Temporal Logic

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Temporal Logic

• CS410

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Remaining Schedule

• Today Temporal Logic
• Thursday Model Checking
• Extra OH Monday, 2pm to 5pm
• Next Tuesday Term Review
• November 30th, projects due at 1am (Dec 1st)
• Dec 1st and Dec 2nd, Final Project Demos
• E-mail cs410 to schedule time, include top 3
  choices
  
• 9am-5pm except not Dec1st from 1pm to 2pm
• Final
• Dec 20th, 830am, DMP 310

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Proof Techniques

• forall X. Y
• Find one X to act as a counter-example to show
  this is false
  
• exists X. Y
• Find one X to act as an example to show this is
  true
  
• How do we show
• forall X. Y is true?
• exists X. Y is false?
• Brute force
• Proof by Contradiction (not covered)

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Emulating Domains Using Properties

• forall X. mazda(X)
• mazda(7) false
• Is this a legitimate counter-example?
• Yes
• forall X. car(X) implies mazda(X)
• not(car(7)) or mazda(7) true
• To find a counterexample, we are restricted to
  objects with have the property of being a car

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Temporal Logic

• Temporal Relating to time
• Hard to reason about events which change over
  time in predicate logic
  
• e.x. classroom(X) and empty(X)
• alternates between true and false depending on
  the time
  
• Solution Model time as a sequence of states. Use
  temporal logic to describe relationships between
  states.

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Temporal Logic
S0
S1
S2
. . .
S
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time

• Divide time into a number of ordered discrete
  states starting from some origin S0
  
• Each object has some number of properties related
  to it in each state?
  
• e.x. empty(object X, state Y)
• Is object X empty in state Y?

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Simplifying Temporal Logic Notation

• Most often we are always talking about one object
  of interest (i.e. a computer program) so we dont
  bother writing it
  
• e.x. object is DMP 301
• empty(Y) ?
• Often we evaluate expression on all states to
  find out for which states it is true
  
• We just write empty, to ask the question in
  which states is the object of interest empty?
  
• i.e. properties dont need to be written as
  functions anymore
  

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Temporal Logic

• Types
• boolean
• properties
• Axioms
• globally(X)
• true at state Sn if X is true for all states Sm
  such that Sm Sn
  
• eventually(X)
• true at state Sn if X is true at some state such
  that Sm Sn
  
• next(X)
• true at state Sn if X is true at state S(n1)
• X until Y
• true at state Sn if X is true at all states, Sm,
  such that Sp Sm
  Sn and Y is first true after Sn at Sp
  
• Note Y must be true at some time in the future
  or else this is false

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globally(X) from S1
S2
S0
S1
. . .
S
8
X must be true at all of these states
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eventually(X) from S1
S2
S0
S1
. . .
S
8
X must be true at one these states
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next(X) from S1
S2
S0
S1
. . .
S
8
X must be true at S2
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(X until Y) from S1
Sn
S2
S0
S1
. . .
S
8

• Y is first true
• at this state, Sp,
• where Sp S1

2. X must be true at all of these states
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Temporal Logic Examples

• Consider time divided into a state for each day
  starting today
  
• thursday true for days that are Thursday
• globally(friday) false at all states
• thursday and next(saturday) false at all states
  
  
• eventually(saturday) true at all states
• eventually(lastDayOfThisClass) true for all
  states before the lastDayOfThisClass
  
• not(dayOfClass) until lastDayOfThisClass false
  at all states

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Example Problem

• At which states are the following true?
• globally(q) Si, i 8
• eventually(next(not(p)) true at all states
• q until globally(not(p)) S4, S5, S6, Si (i
  8)
  
• p until not(q) false at all states

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Example

• The desired requirements on our device are as
  follows
  
• If someone releases the button, the toy says hi
• The hi only happens for one unit of time
• The toy produces at most one hi per button
  push
  
• Such devices are useful in real systems in an
  elevator, for example, even if someone holds the
  button down for several seconds, we only want to
  summon the elevator once

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Transform Transitions into States

• Allows us to talk about input and output at
  certain states
  
• Details are simple and not important
• You wont be required to perform transformation
  on a test
  
• At each state any transition may be taken
• i.e. the choice of transition is
  non-deterministic
  

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Computation Trees

• Properties should hold for all possible paths

Paths for this state
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Example

• If someone releases the button, the toy says hi
• (but_dn and not(push)) implies eventually(hi)
• i.e. eventually(hi) must be true at a state where
  (but_dn and not(push)) is true
  
• The hi only happens for one unit of time
• hi implies next(not(hi))
• The device produces at most one hi per button
  push
  
• hi implies (not(hi) until push)

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Conclusion

• Temporal Logic allows us to ask questions about
  sequences of states
  
• Possible sequences of states for models can be
  derived using a computation tree
  
• Questions are answered using algorithms on
  computation trees
  
• Model Checking, Thursday