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) 1

Shape of an Event = < name > � when < guard > . . . then < assignment > . . . end 2

Assignments < variable > := < expression > Deterministic Non-deterministic any where < variable > < condition > . . . then < variable > := < expression > . . . end 3

Set Theory (1) ∈ set membership operator set of Natural Numbers: { 0 , 1 , 2 , 3 , . . . } N interval from a to b : { a, a + 1 , . . . , b } a .. 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) F 1 ( S ) Non-empty set of finite subsets of S F ( S ) Set of finite subsets of S P 1 ( 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 6

Set Theory (4) set of bijections from S to T S ֌ ։ T S × T Cartesian product of S and T − g overwriting operator for functions f ✁ 7

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 8

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 9

A Small Theory of Parities Constant: pty pty ∈ N → { 0 , 1 } pty (0) = 0 ∀ n · ( n ∈ N ⇒ pty ( n + 1) = 1 − pty ( n ))   x ∈ N     y ∈ N       x ∈ y .. y + 1   ∀ x, y ·   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 ∈ N     y ∈ N     ∀ x, y ·  x � = nxt ( y )      ⇒     itv ( x, nxt ( y )) = itv ( x, y ) ∪ { nxt ( y ) } ) ∀ x · ( x ∈ N ⇒ itv ( nxt ( x ) , x ) = N ) 11

A Small Theory of Trees Set: N Constants: r , f r ∈ N f ∈ N \ { r } → N   S ⊆ N     r ∈ S     f − 1 [ S ] ⊆ S ∀ S ·       ⇒     N ⊆ S 12