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 X’s 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 (cont’d) 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 (cont’d) 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 (cont’d) 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 <= n <= 5} ran f --- The set of values yielded by f(x), where x memberOf dom f ran f = ? f: X -> 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 -|-> 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 -> Person

8. Introduction to Z Functions (cont’d) (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, =>, Quantifiers ? Exists n: N. n = n 2 ? Exists p: Person. P == father (Adam) ? Forall i: N. I 2 >= I ? Forall I, j: N. I > j => I 2 > j 2 ? Forall x, y: Person, x == spouse(y) 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 -|-> Date ----------------------------------------------------------------- | known = dom birthday ----------------------------------------------------------------- One possible state: known = {Adam, Caine, Eve} birthday = {Adam |-> Apr/01, Eve |-> 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 Overview “display Pat’s ECG from 1:01pm to 1:02pm” - 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 W -> T Scale Translate Clip waveform (translated) trace Clipped trace trace

13. waveforms, segments, coordinates, traces A waveform can be modeled as a partial function of time Waveform == Time -|-> 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 |-> 1 Volt, 2 Sec |-> 3 Volt, 3 Sec |-> 2 Volt} wf2 == {1 Sec |-> 1 Volt, 3 Sec |-> 5 Volt, 4 Sec |-> 6 Volt} wf3 == {25 |-> 5, 26 |-> 6, 27 |-> 8, 28 |-> 10, 29 |-> 11, 30 |-> 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 -|-> Point ? {(0, 2.5), (1, 3), (2, 4), (3, 5), (4, 5.5), (5, 6.5)} memberOf Trace ? {25 |-> 5, 26 |-> 6, 27 |-> 8, 28 |-> 10, 29 |-> 11, 30 |-> 13} memberOf Trace ? {25 |-> (0, 2.5), 26 |-> (1, 3), 27 |-> (2, 4), 28 |-> (3, 5), 29 |-> (4, 5.5), 30 |-> (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 |-> 5, 26 |-> 6, 27 |-> 8 28 |-> 10, 29 |-> 11, 30 |-> 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 |-> (0, 2.5), 26 |-> (1, 3), 27 |-> (2, 4), 28 |-> (3, 5), 29 |-> (4, 5.5), 30 |-> (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 |-> (1, 3.5), 26 |-> (2, 4), 27 |-> (3, 5), 28 |-> (4, 6), 29 |-> (5, 6.5), 30 |-> (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 < x < HMax ^ -VMax < y < VMax}. | clipped = moved screen --------------------------------------------------------------------------------------------------- Local definitions within predicates are introduced by the keyword “let” The operator is for range restriction: R S == {a |-> b | (a |-> 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} -----------------------------------------------------------------------------------------