam

Exam#3 April 1, 2002

This is an open-book and open-notes examination. Do all problems in 80 minutes. Make sure you justify your steps using the formal inference rules to earn full credit. (This includes rewriting rules–this will help you to avoid silly mistakes). Left justify assumptions and indent goals – it makes evaluating your proofs easier.

1. (15 points) Relations and Orders (10 points for setting up the proof; 5 points for the proof)

Let R(x, y) be a relation that is

  • Transitive
  • Irreflexive Let S(x, y) be a relation that is
  • Transitive
  • Symmetric
  • Reflexive Let relation Q be defined as ∀ x. y. [Q(x, y)R(x, y)S(x,y)

Prove there exists a case where Q(x, y) and Q(y, x)are both true.

2. (10 points) Prove the following

G1. x.y.z[(y z x z) (y )]

A3.(y x) z y (x z)

A2.x

A1.x e x

A0.x x

1

x

x e

3. (15 points) Prove the following

A1. Whoever cooperates with Deena will hire Cindy. A2. Jenny hires only for friends of Laura. A3. No one is friend of Kelly and has Cindy as a friend. G1. If Laura is a friend of Kelly, Jenny will not cooperate with Deena.

A. ( 10 points of 15 ) Use the following notation to translate the above statements into symbolic form:

  • C(x, y): x cooperates with y
  • H(x, y): x hires y
  • F(x, y): x is a friend of y
  • d: Deena, c: Cindy, j: Jenny, l: Laura, k: Kelly

B. (5 points of 15 ) Proven the goal G1 given assumptions A1, A2, and A3. Make sure you justify each step in the tableau.

CSE607: Exam-3 S-K Chin 2002 Theories with Equality Exam: 2

  • Bonus Problem (5 points) Please prove G1 by using the following assumptions and the knowledge introduced in [Ferraiolo et al. 1999].

Please use the following notation: U(x) : x is a user R(x) : x is a role P(x) : x is a permission RM(r, u): u is a member of role r Ea(i, j): i and j are mutually exclusive roles

Role Membership Inheritance: ( ∀ i , j : role )(∀ u : user ).[( i ≥ j )∧ RM ( i , u )⊃ RM ( j , u )]

Static Separation of Duty: ( ∀ u : user )(∀ i , j : role ).[( RM ( i , u )∧ RM ( j , u ))⊃¬ Ea ( i , j )]

A1. Harry is authorized for role1. A2. Role1 contains role2. A3. Role3 is mutually exclusive with role2. G1. Harry can not be authorized for role3.

Use the following notation: H: Harry, r1:role1, r2, role2, r3:role

Reference: FERRAIOLO, D. F., BARKLEY, J. F., AND KUHN, D. R. 1999. A Role-Based Access Control Model and Reference Implementation Within a Corporate Intranet, ACM Transactions on Information and System Security , Vol. 2, No. 1, February 1999, 34--