Z Predicates and Schemas

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Z Predicates and Schemas

• SE 420, Formal Methods
• Mark Ardis, RIT

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Outline

• Predicate Calculus
• Logical connectives
• Quantifiers
• Schemas
• Definition
• Initialization
• Operations
• Schema calculus

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Logical Connectives
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Universal Quantifiers

• divides Z ? Z
• ? d, n Z d divides n ? n mod d 0
• 2 divides 14
• (2, 14) ? divides

Note underline syntax for infix binary operator
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Existential Quantifiers

• ? i Z i lt 10
• ? x ? y x lt y
• ? y ? x x lt y

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Cartoon of the Day (1/2)
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Cartoon of the Day (2/2)
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Editor Example

• User can type in text
• User can move cursor forward and backward through
  the text
• User can delete the character in front of the
  cursor

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State schemas

• Editor
• left, right TEXT
• ( left ? right ) ? maxsize

left is before the cursor, right is after
closed box means that the declarations are local
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Vending Machine Example
q
d
d
n
n
n
n
n
d
d
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Exercise 1
Write a Z state schema for the vending machine
example, keeping track of the number of nickels,
dimes and quarters. Include a component of the
state that represents the monetary sum of the
coins.
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Initialization schemas
conventional name begins with Init

• InitEditor
• Editor
• left right ltgt

Editor is included
left and right come from Editor
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Exercise 2
Write the Z initialization schema for the vending
machine.
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Operation schemas
global declaration of printing characters

• printing P Char
• Insert
• ? Editor
• ch? CHAR
• ch? ? printing
• left left ? ltch?gt
• right right

? means that the state will change
? means input (! means output)
precondition
left is the value after the operation
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Exercise 3
Specify a version of the Add_Dime operation that
only works if the amount of money in the machine
is less than 20 cents.
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Implicit preconditions

• right_arrow CHAR
• right_arrow ? printing
• Forward
• ? Editor
• ch? CHAR
• ch? right_arrow
• left left ? lthead(right)gt
• right tail(right)

implicit precondition right ? ltgt
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Schema calculus

• EOF
• Editor
• right ltgt
• RightArrow
• ch? CHAR
• ch? right_arrow
• T_Forward Forward ? ( EOF ? RightArrow ? ?
  Editor )

another state schema
an operation schema
? means that the state does not change
a one-line schema definition
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Exercise 4
Specify a total function T_Add_Dime in terms of
Add_Dime.