BPMN and Beyond Business process modelling notation, workflow ...

ASM Foundations of Database Management 7
join, selection, union, difference, intersection). We combine these functions into
an S-algebra.
Databases change over time, We may represent the change history of a
database by a sequence of database states. The changes are defined by an application
of database systems operations to the current database state. We may
require that any change applied to a model shall either transform the model to
a new model or shall not be considered otherwise. This operating requirement
leads to transaction semantics of database operating. We can use the logic L S
for the definition of transition constraints.
A transition constraint consists of a pair of formulas (ψ pre , ψ post ) from L S .
The transition constraint is valid for a database modification from DB to DB ′
if DB |= ψ pre and DB ′ |= ψ post . Static integrity constraints from a finite set
Σ can be mapped to transition constraints ( ∧ α∈Σ α, ∧ α∈Σ α). Let Σ dynamic be
the set of transition constraints.
A sequence of database states DB 0 , DB 1 , ..., DB i+1 = τ A (DB i ), ...
satisfying Σ dynamic is called a run of the database system. The run can be
defined though S-algorithms A or transformation rules that impose a one-step
transformation τA
DB of a database state to its successor in the run.
A database program is given by a rule name r of arity n is an expression
r(x 1 , ..., x n ) = P where P is a database transition operation and the free variables
of P are contained in the list x 1 , ..., x n .
Database transition operations are either basic operations from the S-algebra
or are constructed by inductively applying the following construction rules:
– Skip rule: skip
– Update rule: f(s 1 , ..., s n ) := t
– Parallel execution rule: P par Q
– Conditional rule: if φ then P else Q
– Let rule: let x = t in P
– For all rule: forall x with φ do P
– Choose rule: choose x with φ do P
– Sequence rule: P seq Q
– Call rule: r(t 1 , ..., t n )
A database system consists of a database management system and of a number
of databases. Let us consider only one database.
An abstract database system M consists of a
– a signature S M of the database system that embodies the signature S of the
database,
– a set of initial states for S M ,
– a set of database programs,
– a distinguished program of arity zero called main program of the system.
We denote the current state of the database system by DBS and by DB the
current state of the database. The transition operation P yields the modification
set U in a state DBS under the variable assignment ζ: +yields+(P,DBS,ζ,U).
Semantics of transition operations defined in a calculus by rules:
Premise 1 , ..., Premise n
Conclusion
Condition