SEKE 2012 Proceedings - Knowledge Systems Institute

(1) V □ (A) ⊆ V □ (B), V ♦ (A) ⊆ V ♦ (B).
(2) A\(x 1 , ..., x m ) ≥ B\(y 1 , ..., y n ), where {x 1 , ..., x m } =
V(A)−V □ (A)−V ♦ (A), {y 1 , ..., y n } = V(B)−V □ (A)−V ♦ (A).
For example, Â=[x ♦ : R; y □ : N | y □ =2⌊x ♦ ⌋] B̂=[x ♦ :
N; u ♦ : R; y □ : R; v □ : R; z : N | y □ =2x ♦ ; v □ = z ∗ u ♦ ],
where x ♦ , u ♦ are may variables, y □ , v □ are must variables.
So intuitively, A ≥ B describes the modal refinement of
data properties of schemas A and B if schemas A and B have
the same may and must variables. A B describes the modal
refinement of data properties of schemas A and B in the general
case, i.e., schemas A and B may have different may and must
variables.
The precise definition of modal refinement must take into
account the fact that the internal actions of P and Q are
independent. For this, we need some following preliminary
notions.
We now give the following definition which describes the
set of states after performing a sequence of internal actions
from a given state.
Definition 4. Given a modal ZIA P and a state s ∈ S P , the
set ε − closure P (s) is the smallest set U ⊆ S P such that (1)
s ∈ U and (2) if t ∈ U and (t, a, t ∗ ) ∈ T H P then t∗ ∈ U.
The environment of a modal ZIA P cannot see the internal
actions of P. Consequently if P is at a state s then the
environment cannot distinguish between s and any state in
ε − closure P (s).
The following definition describes the set of states after
performing several internal actions and an external action from
a given state.
Definition 5. Consider a modal ZIA P and a state s ∈ S P .
For an action a, we let
ExtDest P (s, a) ={s ∗ |∃(t, a, t ∗ ) ∈ T P .t ∈ ε − closure P (s)
and s ∗ ∈ ε − closure P (t ∗ )}.
In the following, we give a modal refinement relation
between modal ZIAs. For modal ZIAs, a state has not only
behavioral properties but also data properties. Therefore this
modal refinement relation involves both the modal refinement
relation between behavioral properties and the modal refinement
relation between data properties.
Definition 6. Consider two modal ZIAs P and Q. A binary
relation ≽ m ⊆ S P × S Q is a modal refinement from Q to P if
for all states s ∈ S P , there exists t ∈ S Q such that s ≽ m t the
following conditions hold:
(1) F S P (s) FS Q (t).
(2) For any action a ∈ A □ P , if s∗ ∈ ExtDest P (s, a), then
there is a state t ∗ ∈ ExtDest Q (t, a) such that F S P (s∗ ) F S Q (t∗ )
and s ∗ ≽ m t ∗ .
(3) For any action a ∈ A ♦ Q , if t∗ ∈ ExtDest Q (t, a), then there
is a state s ∗ ∈ ExtDest Q (s, a) such that F S P (s∗ ) F S Q (t∗ ) and
s ∗ ≽ m t ∗ .
Intuitively, must actions (variables) represents the overapproximation
of actions (variables), and may actions (variables)
represents the under-approximation of actions (variables).
The modal refinement relation describes the both overapproximation
and under-approximation for transitions and
variables for modal ZIAs.
We say that modal ZIA P is refined by modal ZIA Q if for
some initial states s in P and t in Q, s is refined by t.
Definition 7. The modal ZIA Q refines the modal ZIA P
written P ≽ m Q if:
there is a modal refinement ≽ m from Q to P, a state s ∈ S i P
and a state t ∈ S i Q such that s ≽ m t.
The above definitions of modal refinement relations can be
extended to the definitions of bisimulation relations by adding
the symmetric condition of relations.
The following lemma states that is a partial order (i.e.,
reflexive and transitive).
Lemma 1. (1) A A.
(2) If A B and B C, then A C.
The following proposition means that ≽ m is a partial order.
Proposition 1. (1) P ≽ m P.
(2) If P ≽ m Q and Q ≽ m R, then P ≽ m R.
IV. A LOGIC FOR MODAL ZIAS
Providing a logical characterization for various refinement/equivalence
relations has been one of the major research
topics in the development of automata theory and process
theories. A logical characterization not only allows us to
reason about behaviors of systems, but also helps to verify
the properties of systems. For modal transition system, a logic
for modal refinement relation was proposed and a logical
characterization was given in [3]. For convariant-contravariant
simulation of labelled transition systems, a logical characterization
for convariant-contravariant simulation relation was
given in [8]. In this section, we give the similar result for
modal ZIAs. In the following, we give a logic for modal ZIAs
named MZIAL and give a logical characterization of modal
refinement relation ≽ m .
A. Syntax of MZIAL
Throughout this paper, we let MZIAL be a language which
is just the set of formulas of interest to us.
Definition 8. The set of formulas called MZIAL, isgiven
by the following rules:
(1) ⊤∈MZIAL.
(2) ⊥∈MZIAL.
(3) If ϕ is in the form of p(x1 ⊙ , ..., x⊙ n ), then ϕ ∈ MZIAL,
where x1 ⊙ , ..., x⊙ n are may or must variables, i.e., x ⊙ i is in the
form of x ♦ i or xi □ , p is a n − ary prediction.
(4) If ϕ i ∈ MZIAL for any i ∈ I, then ∧ i∈I ϕ i ∈ MZIAL.
(5) If ϕ i ∈ MZIAL for any i ∈ I, then ∨ i∈I ϕ i ∈ MZIAL.
(6) If ϕ ∈ MZIAL, then (∀ x ♦ )ϕ ∈ MZIAL.
(7) If ϕ ∈ MZIAL, then (∃ x □ )ϕ ∈ MZIAL.
(8) If ϕ ∈ MZIAL, then [[a ♦ ]]ϕ ∈ MZIAL.
(9) If ϕ ∈ MZIAL, then 〈〈a □ 〉〉ϕ ∈ MZIAL.
B. Semantics of MZIAL
We will describe the semantics of MZIAL, that is, whether
a given formula is true or false.
The satisfaction relation |= is given recursively by the
following definition, where (P, s) |= ϕ means that state s of
modal ZIA P satisfies formula ϕ.
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