Logics

Calvin (Deutschbein)

14 Feb 2024

Announcements

  • Welcome to DATA-599: Cybersecurity!
  • You should have a Kripke Structure to share.
  • Depending on pacing, we'll have new ways of specifying for next time.

Review Question

In a Kripke, what is I?

  1. All possible configurations of a studied system
  2. All possible traits or features of a studied system
  3. All allowable starting configurations of a studied system
  4. All allowable traits or features of a studied system to correctly start

Think about what states and propositions are, and how they fit in here.

Review Question

In a Kripke, R is a relation. What does this mean?

  1. R considers pairs of distinct conditions or traits of a system
  2. R considers pairs of configurations of a systems
  3. R connects configurations to conditions or traits for some system
  4. R connects conditions or traits to configurations of some system

Think about what states and propositions are, and how they fit in here.

Review Question

We said properties were sets of something. What?

  1. States
  2. Sequences
  3. Systems
  4. Times

Review Question

How do Kripke Structures relate to properties?

  1. A Kripke structure defines a property without enumeration
  2. A Kripke structure can create an example of a trace within a property
  3. A Kripke structure is the system that may or may not have some property
  4. A Kripke structure's atomic propositions are properties

How did we motivate the introduction of Kripke structures?

A concretization

Consider the following representation:

  • Take p to be "can guess passwords"
  • Take q to be "system access is denied"

A passwording service must disclose if an entered password is correct.

  • p and q are true initially.
    • No checking, no access
  • Once q becomes false, it never becomes true again.
    • Once logged in, we can no longer guess passwords.
  • Once q becomes false, p stays true forever.
    • Once logged in, access is not revoked.

Is this good? Does this describe any services you use?

Review Question 1

Consider the following Kripke Structure:

Which of the following traces would be accepted by this structure?

  1. {p,q}, {q }, {p,q}, {q }, {p }, {p }, {p }, ...
  2. {p }, {q }, {p,q}, {q }, {p }, {p }, {p }, ...
  3. {q }, {p,q}, {q }, {p,q}, {q }, {p }, {p }, ...

Review Question 2

Consider the following Kripke Structure:

Which of the following traces would be accepted by this structure?

  1. {p,q}, {q }, {p,q}, {q }, {p }, {p }, {p }, ...
  2. {p,q}, {q }, {p, }, {q }, {p }, {p }, {p }, ...
  3. {p,q}, {p }, {q, }, {p,q}, {q }, {p }, {p }, ...

Review Question 3

Consider the following Kripke Structure:

Which of the following traces would be accepted by this structure?

  1. {p,q}, {q }, {p,q}, {q }, {p,q}, {q }, {q }, ...
  2. {p,q}, {q }, {p,q}, {q }, {p }, {q }, {p }, ...
  3. {p,q}, {q }, {p }, {p }, {p }, {p }, {p }, ...

Review Question 4

Consider the following Kripke Structure:

What describes (labels) S ?

  1. { {p,q}, {q }, {p } }
  2. { {p,q} }
  3. { {p }, {q } }

Review Question 5

Consider the following Kripke Structure:

What describes (labels) I ?

  1. { {p,q}, {q }, {p } }
  2. { {p,q} }
  3. { {p }, {q } }

Review Question 6

Consider the following Kripke Structure:

What is AP ?

  1. { {p,q}, {q }, {p } }
  2. { {p,q} }
  3. { {p }, {q } }

Review Question 7

Consider the following Kripke Structure:

What describes (labels) R ?

  1. { {q} → {p }, {p} → {p,q} }
  2. { {p,q} → {{q }}, {q} → {{p,q},{p }}, {p} → {{p }} }

Limitations

Kripke Structures are better than enumeration, but...

  • Representation in text is difficult to understand
  • Showing equivalence to traces can be difficult

We will introduce a better way to write out relations, then look at our Kripke Structures.

An example

Consider the following Kripke Structure:

The following statements are true of traces accepted by this structure:

  • p and q are true initially.
  • Once q becomes false, it never becomes true again.
  • Once q becomes false, p stays true forever.
  • p goes from false to true when q goes from true to true false.

A concretization

Consider the following representation:

  • Take p to be "password checking is blocked/secured"
  • Take q to be "system access is blocked/secured"

A passwording service must disclose if an entered password is correct.

  • p and q are true initially.
    • No checking, no access
  • Once q becomes false, it never becomes true again.
    • Once logged in, we can no longer guess passwords.
  • Once q becomes false, p stays true forever.
    • Once logged in, access is not revoked.

Infinite guesses (bad), but persistent users may not look up passwords (good).

Goals

We have:

  • A way to describe the current state of the system (AP)

We need:

  • A way to describe relations between atomic propositions over time.

Consider:

  • p and q are true initially.
  • Once q becomes false, it never becomes true again.
    • q until not q
  • Once q becomes false, p stays true forever.
    • q until not q
  • p goes from false to true when q goes from true to true false.

Next and Until

It often suffices to define two temporal operators:

  • X: “next”
  • U: “until”

Along with existing logical operators:

  • ¬: “not”
  • ∨: “or”

We need apply this over atomic propositions in AP to describe the security requirements of a system.

>

Composition

With logical negation (not) and disjunction (or) we can generate other logical connectives.

p q ¬p p∨q ¬p∨¬q ¬(¬p∨¬q)≡p∧q
True True False TrueFalse True
TrueFalseFalse True TrueFalse
False True True True TrueFalse
FalseFalse TrueFalse TrueFalse

For example, "and" or conjunction over two atomic propositions is equivalent to the negation or "not" of the "or" or disjunction over the negation or "not" of the two atomic propositions.

Next and Until

X a

  • * → a → * → ...

a U b

  • a∧¬b → b → * → ...
  • a∧¬b → a∧¬b → b → * → ...

Linear Temporal Logic Composition

Name Usage Meaning Equivalencies
neXt X p p holds in the next time X p
Future F p p holds in some future time TrueUp
Global(ly) G p p holds in all future times ¬(TrueU¬p)
Until p U q p holds unless q, and q must hold at some point in the future p U q
Release p R q q holds up to and including when p holds ¬(¬p U ¬q)
Weak until p W q p holds up to when q holds (p U q) ∨ G p
Mighty release p M q q holds up to and including when p holds, and p must hold at some point in the future q U (p ∨q)

Linear Temporal Logic Exercise

Populate this table in groups of n. Likely 15-20 minutes.

Operator Trace 1 Trace 2
X p * → p → * → ...
F p ? ?
G p ?
p U q p∧¬q → p∧¬q → q → * → ...
p R q ? ?
p W q ? ?
p M q ?

Linear Temporal Logic Traces

Operator Trace 1 Trace 2
X p * → p → * → ...
F p * → p → * → ... * → * → p → * → ...
G p p → p → p → ...
p U q p∧¬q → p∧¬q → q → * → ...
p R q q → q → p ∧ q → * → ... q → q → q → ...
p W q p → p → q → * → ... p → p → p → ...
p M q q → q → p ∧ q → * → ...

Homework (Review)

Identify something in your life (an app, a work thing, a building, a company) with security features.

  • It may be the same or a different thing.
  • You may wish to focus on a small part of your previous example, in the case of more complex systems.
  • Update you diagram to be a Kripke Structure.
  • Fully specify each of {S, I, R, L}

Homework (Discussion)

Brave* volunteers** will present their Kripke Structures.

  • In what states may the discussed system be?
  • In what sequence may these states occur?
  • Does the discussed system implement a security policy
  • Does the system change over time?
  • What is a non-trivial example and counterexample of a trace?
  • What Linear Temporal Logic properties partially or completely describe the system.
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