Transcript Formalization of Oscilloscope

Formalization of Oscilloscope
On Formalism
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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?
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| ==
?
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
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
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)
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 taken 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
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 == ?
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)
Introduction to Z
• Schemas
A schema consists of a set of declarations olf 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, Eve}
birthday = {Adam |-> Apr/01, Eve |-> Apr/01}
A Simple Oscilloscope
• Overview