Abstract
Real-Time Logic is a formal notation for reasoning about temporal behaviour. Z is a general purpose specification language, but lacks explicit features for expressing real-time constraints. We show how these complementary methods can be formally unified. An approach to verification of real-time properties by deriving temporal information directly from the specification is then described.
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Abbreviations
- ℕ:
-
The set of natural numbers (non-negative integers)
- ℙS :
-
Powerset: the set of all subsets of S
- seq∞ X :
-
Set of finite or infinite sequences with elements drawn from X
- m ... n:
-
The set of integers between m and n inclusive
- 〈a,...,s〉:
-
Sequence of items from a to s
- tail S:
-
Sequence resulting from removal of first element from sequence S
- X ↔ Y:
-
The set of relations from X to Y
- X → Y:
-
The set of total functions from X to Y
- X → Y:
-
The set of partial functions from X to Y
- X × Y:
-
Cartesian product: the set of all ordered pairs from X and Y
- f t:
-
The function f applied to t (left associative)
- x R y:
-
Relation R as an infix operator: (x,y) ε R
- R + :
-
Non-reflexive transitive closure of R.
- R −1 :
-
Inverse of relation R
- R(|S|):
-
Relational image: all elements in R mapped to by elements of S
- dom R :
-
Domain of relation R (or indices of a sequence)
- ran R :
-
Range of relation R (or items in a sequence)
- R ⊳ T:
-
Range restriction of relation (or sequence) R to T
- S ⊲ R:
-
Domain subtraction: R less all pairs with first elements from S
- R ⊳ T:
-
Range subtraction: R less all pairs with second elements from T
- S ↿ A:
-
Sequence A restricted to items with indices from S
- D ¦ P · t:
-
The set of t's such that P holds given declarations D
- D ¦ P:
-
The set of D's such that P holds
- VD ¦ P · Q:
-
Universal quantification: for all D's such that P holds, Q holds
- ∀ D • P :
-
Universal quantification: for all D's P holds
- ∃ D • P :
-
Existential quantification: there exists D such that P holds
- ∄ D ¦ P · Q :
-
There does not exist D such that P and Q hold
- ∄ D • P :
-
There does not exist D such that P holds
- ∃1 D • P :
-
Unique existence: there is one D such that P holds
References
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© 1991 Springer-Verlag Berlin Heidelberg
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Fidge, C.J. (1991). Specification and verification of real-time behaviour using Z and RTL. In: Vytopil, J. (eds) Formal Techniques in Real-Time and Fault-Tolerant Systems. FTRTFT 1992. Lecture Notes in Computer Science, vol 571. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-55092-5_22
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DOI: https://doi.org/10.1007/3-540-55092-5_22
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