SEKE 2012 Proceedings - Knowledge Systems Institute

II. OVERVIEW OF MODAL TRANSITION SYSTEMS,
INTERFACE AUTOMATA AND ZLANGUAGE
In this section, we give a brief review of modal transition
systems, interface automata and Z language.
Modal transition systems have been proposed in [11]. A
refinement relation and a logical characterization of refinement
were also given in [11]. For a set of actions A, a modal
transition system (MTS) is a triple (P, −→ □ , −→ ♦ ), where
P is a set of states and −→ □ , −→ ♦ ⊆ P × A × P are transition
relations such that −→ □ ⊆−→ ♦ . The transitions in −→ □ are
called the must transitions and those in −→ ♦ are the may
transitions. In an MTS, each must transition is also a may
transition, which intuitively means that any required transition
is also allowed.
An interface automaton (IA) [1], introduced by de Alfaro
and Henzinger, is an automata-based model suitable for specifying
component-based systems. An IA consists of states,
initial states, internal actions, input actions, output actions and
a transition relation. The composition and refinement of two
IAs are proposed in [1].
Z was introduced in the early 80’s in Oxford by Abrial as
a set-theoretic and predicate language for the specification of
data structure, state spaces and state transformations. A boxed
notation called schemas is used for structuring Z specifications.
Z makes use of identifier decorations to encode intended
interpretations. A state variable with no decoration represents
the current (before) state and a state variable ending with a
prime ( ′ ) represents the next (after) state. A variable ending
with a question mark (?) represents an input and a variable
ending with an exclamation mark (!) represents an output. In
Z, there are many schema operators. For example, we write
S ∧ T to denote the conjunction of these two schemas: a new
schema formed by merging the declaration parts of S and T
and conjoining their predicate parts. S ⇒ T (S ⇔ T) is similar
to S∧T except connecting their predicate parts by ⇒ (⇔). The
hiding operation S\(x 1 , ..., x n ) removes from the schema S the
components x 1 , ..., x n explicitly listed, which must exist. The
hiding operation S\(x 1 , ..., x n ) removes from the schema S the
components x 1 , ..., x n explicitly listed, which must exist. Formally,
S\(x 1 , ..., x n ) is equivalent to (∃ x 1 : t 1 ; ...; x n : t n • S),
where x 1 , ..., x n have types t 1 , ..., t n in S. The notation ∃ x : a•S
states that there is some object x in a for which S is true. The
notation ∀ x : a • S states that for each object x in a, S is true.
For the sake of space, more details of Z can be refereed to
some books on Z [15].
III. MODAL INTERFACE AUTOMATA WITH ZNOTATION
This paper is based on modal ZIA, a specification language
which integrates modal transition systems, interface automata
and Z. Modal ZIA is defined such that apart from enabling one
to deal with the modal properties, behavioral properties and
the data properties of a system independently. In this section,
we combine modal transition systems, inference automata and
Z language to give a specification approach for software components.
We first give the definition of such model. Then we
define the modal refinement of modal ZIA. Furthermore, we
propose and prove some properties on the modal refinement
of modal ZIA.
A. Model of Modal ZIA
Interface automata provide a specification approach for interface
behavior properties. But this approach can not describe
data structures specification of states. On the other hand,
Z can specify the state of a system, but is not suitable to
behavioral properties. This section describes modal ZIA as a
conservative extension of both interface automata and Z in
the sense that almost all syntactical and semantical aspects of
interface automata and Z are preserved.
In the original interface automata, states and transitions
are abstract atomic symbols. But in modal ZIA, states and
transitions are described by Z schemas.
In the rest of this paper, we use the following terminology:
(1) A state schema is a schema which does not contain any
variable with decoration ′ .
(2) An input operation schema is an operation schema which
contains input variables.
(3) An output operation schema is an operation schema
which contains output variables.
(4) An internal operation schema is an operation schema
which contains variables with decoration ′ .
Intuitively, a state schema is assigned to a state which
may contain many variables, this state schema describes the
constraint of variables in the state. A variable with decoration
′
denotes a variable at next state. So a state schema does
not contain any variable with decoration ′ . An input (output)
operation schema is assigned to an input (output) action
which may contain many input (output) variables, this input
(output) operation schema describes the constraint of variables
in the input (output) action. So an input (output) operation
schema is an operation schema which contains input (output)
variables. An internal operation schema is assigned to an
internal action, this internal operation schema describes the
change of variables after performing the internal action. So
an internal operation schema is an operation schema which
contains variables with decoration ′ .
In the rest of paper, given an assignment ρ and a schema A,
we write ρ |= A if ρ assigns every variable x in the declaration
part of A to an element of its type set, which satisfies the
predicate part of A; we write |= A if ρ |= A for any assignment
ρ.
Let S to be a Z schema, we use V I (S) (V O (S), V H (S)) to
denote the set of input variables (output variables, internal
variables) in S.
Definition 1. A modal interface automaton with Z notation
(modal ZIA) P = 〈S P , SP i , AI P , AO P , AH P , VI P , VO P , VH P , FS P , FA P ,
G IA
P , GOA P , A□ P , A♦ P , V□ P , V♦ P , T P〉 consists of the following
elements:
(1) S P is a set of states.
(2) SP i ⊆ S P is a set of initial states. If SP i = ∅ then P is
called empty.
(3) A I P , AO P and AH P are disjoint sets of input, output, and
internal actions, respectively. We denote by A P = A I P ∪AO P ∪AH P
526