Formalization of Oscilloscope

1
Formalization of Oscilloscope
2
On Formalism

• High-level (implementation-independent)
  specification
• Recall Larch An Algebraic Formal Spec. Lang.
• Why formal? Precise, consistent, and complete
• Formal semantics
• Formal grammar,
• e.g., syllogism
• All persons die.
• Adam is a person
• ------------------------
• Adam dies.
• Semantics?

3
Introduction to Z

• Based on typed set theory and first order logic
• Sets
• oneTwoThree 1, 2, 3
• Person Adam, Eve
• S P X 2 X --- S is a set of Xs powerset,
  i.e., the set of all subsets of X
• oneTwoThreeSet P oneTwoThree P 1, 2, 3
  ?
• personSet P person P Adam, Eve ?
• P X ?

4
Introduction to Z

• Sets (contd)
• x memberOf S
• ? 1 memberOf 1, 2, 3
• ? 1 memberof P 1, 2, 3
• ? 1 memberof P 1, 2, 3
• ? Adam memberOf P Person
• ? Adam memberOf Person
• ? Adam, Eve memberOf P Person

5
Introduction to Z

• Sets (contd)
• S subsetOf S
• ? 1 subsetOf 1, 2, 3
• ? 1, 2 subsetOf P 1, 2, 3
• ? 1, 2 subsetOf P 1, 2, 3
• ? Adam subsetOf P Person
• ? Adam subsetOf Person
• ? Person subsetOf Person
• ? Person subsetOf P Person

6
Introduction to Z

• Sets (contd)
• S X S (cross/cartesian product)
• oneTwoThree X person 1, 2, 3 X Adam,
  Eve
• 1, Adam, 1, Eve, 2, Adam, 2, Eve,
  3, Adam, 3, Eve
• ? 1, 2 subsetOf 1, 2 X 1, 2
• S U S, S intersect S, S\S, etc. (skip)

7
Introduction to Z

• Functions
• dom f --- The set of values x for which f(x) is
  defined
• f(x) x 2 , dom f n memberOf N 1 lt n lt 5
• ran f --- The set of values yielded by f(x),
• where x memberOf dom f
• ran f ?
• f X -gt Y --- f is a total function from X to Y
• i.e., f is defined for all x memberOf
  dom(f), i.e.,
• dom(f) X
• f X --gt Y --- f is a partial function from X to
  Y
• i.e., f is defined for some values in X
• if f(x) 1/x, ? dom(f) Z
• ? spouse Person -gt Person

8
Introduction to Z

• Functions (contd)
• (lambda x T . t) returns the value of the term t
• (lambda x N . X 2 ) 5 25
• (lambda x N . (X 2 , 1/x) ?
• (lambda x, y N . (X 2 y, y - 1/x) 5 1 ?

9
Introduction to Z

• First Order Logic
• Logical connectives AND, OR, NOT, gt, ltgt
• Quantifiers
• ? Exists n N . n n 2
• ? Exists p Person . P father (Adam)
• ? Forall i N . I 2 gt I
• ? Forall I, j N . I gt j gt I 2 gt j 2
• ? Forall x, y Person, x spouse(y) ltgt y
  spouse(x)

10
Introduction to Z

• Schemas
• A schema consists of a set of declarations of
  variables and a predicate constraining these
  variables (i.e., state space and operations)
• ----- BirthdayBook ------------------------------
  ----------
• known P Person
• birthday Person --gt Date
• -------------------------------------------------
  ----------------
• known dom birthday
• -------------------------------------------------
  ----------------
• One possible state
• known Adam, Caine, Eve
• birthday Adam -gt Apr/01, Eve -gt Apr/01

11
A Simple Oscilloscope

• ? What is a waveform?
• - Engineer 0 a graph
• - Engineer1 a 1-kbyte array of 8-bit samples
• - Engineer3 a set of voltage values
• - Engineer4 a function from time to volts

12
A Simple Oscilloscope
W -gt T
Clip
waveform
(translated) trace
Clipped trace
trace
Scale
Translate

• Overview
• display Pats ECG from 101pm to 102pm
• - Ultimately displaying a trace
• mapping time to a horizontal distance across
  the screen
• voltage to a vertical offset on the screen
• - Scale both horizontal (seconds/meter) and
  vertical (volts/meter) scaling
• to convert a voltage versus time signal to
• a point-on-screen versus time display
  of the trace
• - Translate the trace on the display by
  horizontal and vertical offsets
• - Clip the trace to fit on the screen

13
waveforms, segments, coordinates, traces

• A waveform can be modeled as a partial function
  of time
• Waveform Time --gt Voltage,
• where Voltage Z X Volt / e.g., (1, Volt),
  (2, Volt)
• Time N X Second / e.g., (0, Second),
  (1, Sec
• E.g., wf1 1 Sec -gt 1 Volt, 2 Sec -gt 3
  Volt, 3 Sec -gt 2 Volt
• wf2 1 Sec -gt 1 Volt, 3 Sec -gt 5 Volt, 4
  Sec -gt 6 Volt
• wf3 25 -gt 5, 26 -gt 6, 27 -gt 8, 28 -gt
  10, 29 -gt 11, 30 -gt 13
• A segment corresponds to a waveform over a
  contiguous time interval
• ? wf1
• ? wf2
• ? wf3
• ? wf3 (25)
• ? Wf3 (29)

14
waveforms, segments, coordinates, traces

• A coordinate can be represented by a real and a
  unit of distance
• Coord R X Meter / (1, Meter), (3.5, Meter)
• where Voltage Z X Volt / e.g., (1, Volt),
  (2, Volt)
• A point on the screen by a pair of coordinates
• Point Coord X Coord / e.g., ((1, Metr), (3,
  Metre)), (1, 3)
• A trace is a mapping from time to points
• Trace Time --gt Point
• ? (0, 2.5), (1, 3), (2, 4), (3, 5), (4, 5.5),
  (5, 6.5) memberOf Trace
• ? 25 -gt 5, 26 -gt 6, 27 -gt 8, 28 -gt 10, 29
  -gt 11, 30 -gt 13 memberOf Trace
• ? 25 -gt (0, 2.5), 26 -gt (1, 3), 27 -gt (2, 4),
  28 -gt (3, 5), 29 -gt (4, 5.5), 30 -gt (5, 6.5)

15
Scale

• takes a segment and scales it both horizontally
  and vertically
• adjusts it s.t. the start of the segment
  corresponds to a horizontal offset of zero
• The horizontal scale factor converts the units
  from seconds to metres
• The vertical scale factor converts from a voltage
  to metres
• ------- Scale ------------------------------------
  ---------------------------------------------
• segment Segment / e.g., 25 -gt 5, 26
  -gt 6, 27 -gt 8 28 -gt 10, 29 -gt 11, 30 -gt 13
• HScale R X Second/Metre / e.g., 1
  Second/Metre
• VScale R X Volt/Metre / e.g., 2 Volt/Metre
• scaled Trace
• --------------------------------------------------
  -----------------------------------------------
• scaled (lambda t dom segment . (t min (dom
  segment) / HScale,
• segment (t) / VScale) )
• --------------------------------------------------
  -----------------------------------------------
• ? scaled

16
Translate

• ---- Translate -----------------------------------
  --------------------------------------------------
  
• scaled Trace / e.g., 25 -gt (0, 2.5), 26 -gt
  (1, 3), 27 -gt (2, 4), 28 -gt (3, 5), 29 -gt (4,
  5.5), 30 -gt (5, 6.5)
• HOffset, VOffset Coord / e.g., (1, 1)
• moved Trace
• --------------------------------------------------
  --------------------------------------------------
  --
• moved (lambda t dom scaled . (first (scaled
  (t)) HOffset,
• second (scaled (t)) VOffset)
  )
• --------------------------------------------------
  --------------------------------------------------
  --
• ? moved

17
Clip

• ----- Clip ---------------------------------------
  -------------------------------------------------
• moved Trace / e.g., 25 -gt (1, 3.5), 26 -gt
  (2, 4), 27 -gt (3, 5), 28 -gt (4, 6), 29 -gt (5,
  6.5), 30 -gt (6, 7.5)
• HMax R X Meter / e.g., 4 Metre
• VMax R X Metre / e.g., 6 Metre
• clipped Trace
• --------------------------------------------------
  -------------------------------------------------
• let screen (x, y) Coord 0 lt x lt HMax
  -VMax lt y lt VMax .
• clipped moved screen
• --------------------------------------------------
  -------------------------------------------------
• Local definitions within predicates are
  introduced by the keyword let
• The operator is for range restriction
• R S a -gt b (a -gt b memberOf R) (b
  memberOf S)
• ? Clipped

18
Trace Display

• Schema conjunction the three schemas are
  combined using schema conjunction
• (Scale Translate Clip)
• Displaying only clipped trace hide scaled and
  moved traces, as they are only used as
  intermediate links keep all the constraints,
  but omit intermediate variables.
• DisplayTrace (Scale Translate Clip) \
  (scaled, moved)
• ? fully expanded declaration

19
DisplayKnobs, DisplaySegments

• (skip)
• ----- DisplayKnobs -------------------------------
  -------------------------------
• HScale R X Second/Metre
• VScale R X Volt/Metre
• HOffset, VOffset Coord
• --------------------------------------------------
  --------------------------------------
• ----- DisplaySegments ----------------------------
  ------------------------------
• segment Segment
• DisplayKnobs
• display P Trace
• --------------------------------------------------
  --------------------------------------
• display DisplayTrace dom DisplayTrace
  dom Segment . clipped
• --------------------------------------------------
  ---------------------------------------