Modeling Conventions J-R. Abrial September 2004 Structure of a - - PowerPoint PPT Presentation

Modeling Conventions

J-R. Abrial September 2004

Structure of a Model

• List of Sets (identifiers)
• List of Constants (identifiers)
• List of Properties (predicates built on sets and constants)
• List of Variables (identifiers)
• List of Invariants (predicates built on sets, constants, and variables)
• List of Events (next slide)

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Shape of an Event < name >

• =

when < guard > . . . then < assignment > . . . end

2

Assignments Deterministic < variable > := < expression > Non-deterministic any < variable > where < condition > . . . then < variable > := < expression > . . . end

3

Set Theory (1) ∈ set membership operator

N

set of Natural Numbers: {0, 1, 2, 3, . . .} a .. b interval from a to b: {a, a + 1, . . . , b} S → T set of total functions from S to T S → T set of partial functions from S to T

4

Set Theory (2) → pair constructing operator {. . .} set defined in extension

∅

empty set

5

Set Theory (3)

F1(S)

Non-empty set of finite subsets of S

F(S)

Set of finite subsets of S

P1(S)

Non-empty set of subsets of S

P(S)

Set of subsets of S

max (S)

Maximum of a non-empty finite set of numbers

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Set Theory (4) S ֌

։ T

set of bijections from S to T S × T Cartesian product of S and T f ✁ − g

• verwriting operator for functions

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Set Theory (5)

dom

domain of a function

ran

range of a function ✁ domain restriction operator ✁ − domain subtraction operator

id(S)

identity function built on the set S

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Set Theory (6) S ∪ T set-theoretic union operator S ∩ T set-theoretic intersection operator S \ T set-theoretic difference operator f−1 converse of a function f[S] image of a set under a function

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A Small Theory of Parities Constant: pty pty ∈ N → {0, 1} pty(0) = 0 ∀ n · (n ∈ N ⇒ pty(n + 1) = 1 − pty(n)) ∀ x, y ·

            

x ∈ N y ∈ N x ∈ y .. y + 1 pty(x) = pty(y) ⇒ x = y

            

10

A Small Theory of Rings Set: N Constants: nxt, itv nxt ∈ N ֌

։ N

itv ∈ N × N → P(N) ∀x · ( x ∈ N ⇒ itv(x, x) = {x} ) ∀ x, y ·

         

x ∈ N y ∈ N x = nxt(y) ⇒ itv(x, nxt(y)) = itv(x, y) ∪ {nxt(y)} )

         

∀x · ( x ∈ N ⇒ itv(nxt(x), x) = N )

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A Small Theory of Trees Set: N Constants: r, f r ∈ N f ∈ N \ {r} → N ∀ S ·

         

S ⊆ N r ∈ S f−1[S] ⊆ S ⇒ N ⊆ S

         

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